Intra-Horizon Expected Shortfall and Risk Structure in Models with Jumps
IIntra-Horizon Expected Shortfall and Risk Structurein Models with Jumps
Walter Farkas ∗ Ludovic Mathys † Nikola Vasiljevi´c ‡ Department of Banking and Finance, University of Zurich, Switzerland.Department of Mathematics, ETH Zurich, Switzerland.Swiss Finance Institute, Switzerland.
Abstract
The present article deals with intra-horizon risk in models with jumps. Our general understanding ofintra-horizon risk is along the lines of the approach taken in [BRSW04], [Ro08], [BMK09], [BP10], and[LV20]. In particular, we believe that quantifying market risk by strictly relying on point-in-time measurescannot be deemed a satisfactory approach in general. Instead, we argue that complementing this approachby studying measures of risk that capture the magnitude of losses potentially incurred at any time of atrading horizon is necessary when dealing with (m)any financial position(s). To address this issue, wepropose an intra-horizon analogue of the expected shortfall for general profit and loss processes and discussits key properties. Our intra-horizon expected shortfall is well-defined for (m)any popular class(es) of L´evyprocesses encountered when modeling market dynamics and constitutes a coherent measure of risk, asintroduced in [CDK04]. On the computational side, we provide a simple method to derive the intra-horizonrisk inherent to popular L´evy dynamics. Our general technique relies on results for maturity-randomizedfirst-passage probabilities and allows for a derivation of diffusion and single jump risk contributions.These theoretical results are complemented with an empirical analysis, where popular L´evy dynamics arecalibrated to the S&P 500 index and Brent crude oil data, and an analysis of the resulting intra-horizonrisk is presented.
Keywords:
Intra-Horizon Risk, Value at Risk, Expected Shortfall, L´evy Processes, Hyper-ExponentialDistribution, Risk Decomposition.
MSC (2010) Classification:
JEL Classification:
C32, C63, G01, G51. ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] a r X i v : . [ q -f i n . M F ] J a n For the past 20 years, the (point-in-time) value at risk, defined as quantile of the profit and loss distributionat the end of a predefined trading horizon, has been one of the most widely used measure of market risk forregulatory capital allocation (cf. [BCBS06], [BCBS19]). Despite its popularity, this risk measure has severalmajor drawbacks that are all known to academics since many years. Firstly, it is mainly concerned withthe probability of a loss and not with the actual loss size itself. In particular, when relying on the (point-in-time) value at risk, the distribution of the losses that exceed the quantile of interest is not taken intoaccount. Secondly, it does not satisfy the subadditivity property for monetary risk measures (cf. [ADEH99],[RT02], [AT02], [EPRWB14]) and therefore does not constitute a coherent measure of risk in the sense of[ADEH99]. Lastly, as a measure of market risk, the (point-in-time) value at risk does not capture the fullmagnitude of losses that may be potentially incurred at any time of a trading horizon (cf.[BRSW04], [Ro08],[BMK09]). To address some of these issues, two streams have emerged in the academic literature. Whilecertain authors (cf. [ADEH99], [RT02], [AT02]) introduced the (point-in-time) expected shortfall, a coherentrisk measure that additionally depends on the tail of the underlying profit and loss distribution at the endof a predefined trading horizon, other authors (cf. [BRSW04], [Ro08], [BMK09], [BP10], [LV20]) developeda new path-dependent market risk measure, the intra-horizon value at risk. In light of the recent admissionof the (point-in-time) expected shortfall in the new market risk framework of the Basel Accords and of theconstant demand therein for sufficient conservatism in the risk estimates (cf. [BCBS19]), we believe thatit is high time to reunify these two branches of the academic literature and to combine their advantagesto propose a novel, coherent and path-dependent market risk measure that depends on all extremes in atrading horizon. This is the content of the present article that extensively discusses an intra-horizon versionof the (point-in-time) expected shortfall.Our paper’s contribution is manifold and has both theoretical and practical relevance. On the theoreticalside, we first generalize the current intra-horizon risk quantification approach of the literature (cf. [BRSW04],[Ro08], [BMK09], [BP10], [LV20]) and propose an intra-horizon analogue of the expected shortfall for gen-eral profit and loss processes. The resulting risk measure has several desirable properties and constitutesa coherent measure of risk in the sense of [CDK04]. Additionally, we show that our general ansatz islinked to simple (and maturity-randomized) first-passage probabilities of the underlying profit and loss pro-cess and subsequently use these relations to prove that the intra-horizon expected shortfall is well-definedfor (m)any popular L´evy dynamics encountered in financial modeling. Secondly, we introduce diffusionand jump contributions to first-passage occurrences under L´evy models and present characterizations ofdiffusion and jump contributions to simple and maturity-randomized first-passage probabilities. Thesecharacterizations are then used in the following way: First, diffusion and jump risk contributions to theintra-horizon expected shortfall are inferred. Second, (semi-)analytical results for diffusion and jump contri-butions to maturity-randomized first-passage probabilities are derived under the class of hyper-exponentialjump-diffusion processes by relying on option pricing methods (cf. among others [Ca09], [CCW09], [CK12],[CYY13], [HM13], [AR16], [LV17], [LV20]). On the practical side, we introduce a simple and efficient ansatzto compute the intra-horizon risk inherent to popular L´evy dynamics. Our general approach consists incombining hyper-exponential jump-diffusion approximations to (pure jump) L´evy processes having com-pletely monotone jumps with our (semi-)analytical results in this class of processes to recover approximate,though arbitrarily close intra-horizon risk results for the original process. In doing so, we rely on similarideas to the ones introduced in [AMP07], [JP10], and [LV20] and subsequently use a mix of our resultsfor maturity-randomized first-passage probabilities and a Laplace inversion algorithm (cf. [Co07]) to arriveat intra-horizon expected shortfall results. Lastly, as an application of the techniques developed in thispaper, we calibrate S&P 500 index and Brent crude oil data to popular L´evy dynamics and investigate theintra-horizon risk inherent to a long position in these underlyings from January 1995 to September 2020.Our empirical findings reveal that even for high loss quantiles (i.e., low α ) the intra-horizon value at riskand the intra-horizon expected shortfall add conservatism to their point-in-time estimates. Additionally,they suggest that these risk measures have a very similar structure across jumps/jump clusters and thatalready a high contribution of their risk is due to only few, large – in terms of the absolute jump size – jumpclusters.The remainder of this paper is structured as follows. In Section 2, we introduce our general intra-horizonrisk quantification approach as well as the notation used in the rest of the paper. This section also linksintra-horizon risk to first-passage probabilities of the underlying profit and loss process. Section 3 dealswith intra-horizon risk in models with jumps and is divided into two parts. Firstly, our intra-horizon riskquantification approach is further developed under the assumption of L´evy dynamics and characterizationsof simple (and maturity-randomized) first-passage probabilities are discussed. These characterizations areused in the second step to derive (semi-)analytical results for maturity-randomized first-passage probabilitiesunder the class of hyper-exponential jump-diffusion processes. Section 4 reviews hyper-exponential jump-diffusion approximations to pure jump L´evy processes having a completely monotone jump density as wellas few adaptions. All the theoretical results of Sections 2-4 are lastly combined in Section 5, where popularL´evy dynamics are calibrated to S&P 500 index data as well as to Brent crude oil data and the intra-horizonrisk inherent to a long position in these underlyings is analyzed from January 1995 to September 2020. Thepaper concludes with Section 6. All proofs and complementary results are presented in the Appendices(Appendix A, B and C). We start by discussing the problem of evaluating the intra-horizon risk inherent to a financial position. Tothis end, we fix a time horizon
T > , F , F , P ), whose filtration F = ( F t ) t ∈ [0 ,T ] satisfies the usual conditions. We let ( P & L t ) t ∈ [0 ,T ] be an F -adapted real-valued stochasticprocess and interpret its realizations as possible discounted profit and loss realizations of a given financialposition over the valuation horizon [0 , T ]. Here, we do not necessarily require the process ( P & L t ) t ∈ [0 ,T ] to start at P & L = 0 but allow instead for more flexibility in the choice of its initial value, i.e., we let P & L = z for general z ∈ R . This generalization proves useful, when rolling profits/losses of financialpositions over multiple valuation periods. In this case, P & L represents the profit/loss accumulated fromthe establishment of the position until the start of the valuation horizon under consideration. Our understanding of intra-horizon risk is in line with the ideas presented in [BRSW04], [BP10] and [LV20].As in these papers, our focus is on market risk, i.e., we only deal with risk that arises out of movements inthe market price of financial assets and fully abstract from other risk types, such as e.g., counterparty risk.Additionally, we believe that quantifying market risk by strictly relying on point-in-time measures cannotbe deemed a satisfactory approach in general. Instead, complementing this approach by studying measuresof risk that capture the magnitude of losses potentially incurred at any time of a trading horizon is necessary We choose the Gaver-Stehfest algorithm that has the particularity to allow for an inversion of the transform on the real lineand that has been successfully used by several authors in the option pricing literature (cf. [KW03], [Ki10], [HM13], [LV17]). for many asset types. This motivates the consideration of the minimum (discounted) profit and loss process,( I P & L t ) t ∈ [0 ,T ] , that is defined via I P & L t := inf ≤ u ≤ t P & L u , t ∈ [0 , T ] . (2.1)Under this notation, the following definition of the intra-horizon value at risk was presented in [BP10]. Definition 1 (Intra-Horizon Value at Risk) . Let
T > and α ∈ (0 , be fixed. The level- α intra-horizonvalue at risk associated with the (discounted) profit and loss process ( P & L t ) t ∈ [0 ,T ] over the time inter-val [0 , T ] , iV@R α,T ( P & L ) , is defined asiV@R α,T ( P & L ) := V@R α (cid:0) I P & L T (cid:1) , (2.2) where, for a random variable Y , V@R α ( Y ) denotes the (point-in-time) value at risk to the level α and isdefined as V@R α ( Y ) := − q α ( Y ) . (2.3) Here, q α ( Y ) denotes the upper α -quantile of Y , which is obtained via q α ( Y ) := sup { y ∈ R : P ( Y ≤ y ) ≤ α } . (2.4)Specifying the intra-horizon value at risk in the above sense is closely linked to the theory of ruin that hasreceived a lot of attention in insurance mathematics. Indeed, the above definition can be re-stated in termsof first-passage probabilities, as iV@R α,T ( P & L ) := − sup (cid:110) (cid:96) ∈ R : P z (cid:0) τ P & L , − (cid:96) ≤ T (cid:1) ≤ α (cid:111) , (2.5)where we denote by P z the probability measure under which the process ( P & L ) t ∈ [0 ,T ] starts at z ∈ R . For (cid:96) ∈ R and a given stochastic process ( Y t ) t ∈ [0 ,T ] , we use the notation τ Y, ± (cid:96) := inf { t ≥ ± Y t ≥ ± (cid:96) } , with inf ∅ = ∞ . (2.6)This representation will prove useful, as it will allow us to combine properties of first-passage probabilitiesto subsequently recover intra-horizon value-at-risk results via standard numerical methods. Although the intra-horizon value at risk already accounts for intra-horizon features, it suffers from twomajor drawbacks. Firstly, it is mainly concerned with the probability of a loss and not with the actual losssize itself. In particular, when relying on the intra-horizon value at risk, the distribution of the losses thatexceed the quantile of interest is not taken into account. Secondly, it is not subadditive (cf. [BP10]) andtherefore does not constitute a coherent measure of risk in the sense of [CDK04]. This is due to the factthat the intra-horizon value at risk consists of an adaption of the point-in-time value at risk, which is knownto have the same deficiencies. To address these major shortcomings, we propose a (market) risk measurethat defines an intra-horizon analogue of the expected shortfall. This is the content of the next definition,where we use the notation E P z [ · ] to indicate expectation under the measure P z . Like its point-in-time counterpart, the intra-horizon value at risk may become superadditive for certain profit and lossprocesses and therefore defies the notion of diversification.
Definition 2 (Intra-Horizon Expected Shortfall) . Let
T > and α ∈ (0 , be fixed and assume that E P z (cid:2) | I P & L T | (cid:3) < ∞ . Then, the level- α intra-horizon expected shortfall associated with the (discounted) profitand loss process ( P & L t ) t ∈ [0 ,T ] over the time horizon [0 , T ] , iES α,T ( P & L ) , is defined asiES α,T ( P & L ) := 1 α α (cid:90) iV@R γ,T ( P & L ) dγ. (2.7)It is not hard to see that the above specification of the intra-horizon expected shortfall overcomes bothmajor shortcomings of the intra-horizon value at risk. Indeed, a brief look at equation (2.7) reveals that ourintra-horizon expected shortfall defines a coherent measure of risk in the sense of [CDK04] that additionallydepends on the distributional properties in the tail of the underlying profit and loss process. In fact, cash-invariance, monotonicity and positive homogeneity of (2.7) directly follow from the corresponding propertiesof the point-in-time expected shortfall via the identity iES α,T ( P & L ) = 1 α α (cid:90) V@R γ ( I P & L T ) dγ =: ES α (cid:0) I P & L T (cid:1) . (2.8)Additionally, subadditivity is obtained by relying on the monotonicity and subadditivity properties ofthe point-in-time expected shortfall and the fact that for two profit and loss processes (cid:0) P & L t (cid:1) t ∈ [0 ,T ] and (cid:0) P & L t (cid:1) t ∈ [0 ,T ] the following identity holds I P & L + P & L T ≥ I P & L T + I P & L T . (2.9)Then, iES α,T (cid:0) P & L + P & L (cid:1) = ES α (cid:0) I P & L + P & L T (cid:1) ≤ ES α (cid:0) I P & L T + I P & L T (cid:1) ≤ ES α (cid:0) I P & L T (cid:1) + ES α (cid:0) I P & L T (cid:1) = iES α,T (cid:0) P & L (cid:1) + iES α,T (cid:0) P & L (cid:1) , (2.10)which is the subadditivity property.The next proposition indicates the link between the intra-horizon expected shortfall and the theory of ruin.In particular, it shows that, whenever well-defined, the difference between intra-horizon expected shortfalland intra-horizon value at risk can be computed for any given profit and loss process based on first-passageprobabilities. The proof is provided in Appendix A. Proposition 1.
