aa r X i v : . [ q -f i n . M F ] J un Modern Stochastics: Theory and Applications 7 (2) (2020) 113–134https://doi.org/10.15559/20-VMSTA152
A pure-jump mean-reverting short rate model
Markus Hess
Independent
[email protected] (M. Hess)Received: 23 April 2019, Revised: 27 March 2020, Accepted: 27 March 2020,Published online: 20 April 2020
Abstract
A new multi-factor short rate model is presented which is bounded from belowby a real-valued function of time. The mean-reverting short rate process is modeled by a sumof pure-jump Ornstein–Uhlenbeck processes such that the related bond prices possess affinerepresentations. Also the dynamics of the associated instantaneous forward rate is providedand a condition is derived under which the model can be market-consistently calibrated. Theanalytical tractability of this model is illustrated by the derivation of an explicit plain vanillaoption price formula. With view on practical applications, suitable probability distributions areproposed for the driving jump processes. The paper is concluded by presenting a post-crisisextension of the proposed short and forward rate model.
Keywords
Short rate, forward rate, zero-coupon bond, option pricing, market-consistentcalibration, post-crisis model, Lévy process, multi-factor model, Ornstein–Uhlenbeckprocess, stochastic differential equation
JEL classification
G12, D52
Stochastic interest rate models play an important role in the modeling of financialmarkets. The literature essentially distinguishes between short rate models, forwardrate models and market models. In the sequel, we give a brief survey on the differentclasses of term structure models. For more detailed information, the reader is referredto the respective research articles or the textbooks [7, 21] and [26].Widely applied short rate models are for example the Vasicek model [38], theHull–White model [29] or the Cox–Ingersoll–Ross (CIR) model [10]. In [38] and
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M. Hess [29] the short rate process is modeled by a stochastic differential equation (SDE)of Ornstein–Uhlenbeck (OU) type driven by a Brownian motion (BM). As a conse-quence, the short rate process is normally distributed in these models and may be-come arbitrarily negative. Both features embody severe disadvantages with view onreal-world market behavior, as the distribution of interest rate data frequently deviatesfrom the normal distribution, while interest rates do not take arbitrarily large negativevalues in practice. In the recent years, there indeed appeared negative interest ratesfrom time to time, but the negative values usually were small and stayed above somelower bound. However, in [10] the short rate is modeled by a so-called square-rootprocess. This approach leads to a mean-reverting, strictly positive and chi-square dis-tributed short rate process. In [6] the authors propose a time-homogeneous short ratemodel which is extended by a deterministic shift function in order to allow for neg-ative rates and a perfect fit to the initially observed term structure. A very detailedoverview on short rate models and their properties can be found in [7, 16] and [21].In [26] and [33] short rate models in an extended multiple-curve framework are pre-sented. The probably most famous forward rate model is the Heath–Jarrow–Morton(HJM) model proposed in [27]. Therein, the instantaneous forward rate process ismodeled directly by an arithmetic SDE driven by a BM. In [4] the HJM model isextended to a jump-diffusion setup where the forward rate process is affected by bothdiffusion and random jump noise. HJM type models are also treated in [7] and Chap-ter 7 in [28]. In [12] and [26] HJM forward rate models in an extended multi-curveframework are discussed. The class of the so-called market models was introducedin [5]. For example, the popular LIBOR model belongs to this modeling class. Inmost cases, market models involve geometric SDEs such that the modeled interestrates usually turn out to be strictly positive. In order to allow the modeled rates alsoto take small negative values, shifted market model approaches have been proposedrecently. Numerous properties of affine LIBOR models are provided in [31]. Marketmodels are also presented in [7]. In [26] LIBOR models in an extended multi-curveframework are discussed. In [17] the authors propose a Lévy forward price model ina multi-curve setup which is able to generate negative interest rates. Term structuremodels which are driven by Lévy processes have also been proposed in [18–20].In the present paper, we introduce a new pure-jump multi-factor short rate modelwhich is bounded from below by a real-valued function of time which can be chosenarbitrarily. The short rate process is modeled by a deterministic function plus a sumof pure-jump zero-reverting Ornstein–Uhlenbeck processes. It turns out that the shortrate is mean-reverting and that the related bond price formula possesses an affinerepresentation. We also provide the dynamics of the related instantaneous forwardrate, the latter being of HJM type. We further derive a condition under which the for-ward rate model can be market-consistently calibrated. The analytical tractability ofour model is illustrated by the derivation of a plain-vanilla option price formula withFourier transform methods. With view on practical applications, we make concreteassumptions on the distribution of the jump noises and show how explicit formulascan be deduced in these cases. We conclude the paper by presenting a post-crisisextension of our short and forward rate model.The outline of the paper is as follows: In Section 2 we introduce our new pure-jump multi-factor short rate model which is bounded from below. Section 3 is ded- pure-jump mean-reverting short rate model icated to the derivation of related bond price and forward rate representations. Sec-tion 4 is devoted to option pricing. Section 5 contains guidelines for a practical appli-cation, while putting a special focus on possible distributional choices for the model-ing of the involved jump noises. In Section 6 we consider a post-crisis extension ofthe proposed short and forward rate model.