Let
T > and α ∈ (0 , be fixed and assume that E P z (cid:2) | I P & L T | (cid:3) < ∞ . Then, the level- α intra-horizon expected shortfall can be re-expressed in the formiES α,T ( P & L ) = 1 α − iV @ R α,T ( P & L ) (cid:90) −∞ P z (cid:0) τ P & L , − (cid:96) ≤ T (cid:1) d(cid:96) + iV@R α,T ( P & L ) . (2.11) Remark 1. i) Combining Proposition 1 with Representation (2.5) leads to an important implication – both theintra-horizon value at risk and the intra-horizon expected shortfall can be fully characterized based onfirst-passage probabilities. Therefore, we will study, for
T > z ∈ R and (cid:96) ≤ u ( T , z ; (cid:96) ) := P z (cid:0) τ P & L , − (cid:96) ≤ T (cid:1) = E P z (cid:104) { τ P & L , − (cid:96) ≤T } (cid:105) with T ∈ [0 , T ] , (2.12)and subsequently recover intra-horizon measures of risk via numerical techniques. Here, T ∈ [0 , T ]refers to the remaining time to maturity and is linked to any pair of times ( t, T ) satisfying 0 ≤ t ≤ T via T = T − t .ii) Besides providing an important link to first-passage probabilities, Equation (2.11) formalizes the in-tuitive property that the intra-horizon expected shortfall always exceeds the intra-horizon value atrisk.iii) Although the intra-horizon risk approach considered in this article contributes to the theory of riskmeasures for stochastic processes, we emphasize that, conceptually, it substantially differs from othermethods belonging to this theory, especially from the methods of dynamic risk (and performance)measures (cf. [AP11], [BCZ14], [KR20]). In fact, while dynamic risk measures aim to quantify, for afixed maturity T > X T , at any point in time t ∈ [0 , T ] a conditionalrisk based on the current information F t , our intra-horizon notion of risk aims to statically englobe thefull dynamics within the time interval [0 , T ] under consideration. Nevertheless, since our intra-horizonrisk measurement approach is based on the theory initially developed for point-in-time risk measures,we believe that both concepts can be combined to introduce dynamic intra-horizon risk versions. Thiscould be part of future research. (cid:7) We next turn to a discussion of intra-horizon risk under infinitely divisible distributions, i.e., we fix an F -L´evy process ( X t ) t ∈ [0 ,T ] and consider two different scenarios:– Scenario 1 , where the dynamics of the (discounted) profit and loss process ( P & L t ) t ∈ [0 ,T ] are directlydescribed by ( X t ) t ∈ [0 ,T ] , i.e., where P & L t = X t , t ∈ [0 , T ] . – Scenario 2 , where the (discounted) profit and loss process ( P & L t ) t ∈ [0 ,T ] reflects the intrinsic value ofa long ( + ) or short ( − ) position in an asset of ordinary exponential L´evy type, i.e., where we have that P & L t = ± (cid:0) z e X t − z (cid:1) , t ∈ [0 , T ] , with z , z ∈ R +0 . We recall that a L´evy process ( X t ) t ≥ on a (filtered) probability space (Ω , F , F , P X ) is a c`adl`ag (right-continuous with left limits) process having independent and stationary increments. The L´evy exponentΨ X ( · ) is defined, for θ ∈ R , in terms of its characteristic triplet ( b X , σ X , Π X ) viaΨ X ( θ ) := − log (cid:16) E P X (cid:104) e iθX (cid:105)(cid:17) = − ib X θ + 12 σ X θ + (cid:90) R (cid:0) − e iθy + iθy {| y |≤ } (cid:1) Π X ( dy ) , (3.1)where E P X [ · ] indicates expectation under the measure P X . Well known results in the theory of L´evy processes(cf. [Sa99], [Ap09]) allow to decompose ( X t ) t ≥ in terms of its jump and diffusion parts as X t = b X t + σ X W t + (cid:90) R y ¯ N X ( t, dy ) , t ≥ , (3.2)where ( W t ) t ≥ denotes an F -Brownian motion and N X refers to an independent Poisson random measureon [0 , ∞ ) × R \ { } that has intensity measure given by Π X . Here, we use for any t ≥ A ∈ B ( R \ { } ) the following notation: N X ( t, A ) := N X ((0 , t ] × A ) , ˜ N X ( dt, dy ) := N X ( dt, dy ) − Π X ( dy ) dt, ¯ N X ( dt, dy ) := (cid:26) ˜ N X ( dt, dy ) , if | y | ≤ ,N X ( dt, dy ) , if | y | > . Additionally, we define the Laplace exponent of ( X t ) t ≥ , for any θ ∈ R satisfying E P X (cid:2) e θX (cid:3) < ∞ , via thefollowing identityΦ X ( θ ) := − Ψ X ( − iθ ) = b X θ + 12 σ X θ − (cid:90) R (cid:0) − e θy + θy {| y |≤ } (cid:1) Π X ( dy ) , (3.3)and recall that ( X t ) t ≥ has the (strong) Markov property. Therefore, its infinitesimal generator is a partialintegro-differential operator given, for sufficiently smooth V : [0 , ∞ ) × R → R , by A X V ( T , x ) := lim t ↓ E P X x (cid:2) V ( T , X t ) (cid:3) − V ( T , x ) t = 12 σ X ∂ x V ( T , x ) + b X ∂ x V ( T , x ) + (cid:90) R (cid:2) V ( T , x + y ) − V ( T , x ) − y {| y |≤ } ∂ x V ( T , x ) (cid:3) Π X ( dy ) , (3.4)where the expectation is taken under the measure P Xx having initial distribution X = x . We will extensivelymake use of these notations in the upcoming sections. Dealing with first-passage events in any of
Scenario 1 and
Scenario 2 clearly reduces to the consideration ofcorresponding events for simple L´evy processes. This follows from the properties of the exponential functionas well as from the fact that, under both scenarios, the process ( X t ) t ∈ [0 ,T ] is the only source of uncertainty.Consequently, we only need to study, for x ∈ R and L ∈ R , the first-passage probabilities defined by u ± X ( T , x ; L ) := P Xx (cid:0) τ X, ± L ≤ T (cid:1) = E P X x (cid:104) { τ X, ± L ≤T } (cid:105) for T ∈ [0 , T ] , (3.5)and note that (2.12) and (3.5) are related with each other by means of the following identity u ( T , z ; (cid:96) ) = u − X ( T , z ; (cid:96) ) , for Scenario 1 ,u − X (cid:0) T , log( z + z ); log( z + (cid:96) ) (cid:1) , for a long position in Scenario 2 ,u + X (cid:0) T , log( z − z ); log( z − (cid:96) ) (cid:1) , for a short position in Scenario 2 . (3.6)At this point, we should note that not all parameters z, (cid:96) ∈ R lead to sensible results when dealing with Scenario 2 . Therefore, under this scenario, the above formula should be always understood on the respectiveranges, i.e., we set log(0) := −∞ and only consider the following values for z, (cid:96) :i) z, (cid:96) ∈ [ − z , ∞ ) for a long position,ii) z, (cid:96) ∈ ( −∞ , z ] for a short position.The next lemma proves useful when dealing with intra-horizon risk in models with jumps and providessimple conditions on the Laplace exponent of the underlying L´evy process for which intra-horizon expectedshortfall measures are well-defined. A proof is provided in Appendix A. Lemma 1.
Let ( X t ) t ≥ be a L´evy process and assume that the following condition is satisfied: ∃ θ (cid:63) > E P X (cid:104) e θ (cid:63) | X | (cid:105) < ∞ . (3.7) Then, there exists a constant c > such that for any x ∈ R and T > we have lim (cid:96) ↑∞ e c · (cid:96) P Xx (cid:16) τ X, + (cid:96) ≤ T (cid:17) = 0 and lim (cid:96) ↓−∞ e − c · (cid:96) P Xx (cid:16) τ X, − (cid:96) ≤ T (cid:17) = 0 . (3.8) In particular, these convergence results ensure that E P z (cid:2) | I P & LT | (cid:3) < ∞ holds for any T > , or, equivalently,that the intra-horizon expected shortfall is well-defined under any of Scenario 1 and Scenario 2. Although Condition (3.7) slightly restricts the applicability of Lemma 1, many popular classes of L´evyprocesses encountered in financial applications satisfy this property, at least on a range of parameters thatis suitable for market modeling purposes. Important examples include hyper-exponential jump-diffusionmodels (cf. [Ko02], [Ca09]), Variance Gamma (VG) processes (cf. [MS90], [MCC98]), the Carr-Geman-Madan-Yor (CGMY) model (cf. [CGMY02]) as well as Normal Inverse Gaussian (NIG) processes (cf. [BN97]).We will deal with the intra-horizon risk inherent to some of these models in our numerical analysis ofSection 5.
Remark 2. i) A closer look at the proof of Lemma 1 reveals that under each of the scenarios under consideration,only one of the conditionslim (cid:96) ↑∞ e c · (cid:96) P Xx (cid:16) τ X, + (cid:96) ≤ T (cid:17) = 0 , or lim (cid:96) ↓−∞ e − c · (cid:96) P Xx (cid:16) τ X, − (cid:96) ≤ T (cid:17) = 0 , is sufficient to ensure that E P z (cid:2) | I P & LT | (cid:3) < ∞ holds for T >
0. Additionally, the proof of Lemma 1indicates that these properties are consequences of the corresponding one-sided conditions ∃ θ (cid:63) > E P X (cid:104) e θ (cid:63) X (cid:105) < ∞ , and ∃ θ (cid:63) < − E P X (cid:104) e θ (cid:63) X (cid:105) < ∞ , (3.9)respectively, which therefore even weaken the requirements on the dynamics of the process ( X t ) t ≥ .ii) When considering a long position in Scenario 2 , well-definedness of the intra-horizon expected shortfallis immediate. In this case, the condition E P z (cid:2) | I P & LT | (cid:3) < ∞ directly follows, for any T >
0, from thefact that I P & LT ≥ − z , i.e., that the maximal possible losses do not exceed the value z . We willdedicate our numerical analysis in Section 5 to exactly this scenario and investigate the intra-horizonrisk inherent to a long position in the S&P 500 index as well as in the Brent crude oil using weeklydata ranging from January 1990 to September 2020. (cid:7) For risk management purposes, it may be of great importance to further understand the structure of risk. Inparticular, one may want to know how often certain shortfall barriers are already exceeded at the time theyare breached. When dealing with market risk, this roughly reduces to the question of whether first-passageoccurrences are triggered by diffusion or by jumps. This question can be further investigated under thepresent L´evy framework and a first-passage decomposition can be obtained. This is discussed next. Here,we start from a similar approach to the one taken in [CCW09] and define, for any (cid:96) ∈ R , the following events E ± := (cid:110) X τ X, ± (cid:96) = (cid:96) (cid:111) and E ±J := (cid:110) X τ X, ± (cid:96) (cid:54) = (cid:96) (cid:111) , (3.10)i.e., we essentially decompose the first-passage times in events that are either triggered by the diffusion partof the process ( X t ) t ∈ [0 ,T ] or by jumps. Clearly, the first-passage events E +0 and E + J are disjoint and thesame additionally holds for E − and E −J . Hence, this allows us to obtain, for T ∈ [0 , T ], x ∈ R and (cid:96) ∈ R , adecomposition of the first-passage probabilities as u ± X ( T , x ; (cid:96) ) = u E ± X ( T , x ; (cid:96) ) + u E ±J X ( T , x ; (cid:96) ) , (3.11)where u E ± X ( T , x ; (cid:96) ) and u E ±J X ( T , x ; (cid:96) ) refer to the functions defined by u E ± X ( T , x ; (cid:96) ) := E P X x (cid:104) { τ X, ± (cid:96) ≤T } ∩ E ± (cid:105) and u E ±J X ( T , x ; (cid:96) ) := E P X x (cid:104) { τ X, ± (cid:96) ≤T } ∩ E ±J (cid:105) . (3.12)We now turn to an analysis of these first-passage probabilities and first aim to obtain a PIDE characterizationof the functions u E ± X ( · ) and u E ±J X ( · ). To this end, we define, for (cid:96) ∈ R , the following domains H + (cid:96) := ( (cid:96), ∞ ) and H − (cid:96) := ( −∞ , (cid:96) ) (3.13)and denote by H ± the closure of these sets in R . Under this notation, the next proposition is obtained byrelying on (strong) Markovian arguments. A proof is provided in Appendix B. Proposition 2.
For any level (cid:96) ∈ R , the first-passage probability contributed by the diffusion part, u E ± X ( · ) ,satisfies the following Cauchy problem: − ∂ T u E ± X ( T , x ; (cid:96) ) + A X u E ± X ( T , x ; (cid:96) ) = 0 , on ( T , x ) ∈ (0 , T ] × (cid:16) R \ H ± (cid:96) (cid:17) , (3.14) u E ± X ( T , x ; (cid:96) ) = 1 , on ( T , x ) ∈ [0 , T ] × { (cid:96) } , (3.15) u E ± X ( T , x ; (cid:96) ) = 0 , on ( T , x ) ∈ [0 , T ] × H ± (cid:96) , (3.16) u E ± X (0 , x ; (cid:96) ) = 0 , on x ∈ R \ H ± (cid:96) . (3.17) As we shall see in a moment, the same approach can be adopted for any strong Markov process that is quasi-left-continuous. We emphasize that this interpretation may not be (fully) correct in cases where τ X, ± (cid:96) = ∞ , however, that these cases aresubsequently excluded from our analysis, as seen in (3.12). Additionally, we note that for general jump dynamics parts of theevent E ± could be due to jumps and that it may be therefore more appropriate to speak of a first-passage decomposition inevents with and without overshoot. Nevertheless, since market models usually assume continuous jump distributions, i.e., anintensity measure Π X of the form Π X ( dy ) = π X ( y ) dy , with appropriate jump density π X ( · ), this situation will not occur. Thisjustifies the use of our initial terminology. Similarly, the first-passage probability contributed by jumps, u E ±J X ( · ) , solves, for any (cid:96) ∈ R , the Cauchyproblem − ∂ T u E ±J X ( T , x ; (cid:96) ) + A X u E ±J X ( T , x ; (cid:96) ) = 0 , on ( T , x ) ∈ (0 , T ] × (cid:16) R \ H ± (cid:96) (cid:17) , (3.18) u E ±J X ( T , x ; (cid:96) ) = 0 , on ( T , x ) ∈ [0 , T ] × { (cid:96) } , (3.19) u E ±J X ( T , x ; (cid:96) ) = 1 , on ( T , x ) ∈ [0 , T ] × H ± (cid:96) , (3.20) u E ±J X (0 , x ; (cid:96) ) = 0 , on x ∈ R \ H ± (cid:96) . (3.21)Combining the characterization in Proposition 2 with (standard) numerical techniques already allows fora numerical treatment of the functions u E ± X ( · ) and u E ±J X ( · ). Furthermore, the proof of Proposition 2 revealsthat our derivations are not restricted to the L´evy framework. Indeed, while the decomposition in (3.11) isa simple consequence of the disjointness of the sets defined in (3.10), the proof of Proposition 2 combines(strong) Markovian arguments with the quasi-left-continuity of the process ( X t ) t ∈ [0 ,T ] . As a consequence,relying on this approach is always possible when dealing with processes that satisfy these two properties anda disentanglement of diffusion and jump contributions can be then obtained via the exact same techniques. Remark 3.
In general, the above techniques can be applied to subsequently recover intra-horizon risk measures as wellas corresponding risk contributions. However, even when dealing with the intra-horizon value at risk to asingle level α ∈ (0 ,
1) numerous iterations of the numerical scheme are needed and the computational costsquickly become high. Therefore, relying on these techniques for expected shortfall measures does not seemto be the best approach. Instead, distributional properties inherent to certain distributions sometimes allowto simplify the problem by switching to maturity-randomization. This holds for instance true when dealingwith hyper-exponential jump-diffusion processes that have the particularity to allow for arbitrarily closeapproximations of L´evy processes with completely monotone jumps (cf. [JP10], [CK11], [HK16]). A discus-sion of this approach as well as of approximations of L´evy densities via hyper-exponential jump densities isprovided in the upcoming sections. (cid:7)
We next deal with maturity-randomized first-passage probabilities. To this end, we start by defining for anyfunction g : R + → R satisfying ∞ (cid:90) e − ϑt | g ( t ) | dt < ∞ , ∀ ϑ > , (3.22)the Laplace-Carson transform LC ( g )( · ) via LC ( g )( ϑ ) := ∞ (cid:90) ϑe − ϑt g ( t ) dt, (3.23)1and note that this transform has several desirable properties. First, the Laplace-Carson transform merelycorresponds to a scaled Laplace transform, for which extensive inversion techniques exist (cf. [Co07]). Addi-tionally, as we shall see in a moment, applying the Laplace-Carson transform in the context of mathematicalfinance allows to randomize the maturity of financial contracts, i.e., to switch from objects with deterministicmaturity to corresponding objects with stochastic (exponentially distributed) maturity. This last propertyoffers a range of alternative ways to tackle problems related to the valuation of financial positions and hastherefore led to a wide adoption of the Laplace-Carson transform in the option pricing literature, with [Ca98]being one of the seminal articles in this context.Having computed the transform (either numerically or analytically), the original function g ( · ) can be recov-ered from LC ( g )( · ) using an inversion algorithm. One possible choice is the Gaver-Stehfest algorithm thathas the particularity to allow for an inversion of the transform on the real line and that has been successfullyused by several authors for option pricing (cf. [KW03], [Ki10], [HM13], [LV17], [CV18]). We will also relyon this algorithm, i.e., we set g N ( t ) := N (cid:88) k =1 ζ k,N LC (cid:0) g (cid:1) (cid:18) k log(2) t (cid:19) , N ∈ N , t > , (3.24)where the coefficients are given by ζ k,N := ( − N + k k min { k,N } (cid:88) j = (cid:98) ( k +1) / (cid:99) j N +1 N ! (cid:18) Nj (cid:19)(cid:18) jj (cid:19)(cid:18) jk − j (cid:19) , N ∈ N , ≤ k ≤ N, (3.25)with (cid:98) a (cid:99) := sup { z ∈ Z : z ≤ a } , and will recover the original function g ( · ) by means of the following relationlim N →∞ g N ( t ) = g ( t ) . (3.26)More details on the Gaver-Stehfest algorithm as well as formal proofs of the convergence result (3.26) for“sufficiently well-behaved functions” are provided in [Ku13] and references therein.We now turn to a discussion of Laplace-Carson transformed first-passage probabilities. First, we notethat the boundedness of the functions u ± X ( · ) and u E X ( · ) for E ∈ {E ± , E ±J } ensures that these first-passageprobabilities satisfy Condition (3.22) and so that the resulting Laplace-Carson transform is well-defined.Additionally, one easily sees that the first-passage decompositions obtained in (3.11) are preserved underthe Laplace-Carson operator, i.e., we have for any ϑ > x ∈ R and (cid:96) ∈ R that LC (cid:0) u ± X (cid:1) ( ϑ, x ; (cid:96) ) = LC (cid:0) u E ± X (cid:1) ( ϑ, x ; (cid:96) ) + LC (cid:0) u E ±J X (cid:1) ( ϑ, x ; (cid:96) ) . (3.27)This property is particularly interesting since it implies that switching back and forth between the originalfirst-passage probabilities and their corresponding Laplace-Carson transforms does not alter the structureof risk across the diffusion and jump parts and therefore allows us to fully concentrate on one or theother. Finally, any of the Laplace-Carson transformed first-passage probabilities can be interpreted as theprobability of a respective first-passage occurring before an independent exponentially distributed randomtime of intensity ϑ > T ϑ , has expired or equivalently before the first jump time of an independent Poisson2process ( N t ) t ≥ having intensity ϑ > LC (cid:0) u ± X (cid:1) ( ϑ, x ; (cid:96) ) = ∞ (cid:90) ϑe − ϑt u ± X ( t, x ; (cid:96) ) dt = E P X x (cid:104) E P X x (cid:104) { τ X, ± (cid:96) ≤ T ϑ } (cid:12)(cid:12)(cid:12) T ϑ (cid:105) (cid:105) = E P X x (cid:104) { τ X, ± (cid:96) ≤ T ϑ } (cid:105) , (3.28) LC (cid:0) u E X (cid:1) ( ϑ, x ; (cid:96) ) = ∞ (cid:90) ϑe − ϑt u E X ( t, x ; (cid:96) ) dt = E P X x (cid:104) E P X x (cid:104) { τ X, ± (cid:96) ≤ T ϑ } ∩ E (cid:12)(cid:12)(cid:12) T ϑ (cid:105) (cid:105) = E P X x (cid:104) { τ X, ± (cid:96) ≤ T ϑ } ∩ E (cid:105) , (3.29)where E ∈ (cid:8) E ± , E ±J (cid:9) . Consequently, any application of the Laplace-Carson operator transforms (in thiscontext) first-passage probabilities into corresponding maturity-randomized quantities and combining theseproperties with arguments similarly used in the proof of Proposition 2 allows us to obtain an OIDE character-ization of the maturity-randomized first-passage probabilities contributed by the diffusion part, LC (cid:0) u E ± X (cid:1) ( · ),and by jumps, LC (cid:0) u E ±J X (cid:1) ( · ). This is the content of the next proposition, whose proof is provided in Ap-pendix B. Proposition 3.
For any level (cid:96) ∈ R and intensity ϑ > , the maturity-randomized first-passage probabilitycontributed by the diffusion part, LC (cid:0) u E ± X (cid:1) ( · ) , satisfies the following Cauchy problem: A X LC (cid:0) u E ± X (cid:1) ( ϑ, x ; (cid:96) ) = ϑ LC (cid:0) u E ± X (cid:1) ( ϑ, x ; (cid:96) ) , on x ∈ R \ H ± (cid:96) , (3.30) LC (cid:0) u E ± X (cid:1) ( ϑ, x ; (cid:96) ) = 1 , on x = (cid:96), (3.31) LC (cid:0) u E ± X (cid:1) ( ϑ, x ; (cid:96) ) = 0 , on x ∈ H ± (cid:96) . (3.32) Similarly, the maturity-randomized first-passage probability contributed by jumps, LC (cid:0) u E ±J X (cid:1) ( · ) , solves, forany (cid:96) ∈ R and ϑ > , the Cauchy problem A X LC (cid:0) u E ±J X (cid:1) ( ϑ, x ; (cid:96) ) = ϑ LC (cid:0) u E ±J X (cid:1) ( ϑ, x ; (cid:96) ) , on x ∈ R \ H ± (cid:96) , (3.33) LC (cid:0) u E ±J X (cid:1) ( ϑ, x ; (cid:96) ) = 0 , on x = (cid:96), (3.34) LC (cid:0) u E ±J X (cid:1) ( ϑ, x ; (cid:96) ) = 1 , on x ∈ H ± (cid:96) . (3.35)Compared with the results in Proposition 2, Proposition 3 offers substantially simpler characterizations. Inparticular, applying the Laplace-Carson operator to the first-passage probabilities reduces the complexityof the respective problems by transforming the PIDE characterizations of Proposition 2 into correspondingOIDE characterizations. Under certain L´evy dynamics ( X t ) t ≥ the resulting problems (3.30)-(3.32) and(3.33)-(3.35) even have a simple analytical solution. This is in particular true for the class of hyper-exponential jump-diffusions that is discussed in Section 3.4. The analysis developed in the previous sections provided a decomposition of diffusion and jump contribu-tions embodied in first-passage probabilities. Since both the intra-horizon value at risk and the intra-horizon3expected shortfall can be fully characterized based on first-passage probabilities (cf. Section 2), these lastresults can be further extended to infer diffusion and jump risk contributions to the intra-horizon risk mea-sures under consideration. This is discussed next.We start by introducing risk contributions for the intra-horizon value at risk. Here, we follow the ideasin [LV20] and understand the diffusion and jump risk contributions as the proportions of the iV@R -first-passage probability contributed by the respective components, i.e., we define, for α ∈ (0 ,
1) and a (dis-counted) profit and loss process ( P & L t ) t ∈ [0 ,T ] satisfying the dynamics specified in either Scenario 1 or Scenario 2 , the diffusion and jump risk contribution inherent to the level- α intra-horizon value at risk overthe time interval [0 , T ], R D iV @ R ( P & L ; α, T ) and R J iV @ R ( P & L ; α, T ) respectively, via R D iV @ R ( P & L ; α, T ) := u E− X ( T, z ; − iV @ R α,T ( P & L ) ) u − X ( T, z ; − iV @ R α,T ( P & L ) ) , for Scenario 1 , u E− X ( T, log( z + z ); log ( z − iV @ R α,T ( P & L ) )) u − X ( T, log( z + z ); log ( z − iV @ R α,T ( P & L ) )) , for a long position in Scenario 2 , u E +0 X ( T, log( z − z ); log ( z + iV @ R α,T ( P & L ) )) u + X ( T, log( z − z ); log ( z + iV @ R α,T ( P & L ) )) , for a short position in Scenario 2 , (3.36)and R J iV @ R ( P & L ; α, T ) := u E−J X ( T, z ; − iV @ R α,T ( P & L ) ) u − X ( T, z ; − iV @ R α,T ( P & L ) ) , for Scenario 1 , u E−J X ( T, log( z + z ); log ( z − iV @ R α,T ( P & L ) )) u − X ( T, log( z + z ); log ( z − iV @ R α,T ( P & L ) )) , for a long position in Scenario 2 , u E + J X ( T, log( z − z ); log ( z + iV @ R α,T ( P & L ) )) u + X ( T, log( z − z ); log ( z + iV @ R α,T ( P & L ) )) , for a short position in Scenario 2 . (3.37)Defining risk contributions embodied in the intra-horizon expected shortfall can be done via similar tech-niques and is closely linked to the computation of risk contributions for the intra-horizon value at risk.Indeed, from Proposition 1 we already know that (given the intra-horizon value at risk to a certain level)the difference between intra-horizon expected shortfall and intra-horizon value at risk consists in an integralover first-passage probabilities that is given by iES α,T ( P & L ) − iV@R α,T ( P & L ) = 1 α − iV @ R α,T ( P & L ) (cid:90) −∞ u ( T, z ; (cid:96) ) d(cid:96), (3.38)where u ( T, z ; (cid:96) ) is specified in each scenario according to Relation (3.6). Therefore, the diffusion and jumprisk contributions inherent to the intra-horizon expected shortfall can be divided into two parts: the re-spective risk contributions in the intra-horizon value at risk and those of the remaining integral (3.38). Forthe latter – which can be interpreted as an average conditional excess intra-horizon tail loss – diffusion andjump risk contributions can be defined as the proportions of the integral contributed by the respective com-ponents, i.e., one recovers, for α ∈ (0 ,
1) and a (discounted) profit and loss process ( P & L t ) t ∈ [0 ,T ] satisfyingthe dynamics specified in either Scenario 1 or Scenario 2 , the diffusion and jump risk contribution inherent4to the integral part (3.38), R DI ( P & L ; α, T ) and R JI ( P & L ; α, T ) respectively, via R DI ( P & L ; α, T ) = − iV @ Rα,T ( P & L ) (cid:82) −∞ u E− X ( T, z ; (cid:96) ) d(cid:96) − iV @ Rα,T ( P & L ) (cid:82) −∞ u − X ( T, z ; (cid:96) ) d(cid:96) , for Scenario 1 , − iV @ Rα,T ( P & L ) (cid:82) − z u E− X ( T, log( z + z ); log( z + (cid:96) )) d(cid:96) − iV @ Rα,T ( P & L ) (cid:82) − z u − X ( T, log( z + z ); log( z + (cid:96) )) d(cid:96) , for a long position in Scenario 2 , − iV @ Rα,T ( P & L ) (cid:82) −∞ u E +0 X ( T, log( z − z ); log( z − (cid:96) )) d(cid:96) − iV @ Rα,T ( P & L ) (cid:82) −∞ u + X ( T, log( z − z ); log( z − (cid:96) )) d(cid:96) , for a short position in Scenario 2 , (3.39)and R JI ( P & L ; α, T ) = − iV @ Rα,T ( P & L ) (cid:82) −∞ u E−J X ( T, z ; (cid:96) ) d(cid:96) − iV @ Rα,T ( P & L ) (cid:82) −∞ u − X ( T, z ; (cid:96) ) d(cid:96) , for Scenario 1 , − iV @ Rα,T ( P & L ) (cid:82) − z u E−J X ( T, log( z + z ); log( z + (cid:96) )) d(cid:96) − iV @ Rα,T ( P & L ) (cid:82) − z u − X ( T, log( z + z ); log( z + (cid:96) )) d(cid:96) , for a long position in Scenario 2 , − iV @ Rα,T ( P & L ) (cid:82) −∞ u E + J X ( T, log( z − z ); log( z − (cid:96) )) d(cid:96) − iV @ Rα,T ( P & L ) (cid:82) −∞ u + X ( T, log( z − z ); log( z − (cid:96) )) d(cid:96) , for a short position in Scenario 2 . (3.40)Finally, using these definitions, the diffusion and jump risk contributions inherent to the level- α intra-horizonexpected shortfall over the time interval [0 , T ], R D iES ( P & L ; α, T ) and R J iES ( P & L ; α, T ) respectively, can berecovered as weighted sums of the corresponding contributions for the intra-horizon value at risk and theintegral part (3.38), i.e., as R D iES ( P & L ; α, T ) = (cid:0) − ω α,T ( P & L ) (cid:1) · R DI ( P & L ; α, T ) + ω α,T ( P & L ) · R D iV @ R ( P & L ; α, T ) , (3.41) R J iES ( P & L ; α, T ) = (cid:0) − ω α,T ( P & L ) (cid:1) · R JI ( P & L ; α, T ) + ω α,T ( P & L ) · R J iV @ R ( P & L ; α, T ) , (3.42)where ω α,T ( P & L ) := iV @ R α,T ( P & L ) iES α,T ( P & L ) denotes the contribution of the intra-horizon value at risk to the intra-horizon expected shortfall. We will come back to this decomposition when discussing numerical results inSection 5. Having elaborated on our core intra-horizon risk measurement approach under the general L´evy framework,we next discuss (semi-)analytical expressions for hyper-exponential jump-diffusion processes. These resultsare particularly interesting since they subsequently allow for an approximate, though arbitrarily precisesemi-analytical measurement of intra-horizon risk within the important class of L´evy processes having acompletely monotone jump density. We will further develop this point in Section 4 and lastly provide anapplication of this approximate approach in the numerical analysis of Section 5.5
We recall that a hyper-exponential jump-diffusion process ( X t ) t ≥ is a L´evy process that combines a Brow-nian diffusion with hyper-exponentially distributed jumps. This process has the usual jump-diffusion struc-ture, i.e., it can be characterized on a filtered probability space (Ω , F , F , P X ) via X t = µt + σ X W t + N t (cid:88) i =1 J i , for t ≥ , (3.43)where ( W t ) t ≥ denotes an F -Brownian motion and ( N t ) t ≥ is an F -Poisson process that has intensity pa-rameter λ >
0. The constants µ ∈ R and σ X ≥ J i ) i ∈ N are assumed to be independent of ( N t ) t ≥ and to form asequence of independent and identically distributed random variables following a hyper-exponential distri-bution, i.e., their (common) density function f J ( · ) is given by f J ( y ) = m (cid:88) i =1 p i ξ i e − ξ i y { y ≥ } + n (cid:88) j =1 q j η j e η j y { y< } , (3.44)where p i > ξ i > i ∈ { , . . . , m } and q j > η j > j ∈ { , . . . , n } . Here, the parameters( p i ) i ∈{ ,...,m } and ( q j ) j ∈{ ,...,n } represent the proportion of jumps that are attributed to particular jump typesand are therefore assumed to satisfy the condition m (cid:80) i =1 p i + n (cid:80) j =1 q j = 1. Finally, we will always assume thatthe intensity parameters ( ξ i ) i ∈{ ,...,m } and ( η j ) j ∈{ ,...,n } are ordered in the sense that ξ < ξ < · · · < ξ m and η < η < · · · < η n (3.45)and note that this does not consist in a loss of generality.As special class of L´evy processes, hyper-exponential jump-diffusions can be equivalently characterized interms of their L´evy triplet (cid:0) b X , σ X , Π X (cid:1) , where b X and Π X are then obtained as b X := µ + (cid:90) {| y |≤ } y Π X ( dy ) and Π X ( dy ) := λf J ( y ) dy. (3.46)Using these results, their L´evy exponent, Ψ X ( · ), is easily obtained via (3.1), asΨ X ( θ ) = − iµθ + 12 σ X θ − λ m (cid:88) i =1 p i ξ i ξ i − iθ + n (cid:88) j =1 q j η j η j + iθ − . (3.47)Similarly, the corresponding Laplace exponent, Φ X ( · ), is well-defined for θ ∈ ( − η , ξ ) and equalsΦ X ( θ ) = µθ + 12 σ X θ + λ m (cid:88) i =1 p i ξ i ξ i − θ + n (cid:88) j =1 q j η j η j + θ − . (3.48)In what follows, we will consider the Laplace exponent as standalone function on the extended real domainΦ X : R \ { ξ , . . . , ξ m , − η , . . . , − η n } → R . This quantity will play a central role in the upcoming deriva-tions. In fact, many distributional properties of hyper-exponential jump-diffusion processes (and of their6generalizations) are closely linked to the roots of the equation Φ X ( θ ) = α , for α ≥
0. This was already usedin diverse articles dealing with option pricing and risk management within the class of mixed-exponentialjump-diffusion processes (cf. among others [Ca09], [CCW09], [CK11], [CK12]). In this context, the followingimportant lemma was partly derived in [Ca09] under hyper-exponential jump-diffusion models. Since theproof of all the remaining statements do not substantially differ from the results derived in [Ca09], the readeris referred to the arguments provided in this article.
Lemma 2.
For Φ X ( · ) defined as in (3.48) and any α > , the following holds:1. If σ X (cid:54) = 0 , the equation Φ X ( θ ) = α has ( m + n + 2) real roots β ,α , . . . , β m +1 ,α and γ ,α , . . . , γ n +1 ,α that satisfy −∞ < γ n +1 ,α < − η n < γ n,α < − η n − < · · · < γ ,α < − η < γ ,α < , (3.49)0 < β ,α < ξ < β ,α < · · · < ξ m − < β m,α < ξ m < β m +1 ,α < ∞ . (3.50)
2. If σ X = 0 and µ (cid:54) = 0 the equation Φ X ( θ ) = α has ( m + n + 1) real roots. Specifically, • if µ > , there are m + 1 positive roots β ,α , . . . , β m +1 ,α and n negative roots γ ,α , . . . , γ n,α thatsatisfy −∞ < − η n < γ n,α < − η n − < · · · < γ ,α < − η < γ ,α < , (3.51)0 < β ,α < ξ < β ,α < · · · < ξ m − < β m,α < ξ m < β m +1 ,α < ∞ . (3.52) • if µ < , there are m positive roots β ,α , . . . , β m,α and n + 1 negative roots γ ,α , . . . , γ n +1 ,α thatsatisfy −∞ < γ n +1 ,α < − η n < γ n,α < − η n − < · · · < γ ,α < − η < γ ,α < , (3.53)0 < β ,α < ξ < β ,α < · · · < ξ m − < β m,α < ξ m < ∞ . (3.54)
3. If σ X = 0 and µ = 0 the equation Φ X ( θ ) = α has ( m + n ) real roots β ,α , . . . , β m,α and γ ,α , . . . , γ n,α that satisfy −∞ < − η n < γ n,α < − η n − < · · · < γ ,α < − η < γ ,α < , (3.55)0 < β ,α < ξ < β ,α < · · · < ξ m − < β m,α < ξ m < ∞ . (3.56)At this point, we should mention that the roots in Lemma 2 are only known in analytical form in very fewcases. Nevertheless, this does not impact the importance and practicability of Lemma 2 since the roots canbe anyway recovered using standard numerical techniques. We turn back to the OIDE characterizations of Proposition 3 and consider the respective problems (3.30)-(3.32) and (3.33)-(3.35) under hyper-exponential jump-diffusion processes with non-zero volatility parameter σ X (cid:54) = 0. Switching to the case where σ X = 0 does not fundamentally change the approach and only few,slight adaptions are needed. We will address some of these adaptions in Section 4, when discussing hyper-exponential jump-diffusion approximations to infinite-activity pure jump processes.To start, we note that the infinitesimal generator (3.4) simplifies in this case to A X V ( T , x ) = 12 σ X ∂ x V ( T , x ) + µ∂ x V ( T , x ) + λ (cid:90) R (cid:0) V ( T , x + y ) − V ( T , x ) (cid:1) f J ( y ) dy, (3.57)7which allows us, together with the properties of the hyper-exponential density f J ( · ), to uniquely solveProblems (3.30)-(3.32) and (3.33)-(3.35) and to derive closed-form expressions for the maturity-randomizedfirst-passage probabilities LC (cid:0) u E ± X (cid:1) ( · ) and LC (cid:0) u E ±J X (cid:1) ( · ). Specifically, we define for any ϑ > m + 1) × ( m + 1) and ( n + 1) × ( n + 1) matrices A ϑ and A ϑ respectively via A ϑ := . . . ξ ξ − β ,ϑ ξ ξ − β ,ϑ · · · ξ ξ − β m +1 ,ϑ ξ ξ − β ,ϑ ξ ξ − β ,ϑ · · · ξ ξ − β m +1 ,ϑ ... ... . . . ... ξ m ξ m − β ,ϑ ξ m ξ m − β ,ϑ · · · ξ m ξ m − β m +1 ,ϑ , A ϑ := . . . η η + γ ,ϑ η η + γ ,ϑ · · · η η + γ n +1 ,ϑ η η + γ ,ϑ η η + γ ,ϑ · · · η η + γ n +1 ,ϑ ... ... . . . ... η n η n + γ ,ϑ η n η n + γ ,ϑ · · · η n η n + γ n +1 ,ϑ , (3.58)and observe that these matrices are invertible. Additionally, we denote for any k ∈ N the 1 × k vectors ofzeros and ones as k := (0 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) k ) and k := (1 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) k ) . (3.59)Using the above notation, the following proposition can be derived. The proof is presented in Appendix C. Proposition 4.