Let (cid:16) Ω , F , F = ( F t ) t ∈ [0 ,T ] , Q (cid:17) be a filtered probability space satisfying the usualhypotheses, i.e. F t = F t + := ∩ s>t F s constitutes a right-continuous filtration and F denotes the sigma-algebra augmented by all Q -null sets (cf. [30, 35]). Here, Q is arisk-neutral probability measure and T > denotes a fixed finite time horizon. In thissetup, for arbitrary n ∈ N we define the stochastic short rate process r = ( r t ) t ∈ [0 ,T ] via r t := µ ( t ) + n X k =1 X kt (2.1)where µ ( t ) is a differentiable real-valued deterministic L -function and X kt con-stitute pure-jump zero-reverting Ornstein–Uhlenbeck (OU) processes satisfying theSDE dX kt = − λ k X kt dt + σ k dL kt (2.2)with deterministic initial values X k := x k ≥ , constant mean-reversion velocities λ k > and constant volatility coefficients σ k > . Herein, the independent, càdlàg,increasing, pure-jump, compound Poisson Lévy processes L kt are defined by L kt := Z t Z D k zdN k ( s, z ) (2.3)where D k ⊆ R + := ]0 , ∞ [ ⊂ R denote jump amplitude sets and N k constitutePoisson random measures (PRMs). Note that the processes X kt and L kt always jumpsimultaneously, while X kt decays exponentially between its jumps due to the damp-ening linear drift term appearing in (2.2). A typical trajectory of a Lévy-driven OUprocess is shown in Figure 15.1 in [9]. Further note that the background-driving time-homogeneous Lévy processes L kt are increasing and thus, constitute so-called subor-dinators. Moreover, for all k ∈ { , . . . , n } and ( s, z ) ∈ [0 , T ] × D k we define the Q -compensated PRMs d ˜ N Q k ( s, z ) := dN k ( s, z ) − dν k ( z ) ds (2.4)which constitute ( F , Q ) -martingale integrators. Herein, the positive and σ -finite Lévymeasures ν k satisfy the integrability conditions Z D k (1 ∧ z ) dν k ( z ) < ∞ , Z z> e ̟z dν k ( z ) < ∞ (2.5)for an arbitrary constant ̟ ∈ R (cf. [9, 17]). For all k ∈ { , . . . , n } and t ∈ [0 , T ] we obtain E Q (cid:2) L kt (cid:3) = t Z D k zdν k ( z ) , Var Q (cid:2) L kt (cid:3) = t Z D k z dν k ( z ) M. Hess both being finite entities due to (2.5) (cf. Section 1 in [17]). We remark that the cur-rently proposed multi-factor short rate model (2.1) has been inspired by the electricityspot price model introduced in [2]. Arithmetic multi-factor models of this type havealso been investigated in [28] and Section 3.2.2 in [3].
Remark 2.1. (a)
Since L kt is increasing and X kt is zero-reverting from above, thefunction µ ( t ) is the mean-reversion floor or lower bound of the short rate process r t ,i.e. it holds r t ≥ µ ( t ) Q -a.s. for all t ∈ [0 , T ] , while r t is mean-reverting from aboveto µ ( t ) . Also note that the presence of a Brownian motion (BM) as driving noisein one of the processes X t , . . . , X nt would destroy the lower boundedness of r t . Incontrast to the presented pure-jump approach, it appears difficult to set up (lower-)bounded processes in arithmetic BM approaches. Moreover, we recall that negativerates have been observed in real-world post-crisis interest rate markets. Such scenar-ios can easily be captured by our model by choosing, e.g. µ ( t ) ≡ c , where c < isan arbitrary constant. (In practical applications, it may happen that the floor function µ ( t ) needs to be readjusted, if interest rates evolve lower than anticipated. This issuehas been discussed in [1] in the context of the SABR model.) (b) Our pure-jump model (2.1) is able to generate short rate trajectories whichclosely resemble those stemming from common Brownian motion approaches, if weallow for small jump sizes only, i.e. D k = (cid:2) ǫ k , ǫ k (cid:3) with small constants < ǫ k < ǫ k .In this context, we emphasize that the well-established pure-jump variance gammamodel is likewise able to generate suitable price trajectories, although there is neitherany diffusion component involved (cf. Section 2.6.3 in [3], Table 4.5 in [9], Sec-tion 5.3.7 in [37]). On top of that, our pure-jump model might even provide moreflexibility concerning the modeling of distributional properties than common BM ap-proaches, since we are able to implement tailor-made distributions via an appropriatechoice of the Lévy measures ν k which fit the empirical behavior of the rates in abest possible manner. This topic is further discussed in Section 5 below. For instance,(generalized) inverse Gaussian, tempered stable or gamma distributions might em-body suitable choices (recall Appendix B.1.2 on p. 151 in [37]). We finally recall thata model of the type (2.1) has been fitted to real market data in [2] (yet in an electricitymarket context).For a time partition ≤ t ≤ s ≤ T the solution of (2.2) under Q can be expressedas X ks = X kt e − λ k ( s − t ) + σ k Z st Z D k e − λ k ( s − u ) zdN k ( u, z ) (2.6)where we used (2.3). The representation (2.6) implies X kt = x k e − λ k t + σ k Z t Z D k e − λ k ( t − s ) zdN k ( s, z ) (2.7)where ≤ t ≤ T . For all t ∈ [0 , T ] we next define the historical filtration F t := σ { L s , . . . , L ns : 0 ≤ s ≤ t } . Proposition 2.2.
For ≤ u ≤ t ≤ T we have E Q ( r t |F u ) = µ ( t ) + n X k =1 (cid:18) X ku e − λ k ( t − u ) + σ k − e − λ k ( t − u ) λ k Z D k zdν k ( z ) (cid:19) , pure-jump mean-reverting short rate model Var Q ( r t |F u ) = n X k =1 σ k − e − λ k ( t − u ) λ k Z D k z dν k ( z ) where the short rate process r t satisfies (2.1). Both entities are finite due to (2.5). (Here and in what follows, we omit all proofs which are straightforward.) Taking u = 0 in Proposition 2.2, we find for all t ∈ [0 , T ] E Q [ r t ] = µ ( t ) + n X k =1 (cid:18) x k e − λ k t + σ k − e − λ k t λ k Z D k zdν k ( z ) (cid:19) , Var Q [ r t ] = n X k =1 σ k − e − λ k t λ k Z D k z dν k ( z ) . Note that it is possible to identify the entities E Q (cid:2) X kt (cid:3) and Var Q (cid:2) X kt (cid:3) inside thelatter equations due to (2.1). Moreover, suppose that µ ( t ) → ˜ µ for t → ∞ where ˜ µ ∈ R is a finite constant. Then we observe lim t →∞ E Q [ r t ] = ˜ µ + n X k =1 σ k λ k Z D k zdν k ( z ) , lim t →∞ Var Q [ r t ] = n X k =1 σ k λ k Z D k z dν k ( z ) which both constitute finite constants. This limit behavior entirely stands in line withthe requirements imposed on short rate models claimed on p. 46 in [7]. In the nextstep, we investigate the characteristic function of r t which is defined via Φ r t ( u ) := E Q (cid:2) e iur t (cid:3) where u ∈ R and t ∈ [0 , T ] . Proposition 2.3.