Assume that ( X t ) t ≥ follows a hyper-exponential jump-diffusion process with non-zero dif-fusion component, as described in (3.43), (3.44) with σ X (cid:54) = 0 . Then, for any level (cid:96) ∈ R and intensity ϑ > ,the maturity-randomized upside first-passage probabilities LC (cid:0) u E +0 X (cid:1) ( · ) and LC (cid:0) u E + J X (cid:1) ( · ) take the form LC (cid:0) u E +0 X (cid:1) ( ϑ, x ; (cid:96) ) = , x > (cid:96), , x = (cid:96), m +1 (cid:80) k =1 v ,k e β k,ϑ · ( x − (cid:96) ) , x < (cid:96), and LC (cid:0) u E + J X (cid:1) ( ϑ, x ; (cid:96) ) = , x > (cid:96), , x = (cid:96), m +1 (cid:80) k =1 v J ,k e β k,ϑ · ( x − (cid:96) ) , x < (cid:96). (3.60) Here, v := (cid:0) v , , . . . , v ,m +1 (cid:1) (cid:124) and v J := (cid:0) v J , , . . . , v J ,m +1 (cid:1) (cid:124) are weight vectors uniquely determined bythe system of linear equations A ϑ v = (1 , m ) (cid:124) and A ϑ v J = (0 , m ) (cid:124) . (3.61) Similarly, for (cid:96) ∈ R and ϑ > , the maturity-randomized downside first-passage probabilities LC (cid:0) u E − X (cid:1) ( · ) and LC (cid:0) u E −J X (cid:1) ( · ) are given by LC (cid:0) u E − X (cid:1) ( ϑ, x ; (cid:96) ) = n +1 (cid:80) k =1 v ,k e γ k,ϑ · ( x − (cid:96) ) , x > (cid:96), , x = (cid:96), , x < (cid:96), and LC (cid:0) u E −J X (cid:1) ( ϑ, x ; (cid:96) ) = n +1 (cid:80) k =1 v J ,k e γ k,ϑ · ( x − (cid:96) ) , x > (cid:96), , x = (cid:96), , x < (cid:96), (3.62) where, v := ( v , , . . . , v ,n +1 ) (cid:124) and v J := ( v J , , . . . , v J ,n +1 ) (cid:124) are uniquely determined by the system oflinear equations A ϑ v = (1 , n ) (cid:124) and A ϑ v J = (0 , n ) (cid:124) . (3.63) The invertibility of A ϑ and A ϑ can be proved as in [CK11] and the reader is referred to this article. i ∈ { , . . . , m } and j ∈ { , . . . , n } the following jump-events E + i := (cid:26) X τ X, + (cid:96) > (cid:96), J N τX, + (cid:96) ∼ Exp( ξ i ) (cid:27) and E − j := (cid:26) X τ X, − (cid:96) < (cid:96), J N τX, − (cid:96) ∼ Exp( η j ) (cid:27) , (3.64)and see that E + J \ (cid:16) { X > (cid:96) } ∪ { τ X, + (cid:96) = ∞} (cid:17) = m (cid:91) i =1 E + i and E −J \ (cid:16) { X < (cid:96) } ∪ { τ X, − (cid:96) = ∞} (cid:17) = n (cid:91) j =1 E − j , (3.65)i.e., we essentially decompose the first-passage events contributed by jumps in events that are triggered byjumps of certain types. Additionally, we note that the first-passage events E + i and E − j for i ∈ { , . . . , m } and j ∈ { , . . . , n } are disjoint among each others. In particular, this gives that the first-passage probabilitiescontributed by jumps, u E ±J X ( · ), and the respective maturity-randomized quantities, LC (cid:0) u E ±J X (cid:1) ( · ), have, for (cid:96) ∈ R , x ∈ R \ H ± (cid:96) , and T ∈ [0 , T ] and ϑ > u E + J X ( T , x ; (cid:96) ) = m (cid:88) i =1 u E + i X ( T , x ; (cid:96) ) , u E −J X ( T , x ; (cid:96) ) = n (cid:88) j =1 u E − j X ( T , x ; (cid:96) ) , (3.66) LC (cid:0) u E + J X (cid:1) ( ϑ, x ; (cid:96) ) = m (cid:88) i =1 LC (cid:0) u E + i X (cid:1) ( ϑ, x ; (cid:96) ) , LC (cid:0) u E −J X (cid:1) ( ϑ, x ; (cid:96) ) = n (cid:88) j =1 LC (cid:0) u E − j X (cid:1) ( ϑ, x ; (cid:96) ) , (3.67)where u E + i X ( T , x ; (cid:96) ) for i ∈ { , . . . , m } and u E − j X ( T , x ; (cid:96) ) for j ∈ { , . . . , n } refer to the functions defined by u E + i X ( T , x ; (cid:96) ) := E P X x (cid:104) { τ X, + (cid:96) ≤T } ∩ E + i (cid:105) and u E − j X ( T , x ; (cid:96) ) := E P X x (cid:104) { τ X, − (cid:96) ≤T } ∩ E − j (cid:105) . (3.68)Next, we note as in [KW03] that the monotonicity of the cumulative distribution functions t (cid:55)→ u E X ( t, x ; (cid:96) ) for E ∈ {E ± , E ±J , E +1 , . . . , E + m , E − , . . . , E − n } implies that we can rewrite each of the maturity-randomized versions LC ( u E X )( · ), for (cid:96) ∈ R , x ∈ R \ H ± (cid:96) and ϑ >
0, in the form LC ( u E X )( ϑ, x ; (cid:96) ) := ∞ (cid:90) ϑe − ϑt u E X ( t, x ; (cid:96) ) dt = ∞ (cid:90) e − ϑt ∂ t u E X ( t, x ; (cid:96) ) dt (3.69)= ∞ (cid:90) e − ϑt u E X ( dt, x ; (cid:96) ) = E P X x (cid:104) e − ϑ τ X, ± (cid:96) E (cid:105) , where τ X, ± (cid:96) = τ X, + (cid:96) for E ∈ {E +1 , . . . , E + m } and τ X, ± (cid:96) = τ X, − (cid:96) for E ∈ {E − , . . . , E − n } . Combining this repre-sentation with the fact that the overshoot distribution is conditionally memoryless and independent of thefirst-passage time, given that the overshoot is greater than zero and that the exponential type of the jumpdistribution is specified (cf. [Ca09]), finally allows us to arrive at the next proposition. The proof is providedin Appendix C.9 Proposition 5.
Assume that ( X t ) t ≥ follows a hyper-exponential jump-diffusion process with non-zero dif-fusion component, as described in (3.43), (3.44) with σ X (cid:54) = 0 and define for any level (cid:96) ∈ R , x ∈ R andintensity ϑ > the following vectors LC ϑ,(cid:96) ( x ) := (cid:16) LC ( u E +0 X )( ϑ, x ; (cid:96) ) , . . . , LC ( u E + m X )( ϑ, x ; (cid:96) ) (cid:17) (cid:124) , e ϑ,(cid:96) ( x ) := (cid:0) e β ,ϑ · ( x − (cid:96) ) , . . . , e β m +1 ,ϑ · ( x − (cid:96) ) (cid:1) (cid:124) , (3.70) LC ϑ,(cid:96) ( x ) := (cid:16) LC ( u E − X )( ϑ, x ; (cid:96) ) , . . . , LC ( u E − n X )( ϑ, x ; (cid:96) ) (cid:17) (cid:124) , e ϑ,(cid:96) ( x ) := (cid:0) e γ ,ϑ · ( x − (cid:96) ) , . . . , e γ n +1 ,ϑ · ( x − (cid:96) ) (cid:1) (cid:124) . (3.71) Then, for x ∈ R \ H + (cid:96) , the vector of maturity-randomized upside jump contributions, LC ϑ,(cid:96) ( x ) , is uniquelydetermined by the system of linear equations A (cid:124) ϑ LC ϑ,(cid:96) ( x ) = e ϑ,(cid:96) ( x ) . (3.72) Similarly, for x ∈ R \ H − (cid:96) the vector of maturity-randomized downside jump contributions, LC ϑ,(cid:96) ( x ) , isuniquely determined by the system of linear equations A (cid:124) ϑ LC ϑ,(cid:96) ( x ) = e ϑ,(cid:96) ( x ) . (3.73)Proposition 5 provides an implicit characterization of diffusion and jump contributions underlying (maturity-randomized) first-passage probabilities and already allows for a derivation of the full vectors LC ϑ,(cid:96) ( x ) or LC ϑ,(cid:96) ( x ) using standard numerical methods. Nevertheless, the systems (3.72) and (3.73) can be explicitlysolved to derive analytical expressions for each of the functions LC ( u E X )( · ) with E ∈ {E +0 , E +1 , . . . , E + m } or E ∈ {E − , E − , . . . , E − n } . This was already derived in a different context in [CYY13]. In particular, theirresults can be refined to arrive at the following useful proposition. Proposition 6.
Assume that ( X t ) t ≥ follows a hyper-exponential jump-diffusion process with non-zero dif-fusion component, as described in (3.43), (3.44) with σ X (cid:54) = 0 . Then, for any level (cid:96) ∈ R , x ∈ R \ H + (cid:96) andintensity ϑ > we have that LC (cid:0) u E + i X (cid:1) ( ϑ, x ; (cid:96) ) = m +1 (cid:88) k =1 v E + i ,k · e β k,ϑ · ( x − (cid:96) ) , i ∈ { , . . . , m } , (3.74) where, for k ∈ { , . . . , m + 1 } , the coefficients v E + i ,k can be expressed in terms of the coefficients v ,k by v E + i , = − ξ i C + ϑ ( ξ i ) (cid:0) B + (cid:1) (cid:48) ( ξ i ) v , , and v E + i ,k = − ξ i C + ϑ ( ξ i ) (cid:0) B + (cid:1) (cid:48) ( ξ i ) v ,k + 1 ξ i d + k,i , k ∈ { , . . . , m + 1 } , (3.75) with B + ( x ) := m (cid:89) s =1 (cid:0) ξ s − x (cid:1) , C + ϑ ( x ) := m +1 (cid:89) s =2 (cid:0) β s,ϑ − x (cid:1) , (3.76) and d + i,j := − B + ( β i,ϑ ) C + ϑ ( ξ j )( ξ j − β i,ϑ ) (cid:0) B + (cid:1) (cid:48) ( ξ j ) (cid:0) C + ϑ (cid:1) (cid:48) ( β i,ϑ ) , i ∈ { , . . . , m + 1 } , j ∈ { , . . . , m } . (3.77) Similarly, for (cid:96) ∈ R , x ∈ R \ H − (cid:96) and ϑ > we have that LC (cid:0) u E − j X (cid:1) ( ϑ, x ; (cid:96) ) = n +1 (cid:88) k =1 v E − j ,k · e γ k,ϑ · ( x − (cid:96) ) , j ∈ { , . . . , n } , (3.78)0 where the coefficients v E − j ,k are given, for k ∈ { , . . . , n + 1 } , by v E − j , = ( − n η j C − ϑ ( η j ) (cid:0) B − (cid:1) (cid:48) ( − η j ) v , , and v E − j ,k = ( − n η j C − ϑ ( η j ) (cid:0) B − (cid:1) (cid:48) ( − η j ) v ,k + 1 η j d − k,j , k ∈ { , . . . , n +1 } , (3.79) with B − ( x ) := n (cid:89) s =1 (cid:0) η s + x (cid:1) , C − ϑ ( x ) := n +1 (cid:89) s =2 (cid:0) γ s,ϑ + x (cid:1) , (3.80) and d − i,j := B − ( γ i,ϑ ) C − ϑ ( η j )( η j + γ i,ϑ ) (cid:0) B − (cid:1) (cid:48) ( − η j ) (cid:0) C − ϑ (cid:1) (cid:48) ( − γ i,ϑ ) , i ∈ { , . . . , n + 1 } , j ∈ { , . . . , n } . (3.81) Remark 4. i) We re-emphasize that the (full) results in Proposition 4, Proposition 5 and Proposition 6 only holdunder non-zero diffusion component. In fact, when σ X = 0 the maturity-randomized first-passageprobabilities LC (cid:0) u ± X (cid:1) ( · ) reduce (for x (cid:54) = (cid:96) ) to the jump contributions LC (cid:0) u E ±J X (cid:1) ( · ) and the finite activityof the underlying jump process implies that the continuous-fit conditions LC (cid:0) u E + J X (cid:1) ( ϑ, (cid:96) − ; (cid:96) ) = 0 and LC (cid:0) u E −J X (cid:1) ( ϑ, (cid:96) +; (cid:96) ) = 0do not anymore hold. In addition, in view of Lemma 2, the matrices defined in (3.58) may have to bereplaced by corresponding ( m + 1) × m , ( n + 1) × n , m × m or n × n matrices. Therefore, the resultingsystems of equations (3.61), (3.63) and (3.72), (3.73) need to be adjusted accordingly and this mayfinally impact our derivations in Proposition 6. We will deal with these adaptions in more details inSection 4.ii) In addition to obtaining (semi-)analytical expressions for LC ( u E X )( · ) with E ∈ {E +0 , E +1 , . . . , E + m } or E ∈ {E − , E − , . . . , E − n } , Proposition 6 reveals, together with (3.60) and (3.62), that for any values x ∈ R \ H ± (cid:96) and intensity ϑ > (cid:96) (cid:55)→ LC ( u E X )( ϑ, x ; (cid:96) ) with E ∈ {E +0 , E +1 , . . . , E + m } or E ∈ {E − , E − , . . . , E − n } consist in linear combinations of exponentials. We will combine this particularlysimple form with the structure of the Gaver-Stehfest algorithm in Section 5 to derive a simple inversionalgorithm for the integral part of Proposition 1. (cid:7) In this section, we complement the theory developed in the previous parts by discussing hyper-exponentialapproximations to (infinite-activity) pure jump processes having a completely monotone jump density. Weslightly adapt the approach followed in [AMP07], [JP10], briefly discuss the resulting approximations andcomment on how they relate to our final aim of intra-horizon risk quantification. The results will play acentral role in the upcoming numerical analysis of Section 5.1
To start, we recall that a one-sided density f : [0 , ∞ ) → R is said to be completely monotone if for any k ∈ N its k -th derivative f ( k ) ( · ) exists and ( − k f ( k ) ( x ) ≥ x ∈ (0 , ∞ ). Additionally, we recallthat when dealing instead with a two-sided density g : R → R this definition naturally extends by requiringthat both of the functions x (cid:55)→ g ( x ) [0 , ∞ ) ( x ) and x (cid:55)→ g ( − x ) [0 , ∞ ) ( x ) are completely monotone. It can beshown that many L´evy processes employed in financial modeling have a completely monotone jump density π X ( · ), i.e., that their intensity measure Π X takes the particular formΠ X ( dy ) = π X ( y ) dy (4.1)with π X ( · ) being a (two-sided) completely monotone density. This includes among others hyper-exponentialjump-diffusion models (cf. [Ko02], [Ca09]), Normal Inverse Gaussian (NIG) processes (cf. [BN97]) as well asthe whole class of stable and tempered stable processes (cf. [KT13], [HK16]), containing the very popularVariance-Gamma (VG) (cf. [MS90], [MCC98]) and Carr-Geman-Madan-Yor (CGMY) models (cf. [CGMY02]).We are going to deal with some of these dynamics in Section 5.In view of Bernstein’s theorem, a L´evy process ( X t ) t ≥ has a completely monotone jump density if and onlyif its density can be decomposed as π X ( y ) = (0 , ∞ ) ( y ) ∞ (cid:90) e − uy µ + ( du ) + ( −∞ , ( y ) (cid:90) −∞ e −| vy | µ − ( dv ) , (4.2)where µ + ( · ) and µ − ( · ) are (non-negative and finite) measures defined on (0 , ∞ ) and ( −∞ , π X ( · ) bya sequence (cid:0) π ( n ) X ( · ) (cid:1) n ∈ N of densities having the form π ( n ) X ( y ) := Λ n f n ( y ) , n ∈ N , (4.3)where f n ( · ) , n ∈ N , are hyper-exponential densities defined, for partitions ( u ( n ) i ) i ∈{ ,...,N n } and ( v ( n ) j ) j ∈{ ,...,M n } of the sets (0 , ∞ ) and ( −∞ , , having vanishing mesh , by f n ( y ) := N n (cid:88) i =1 p ( n ) i ˜ u ( n ) i e − ˜ u ( n ) i y { y ≥ } + M n (cid:88) j =1 q ( n ) j | ˜ v ( n ) j | e | ˜ v ( n ) j | y { y< } , (4.4)and, for n ∈ N ,˜ u ( n ) i := 12 (cid:16) u ( n ) i − + u ( n ) i (cid:17) , i = 1 , . . . , N n , ˜ v ( n ) j := 12 (cid:16) v ( n ) j − + v ( n ) j (cid:17) , j = 1 , . . . , M n , (4.5) p ( n ) i := µ + (cid:0)(cid:2) u ( n ) i − , u ( n ) i (cid:1)(cid:1) Λ n · ˜ u ( n ) i , i = 1 , . . . , N n , q ( n ) j := µ − (cid:0)(cid:0) v ( n ) j − , v ( n ) j (cid:3)(cid:1) Λ n · | ˜ v ( n ) j | , j = 1 , . . . , M n , (4.6)Λ n := N n (cid:88) i =1 µ + (cid:0)(cid:2) u ( n ) i − , u ( n ) i (cid:1)(cid:1) ˜ u ( n ) i + M n (cid:88) j =1 µ − (cid:0)(cid:0) v ( n ) j − , v ( n ) j (cid:3)(cid:1) | ˜ v ( n ) j | . (4.7) Our convention is that the elements of the partitions are increasing in their index, i.e., we assume that for any n ∈ N therelations 0 < | v ( n ) M n | < . . . < | v ( n )0 | < ∞ and 0 < u ( n )0 < . . . < u ( n ) N n < ∞ hold. Recall that the mesh of a partition ( u ( n ) i ) i ∈{ ,...,N n } is defined by max ≤ i ≤ N n | u ( n ) i − u ( n ) i − | . X t ) t ≥ be such aprocess whose L´evy triplet is denoted by (cid:0) b X , σ X , Π X (cid:1) . Then, for any sequence ( (cid:15) n ) n ∈ N of positive numbersconverging to zero corresponding sequences of partitions (cid:0) ( u ( n ) i ) i ∈{ ,...,N n } (cid:1) n ∈ N and (cid:0) ( v ( n ) j ) j ∈{ ,...,M n } (cid:1) n ∈ N canbe constructed such that the following conditions hold for each n ∈ N : (cid:90) ( −∞ ,v ( n )0 ] ∪ [ u ( n ) Nn , ∞ ) π X ( y ) dy < (cid:15) n , (cid:90) ( v ( n )0 ,u ( n ) Nn ) \ ( v ( n ) Mn ,u ( n )0 ) (cid:0) π X ( y ) − π ( n ) X ( y ) (cid:1) dy < (cid:15) n , (4.8) u ( n )0 (cid:90) v ( n ) Mn y (cid:0) π X ( y ) − π ( n ) X ( y ) (cid:1) dy < (cid:15) n . (4.9)In particular, Requirement (4.9) can be fulfilled since any L´evy process’ intensity measure satisfies (cid:90) R (cid:0) ∧ y (cid:1) Π X ( dy ) < ∞ . (4.10)We denote by ( X nt ) t ≥ the resulting, approximating L´evy process having jump density π ( n ) X ( · ), Gaussianparameter σ n := σ X and drift b n ∈ R defined by the identity Φ X n (1) = Φ X (1). Then, following the linesof argument in [JP10] allows to see that the sequence of approximating processes constructed this way, (cid:0) ( X nt ) t ≥ (cid:1) n ∈ N , converges weakly, in the Skorokhod topology, to the true process ( X t ) t ≥ and additionallythat for any T ≥ x ∈ R and (cid:96) ∈ RP X n x (cid:16) τ X n , ± (cid:96) ≤ T (cid:17) → P Xx (cid:16) τ X, ± (cid:96) ≤ T (cid:17) , n → ∞ . (4.11)In view of our general intra-horizon risk measurement approach, the latter convergence has an importantimplication. Not only can L´evy processes with completely monotone jumps be approximated by hyper-exponential jump-diffusion models, but the same approximating sequence can be also used for intra-horizonrisk quantification. We will rely on this approach in Section 5, when discussing the intra-horizon risk inherentto certain infinite-activity pure jump L´evy dynamics, i.e., we will derive intra-horizon risk results to theseL´evy models by relying on their hyper-exponential approximations ( X nt ) t ≥ with σ n = σ X = 0. However,since our main results in Section 3.4 made explicitly use of the assumption σ X (cid:54) = 0, a few adaptions need tobe discussed. This is the content of the next section. Remark 5.