For k ∈ { , . . . , n } we define the deterministic functions Λ k ( s, z ) := σ k e − λ k ( t − s ) z, ψ k ( t, u ) := iue − λ k t ,ρ k ( t, u ) := Z t Z D k (cid:2) e iu Λ k ( s,z ) − (cid:3) dν k ( z ) ds. Then for any u ∈ R and t ∈ [0 , T ] the characteristic function of r t can be decomposedas Φ r t ( u ) = e iuµ ( t ) n Y k =1 Φ X kt ( u ) where the characteristic function of X kt is given by Φ X kt ( u ) = e ψ k ( t,u ) x k + ρ k ( t,u ) with deterministic and affine characteristic exponent. M. Hess
An immediate consequence of Proposition 2.3 is the subsequent affine represen-tation Φ r t ( u ) = n Y k =1 e ψ k ( t,u ) x k + φ k ( t,u ) (2.8)where we introduced the deterministic functions φ k ( t, u ) := ρ k ( t, u ) + iuµ ( t ) /n. We emphasize that r t is an affine function of the factors X t , . . . , X nt such that ourmodel turns out to be a special case of the affine short rate models considered inSection 3.3 in [14]. To read more on affine processes we refer to [7, 14, 16, 26] and[31]. We next define the moment generating function of r t via κ r t ( v ) := E Q [ e vr t ] (2.9)which implies the well-known equalities Φ r t ( u ) = κ r t ( iu ) and κ r t ( v ) = Φ r t ( − iv ) .Note that the moment generating function κ r t ( v ) is well-defined due to (2.5). In thesequel, we derive the time dynamics of the short rate process. Proposition 2.4.
For all t ∈ [0 , T ] the short rate process follows the dynamics dr t = µ ′ ( t ) − n X k =1 λ k X kt ! dt + n X k =1 σ k Z D k zdN k ( t, z ) . (2.10) Remark 2.5.
We recall that our model constitutes an extension of the short rate modelproposed in [6], whereas we work with multiple pure-jump processes L t , . . . , L nt asdriving noises instead of the single Brownian motion W t appearing in Eq. (1) in [6].Moreover, comparing Eq. (3) in [6] with Eq. (2.1) above, we see that x t and ϕ ( t ; α ) in [6] correspond in our setup to P nk =1 X kt and µ ( t ) , respectively. In this section, we derive representations for zero-coupon bond prices, forward ratesand the interest rate curve related to the short rate model introduced in Section 2. Tobegin with, we introduce a bank account with stochastic interest rate r t satisfying dβ t = r t β t dt (3.1)with normalized initial capital β = 1 . The solution of (3.1) reads as β t = exp (cid:26)Z t r s ds (cid:27) (3.2)where t ∈ [0 , T ] . In this setup, the (zero-coupon) bond price at time t ≤ T withmaturity T is given by P ( t, T ) := β t E Q (cid:0) β − T |F t (cid:1) = E Q exp ( − Z Tt r s ds ) (cid:12)(cid:12)(cid:12)(cid:12) F t ! (3.3) pure-jump mean-reverting short rate model where t ∈ [0 , T ] (cf. [6, 7, 26]). Note that P ( t, T ) > Q -a.s. ∀ t ∈ [0 , T ] byconstruction. Since r t ≥ µ ( t ) Q -a.s. ∀ t ∈ [0 , T ] [recall Remark 2.1 (a)], we observe P ( t, T ) ≤ M t,T := exp ( − Z Tt µ ( s ) ds ) (3.4) Q -a.s. ∀ t ∈ [0 , T ] due to (3.3) and the monotonicity of conditional expectations.The upper bound M t,T appearing in (3.4) is deterministic and strictly positive for all ≤ t ≤ T . If µ ( t ) ≥ , then it holds P ( t, T ) ≤ Q -a.s. ∀ t ∈ [0 , T ] (similar to,e.g., the CIR model [10]; also see [7, 21]). On the other hand, if µ ( t ) < , then weonly know that M t,T > . Proposition 3.1.
For k ∈ { , . . . , n } and t ∈ [0 , T ] we define the deterministicfunctions A k ( t, T ) := Z Tt (cid:18) − µ ( s ) n + Z D k (cid:2) e σ k B k ( s,T ) z − (cid:3) dν k ( z ) (cid:19) ds,B k ( t, T ) := e − λ k ( T − t ) − λ k ≤ . (3.5) Then the bond price at time t ≤ T with maturity T possesses the affine representation P ( t, T ) = n Y k =1 e A k ( t,T )+ B k ( t,T ) X kt (3.6) where the factors X kt satisfy (2.7). Proof.
First of all, we put (2.6) into (2.1) and obtain r s = µ ( s ) + n X k =1 X kt e − λ k ( s − t ) + n X k =1 Z st Z D k σ k e − λ k ( s − u ) zdN k ( u, z ) where ≤ t ≤ s ≤ T . We next substitute the latter equation into (3.3), hereafterapply Fubini’s theorem and identify the functions B k ( · , T ) . This procedure yields P ( t, T ) = exp ( − Z Tt µ ( s ) ds + n X k =1 B k ( t, T ) X kt ) × E Q exp ( n X k =1 Z Tt Z D k σ k B k ( s, T ) zdN k ( s, z ) ) (cid:12)(cid:12)(cid:12)(cid:12) F t ! . Taking the independent increment property of the ( Q -independent) Lévy processes L , . . . , L n into account, we obtain E Q exp ( n X k =1 Z Tt Z D k σ k B k ( s, T ) zdN k ( s, z ) ) (cid:12)(cid:12)(cid:12)(cid:12) F t ! M. Hess = n Y k =1 E Q " exp (Z Tt Z D k σ k B k ( s, T ) zdN k ( s, z ) ) where t ∈ [0 , T ] . The usual expectations appearing here can be handled by the Lévy–Khinchin formula for additive processes (see, e.g., [9, 30, 36]) which leads us to E Q " exp (Z Tt Z D k σ k B k ( s, T ) zdN k ( s, z ) ) = exp (Z Tt Z D k (cid:2) e σ k B k ( s,T ) z − (cid:3) dν k ( z ) ds ) . Putting the latter equations together and identifying the functions A k ( · , T ) , we endup with the asserted representation (3.6).Recall that the bond price in (3.6) is the product of exponential affine functionsof the factors X t , . . . , X nt (but not of r t ). Also note that for all k ∈ { , . . . , n } and t ∈ [0 , T ] it holds A k ( t, t ) = B k ( t, t ) = 0 . (3.7)We remark that the functions B k ( t, T ) in (3.5) possess the same structure as thecorresponding ones in the Vasicek model (cf. [38], or [7, 16, 21]). For all t ∈ [0 , T ] Eq. (3.6) can be rewritten as P ( t, T ) = exp ( n X k =1 (cid:2) A k ( t, T ) + B k ( t, T ) X kt (cid:3)) (3.8)which implies P ( T, T ) = 1 due to (3.7). Moreover, from (3.5) we infer the timederivatives A ′ k ( t, T ) = µ ( t ) n − Z D k (cid:2) e σ k B k ( t,T ) z − (cid:3) dν k ( z ) , B ′ k ( t, T ) = e − λ k ( T − t ) > (3.9)where A ′ k := ∂ t A k and B ′ k := ∂ t B k . Hence, the functions B k ( t, T ) ≤ are strictlymonotone increasing in t . Also note that the formulas found in (3.9) entirely stand inline with those claimed in (4.4)–(4.5) in [31]. From (3.5), (3.7) and (3.9) we deducethe following system of ordinary differential equations (ODEs) A k ( t, T ) = − Z Tt A ′ k ( s, T ) ds, B ′ k ( t, T ) = 1 + λ k B k ( t, T ) ,A k ( T, T ) = B k ( T, T ) = 0 where t ∈ [0 , T ] and k ∈ { , . . . , n } . We are now prepared to derive the time dynam-ics of the bond price process ( P ( t, T )) t ∈ [0 ,T ] . Proposition 3.2.
For k ∈ { , . . . , n } , t ∈ [0 , T ] and z ∈ D k we define the determin-istic functions ζ k ( t, T, z ) := e σ k B k ( t,T ) z − (3.10) pure-jump mean-reverting short rate model with B k ( t, T ) as in (3.5). Then the bond price satisfies the t -dynamics under Q dP ( t, T ) P ( t − , T ) = r t dt + n X k =1 Z D k ζ k ( t, T, z ) d ˜ N Q k ( t, z ) . (3.11)Recall that it holds ζ k ( t, T, z ) ≤ , since σ k B k ( t, T ) z ≤ for all k , t and z .We stress that (3.11) possesses the same structure as the corresponding Eq. (10.9) in[37], whereas the latter stems from a Brownian motion model without jumps. In thenext step, we provide the solution of the SDE (3.11). Proposition 3.3.
For all t ∈ [0 , T ] the solution of (3.11) under Q reads as P ( t, T ) = P (0 , T ) exp (Z t r s ds − n X k =1 Z t Z D k ζ k ( s, T, z ) dν k ( z ) ds + n X k =1 Z t Z D k σ k B k ( s, T ) zdN k ( s, z ) ) (3.12) where the initial value P (0 , T ) is deterministic and fulfills P (0 , T ) > . Furthermore, for all t ∈ [0 , T ] let us introduce the discontinuous Doléans-Dadeexponential Ξ kt := E (cid:0) h k ∗ ˜ N Q k (cid:1) t := exp (Z t Z D k h k ( s, z ) d ˜ N Q k ( s, z ) − Z t Z D k (cid:2) e h k ( s,z ) − − h k ( s, z ) (cid:3) dν k ( z ) ds ) (3.13)where h k ( s, z ) is an arbitrary integrable deterministic function (which may alsodepend on T ). We recall that Ξ k = 1 and that (cid:0) Ξ kt (cid:1) t ∈ [0 ,T ] constitutes a local Q -martingale. With definition (3.13) at hand, we can express Eq. (3.12) as follows. Corollary 3.4.
For all ≤ t ≤ T the bond price satisfies P ( t, T ) = P (0 , T ) β t n Y k =1 E (cid:0) ξ k ∗ ˜ N Q k (cid:1) t (3.14) where β t is the bank account process given in (3.2), E denotes the Doleáns-Dadeexponential defined in (3.13) and ξ k ( s, z ) := σ k B k ( s, T ) z = log (1 + ζ k ( s, T, z )) is a deterministic function. Moreover, for all ≤ t ≤ T we define the discounted bond price ˆ P ( t, T ) := P ( t, T ) β t (3.15)where ˆ P (0 , T ) = P (0 , T ) . From (3.3) we deduce ˆ P ( t, T ) = E Q ( β − T | F t ) suchthat ˆ P ( t, T ) constitutes an F t -adapted (true) martingale under Q , as required by therisk-neutral pricing theory. Plugging (3.14) into (3.15), for all t ∈ [0 , T ] we obtain ˆ P ( t, T ) = P (0 , T ) n Y k =1 E (cid:0) ξ k ∗ ˜ N Q k (cid:1) t (3.16) M. Hess where P (0 , T ) is deterministic and ξ k is such as defined in Corollary 3.4. We obtainthe following result. Proposition 3.5.
For all t ∈ [0 , T ] the discounted bond price satisfies the Q -mar-tingale dynamics d ˆ P ( t, T )ˆ P ( t − , T ) = n X k =1 Z D k ζ k ( t, T, z ) d ˜ N Q k ( t, z ) where the deterministic functions ζ k ( t, T, z ) are such as defined in (3.10). With reference to [7], we define the instantaneous forward rate at time t withmaturity T via f ( t, T ) := − ∂ T log P ( t, T ) (3.17)where t ∈ [0 , T ] and ∂ T denotes the differential operator with respect to T . Equation(3.17) is equivalent to the representation P ( t, T ) = exp ( − Z Tt f ( t, u ) du ) . (3.18) Lemma 3.6.
For all k ∈ { , . . . , n } and t ∈ [0 , T ] it holds ∂ T A k ( t, T ) = − µ ( T ) n − σ k Z Tt Z D k ze σ k B k ( s,T ) z − λ k ( T − s ) dν k ( z ) ds,∂ T B k ( t, T ) = − e − λ k ( T − t ) . Proof.