Although our general approximation scheme relies on ideas similarly employed in [AMP07] and [JP10],the resulting approximating processes (cid:0) ( X nt ) t ≥ (cid:1) n ∈ N substantially deviate in their structure from the onespresented in these papers. This comes from the fact that the authors in [AMP07] and [JP10] choose toaggregate small jumps into an additional diffusion factor by taking σ n := σ X + u ( n )0 (cid:90) v ( n ) Mn y (cid:0) π X ( y ) − π ( n ) X ( y ) (cid:1) + dy, (4.12)while we prefer to stick with the diffusion coefficient of the original process ( X t ) t ≥ and rely instead on(4.3), (4.4), and σ n := σ X . Since the difference between the two diffusion coefficients does not exceed (cid:15) n ,3choosing one or the other approximation scheme may seem equivalent. However, aggregating small jumpsinto an additional diffusion factor transforms in particular infinite-activity pure jump processes into approxi-mating hyper-exponential jump-diffusion processes with non-zero diffusion. When dealing with first-passageproblems, this additional diffusion factor is known to artificially imply a smooth-pasting condition at thebarrier level, which subsequently leads, near the barrier, to qualitative differences in the solutions to thefirst-passage problem under the original process and under the approximating processes (cf. [BL09], [BL12]).As we are particularly interested in quantifying intra-horizon risk for small α , we only need to compute first-passage probabilities for starting values far from the barrier. Consequently, relying on the same approachused in [AMP07] and [JP10] may still provide reasonable results (cf. [LV20]). Nevertheless, we prefer tofollow a more natural approach and keep the pure jump structure of the original process by relying on (4.3),(4.4), and σ n := σ X . (cid:7) At this point, we have already emphasized that the structure of pure jump processes implies for the maturity-randomized first-passage probabilities LC (cid:0) u E ± X (cid:1) ( · ) and LC (cid:0) u E ±J X (cid:1) ( · ) that the continuous-pasting conditions LC (cid:0) u E ± X (cid:1) ( ϑ, (cid:96) ∓ ; (cid:96) ) = LC (cid:0) u E ± X (cid:1) ( ϑ, (cid:96) ; (cid:96) ) and LC (cid:0) u E ±J X (cid:1) ( ϑ, (cid:96) ∓ ; (cid:96) ) = LC (cid:0) u E ±J X (cid:1) ( ϑ, (cid:96) ; (cid:96) ) (4.13)do not anymore hold. Instead, when dealing with pure jump processes of infinite variation, these conditionsneed to be replaced by LC (cid:0) u E ± X (cid:1) ( ϑ, (cid:96) ∓ ; (cid:96) ) = LC (cid:0) u E ± X (cid:1) ( ϑ, (cid:96) ± ; (cid:96) ) and LC (cid:0) u E ±J X (cid:1) ( ϑ, (cid:96) ∓ ; (cid:96) ) = LC (cid:0) u E ±J X (cid:1) ( ϑ, (cid:96) ± ; (cid:96) ) . (4.14)Although (4.14) is not anymore satisfied under hyper-exponential jump-diffusion approximations to purejump processes of infinite variation, one may want to impose them anyway and analogous results to the onesin Proposition 4 can be derived under (4.14). Alternatively, jump contributions to (maturity-randomized)first-passage probabilities can be obtained by following the approach taken in Proposition 5 and subsequentlysolving the resulting systems of equations. This leads to the following analogue of Proposition 6. The readeris referred for a proof in a slightly different context to [CYY13]. Proposition 7.
Assume that ( X t ) t ≥ follows a hyper-exponential jump-diffusion process, as described in(3.43), (3.44) and with σ X = 0 and µ ≤ . Then, for any level (cid:96) ∈ R , x ∈ R \ H + (cid:96) and intensity ϑ > wehave that LC (cid:0) u E + i X (cid:1) ( ϑ, x ; (cid:96) ) = m (cid:88) k =1 ˜ v E + i ,k · e β k,ϑ · ( x − (cid:96) ) , i ∈ { , . . . , m } , (4.15) where the coefficients ˜ v E + i ,k are given by ˜ v E + i ,k = − ˜B + ( β k,ϑ ) ˜C + ϑ ( ξ i ) ξ i ( ξ i − β k,ϑ ) (cid:0) ˜B + (cid:1) (cid:48) ( ξ i ) (cid:0) ˜C + ϑ (cid:1) (cid:48) ( β k,ϑ ) , k ∈ { , . . . , m } , (4.16) with ˜B + ( x ) := m (cid:89) s =1 (cid:0) ξ s − x (cid:1) , ˜C + ϑ ( x ) := m (cid:89) s =1 (cid:0) β s,ϑ − x (cid:1) . (4.17)4 Similarly, if σ X = 0 and µ ≥ , we obtain, for (cid:96) ∈ R , x ∈ R \ H − (cid:96) and ϑ > , that LC (cid:0) u E − j X (cid:1) ( ϑ, x ; (cid:96) ) = n (cid:88) k =1 ˜ v E − j ,k · e γ k,ϑ · ( x − (cid:96) ) , j ∈ { , . . . , n } (4.18) where the coefficients ˜ v E − j ,k are given by ˜ v E − j ,k = ˜B − ( γ k,ϑ ) ˜C − ϑ ( η j ) η j ( η j + γ k,ϑ ) (cid:0) ˜B − (cid:1) (cid:48) ( − η j ) (cid:0) ˜C − ϑ (cid:1) (cid:48) ( − γ k,ϑ ) , k ∈ { , . . . , n } , (4.19) with ˜B − ( x ) := n (cid:89) s =1 (cid:0) η s + x (cid:1) , ˜C − ϑ ( x ) := n (cid:89) s =1 (cid:0) γ s,ϑ + x (cid:1) . (4.20) Remark 6. i) We note that the coefficients v E + i ,k , ˜ v E + i ,k and v E − j ,k , ˜ v E − j ,k in Proposition 6 and Proposition 7 dependon the intensity ϑ > ϑ > v E + i ,k ( ϑ ), ˜ v E + i ,k ( ϑ ), and v E − j ,k ( ϑ ), ˜ v E − j ,k ( ϑ ) instead.ii) To conclude our analysis, we observe that, for σ X = 0 and µ > σ X = 0 and µ < (cid:96) ∈ R , x ∈ R \ H + (cid:96) (or x ∈ R \ H − (cid:96) ) and ϑ >
0, the functions LC (cid:0) u E +0 X (cid:1) ( · ) and LC (cid:0) u E + i X (cid:1) ( · ), for i ∈ { , . . . , m } , (or LC (cid:0) u E − X (cid:1) ( · ) and LC (cid:0) u E − j X (cid:1) ( · ),for j ∈ { , . . . , n } ,) are given via (A.54) and (3.74), (3.75) (or (A.57) and (3.78), (3.79) respectively).This follows by combining the results in Lemma 2 with Proposition 5, since considering these two casesdo not alter the number of positive (or negative) roots to the equation Φ X ( θ ) = ϑ and, therefore, thesystem of equations (3.72) (or (3.73)). Further details can be also found in [CYY13]. (cid:7) To illustrate the practicability of the intra-horizon risk measurement approach developed in the previoussections, we lastly analyze the 10-days intra-horizon risk inherent to a long position in the S&P 500 indexas well as in the Brent crude oil over (slightly more than) 25 years. More specifically, we focus on the casewhere the (discounted) profit and loss process reflects the intrinsic value of a long position in either of theseunderlyings (cf.
Scenario 2 in Section 3) and derive historical intra-horizon risk results by calibrating double-exponential (Kou), Variance-Gamma (VG), and Carr-Geman-Madan-Yor (CGMY) dynamics to S&P 500index and Brent crude oil data and subsequently approximating the intra-horizon risk in these models bycombining the general approach of Section 3 with the methods presented in Section 4.5
Our data set comprises historical returns of the S&P 500 index and the Brent crude oil from January 1990until September 2020, therefore spanning more than three decades. During this period, a wide variety ofmacroeconomic, financial, and political risk factors have influenced the performance of the US equity andcommodity market. Following [BP10], we consider weekly frequency in our empirical study, which gives usin total 1,344 calibration dates starting from January 1995 (i.e., the initial five years of weekly returns areused to construct the first rolling-window sample). As discussed in [LV20], weekly returns are suboptimal inthe sense that there is a mismatch between the sampling frequency and the 10-days horizon that is typicallyconsidered in risk management applications. However, using biweekly returns would halve the number ofobservations, which would further exacerbate estimation problems regarding the risk measures consideredin this paper. Therefore, similarly to the previous studies, our decision to rely on weekly returns representsa trade-off between the quality of our estimation results and the accuracy of the sampling frequency.
Our approach closely follows [BP10]. We estimate parameters of certain L´evy models on a rolling-windowbasis using a maximum likelihood estimation (MLE) procedure. The only difference is that we rely on theFourier cosine method of [FO08] to estimate the probability density function of the weekly index returns.This method is very fast and has been already recommended for a similar application in [LV20]. We consider the Kou (double-exponential jump-diffusion) model as well as two L´evy models that are wellestablished in the literature and widely applied in practice: the Variance-Gamma (VG) model and the Carr-Geman-Madan-Yor (CGMY) model. In the case of the CGMY model, we set the fine structure parameter, Y , to Y = 0 .
5. This particular choice was proposed in [BP10], mainly for two reasons. First, the resultingmodel has an infinite-activity-and-finite-variation property which is known to describe time series of equityreturns very well. Second, having this parameter fixed allows for a better identification of the remainingmodel parameters, i.e., the jump arrival rate C , and the exponential decay parameters G and M . For theVG model, we note that the resulting dynamics corresponds to a special case of the CGMY dynamics – thefine structure parameter is given by Y = 0.The next table summarizes our calibration results for the two models under consideration. Besides reportingaverage values and standard deviations for all model parameters, we have chosen to include the median aswell as the median absolute deviation. This is important as the mean may be influenced by outliers and thenon-normality of the data.Several patterns can be observed over time and across the estimates for the S&P 500 index (Bloombergticker: SPX). First, the jump intensity parameter – denoted C for the VG and CGMY models, and λ forthe Kou model – spiked during the 1997 Asian Financial Crisis and the 2008-2009 Global Financial Crisis.Interestingly, the jump intensity was rather high during the 2002-2007 Bull Market, however it was notaccompanied by elevated positive and/or negative jumps. Second, the expected size of positive jumps –i.e., the inverse of the parameter M for the VG and CGMY model, and the inverse of the parameter η for the Kou model – is rather stable at the level of about 1-2% for the whole sample and tends to becomesmaller during the crisis periods. Third, the expected size of negative jumps – i.e., the inverse of theparameter G for the VG and CGMY model, and the inverse of the parameter θ for the Kou model – wasparticularly high (around 3.5-4.0%) during the 2000 Dot-Com Bubble Burst and following the 2008-2009 Details around the calibration are thoroughly discussed in the two cited papers, hence we do not elaborate here further onthe estimation procedure.
M > G and η > θ for the respective models. Last but not least, we stress that the key driver of the differences between thereported parameter estimates is the fine structure parameter Y which is fixed at Y = 0 and Y = 0 . Table 1: Summary statistics for the calibrated parameters.
We estimate parameters of the Kou,Variance-Gamma (VG) and, Carr-Geman-Madan-Yor (CGMY) models using weekly historical returns ofthe S&P 500 index and the Brent crude oil from January 1995 until September 2020 (the total number ofcalibration dates is 1,344) on a five-years rolling-window basis. The table reports the average and medianvalues, the standard deviations and the mean absolute deviations (MAD) for the estimated parameters andthe negative log-likelihood (MLE) over time. The parameters C , G and M are based on unconstrainedcalibrations. We set the values of the parameter Y to Y = 0 and Y = 0 . Panel A: Summary Statistics for the Kou model σ (%) λ p η θ MLE
SPX CO1 SPX CO1 SPX CO1 SPX CO1 SPX CO1 SPX CO1
Mean 6 .
94 16 .
41 127 .
89 264 .
78 0 .
40 0 .
25 163 .
43 151 .
23 90 .
94 79 .
41 4 .
41 5 . .
23 19 .
83 103 .
72 169 .
26 0 .
32 0 .
12 100 .
08 61 .
52 77 .
00 64 .
10 4 .
39 5 . .
83 11 .
60 116 .
68 319 .
41 0 .
25 0 .
28 249 .
24 195 .
83 44 .
13 86 .
44 0 .
23 0 . .
36 10 .
73 72 .
19 250 .
93 0 .
22 0 .
24 111 .
29 145 .
74 34 .
37 54 .
47 0 .
20 0 . Panel B: Summary Statistics for the VG model
C G M Y
MLE
SPX CO1 SPX CO1 SPX CO1 SPX CO1 SPX CO1
Mean 92 .
56 243 .
60 74 .
80 55 .
23 112 .
62 96 .
94 0 0 4 .
41 5 . .
21 136 .
42 72 .
85 57 .
81 105 .
41 68 .
88 0 0 4 .
39 5 . .
09 190 .
60 23 .
85 18 .
95 46 .
05 72 . – – .
23 0 . .
33 169 .
78 19 .
81 27 .
04 31 .
37 58 . – – .
20 0 . Panel C: Summary Statistics for the CGMY model
C G M Y
MLE
SPX CO1 SPX CO1 SPX CO1 SPX CO1 SPX CO1
Mean 6 .
59 39 .
69 48 .
56 52 .
88 85 .
00 196 .
87 0 . . .
41 5 . .
23 14 .
77 44 .
84 39 .
95 77 .
05 55 .
36 0 . . .
39 5 . .
32 66 .
25 19 .
03 51 .
26 38 .
32 745 . – – .
23 0 . .
53 38 .
67 16 .
09 29 .
28 26 .
89 216 . – – .
20 0 . In addition to the US large-cap equity index, we calibrate the models to Brent crude oil data (Bloombergticker: CO1). First, we note that the Brent crude oil historical returns exhibit annualized volatility of about736 percent, which is approximately twice the volatility of the S&P 500 index. Due to more frequent andviolent negative jumps, the distribution of the Brent crude oil returns features more pronounced negativeskewness and fatter tails. These stylized facts make the Brent crude oil an interesting candidate for ourempirical exercise. Depending on the model, we obtain a jump intensity parameter that is two to six timeshigher than in the case of the equity index. The average magnitude of the positive (negative) jumps issomewhat smaller (larger) for the Brent crude oil. Moreover, these parameters were subject to pronouncedswings over time. The most dramatic episodes were observed during the 2008-2009 Global Financial Crisis,then between 2014 and 2016 and in November 2018 due to the excess crude oil supply, and finally in 2020as a result of the demand shock driven by the pandemic-induced economic slowdown. Last but not least, weremark that the MLE score is higher for the Brent crude oil than for the S&P 500 index, which is indicativeof somewhat elevated estimation and model risk. This result is not surprising given the extreme marketmoves in the oil market over the last three decades.
Having calibrated the Kou, VG, and CGMY dynamics to S&P 500 index and to Brent crude oil data, bothranging from January 1990 to September 2020, we next turn to a derivation of the 10-days intra-horizon riskin these models. While the Kou model allows for a direct application of the results derived in Section 3, wecompute intra-horizon risk results for VG and CGMY dynamics based on hyper-exponential jump-diffusionapproximations and the combination of the theory developed in Section 3, Proposition 7, and the Gaver-Stehfest inversion algorithm. More specifically, we follow the ideas of [AMP07] (cf. also [LV20]), i.e., we fixin advance the number of exponentials N n and M n in the approximating density (4.3), (4.4) and minimizethe distance between the approximating and the true L´evy densities by optimally choosing the partitionof the integration intervals. This slightly differs from the approach used in [JP10], where the authorsadditionally fix all mean jump sizes ( ξ i ) − and ( η j ) − and subsequently use least-squares optimization todetermine the values of the remaining (mixing) parameters. However, while these authors only work withfew exponentials, we choose N n = 100 and M n = 100 and incorporate this way 200 exponentials. Thisadditionally ensures that we approximate small jumps sufficiently well, as we have decided to keep thepure jump structure of the approximated processes by refraining from converting small jumps into an extradiffusion factor (cf. Remark 5 in Section 4). As soon as hyper-exponential jump-diffusion processes are fixed, we make use of the derivations in theprevious sections to derive intra-horizon risk results in the following way. First, 10-days intra-horizonvalue-at-risk measures as well as corresponding risk contributions per jump type are obtained by combiningProposition 7 with Relations (2.5), (3.6) and (3.26), i.e., we invert the functions LC (cid:0) u E − j X (cid:1) ( · ), for any j ∈ The authors in [JP10] only use a total of 14 exponentials, i.e., 7 exponentials for both the positive and the negative partsof the distribution. Our numerical tests show that the suggested procedure is fast and stable. However, we emphasize that other approachesexist in the literature (cf. for instance [CLM10]) and that investigating the performance of all these algorithms is not the sakeof this article, but constitutes a separate research topic. (a) Evolution of Absolute Risk (Kou) (b)
Intra-Horizon to Point-in-Time Risk Ratio (Kou) (c)
Evolution of Absolute Risk (VG) (d)
Intra-Horizon to Point-in-Time Risk Ratio (VG) (e)
Evolution of Absolute Risk (CGMY) (f )
Intra-Horizon to Point-in-Time Risk Ratio (CGMY)
Figure 1: Comparison of the intra-horizon and point-in-time risk for the S&P 500 index.
Fig-ure 1a, Figure 1c, and Figure 1e show the time evolution of the 10-days intra-horizon risk (cid:0) intra-horizon valueat risk, iV@R , ( P & L ), and intra-horizon expected shortfall, iES , ( P & L ) (cid:1) and 10-days point-in-timerisk (cid:0) (standard) value at risk, V@R (cid:0) P & L (cid:1) , and (standard) expected shortfall, ES (cid:0) P & L (cid:1)(cid:1) to the99% loss quantile from January 1995 until September 2020. The resulting absolute risk levels correspondto negative return levels under the respective dynamics. Additionally, Figure 1b, Figure 1d, and Figure 1fprovide the risk ratio of intra-horizon risk to point-in-time risk under the respective L´evy models.9 (a) Evolution of Absolute Risk (Kou) (b)
Intra-Horizon to Point-in-Time Risk Ratio (Kou) (c)
Evolution of Absolute Risk (VG) (d)
Intra-Horizon to Point-in-Time Risk Ratio (VG) (e)
Evolution of Absolute Risk (CGMY) (f )
Intra-Horizon to Point-in-Time Risk Ratio (CGMY)
Figure 2: Comparison of the intra-horizon and point-in-time risk for the Brent crude oil.
Figure 2a, Figure 2c, and Figure 2e show the time evolution of the 10-days intra-horizon risk (cid:0) intra-horizonvalue at risk, iV@R , ( P & L ), and intra-horizon expected shortfall, iES , ( P & L ) (cid:1) and 10-days point-in-time risk (cid:0) (standard) value at risk, V@R (cid:0) P & L (cid:1) , and (standard) expected shortfall, ES (cid:0) P & L (cid:1)(cid:1) to the 99% loss quantile from January 1995 until September 2020. The resulting absolute risk levels cor-respond to negative return levels under the respective dynamics. Additionally, Figure 2b, Figure 2d, andFigure 2f provide the risk ratio of intra-horizon risk to point-in-time risk under the respective L´evy models.0 { , . . . , n } , via u E − j X ( T, x ; (cid:96) ) = lim N →∞ (cid:16) u E − j X (cid:17) N ( T, x ; (cid:96) ) , (5.1) (cid:16) u E − j X (cid:17) N ( T, x ; (cid:96) ) := N (cid:88) k =1 ζ k,N LC (cid:0) u E − j X (cid:1)(cid:16) k log(2) T , x ; (cid:96) (cid:17) = N (cid:88) k =1 n (cid:88) s =1 ζ k,N ˜ v E − j ,s (cid:16) k log(2) T (cid:17) exp (cid:110) γ s, k log(2) T · ( x − (cid:96) ) (cid:111) , (5.2)with ζ k,N defined as in (3.25), and derive the respective results based on Relations (2.5), (3.6) and the ideasintroduced in Section 3.3. Once these quantities are obtained, recovering 10-days intra-horizon expectedshortfall results reduces to the evaluation of integrals and of fractions of integrals of the form of (3.38) and(3.42). Here, combining (5.1), (5.2) with the monotonicity of the function (cid:96) (cid:55)→ u E − j X ( T, x ; (cid:96) ) allows us toderive, for each j ∈ { , . . . , n } , that − iV @ R α,T ( P & L ) (cid:90) − z u E − j X (cid:0) T, log( z + z ); log( z + (cid:96) ) (cid:1) d(cid:96) = lim N →∞ (cid:16) I E − j α,T (cid:17) N ( z, z ) , (5.3)where (cid:16) I E − j α,T (cid:17) N ( z, z ) is given, for any N ∈ N , via (cid:16) I E − j α,T (cid:17) N ( z, z ) := − iV @ R α,T ( P & L ) (cid:90) − z (cid:16) u E − j X (cid:17) N (cid:0) T, log( z + z ); log( z + (cid:96) ) (cid:1) d(cid:96) = N (cid:88) k =1 n (cid:88) s =1 ζ k,N ˜ v E − j ,s (cid:16) k log(2) T (cid:17) − iV @ R α,T ( P & L ) (cid:90) − z (cid:18) z + zz + (cid:96) (cid:19) γ s, k log(2) T d(cid:96) = N (cid:88) k =1 n (cid:88) s =1 ζ k,N ˜ v E − j ,s (cid:16) k log(2) T (cid:17) z + z − γ s, k log(2) T (cid:18) z − iV @ R α,T ( P & L ) z + z (cid:19) − γ s, k log(2) T . (5.4)This finally provides us with a simple numerical scheme to compute 10-days intra-horizon expected shortfallmeasures as well as, based on the ideas outlined in Section 3.3, corresponding risk contributions per jumptype inherent to any long position in either the S&P 500 index or the Brent crude oil. We now turn to the empirical risk results and start by providing a comparison of the intra-horizon andpoint-in-time risks inherent to a long position in the S&P 500 index as well in the Brent crude oil fromJanuary 1995 to September 2020. To this end, we have plotted in Figure 1a, Figure 1c, Figure 1e andFigure 2a, Figure 2c, Figure 2e, the time evolution of the absolute 10-days intra-horizon and point-in-time risks to the 99% quantile of the loss distribution calculated under the respective L´evy dynamicsfor the S&P 500 index and for the Brent crude oil, respectively. These results express intra-horizon andpoint-in-time risks in terms of (negative) return levels, i.e., the graphs were obtained by computing the In our notation, this corresponds to fixing α = 1%. (a) Ratios for the S&P 500 index (b)
Ratios for the Brent crude oil
Figure 3: Time evolution of the intra-horizon/point-in-time value at risk to expected short-fall ratios.
We have plotted for Kou, VG, and CGMY the time evolution of the intra-horizon andpoint-in-time risk ratios ω , ( P & L ) := iV@R , ( P & L ) (cid:14) iES , ( P & L ) and ω (cid:0) P & L (cid:1) := V@R (cid:0) P & L (cid:1)(cid:14) ES (cid:0) P & L (cid:1) , respectively.These ratios give the relative contribution of the 10-days intra-horizon/point-in-time value at risk to the10-days intra-horizon/point-in-time expected shortfall to the 99% quantile of the loss distribution under therespective L´evy dynamics.respective risk measures while fixing z = z = 1 in Scenario 2 (cf. Section 3). To complement theseresults, we have also provided in Figure 1b, Figure 1d, Figure 1f and Figure 2b, Figure 2d, Figure 2f thetime evolution of the intra-horizon to point-in-time risk ratio. Finally, Figure 3 presents the evolutionof the intra-horizon and point-in-time ratios ω , ( P & L ) := iV@R , ( P & L ) (cid:14) iES , ( P & L ) and ω (cid:0) P & L (cid:1) := V@R (cid:0) P & L (cid:1)(cid:14) ES (cid:0) P & L (cid:1) for all model dynamics and both underlyings. Whilewe have chosen to follow the framework of the Basel Accords (cf. [BCBS19]) and to provide results fora 10-days horizon, we note that we do not rely on the 97.5% quantile of the loss distribution prescribedin [BCBS19], but prefer to investigate the 99% level. This is mainly to stay consistent with the existingliterature on intra-horizon risk quantification (cf. [BRSW04], [Ro08], [BMK09], [BP10], [LV20]) and to allowfor a direct comparability of our results with other articles.The results in Figure 1, Figure 2, and Figure 3 are in line with our intuition: First, the 10-days intra-horizon expected shortfall to the 99% loss quantile, iES , ( P & L ), exceeds at any time the intra-horizonvalue at risk at the same level, iV@R , ( P & L ), and the same additionally holds true for the point-in-timemeasures. Moreover, intra-horizon risk measures always exceed their point-in-time equivalent. This becomesevident when looking at Figures 1a-1f and Figures 2a-2f, where the intra-horizon risk curve is always higherthan its point-in-time reference and the intra-horizon to point-in-time risk ratio never falls below 1.0. Inparticular, Figure 1b, Figure 1d, Figure 1f, and Figure 2b, Figure 2d, Figure 2f show that this ratio hasa similar structure for both (intra-horizon) value at risk and (intra-horizon) expected shortfall, however,that it seems to be greater for the (intra-horizon) value at risk. Finally, we also note that intra-horizonrisk is generally 5-10% higher than point-in-time risk. Next, when investigating any of Figure 1, Figure 2,and Figure 3, one sees that all the intra-horizon and point-in-time measures behave similarly. In particular,all the lines inFigure 1a, Figure 1c, Figure 1e, and Figure 2a, Figure 2c, Figure 2e exhibit an (almost)2 (a) Evolution of Jump Contributions (VG) (b)
Evolution of Jump Contributions (CGMY) (c)
Average Jump Size and Absolute Risk (VG) (d)
Average Jump Size and Absolute Risk (CGMY)
Figure 4: Comparison of the intra-horizon risk and the contribution of certain jumps for theS&P 500 index.
Figure 4a and Figure 4b show the time evolution of the 10-days intra-horizon risk con-tributions to the 99% loss quantile for the greatest – in absolute size – 3 down jumps, the greatest 5 downjumps, and the greatest 10 down jumps in the hyper-exponential jump-diffusion approximations. Addition-ally, Figure 4c and Figure 4d present the relation of the (absolute) average (down) jump size – weightedby the probability of occurrence of each jump in the hyper-exponential jump-diffusion approximations – tothe absolute intra-horizon risk level. As earlier, the absolute risk levels correspond to negative return levelsunder the respective dynamics.3 (a)
Evolution of Jump Contributions (VG) (b)
Evolution of Jump Contributions (CGMY) (c)
Average Jump Size and Absolute Risk (VG) (d)
Average Jump Size and Absolute Risk (CGMY)
Figure 5: Comparison of the intra-horizon risk and the contribution of certain jumps for theBrent crude oil.
Figure 5a and Figure 5b show the time evolution of the 10-days intra-horizon risk con-tributions to the 99% loss quantile for the greatest – in absolute size – 3 down jumps, the greatest 5 downjumps, and the greatest 10 down jumps in the hyper-exponential jump-diffusion approximations. Addition-ally, Figure 5c and Figure 5d present the relation of the (absolute) average (down) jump size – weightedby the probability of occurrence of each jump in the hyper-exponential jump-diffusion approximations – tothe absolute intra-horizon risk level. As earlier, the absolute risk levels correspond to negative return levelsunder the respective dynamics.4identical shape and seem to be obtained via a parallel shift of anyone of them. However, a closer look atthese graphs reveals that the absolute difference between intra-horizon/point-in-time expected shortfall andintra-horizon/point-in-time value at risk tends to substantially increase in more severe times. That thisbehavior does not seem to only hold at an absolute level but also in relative terms can be seen in Figure 3where the intra-horizon/point-in-time value-at-risk contribution to the intra-horizon/point-in-time expectedshortfall takes its lowest values in crisis periods.
To finalize our discussion, we investigate the structure of intra-horizon risk across jumps. To this end, wepresent in Figure 4 and Figure 5 a comparison of intra-horizon risk and jump contributions. In particular, wehave plotted in Figure 4a, Figure 4b and Figure 5a, Figure 5b the time evolution of the 10-days intra-horizonrisk contributions to the 99% loss quantile for the greatest – in absolute size – 3 down jumps, the greatest5 down jumps, and the greatest 10 down jumps in the hyper-exponential jump-diffusion approximationsof VG and CGMY. Additionally, Figure 4c, Figure 4d and Figure 5c, Figure 5d show the relation of the(absolute) average (down) jump size – weighted by the probability of occurrence of each jump – to theabsolute intra-horizon risk level. The results are in line with the existing literature (cf. e.g., [LV20]) aswell as with the observations in the previous section. First, we note that the greatest 3, 5, and 10 jumpsin the hyper-exponential jump-diffusion approximations already provide a high contribution to both intra-horizon value at risk and intra-horizon expected shortfall, with a slightly higher contribution for VG thanfor CGMY. Additionally, looking at Figures 4a-4d and Figures 5a-5d reveals that the structure of risk acrossjumps does not differ for both of these risk measures. Indeed, while the risk contributions per jump types toboth intra-horizon value at risk and intra-horizon expected shortfall are almost identical, the intra-horizonexpected shortfall results in Figures 4c-4d and Figures 5c-5d merely replicate the shape of the intra-horizonvalue-at-risk results at a slightly higher risk level. This is due to the fact that the intra-horizon expectedshortfall always exceeds the intra-horizon value at risk for the same time horizon and quantile. Lastly,we emphasize that Figures 4c-4d and Figures 5c-5d additionally present evidence of the fact that higher(absolute) average jumps generally lead to higher absolute risk levels. This is intuitively clear, since greater(absolute) average jumps immediately increase the tail of the jump distribution which likewise impacts thetail of the overall profit and loss distribution.
The present article extended the current literature on intra-horizon risk quantification in several directions.First, we proposed an intra-horizon analogue of the expected shortfall and discussed some of its key propertiesunder general L´evy dynamics. The resulting (intra-horizon) risk measure is well-defined for (m)any popularclass(es) of L´evy processes encountered in financial modeling and constitutes a coherent measure of risk inthe sense of [CDK04]. Secondly, we linked our intra-horizon expected shortfall to first-passage occurrencesand derived a characterization of diffusion and jump contributions to simple and maturity-randomized first-passage probabilities. These results were subsequently used to infer diffusion and jump risk contributionsto the intra-horizon expected shortfall and additionally allowed us to obtain (semi-)analytical results formaturity-randomized first-passage probabilities under hyper-exponential jump-diffusion dynamics. Next, wereviewed hyper-exponential jump-diffusion approximations to L´evy processes having completely monotonejumps and proposed an adaption of the results in [AMP07], [JP10] that naturally preserves the diffusionvs. jump structure of the approximated processes. We then calibrated popular L´evy processes to S&P 500 We emphasize that similar results can be derived for point-in-time risk. However, due to the focus of the paper, we onlyprovide intra-horizon risk results. α ) the intra-horizon value at riskand the intra-horizon expected shortfall add conservatism to their point-in-time estimates. Additionally,they suggested that these risk measures have a very similar structure across jumps/jump clusters and thatalready a high contribution of their risk is due to only few, great – in terms of the absolute jump size –jump clusters.We are convinced that the techniques and ideas presented in this paper can serve as a basis for severalextensions of the theory of risk measures for stochastic processes and hope that they will further stimulate thework of the community on this topic. First, we believe that several of our ideas could be extended to introduceintra-horizon versions of spectral (point-in-time) risk measures, as introduced in [Ac02]. Second, the recentlyincreasing interest in the use of expectile-based measures – especially of the expectile value at risk – for riskmanagement purposes (cf. [BKMR14], [BD17]) may encourage the investigation of intra-horizon expectile-based risk in future research. Third, we recognize that the ideas considered in this article substantially differfrom the methods of dynamic risk measures. Nevertheless, we believe that both concepts could be combinedto introduce dynamic intra-horizon risk versions and we hope that this will help complementing the not yetunified theory of risk measures for stochastic processes. Finally, we think that embedding the estimationand model risk into our framework would be an interesting avenue for future reseach as well. Acknowledgements:
The authors are grateful to the associate editor and two anonymous referees for theirconstructive comments. Additionally, the authors would like to thank Sergei Levendorskii, Tadeusz Czernik,Max Nendel, Carlo Sala, Giovanni Barone-Adesi, Matteo Burzoni, Johannes Wiesel, Felix-Benedikt Liebrichand the participants of the 9th General AMaMeF Conference, the 2019 Vienna Congress on MathematicalFinance (VCMF 2019), and the 2019 Quantitative Methods in Finance Conference (QMF 2019) for theirvaluable comments and suggestions.
Appendices
Appendix A: Proofs - General Results
Proof of Proposition 1.