By the definition of B k ( t, T ) claimed in (3.5) we find ∂ T B k ( t, T ) = − e − λ k ( T − t ) (3.19)so that the functions B k ( t, T ) are strictly monotone decreasing in T . From (3.5) and(3.10) we further deduce ∂ T A k ( t, T ) = − µ ( T ) n + ∂ T Z Tt Z D k ζ k ( s, T, z ) dν k ( z ) ds ! whereas Fubini’s theorem (see, e.g., Theorem 2.2 in [3]) leads us to ∂ T Z Tt Z D k ζ k ( s, T, z ) dν k ( z ) ds ! = Z D k ∂ T Z Tt ζ k ( s, T, z ) ds ! dν k ( z ) . (We are able to apply Fubini’s theorem here, since the deterministic function ζ k ( s,T,z ) is measurable and square-integrable with respect to s ∈ [0 , T ] and z ∈ D k .) TheLeibniz formula for parameter integrals (see Lemma 2.4.1 on p. 13 in [28]) yields ∂ T Z Tt ζ k ( s, T, z ) ds ! = ζ k ( T, T, z ) + Z Tt ∂ T ζ k ( s, T, z ) ds = − σ k Z Tt ze σ k B k ( s,T ) z − λ k ( T − s ) ds where we used (3.10), (3.7) and (3.19). Putting these formulas together, the proof iscomplete. pure-jump mean-reverting short rate model Proposition 3.7.
For all t ∈ [0 , T ] the instantaneous forward rate can be representedas f ( t, T ) = µ ( T )+ n X k =1 Z Tt Z D k σ k ze σ k B k ( s,T ) z − λ k ( T − s ) dν k ( z ) ds + n X k =1 X kt e − λ k ( T − t ) (3.20) where the factor processes X kt satisfy (2.7) and B k ( s, T ) is like defined in (3.5). Proof.
We substitute (3.8) into (3.17) and obtain f ( t, T ) = − n X k =1 (cid:2) ∂ T A k ( t, T ) + X kt ∂ T B k ( t, T ) (cid:3) . Combining this equality with Lemma 3.6, we derive the claimed representation (3.20).Replacing T by t in (3.20), we immediately find f ( t, t ) = r t due to (2.1). Thisequality stands in line with the usual conventions in interest rate theory (see, e.g.,[7, 16, 21]). Proposition 3.8.
For all t ∈ [0 , T ] the instantaneous forward rate fulfills the pure-jump multi-factor HJM type equation f ( t, T ) = f (0 , T )+ n X k =1 Z t Z D k σ k ze − λ k ( T − s ) (cid:8) dN k ( s, z ) − e σ k B k ( s,T ) z dν k ( z ) ds (cid:9) (3.21) where the deterministic initial value is given by f (0 , T ) = − ∂ T log P (0 , T ) . In what follows, we illustrate how our forward rate model can be fitted to the ini-tially observed term structure. This procedure is often called market-consistent cali-bration in the literature. For this purpose, we denote by f M (0 , T ) the deterministicinitial forward rate. If f (0 , T ) = f M (0 , T ) and hence, if P (0 , T ) = P M (0 , T ) forall maturity times T > , then the underlying model is called market-consistent . Proposition 3.9.
The forward rate model (3.20)–(3.21) can be market-consistentlycalibrated to a given term structure f M (0 , T ) by choosing the floor function µ ( · ) in (3.20) according to µ ( T ) = f M (0 , T ) − n X k =1 x k e − λ k T + Z T Z D k σ k ze σ k B k ( s,T ) z − λ k ( T − s ) dν k ( z ) ds ! (3.22) for all maturity times T > . Note that the floor function µ ( t ) for all t ∈ [0 , T ] can be obtained from (3.22) byreplacing T by t therein. Moreover, we define the interest rate curve at time t < T with maturity T via R ( t, T ) := log P ( t, T ) t − T . (3.23) M. Hess
This object is called continuously-compounded spot rate on p. 60 in [7]. It obviouslyholds P ( t, T ) = e − ( T − t ) R ( t,T ) (3.24)where t ∈ [0 , T [ . Comparing the exponent in (3.24) with that in (3.8), we infer R ( t, T ) = 1 t − T n X k =1 (cid:2) A k ( t, T ) + B k ( t, T ) X kt (cid:3) where A k and B k are such as defined in (3.5). Hence, it turns out that the interestrate curve R ( t, T ) can be represented as a sum of affine functions of the pure-jumpOU factors X t , . . . , X nt . In this sense, our short rate model possesses an affine termstructure (cf. Section 3.2.4 in [7], or [14, 16, 21]). The latter observation entirelystands in line with (3.8). Proposition 3.10.
For all t ∈ [0 , T [ the interest rate curve possesses the representa-tion R ( t, T ) = 1 t − T log P (0 , T ) + Z t r s ds − n X k =1 Z t Z D k ζ k ( s, T, z ) dν k ( z ) ds + n X k =1 Z t Z D k σ k B k ( s, T ) zdN k ( s, z ) ! where ζ k and B k are such as defined in (3.10) and (3.5), respectively. Proposition 3.11.
For all t ∈ [0 , T ] the integrated short rate process can be repre-sented as I t := Z t r s ds = Z t µ ( s ) ds − n X k =1 x k B k (0 , t ) − n X k =1 Z t Z D k σ k B k ( s, t ) zdN k ( s, z ) (3.25) where the deterministic functions B k are such as defined in (3.5). Proof.
We substitute (2.1) and (2.7) into the definition of I t and obtain I t = Z t µ ( s ) ds − n X k =1 x k B k (0 , t ) + n X k =1 Z t Z s Z D k σ k e − λ k ( s − u ) zdN k ( u, z ) ds where B k is like defined in (3.5). An application of Fubini’s theorem (see Theorem2.2 in [3]) yields Z t Z s Z D k σ k e − λ k ( s − u ) zdN k ( u, z ) ds = − Z t Z D k σ k B k ( u, t ) zdN k ( u, z ) , so that the proof is complete.Recall that the last jump integral in (3.25) constitutes a so-called Volterra inte-gral, as the time parameter t appears both inside the integrand and inside the upperintegration bound. Also note that it holds I t = log β t with I = 0 due to (3.2). pure-jump mean-reverting short rate model In this section, we investigate the evaluation of a plain vanilla option written on thezero-coupon bond price P ( · , T ) . With reference to the risk-neutral pricing theory, theprice at time t ≤ τ of an option with payoff H τ at the maturity time τ reads as C t = β t E Q (cid:0) β − τ H τ |F t (cid:1) = E Q (cid:0) e − R τt r s ds H τ |F t (cid:1) (4.1)where β is the bank account process given in (3.2) and Q denotes a risk-neutral pric-ing measure (cf. Eq. (3.1) in [7]). We now consider a call option written on the bondprice P ( · , T ) with maturity time T satisfying T ≥ τ . The payoff of the call optionwritten on P ( τ, T ) with deterministic strike price K > and maturity time τ is thengiven by H τ = [ P ( τ, T ) − K ] + := max { , P ( τ, T ) − K } . (4.2)In what follows, we define the Fourier transform, respectively inverse Fourier trans-form, of a real-valued deterministic function p ( · ) ∈ L ( R ) via ˆ p ( y ) := 12 π Z R p ( u ) e − iyu du, p ( u ) = Z R ˆ p ( y ) e iyu dy. Proposition 4.1. [call option on bond price] For all ≤ t ≤ τ ≤ T the price of acall option with payoff H τ given in (4.2), strike price K > and maturity time τ canbe expressed as C t = Z R ˆ q ( y ) exp ( I t + θ ( τ, y ) + n X k =1 ψ k ( t, τ, y )+ n X k =1 Z t Z D k η k ( s, z, y ) dN k ( s, z ) ) dy (4.3) where the integrated short rate process I t is such as defined in (3.25) and η k ( s, z, y ) := η k ( s, z, T, τ, y ):= σ k [( a + iy ) B k ( s, T ) − ( a + iy − B k ( s, τ )] z, ˆ q ( y ) := P (0 , T ) a + iy π ( a + iy ) ( a + iy − K a + iy − ,ψ k ( t, τ, y ) := Z τt Z D k (cid:2) e η k ( s,z,y ) − (cid:3) dν k ( z ) ds, (4.4) θ ( τ, y ) := ( a + iy − Z τ µ ( s ) ds − n X k =1 x k B k (0 , τ ) ! − ( a + iy ) n X k =1 Z τ Z D k ζ k ( s, T, z ) dν k ( z ) ds constitute deterministic functions, while a > is an arbitrary real-valued constant.Herein, the functions ζ k and B k are such as defined in (3.10) and (3.5), respectively,while P (0 , T ) denotes the deterministic initial bond price. M. Hess
Proof.
We substitute (4.2) and (3.12) into (4.1) and obtain C t = E Q (cid:16) e I t − I τ (cid:2) P (0 , T ) e G τ − K (cid:3) + |F t (cid:17) where I t denotes the integrated short rate process defined in (3.25) and G τ := I τ − n X k =1 Z τ Z D k ζ k ( s, T, z ) dν k ( z ) ds + n X k =1 Z τ Z D k σ k B k ( s, T ) zdN k ( s, z ) is a real-valued stochastic process. For u ∈ R we introduce the deterministic function q ( u ) := e − au [ P (0 , T ) e u − K ] + where a > is a constant real-valued dampening parameter ensuring the integrabilityof the payoff function. Indeed, it holds q ( · ) ∈ L ( R ) . With the latter definition athand, we obtain C t = E Q (cid:0) e I t − I τ + aG τ q ( G τ ) |F t (cid:1) . With reference to [8] (also see [18]), we apply the inverse Fourier transform on thelatter equation and hereafter, use Fubini’s theorem which leads us to C t = Z R ˆ q ( y ) E Q (cid:0) e Z t,τ |F t (cid:1) dy where we have set Z t,τ := I t − I τ + ( a + iy ) G τ for all ≤ t ≤ τ . By merging the definition of G τ and (3.25) into the definition of Z t,τ we deduce Z t,τ = I t + θ ( τ, y ) + n X k =1 Z τ Z D k η k ( s, z, y ) dN k ( s, z ) where we identified the deterministic functions θ ( τ, y ) and η k ( s, z, y ) defined in(4.4). Hence, E Q (cid:0) e Z t,τ |F t (cid:1) = exp ( I t + θ ( τ, y ) + n X k =1 Z t Z D k η k ( s, z, y ) dN k ( s, z ) ) × E Q " exp ( n X k =1 Z τt Z D k η k ( s, z, y ) dN k ( s, z ) ) since I t is F t -adapted and θ ( τ, y ) is deterministic. In the derivation of the latterequation, we used the independent increment property under Q of the involved pure-jump integrals. We next apply the Lévy–Khinchin formula for additive processes (see,e.g., [9, 30, 36]) and derive E Q " exp ( n X k =1 Z τt Z D k η k ( s, z, y ) dN k ( s, z ) ) = exp ( n X k =1 ψ k ( t, τ, y ) ) pure-jump mean-reverting short rate model where the characteristic exponents ψ k ( t, τ, y ) are such as defined in (4.4). Puttingthe latter equations together, we eventually end up with (4.3). The expression for theFourier transform ˆ q ( y ) is obtained by straightforward calculations using the defini-tion of the function q ( u ) . Corollary 4.2.