To start, we note that Proposition 3.2 in [AT02] implies that for any
T > α ∈ (0 ,
1) the intra-horizon expected shortfall associated to the profit and loss process ( P & L t ) t ∈ [0 ,T ] , iES α,T ( P & L ), is given by iES α,T ( P & L ) = − α (cid:18) E P z (cid:104) I P & L T { I P & L T ≤ q α ( I P & L T ) } (cid:105) − q α ( I P & L T ) (cid:104) P z (cid:16) I P & L T ≤ q α (cid:0) I P & L T (cid:1)(cid:17) − α (cid:105) (cid:19) . (A.1)Therefore, we next derive an expression for the integral/expectation part in (A.1) and will subsequently usethe result to recover (2.11). Here, noting that, under P z , the inequality I P & L T ≤ z holds for any T ≥ E P z (cid:104) I P & L T { I P & L T ≤ q α ( I P & L T ) } (cid:105) = E P z (cid:20) − z (cid:90) I P & L T { I P & L T ≤ q α ( I P & L T ) } d(cid:96) (cid:21) + z P z (cid:16) I P & L T ≤ q α (cid:0) I P & L T (cid:1)(cid:17) = − z (cid:90) −∞ P z (cid:16) I P & L T ≤ (cid:96), I P & L T ≤ q α (cid:0) I P & L T (cid:1)(cid:17) d(cid:96) + z P z (cid:16) I P & L T ≤ q α (cid:0) I P & L T (cid:1)(cid:17) = − z (cid:90) q α (cid:0) I P & L T (cid:1) P z (cid:16) I P & L T ≤ q α (cid:0) I P & L T (cid:1)(cid:17) d(cid:96) − q α (cid:0) I P & L T (cid:1)(cid:90) −∞ P z (cid:16) I P & L T ≤ (cid:96) (cid:17) d(cid:96) + z P z (cid:16) I P & L T ≤ q α (cid:0) I P & L T (cid:1)(cid:17) = q α (cid:0) I P & L T (cid:1) P z (cid:16) I P & L T ≤ q α (cid:0) I P & L T (cid:1)(cid:17) − q α (cid:0) I P & L T (cid:1)(cid:90) −∞ P z (cid:16) I P & L T ≤ (cid:96) (cid:17) d(cid:96). (A.2)Hence, combining (A.1) and (A.2) with the relation P z (cid:16) I P & L T ≤ (cid:96) (cid:17) = P z (cid:16) τ P & L , − (cid:96) ≤ T (cid:17) gives that theintra-horizon expected shortfall can be expressed in terms of first-passage probabilities, as iES α,T ( P & L ) = 1 α q α (cid:0) I P & L T (cid:1)(cid:90) −∞ P z (cid:16) τ P & L , − (cid:96) ≤ T (cid:17) d(cid:96) − q α (cid:0) I P & L T (cid:1) , (A.3)which finally provides Equation (2.11). Proof of Lemma 1.
To show (3.8), we rely on similar arguments to the ones used in [CK11] (cf. also [HM13]).Here, we focus on the result for the upside first-passage probabilities and show that there exists a constant c > x ∈ R and T > (cid:96) ↑∞ e c · (cid:96) P Xx (cid:16) M X T ≥ (cid:96) (cid:17) = 0 , (A.4)where ( M Xt ) t ≥ denotes the maximum process associated to ( X t ) t ≥ , i.e., the process defined by M Xt := sup ≤ u ≤ t X u , t ≥ . (A.5)Once (A.4) is established, the respective convergence result for the downside first-passage probabilities iseasily obtained by noting that the minimum process ( I Xt ) t ≥ satisfies M ˜ Xt = − I Xt , t ≥ , (A.6)where we have denoted by ( ˜ X t ) t ≥ the dual process to ( X t ) t ≥ , i.e., the process that is defined by ˜ X t := − X t , t ≥
0. Therefore, we only have to prove (A.4). Here, we start by recalling that the process ( Z θt ) t ≥ definedvia Z θt := e θX t − t Φ X ( θ ) , t ≥ , (A.7)7is, for any θ ∈ R satisfying E P X (cid:2) e θX (cid:3) < ∞ , a well-defined martingale. Using the optional sampling theorem,this allows us to derive, in particular, that for θ (cid:63) > x ∈ R , T > e θ (cid:63) (cid:96) P Xx (cid:16) M X T ≥ (cid:96) (cid:17) ≤ E P X x (cid:104) exp (cid:8) θ (cid:63) X ( τ X, + (cid:96) ∧T ) (cid:9)(cid:105) ≤ (cid:26) e Φ X ( θ (cid:63) ) T + θ (cid:63) x , if Φ X ( θ (cid:63) ) > ,e θ (cid:63) x , if Φ X ( θ (cid:63) ) ≤ . (A.8)Therefore, we obtain in any case for x ∈ R and T > e θ (cid:63) (cid:96) P Xx (cid:16) M X T ≥ (cid:96) (cid:17) ≤ C for some constant C > θ (cid:63) > θ > c > c θ = θ (cid:63) ,that for any x ∈ R and T > e c · (cid:96) P Xx (cid:16) M X T ≥ (cid:96) (cid:17) = e c (1 − θ ) (cid:96) e c θ · (cid:96) P Xx (cid:16) M X T ≥ (cid:96) (cid:17) = e c (1 − θ ) (cid:96) e θ (cid:63) (cid:96) P Xx (cid:16) M X T ≥ (cid:96) (cid:17) → , as (cid:96) ↑ ∞ , (A.9)holds, hence (A.4).To prove that E P z (cid:2) | I P & LT | (cid:3) < ∞ holds for any T >
0, we combine the convergence results (3.8) with standardtechniques. First, we note that E P z (cid:104) | I P & LT | (cid:105) = E P z (cid:20) ∞ (cid:90) {| I P & LT |≥ (cid:96) } d(cid:96) (cid:21) ≤ ∞ (cid:90) P z (cid:16) I P & LT ≥ (cid:96) (cid:17) d(cid:96) + ∞ (cid:90) P z (cid:16) I P & LT ≤ − (cid:96) (cid:17) d(cid:96) (A.10) ≤ | z | + ∞ (cid:90) P z (cid:16) I P & LT ≤ − (cid:96) (cid:17) d(cid:96). Therefore, we only need to show the finiteness of the integral on the right hand side. Under
Scenario 1 , theboundedness of (cid:96) (cid:55)→ e c · (cid:96) P Xz (cid:16) τ X, −− (cid:96) ≤ T (cid:17) on [0 , ∞ ) implies that ∞ (cid:90) P z (cid:16) I P & LT ≤ − (cid:96) (cid:17) d(cid:96) = ∞ (cid:90) e − c · (cid:96) e c · (cid:96) P Xz (cid:16) τ X, −− (cid:96) ≤ T (cid:17) d(cid:96) ≤ K ∞ (cid:90) e − c · (cid:96) d(cid:96) < ∞ , (A.11)and this already provides the required result. Therefore, we next focus on Scenario 2 . Here, we first notethat for a long position the finiteness of the integral directly follows from the fact that I P & LT ≥ − z . Hence,we are left with the case of a short position under Scenario 2 . In this case, similar arguments as in (A.11)give that ∞ (cid:90) P z (cid:16) I P & LT ≤ − (cid:96) (cid:17) d(cid:96) = ∞ (cid:90) e − c log( z + (cid:96) ) e c log( z + (cid:96) ) P X log( z − z ) (cid:16) τ X, +log( z + (cid:96) ) ≤ T (cid:17) d(cid:96) ≤ K ∞ (cid:90) z + (cid:96) ) c d(cid:96) < ∞ , (A.12)where the finiteness follows since c >
1. This finally gives the claim.8
Appendix B: Proofs - First-Passage Probabilities
Proof of Proposition 2.
We start by noting that, for any ( t , x, δ ) ∈ [0 , T ] × R × R , the process ( Z t ) t ∈ [0 , t ] defined via Z t := ( t − t, x + X t , δ + ∆ X t ) is a strong Markov process with state domain given by D t :=[0 , t ] × R × R and define, for any (cid:96) ∈ R , the following stopping domains S + (cid:96) := S + (cid:96), ∪ S + (cid:96), , with S + (cid:96), := { }× R × R and S + (cid:96), := [0 , T ] × [ (cid:96), ∞ ) × [ δ, ∞ ) , S J , + (cid:96) := [0 , T ] × ( (cid:96), ∞ ) × [ δ, ∞ ) , S , + (cid:96) := S + (cid:96) \ S J , + (cid:96) = S , + (cid:96), ∪ S , + (cid:96), , (A.13)with S , + (cid:96), = S + (cid:96), \ S J , + (cid:96) and S , + (cid:96), = S + (cid:96), \ S J , + (cid:96) , S − (cid:96) := S − (cid:96), ∪ S − (cid:96), , with S − (cid:96), := { }× R × R and S − (cid:96), := [0 , T ] × ( −∞ , (cid:96) ] × ( −∞ , δ ] , S J , − (cid:96) := [0 , T ] × ( −∞ , (cid:96) ) × ( −∞ , δ ] , S , − (cid:96) := S − (cid:96) \ S J , − (cid:96) = S , − (cid:96), ∪ S , − (cid:96), , (A.14)with S , − (cid:96), = S − (cid:96), \ S J , − (cid:96) and S , − (cid:96), = S − (cid:96), \ S J , − (cid:96) . Clearly, both S + (cid:96) and S − (cid:96) are closed in the state space D T . We therefore obtain that, for each of thesesdomains, the first entry times τ S ± (cid:96) defined via τ S ± (cid:96) := inf (cid:8) t ≥ Z t ∈ S ± (cid:96) (cid:9) (A.15)is a stopping time that satisfies τ S ± (cid:96) ≤ t , under P Zz , the measure having initial distribution Z = z = ( t , x, δ ).Using this notation, we can now re-express the first-passage probabilities u E ± X ( · ) and u E ±J X ( · ) as solutions ofappropriate stopping problems. Indeed, it is easily seen that u E ± X ( T , x ; (cid:96) ) = V ± (cid:0) ( T , x, (cid:1) and u E ±J X ( T , x ; (cid:96) ) = V ±J (cid:0) ( T , x, (cid:1) , (A.16)where the value functions V ± ( · ) and V ±J ( · ) have the following probabilistic representations: V ± ( z ) = E P Z z (cid:20) S , ± (cid:96), (cid:16) Z τ S± (cid:96) (cid:17)(cid:21) and V ±J ( z ) = E P Z z (cid:20) S J , ± (cid:96) (cid:16) Z τ S± (cid:96) (cid:17)(cid:21) . (A.17)Additionally, standard arguments based on the strong Markov property (cf. [PS06], [Ma19a], [Ma19b]) implythat, for any (cid:96) ∈ R , the functions V ± ( · ) and V ±J ( · ) satisfy ∂ t V ± (cid:0) ( t , x, δ ) (cid:1) = A X V ± (cid:0) ( t , x, δ ) (cid:1) , on D T \ S ± (cid:96) , (A.18) V ± (cid:0) ( t , x, δ ) (cid:1) = S , ± (cid:96), (cid:0) ( t , x, δ ) (cid:1) , on S ± (cid:96) , (A.19)and ∂ t V ±J (cid:0) ( t , x, δ ) (cid:1) = A X V ±J (cid:0) ( t , x, δ ) (cid:1) , on D T \ S ± (cid:96) , (A.20) V ±J (cid:0) ( t , x, δ ) (cid:1) = S J , ± (cid:96) (cid:0) ( t , x, δ ) (cid:1) , on S ± (cid:96) . (A.21)Therefore, recovering u E ± X ( · ) and u E ±J X ( · ) via (A.16) directly gives Equations (3.14)-(3.16) and (3.18)-(3.20),respectively. Since Equations (3.17) and (3.21) are naturally satisfied, Proposition 2 follows. Note that we implicitly use the fact that the infinitesimal generator of the process (∆ X t ) t ∈ [0 , t ] vanishes. This follows since( X t ) t ∈ [0 , t ] is, as Feller process, quasi-left-continuous. Proof of Proposition 3.
We prove Proposition 3 using a similar approach to the one adopted in the proofof Proposition 2. First, we recall from (3.29) that any of the maturity-randomized first-passage probabilities LC (cid:0) u E ± X (cid:1) ( · ) and LC (cid:0) u E ±J X (cid:1) ( · ) can be seen as the probability of a respective first-passage event occurring beforethe first jump time of an independent Poisson process ( N t ) t ≥ with intensity ϑ >
0. Therefore, we consider,for any ( n, x, δ ) ∈ N × R × R , the process ( Z t ) t ≥ defined via Z t := ( n + N t , x + X t , δ + ∆ X t ) and notethat it is a strong Markov process on the state domain D := N × R × R . Additionally, we define, for (cid:96) ∈ R ,the following stopping domains S + (cid:96) := S + (cid:96), ∪ S + (cid:96), , with S + (cid:96), := N × R × R and S + (cid:96), := { } × [ (cid:96), ∞ ) × [ δ, ∞ ) , S J , + (cid:96) := { } × ( (cid:96), ∞ ) × [ δ, ∞ ) , S , + (cid:96) := S + (cid:96) \ S J , + (cid:96) = S , + (cid:96), ∪ S , + (cid:96), , (A.22)with S , + (cid:96), = S + (cid:96), \ S J , + (cid:96) and S , + (cid:96), = S + (cid:96), \ S J , + (cid:96) , S − (cid:96) := S − (cid:96), ∪ S − (cid:96), , with S − (cid:96), := N × R × R and S − (cid:96), := { } × ( −∞ , (cid:96) ] × ( −∞ , δ ] , S J , − (cid:96) := { } × ( −∞ , (cid:96) ) × ( −∞ , δ ] , S , − (cid:96) := S − (cid:96) \ S J , − (cid:96) = S , − (cid:96), ∪ S , − (cid:96), , (A.23)with S , − (cid:96), = S − (cid:96), \ S J , − (cid:96) and S , − (cid:96), = S − (cid:96), \ S J , − (cid:96) , and see that both S + (cid:96) and S − (cid:96) form a closed set in D . Consequently, for each of these domains, the firstentry time defined by τ S ± (cid:96) := inf (cid:8) t ≥ Z t ∈ S ± (cid:96) (cid:9) (A.24)is a stopping time. Furthermore, the finiteness of the first moment of the exponential distribution for anyintensity parameter ϑ > P Zz -almost sure finiteness of τ S ± (cid:96) for any z = ( n, x, δ ), where P Zz refersto the measure having initial distribution Z = z . Using this notation, we can therefore follow the line ofthe arguments developed in the proof of Proposition 2 and re-express the maturity-randomized first-passageprobabilities LC (cid:0) u E ± X (cid:1) ( · ) and LC (cid:0) u E ±J X (cid:1) ( · ) as solutions of appropriate stopping problems: LC (cid:0) u E ± X (cid:1) ( ϑ, x ; (cid:96) ) = (cid:98) V ± (cid:0) (0 , x, (cid:1) and LC (cid:0) u E ±J X (cid:1) ( ϑ, x ; (cid:96) ) = (cid:98) V ±J (cid:0) (0 , x, (cid:1) , (A.25)where the value functions (cid:98) V ± ( · ) and (cid:98) V ±J ( · ) have the following probabilistic representations: (cid:98) V ± ( z ) = E P Z z (cid:20) S , ± (cid:96), (cid:16) Z τ S± (cid:96) (cid:17)(cid:21) and (cid:98) V ±J ( z ) = E P Z z (cid:20) S J , ± (cid:96) (cid:16) Z τ S± (cid:96) (cid:17)(cid:21) . (A.26)Additionally, standard arguments based on the strong Markov property (cf. [PS06], [Ma19a], [Ma19b]) implythat, for any (cid:96) ∈ R , the functions (cid:98) V ± ( · ) and (cid:98) V ±J ( · ) satisfy the following problems A Z (cid:98) V ± (cid:0) ( n, x, δ ) (cid:1) = 0 , on D \ S ± (cid:96) , (A.27) (cid:98) V ± (cid:0) ( n, x, δ ) (cid:1) = S , ± (cid:96), (cid:0) ( n, x, δ ) (cid:1) , on S ± (cid:96) , (A.28) We emphasize that several choices of a product-metric on D give the closedness of the set S + (cid:96) and S − (cid:96) . In particular, onemay choose on N the following metric d N ( m, n ) := (cid:26) | − m − − n | , m (cid:54) = n, , m = n, and consider the product-metric on D obtained by combining d N ( · , · ) on N with the Euclidean metric on R . A Z (cid:98) V ±J (cid:0) ( n, x, δ ) (cid:1) = 0 , on D \ S ± (cid:96) , (A.29) (cid:98) V ±J (cid:0) ( n, x, δ ) (cid:1) = S J , ± (cid:96) (cid:0) ( n, x, δ ) (cid:1) , on S ± (cid:96) . (A.30)where A Z denotes the infinitesimal generator of the process ( Z t ) t ≥ . To complete the proof, it thereforesuffices to note that (for any suitable function V : D → R ) the infinitesimal generator A Z can be re-expressedas A Z V ( z ) = A nN V (cid:0) ( n, x, δ ) (cid:1) + A xX V (cid:0) ( n, x, δ ) (cid:1) (A.31)= ϑ (cid:0) V (cid:0) ( n + 1 , x, δ ) (cid:1) − V (cid:0) ( n, x, δ ) (cid:1)(cid:1) + A xX V (cid:0) ( n, x, δ ) (cid:1) , (A.32)where A N denotes the infinitesimal generator of the Poisson process ( N t ) t ≥ and the notation A nN and A xX is used to indicate that the generators are applied to n and x respectively. Indeed, recovering LC (cid:0) u E ± X (cid:1) ( · )and LC (cid:0) u E ±J X (cid:1) ( · ) via (A.25) while noting Relation (A.32) and the fact that for any x ∈ R and δ ∈ R we have (cid:98) V ± (cid:0) (1 , x, δ ) (cid:1) = 0 and (cid:98) V ±J (cid:0) (1 , x, δ ) (cid:1) = 0 (A.33)finally leads to Problems (3.30)-(3.32) and (3.33)-(3.35), respectively. Appendix C: Proofs - Hyper-Exponential Jump-Diffusions
Proof of Proposition 4.