In the special case t = 0 , the call option price formula (4.3) simplifiesto C = Z R ˆ q ( y ) exp ( θ ( τ, y ) + n X k =1 ψ k (0 , τ, y ) ) dy which is deterministic. In this section, we show how the short rate model introduced in Section 2 can beimplemented in practical applications. For this purpose, we now present more detailedexpressions in order to prepare our model for a possible calibration of the involvedparameters. First of all, let us recall that the increasing compound Poisson processes L kt defined in (2.3) for every k ∈ { , . . . , n } and t ∈ [0 , T ] can be expressed as L kt = N kt X j =1 Y kj (5.1)(cf. Section 5.3.2 in [37]) where N kt constitutes a standard Poisson process under Q with deterministic jump intensity α k > . That is, N kt ∼ P oi ( α k t ) such that for all m ∈ N it holds Q (cid:0) N kt = m (cid:1) = ( α k t ) m m ! e − α k t . The strictly positive jump amplitudes of the Lévy process L kt are modeled by the i.i.d.random variables Y k , Y k , . . . which take values in the set D k ⊆ ]0 , ∞ [ . We recallthat the random variables Y k , Y k , . . . are independent of the Poisson processes N kt for all combinations of indices k, k ∈ { , . . . , n } . We further put c k := E Q [ Y k ] ∈ D k and recall that the compensated compound Poisson process (cid:0) L kt − c k α k t (cid:1) t ∈ [0 ,T ] constitutes an ( F t , Q ) -martingale for each k which implies c k α k = Z D k zdν k ( z ) due to (2.3) and (2.4). We stress that the Poisson processes N kt shall not be mixed upwith the Poisson random measures dN k ( t, z ) .In the following, we propose a number of probability distributions living on thepositive half-line (recall Section B.1.2 in [37]) which constitute suitable candidatesfor the modeling of the jump size distribution in our new short rate model. As a firstexample, we propose to work with the gamma distribution and thus, assume that each M. Hess random variable Y kj is exponentially distributed under Q with parameter ε k > forall j and k . In this case, the related Lévy measure possesses the Lebesgue density dν k ( z ) = α k ε k e − ε k z dz (5.2)where z ∈ D k = ]0 , ∞ [ and k ∈ { , . . . , n } . We find c k = 1 /ε k and Y kj ∼ Γ (1 , ε k ) .Hence, following the notation used in Section 5.5.1 in [37], we state that we presentlyare in a Γ ( α k , ε k ) -Ornstein–Uhlenbeck process setup (also see Section 8.2 in [31]and Example 15.1 in [9] in this context). Proposition 5.1.
Suppose that the random variables Y kj in (5.1) are exponentiallydistributed (i.e. Γ (1 , ε k ) -distributed) under Q with parameters ε k > for all j and k . Then, for u ∈ R and t ∈ [0 , T ] the characteristic function of L kt is given by Φ L kt ( u ) = exp (cid:26) iuα k tε k − iu (cid:27) where α k denotes the jump intensity of the standard Poisson process N kt appearingin (5.1). Proof.
Successively applying the definition of the characteristic function, (2.3), theLévy–Khinchin formula and (5.2), for u ∈ R and t ∈ [0 , T ] we obtain Φ L kt ( u ) = E Q (cid:2) e iuL kt (cid:3) = exp (cid:26) α k ε k t Z ∞ (cid:2) e iuz − (cid:3) e − ε k z dz (cid:27) . We eventually perform the integration and end up with the asserted equality.An immediate consequence of Proposition 5.1 is the following representation forthe moment generating function of L kt being valid for all v ∈ R \ { ε k } κ L kt ( v ) = Φ L kt ( − iv ) = exp (cid:26) vα k tε k − v (cid:27) . Proposition 5.2.
Assume that the random variables Y kj in (5.1) are exponentiallydistributed (i.e. Γ (1 , ε k ) -distributed) under Q with parameters ε k > for all j and k . Then, for all t ∈ [0 , T ] , k ∈ { , . . . , n } and x ∈ R the probability density functionof L kt under Q takes the form f L kt ( x ) = 12 π Z ∞ exp (cid:26) iu (cid:18) x − α k tε k + iu (cid:19)(cid:27) du. Proof.
First, note that it holds Φ L kt ( − u ) = Z ∞ e − iux f L kt ( x ) dx = 2 π ˆ f L kt ( u ) due to the definitions of the characteristic function and the Fourier transform claimedin the sequel of (4.2). We next apply the inverse Fourier transform which yields thedensity function f L kt ( x ) = 12 π Z ∞ Φ L kt ( − u ) e iux du. We finally plug the result of Proposition 5.1 into the latter equation which completesthe proof. pure-jump mean-reverting short rate model
We stress that Eq. (5.2) can be substituted into the corresponding formulas ap-pearing in the previous Propositions 2.2, 2.3, 3.1, 3.3, 3.7–3.10 and 4.1 yielding moreexplicit expressions for the involved entities, yet associated with gamma-distributedjump amplitudes in the underlying short rate model. We illustrate this statement byan application of Eq. (5.2) on Proposition 2.3. The precise result reads as follows.
Proposition 5.3.
Suppose that the random variables Y kj in (5.1) are exponentiallydistributed (i.e. Γ (1 , ε k ) -distributed) under Q with parameters ε k > for all j and k . Let σ k > be the constant volatility coefficients introduced in (2.2). Then, forall t ∈ [0 , T ] and v ∈ R with v < min k { ε k /σ k } , k ∈ { , . . . , n } , the momentgenerating function under Q of the short rate process r t reads as κ r t ( v ) = Φ r t ( − iv ) = exp ( vµ ( t ) + n X k =1 ρ k ( t, − iv ) + n X k =1 ψ k ( t, − iv ) x k ) with deterministic functions ψ k ( t, − iv ) = ve − λ k t , ρ k ( t, − iv ) = α k λ k log (cid:12)(cid:12)(cid:12)(cid:12) ε k − vσ k e − λ k t ε k − vσ k (cid:12)(cid:12)(cid:12)(cid:12) . Proof.
For each k ∈ { , . . . , n } we define the deterministic functions b k ( s, v ) := v σ k e − λ k ( t − s ) − ε k which satisfy b k ( s, v ) < whenever s ∈ [0 , t ] and v < min k { ε k /σ k } . In this setting, we combine Eq. (5.2) with the definitions of ρ k and Λ k given in Proposition 2.3 and obtain ρ k ( t, − iv ) = − α k Z t ε k + b k ( s, v ) b k ( s, v ) ds. We perform the integration and obtain the formula for ρ k claimed in the proposition.The representation for the moment generating function κ r t ( v ) finally follows fromProposition 2.3.Other distributional choices for the random variables Y kj modeling the jump am-plitudes would be, for example, the inverse Gaussian distribution (see Section 5.5.2in [37]), the generalized inverse Gaussian distribution (see Section 5.3.5 in [37]) orthe tempered stable distribution (see Section 5.3.6 in [37]). The related formulas forthe Lebesgue density of the Lévy measure dν k ( z ) corresponding to Eq. (5.2) can befound in [37]. Remark 5.4.