We only provide a proof for the upside first-passage probabilities, i.e., for thefunctions LC (cid:0) u E +0 X (cid:1) ( · ) and LC (cid:0) u E + J X (cid:1) ( · ), and note that the downside first-passage probabilities can be derivedanalogously.We begin by fixing ϑ > (cid:96) ∈ R and showing that the maturity-randomized first-passage probabilities LC (cid:0) u E +0 X (cid:1) ( · ) and LC (cid:0) u E + J X (cid:1) ( · ) have the structure described in (3.60) with coefficients ( v ,k ) k ∈{ ,...,m +1 } and( v J ,k ) k ∈{ ,...,m +1 } satisfying the linear equations in (3.61). Here, we first note that the same arguments asin the proof of Theorem 3.2. in [CK11] imply that the general solution to OIDE (3.30) takes the form˜ V ( ϑ, x ; (cid:96) ) = m +1 (cid:88) k =1 w (cid:63)k e β k,ϑ · x + n +1 (cid:88) h =1 w (cid:63)h e γ h,ϑ · x , (A.34)or, equivalently, V ( ϑ, x ; (cid:96) ) = m +1 (cid:88) k =1 w k e β k,ϑ · ( x − (cid:96) ) + n +1 (cid:88) h =1 w h e γ h,ϑ · ( x − (cid:96) ) . (A.35)Although both expressions (A.34) and (A.35) lead to equivalent results, we choose to follow Ansatz (A.35)since it will allow us to separate the dependency of the first-passage level (cid:96) from the remaining parts. Thislast property will prove useful in subsequent computations discussed, for instance, in Section 5.Next, the boundedness of the functions LC (cid:0) u E +0 X (cid:1) ( · ) and LC (cid:0) u E + J X (cid:1) ( · ) gives that for both of these functionsone must have w = . . . = w n +1 = 0 . (A.36) Here again, we have implicitly used the quasi-left-continuity of the process ( X t ) t ≥ , which follows from the Feller property. v ,k ) k ∈{ ,...,m +1 } and( v J ,k ) k ∈{ ,...,m +1 } . First, the continuous-fit conditions LC (cid:0) u E +0 X (cid:1) ( ϑ, (cid:96) − ; (cid:96) ) = LC (cid:0) u E +0 X (cid:1) ( ϑ, (cid:96) ; (cid:96) ) and LC (cid:0) u E + J X (cid:1) ( ϑ, (cid:96) − ; (cid:96) ) = LC (cid:0) u E + J X (cid:1) ( ϑ, (cid:96) ; (cid:96) ) (A.37)imply that m +1 (cid:88) k =1 v ,k = 1 and m +1 (cid:88) k =1 v J ,k = 0 (A.38)must hold, respectively. Therefore, to fully determine the coefficients ( v ,k ) k ∈{ ,...,m +1 } and ( v J ,k ) k ∈{ ,...,m +1 } at least m additional equations are required in each case. We derive these equations by substituting (3.60)back in OIDE (3.30). Here, we focus on LC (cid:0) u E +0 X (cid:1) ( · ) and will briefly comment on LC (cid:0) u E + J X (cid:1) ( · ) afterwards. Tostart, we derive for x < (cid:96) that (cid:90) R LC (cid:0) u E +0 X (cid:1) ( ϑ, x + y ; (cid:96) ) f J ( y ) dy = (cid:90) −∞ LC (cid:0) u E +0 X (cid:1) ( ϑ, x + y ; (cid:96) ) n (cid:88) j =1 q j η j e η j y dy + (cid:96) − x (cid:90) LC (cid:0) u E +0 X (cid:1) ( ϑ, x + y ; (cid:96) ) (cid:32) m (cid:88) i =1 p i ξ i e − ξ i y (cid:33) dy + ∞ (cid:90) (cid:96) − x LC (cid:0) u E +0 X (cid:1) ( ϑ, x + y ; (cid:96) ) (cid:32) m (cid:88) i =1 p i ξ i e − ξ i y (cid:33) dy = n (cid:88) j =1 q j η j e − η j x x (cid:90) −∞ LC (cid:0) u E +0 X (cid:1) ( ϑ, z ; (cid:96) ) e η j z dz + m (cid:88) i =1 p i ξ i e ξ i x (cid:96) (cid:90) x LC (cid:0) u E +0 X (cid:1) ( ϑ, z ; (cid:96) ) e − ξ i z dz + m (cid:88) i =1 p i ξ i e ξ i x ∞ (cid:90) (cid:96) LC (cid:0) u E +0 X (cid:1) ( ϑ, z ; (cid:96) ) e − ξ i z dz = m +1 (cid:88) k =1 n (cid:88) j =1 q j η j e − η j x v ,k x (cid:90) −∞ e ( η j + β k,ϑ ) z e − β k,ϑ · (cid:96) dz + m +1 (cid:88) k =1 m (cid:88) i =1 p i ξ i e ξ i x v ,k (cid:96) (cid:90) x e − ( ξ i − β k,ϑ ) z e − β k,ϑ · (cid:96) dz (A.39)= m +1 (cid:88) k =1 n (cid:88) j =1 q j η j η j + β k,ϑ v ,k e β k,ϑ · ( x − (cid:96) ) + m +1 (cid:88) k =1 m (cid:88) i =1 p i ξ i ξ i − β k,ϑ v ,k e β k,ϑ · ( x − (cid:96) ) − m +1 (cid:88) k =1 m (cid:88) i =1 p i ξ i ξ i − β k,ϑ v ,k e − ξ i ( (cid:96) − x ) . (A.40)Therefore, using the relations ∂ x LC (cid:0) u E +0 X (cid:1) ( ϑ, x ; (cid:96) ) = m +1 (cid:88) k =1 v ,k β k,ϑ e β k,ϑ · ( x − (cid:96) ) and ∂ x LC (cid:0) u E +0 X (cid:1) ( ϑ, x ; (cid:96) ) = m +1 (cid:88) k =1 v ,k ( β k,ϑ ) e β k,ϑ · ( x − (cid:96) ) (A.41) Note that these continuous-fit conditions are guaranteed by σ X > x < (cid:96) :0 = A X LC (cid:0) u E +0 X (cid:1) ( ϑ, x ; (cid:96) ) − ϑ LC (cid:0) u E +0 X (cid:1) ( ϑ, x ; (cid:96) )= m +1 (cid:88) k =1 v ,k e β k,ϑ · ( x − (cid:96) ) (cid:0) Φ X ( β k,ϑ ) − ϑ (cid:1) − λ m (cid:88) i =1 p i e − ξ i ( (cid:96) − x ) (cid:32) m +1 (cid:88) k =1 ξ i ξ i − β k,ϑ v ,k (cid:33) = − λ m (cid:88) i =1 p i e − ξ i ( (cid:96) − x ) (cid:32) m +1 (cid:88) k =1 ξ i ξ i − β k,ϑ v ,k (cid:33) . (A.42)Since ξ , . . . , ξ m are different from each other, we conclude that the following equation must hold: m +1 (cid:88) k =1 ξ i ξ i − β k,ϑ v ,k = 0 , ∀ i ∈ { , . . . , m } . (A.43)Hence, combining (A.38) and (A.43) results in A ϑ v = (1 , m ) (cid:124) immediately.To derive the corresponding system of equations for LC (cid:0) u E + J X (cid:1) ( · ), we proceed as above. First, reproducingthe derivation of the integral term, provides the following result for x < (cid:96) : (cid:90) R LC (cid:0) u E + J X (cid:1) ( ϑ, x + y ; (cid:96) ) f J ( y ) dy = m +1 (cid:88) k =1 n (cid:88) j =1 q j η j η j + β k,ϑ v J ,k e β k,ϑ · ( x − (cid:96) ) + m +1 (cid:88) k =1 m (cid:88) i =1 p i ξ i ξ i − β k,ϑ v J ,k e β k,ϑ · ( x − (cid:96) ) − m +1 (cid:88) k =1 m (cid:88) i =1 p i ξ i ξ i − β k,ϑ v J ,k e − ξ i ( (cid:96) − x ) + m (cid:88) i =1 p i e − ξ i ( (cid:96) − x ) . (A.44)Then, inserting the latter expression in OIDE (3.30) gives that0 = − λ m (cid:88) i =1 p i e − ξ i ( (cid:96) − x ) (cid:32) m +1 (cid:88) k =1 ξ i ξ i − β k,ϑ v J ,k − (cid:33) . (A.45)This finally implies that m +1 (cid:88) k =1 ξ i ξ i − β k,ϑ v J ,k = 1 , ∀ i ∈ { , . . . , m } , (A.46)which immediately results, together with (A.38), in A ϑ v J = (0 , m ) (cid:124) .To conclude, we note that the uniqueness of all the vectors v , v J , v and v J follows from the invertibilityof the matrices A ϑ and A ϑ (cf. [CK11]). Proof of Proposition 5.
Our proof mainly relies on arguments introduced in [CCW09] (cf. also [CYY13])and focuses, as in the proof of Propositions 4, on the upside first-passage probabilities. However, we notethat the same techniques can be applied to derive the corresponding downside results.We start by considering, for (cid:96) ∈ R , x ∈ R \ H + (cid:96) , and ϑ > b (cid:55)→ E P X x (cid:104) exp (cid:110) − ϑτ X, + (cid:96) + bX τ X, + (cid:96) (cid:111)(cid:105) (A.47)3on the domain D := { b ∈ C : Re( b ) ≥ } and recall that the overshoot distribution is conditionallymemoryless and independent of the first-passage time provided the overshoot is greater than zero and theexponential type of the jump distribution is specified, i.e., we have, for (cid:96) ∈ R , x ∈ R \ H + (cid:96) and any t > y > P Xx (cid:16) X τ X, + (cid:96) − (cid:96) ≥ y (cid:12)(cid:12)(cid:12) X τ X, + (cid:96) > (cid:96), J N τX, + (cid:96) ∼ Exp( ξ i ) (cid:17) = e − ξ i y , (A.48)and P Xx (cid:16) τ X, + (cid:96) ≤ t, X τ X, + (cid:96) − (cid:96) ≥ y (cid:12)(cid:12)(cid:12) X τ X, + (cid:96) > (cid:96), J N τX, + (cid:96) ∼ Exp( ξ i ) (cid:17) = P Xx (cid:16) τ X, + (cid:96) ≤ t (cid:12)(cid:12)(cid:12) X τ X, + (cid:96) > (cid:96), J N τX, + (cid:96) ∼ Exp( ξ i ) (cid:17) · P Xx (cid:16) X τ X, + (cid:96) − (cid:96) ≥ y (cid:12)(cid:12)(cid:12) X τ X, + (cid:96) > (cid:96), J N τX, + (cid:96) ∼ Exp( ξ i ) (cid:17) . (A.49)This was already derived in [Ca09] and implies, in particular, that for any purely imaginary number b ∈ CE P X x (cid:104) exp (cid:110) − ϑτ X, + (cid:96) + bX τ X, + (cid:96) (cid:111)(cid:105) = e b · (cid:96) E P X x (cid:104) e − ϑτ X, + (cid:96) E +0 (cid:105) + e b · (cid:96) m (cid:88) i =1 E P X x (cid:104) e − ϑτ X, + (cid:96) E + i (cid:105) ∞ (cid:90) ξ i e − ( ξ i − b ) y dy, = e b · (cid:96) E P X x (cid:104) e − ϑτ X, + (cid:96) E +0 (cid:105) + e b · (cid:96) m (cid:88) i =1 ξ i ξ i − b E P X x (cid:104) e − ϑτ X, + (cid:96) E + i (cid:105) . (A.50)Combining this identity with an analytic continuation argument will allow us to derive the required systemof equations. Indeed, we first note that, for any x ∈ R , ϑ > b ∈ C , theprocess ( M t ) t ≥ defined via M t := e − ϑt + bX t − e bx − (cid:0) Φ X ( b ) − ϑ (cid:1) t (cid:90) e − ϑs + bX s ds (A.51)is a zero-mean P Xx -martingale and obtain, for x ∈ R \ H + (cid:96) , via the optional sampling theorem that0 = E P X x (cid:104) exp (cid:110) − ϑτ X, + (cid:96) + bX τ X, + (cid:96) (cid:111)(cid:105) − e bx − (cid:0) Φ X ( b ) − ϑ (cid:1) E P X x (cid:34) τ X, + (cid:96) (cid:90) e − ϑs + bX s ds (cid:35) = e b · (cid:96) E P X x (cid:104) e − ϑτ X, + (cid:96) E +0 (cid:105) + e b · (cid:96) m (cid:88) i =1 ξ i ξ i − b E P X x (cid:104) e − ϑτ X, + (cid:96) E + i (cid:105) − e bx − (cid:0) Φ X ( b ) − ϑ (cid:1) E P X x (cid:34) τ X, + (cid:96) (cid:90) e − ϑs + bX s ds (cid:35) . (cid:124) (cid:123)(cid:122) (cid:125) =: g ( b ) (A.52)Hence, if one defines a new function by G ( b ) := m (cid:81) i =1 ( ξ i − b ) · g ( b ), one easily sees that G ( · ) is well-definedand (as a function of b ) analytic on the full domain D . Therefore, by the identity theorem for analyticfunctions (cf. [Ru87]), we must have that G ( b ) ≡ b ∈ D . Accordingly, we must have that g ( b ) ≡ b ∈ D \ { ξ , . . . , ξ m } . This finally allows us to replace b in Equation (A.52) by the positive roots β ,ϑ , . . . , β m +1 ,ϑ to obtain that e β k,ϑ · ( x − (cid:96) ) = E P X x (cid:104) e − ϑτ X, + (cid:96) E +0 (cid:105) + m (cid:88) i =1 ξ i ξ i − β k,ϑ E P X x (cid:104) e − ϑτ X, + (cid:96) E + i (cid:105) , ∀ k ∈ { , . . . , m + 1 } , (A.53)which gives, in view of (3.69), the required system of equations. Proof of Proposition 6.
Our derivations are based on the results obtained in Theorem 2.1 and Theo-rem 2.2 in [CYY13]. Indeed, combining first the results of Theorem 2.2 with (3.69) and Proposition 5 givesthat for (cid:96) ∈ R , x ∈ R \ H + (cid:96) and ϑ > LC ( u E +0 X )( ϑ, x ; (cid:96) ) = G + e β ,ϑ · ( x − (cid:96) ) − G + · m +1 (cid:88) k =2 (cid:32) m (cid:88) s =1 d + k,s ξ s − β ,ϑ (cid:33) · e β k,ϑ · ( x − (cid:96) ) , (A.54)and, for i ∈ { , . . . , m } , LC ( u E + i X )( ϑ, x ; (cid:96) ) = − G + ξ i C + ϑ ( ξ i ) (cid:0) B + (cid:1) (cid:48) ( ξ i ) e β ,ϑ · ( x − (cid:96) ) + 1 ξ i m +1 (cid:88) k =2 (cid:32) d + k,i + G + C + ϑ ( ξ i ) (cid:0) B + (cid:1) (cid:48) ( ξ i ) m (cid:88) s =1 d + k,s ξ s − β ,ϑ (cid:33) · e β k,ϑ · ( x − (cid:96) ) , (A.55)with G + := B + ( β ,ϑ ) C + ϑ ( β ,ϑ ) and B + ( · ), C + ϑ ( · ) and d + i,j defined as in (3.76) and (3.77). Therefore, comparing Result(A.54) with the corresponding results in Proposition 4 already gives that v , = G + and v ,k = − G + · m (cid:88) s =1 d + k,s ξ s − β ,ϑ , k ∈ { , . . . , m + 1 } , (A.56)and substituting this identity back in (A.55) finally gives the results (3.74) and (3.75).Deriving the results for the downside case can be done in the same way. First, combining Theorem 2.1in [CYY13] with (3.69) and Proposition 5 gives, for (cid:96) ∈ R , x ∈ R \ H − (cid:96) and ϑ >
0, that LC ( u E − X )( ϑ, x ; (cid:96) ) = G − e γ ,ϑ · ( x − (cid:96) ) − G − · n +1 (cid:88) k =2 (cid:32) n (cid:88) s =1 d − k,s η s + γ ,ϑ (cid:33) · e γ k,ϑ · ( x − (cid:96) ) (A.57)and, for j ∈ { , . . . , n } , LC ( u E − j X )( ϑ, x ; (cid:96) ) = ( − n G − η j C − ϑ ( η j ) (cid:0) B − (cid:1) (cid:48) ( − η j ) e γ ,ϑ · ( x − (cid:96) ) + 1 η j n +1 (cid:88) k =2 (cid:32) d − k,j + ( − n − G − C − ϑ ( η j ) (cid:0) B − (cid:1) (cid:48) ( − η j ) n (cid:88) s =1 d − k,s η s + γ ,ϑ (cid:33) · e γ k,ϑ · ( x − (cid:96) ) (A.58)with G − := ( − n B − ( γ ,ϑ ) C − ϑ ( − γ ,ϑ ) and B − ( · ), C − ϑ ( · ) and d − i,j defined as in (3.80) and (3.81). Therefore, comparingResult (A.54) with the corresponding results in Proposition 4 already gives that v , = G − and v ,k = − G − · n (cid:88) s =1 d − k,s η s + γ ,ϑ , k ∈ { , . . . , n + 1 } (A.59)holds and substituting this identity back in (A.58) finally gives (3.78) and (3.79).5 References [ADEH99] Artzner Philippe, Delbaen Freddy, Eber Jean-Marc and Heath David,
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