We recall that the time-homogeneous compound Poisson processes L kt introduced in (2.3) can be simulated according to Algorithms 6.1 and 6.2 in [9].Further, in our model it is easily possible to calculate the moments of X kt and r t (see the sequel of Proposition 2.2) so that our model can be fitted to any yield curveobserved in the market by using the moment estimation method described in Section7.2.2 in [9]. This procedure is also called moment matching , as the underlying idea isto make the empirical moments match with the theoretical moments of the model byfinding a suitable parameter set. M. Hess
In this section, we propose a post-crisis extension of the pure-jump lower-boundedshort rate model introduced in Section 2. (To read more on post-crisis interest ratemodels, the reader is referred to [11–14, 17, 22–26, 32–34].) Inspired by the modelingsetups presented in [33] and Chapter 2 in [26], for all t ∈ [0 , T ] we define the shortrate spread under Q by the stochastic process s t := µ ∗ ( t ) + l X k = n +1 X kt showing a similar structure as (2.1). Herein, µ ∗ ( t ) ≥ constitutes an integrable real-valued deterministic function and the factors X kt satisfy the SDE (2.2), but presentlyfor indices k ∈ { n + 1 , . . . , l } where l ∈ N with l > n . Note that it holds s t ≥ µ ∗ ( t ) Q -a.s. for all t ∈ [0 , T ] such that the short rate spread is bounded from below– similar to the short rate itself [recall Remark 2.1 (a)]. We interpret s t as an additive spread and therefore set for all t ∈ [0 , T ] r t := r t + s t (6.1)(cf. [12, 33]) where r t denotes the short rate process and r t is called fictitious shortrate, similarly to [26]. With reference to p. 46 in [26], we recall that the short ratespread s t not only incorporates credit risks, but also various other risks in the inter-bank sector which affect the evolution of the LIBOR rates. Let us moreover mentionthat the short rate r t defined in (2.1) and the short rate spread s t can be ‘correlated’by allowing for (at least) one common factor in their respective definitions. Moreprecisely, if the sum in the definition of s t started running from k = n (instead of k = n + 1 ), then the factor X nt would appear both in the definition of r t and inthe definition of s t such that the two latter stochastic processes would no longer beindependent.We next substitute (2.1) as well as the definition of s t into (6.1) and deduce r t = µ ( t ) + l X k =1 X kt (6.2)where we introduced the real-valued deterministic function µ ( t ) := µ ( t ) + µ ∗ ( t ) . Itobviously holds r t ≥ µ ( t ) Q -a.s. for all t ∈ [0 , T ] . In accordance to Section 3.4.1in [14], Eq. (2.35) in [26] and Section 1 in [33], we define the fictitious bond price inour post-crisis short rate model via P ( t, T ) := E Q exp ( − Z Tt r u du ) (cid:12)(cid:12)(cid:12)(cid:12) F t ! (6.3)where t ∈ [0 , T ] . The object P ( t, T ) is sometimes called artificial bond price in theliterature, as it is not physically traded in the market. Evidently, P ( T, T ) = 1 . Com-paring (6.2) with (2.1) and (6.3) with (3.3), we see that all our single-curve results pure-jump mean-reverting short rate model presented in the previous sections carry over to the currently considered post-crisiscase. More precisely, the entities µ ( t ) , r t , P ( t, T ) and n emerging in the previoussingle-curve equations presently have to be replaced by µ ( t ) , r t , P ( t, T ) and l , re-spectively. Moreover, in the present case, we observe Q -a.s. for all t ∈ [0 , T ]0 < P ( t, T ) ≤ exp ( − Z Tt µ ( u ) du ) due to (6.3), the lower boundedness of r t and the monotonicity of conditional expec-tations. Proposition 6.1.
It holds Q -a.s. for all t ∈ [0 , T ] P ( t, T ) ≤ P ( t, T ) (6.4) where P ( t, T ) constitutes the bond price defined in (6.3) and P ( t, T ) is given in (3.3). Proof.
Note that taking µ ∗ ( t ) ≥ ∀ t ∈ [0 , T ] implies s t ≥ Q -a.s. for all t ∈ [0 , T ] . In this case, we deduce r t ≥ r t Q -a.s. for all t ∈ [0 , T ] due to (6.1). Hence,we find exp ( − Z Tt r u du ) ≤ exp ( − Z Tt r u du ) Q -a.s. for all t ∈ [0 , T ] . We next take conditional expectations with respect to F t and Q in the latter inequality, hereafter apply the monotonicity of conditional expectationsand finally identify (6.3) and (3.3) in the resulting inequality.The result obtained in Proposition 6.1 possesses the following economical in-terpretation: The inequality (6.4) shows that nontraded bonds are cheaper than theirnonfictitious counterparts which are physically traded in the market. This feature ap-pears economically reasonable and stands in accordance with the modeling assump-tions and statements in [12], Section 2.1.3 in [26] and Section 2.1 in [33]. Moreover,combining (6.3) and (3.18), we obtain (parallel to [12]) P ( t, T ) = exp ( − Z Tt f ( t, u ) du ) where f is sometimes called fictitious/artificial forward rate in the literature. It holds f ( t, t ) = r t for all t ∈ [0 , T ] . With reference to [11] and [12], for all t ∈ [0 , T ] weintroduce the forward rate spread via g ( t, T ) := f ( t, T ) − f ( t, T ) so that we have not only set up a new pure-jump post-crisis short rate model, butsimultaneously a new pure-jump post-crisis forward rate model of HJM-type in thecurrent section. Recall that (6.4) is equivalent to f ( t, u ) ≤ f ( t, u ) Q -a.s. for all ≤ t ≤ u ≤ T . From this, we conclude that g ( t, T ) ≥ Q -a.s. for all t ∈ [0 , T ] . Itfurther holds g ( t, t ) = s t for all t ∈ [0 , T ] due to (6.1). M. Hess
Furthermore, in the present post-crisis setting, for a time partition ≤ t ≤ T