Approximate XVA for European claims
AApproximate XVA for European claims
F. Antonelli ∗ , A. Ramponi † , S. Scarlatti ‡ July 16, 2020
Abstract
We consider the problem of computing the Value Adjustment of European contingentclaims when default of either party is considered, possibly including also funding and collat-eralization requirements.As shown in Brigo et al. ([12], [13]), this leads to a more articulate variety of ValueAdjustments (XVA) that introduce some nonlinear features. When exploiting a reduced-form approach for the default times, the adjusted price can be characterized as the solutionto a possibly nonlinear Backward Stochastic Differential Equation (BSDE). The expectationrepresenting the solution of the BSDE is usually quite hard to compute even in a Markoviansetting, and one might resort either to the discretization of the Partial Differential Equationcharacterizing it or to Monte Carlo Simulations. Both choices are computationally veryexpensive and in this paper we suggest an approximation method based on an appropriatechange of numeraire and on a Taylor’s polynomial expansion when intensities are representedby means of affine processes correlated with the asset’s price. The numerical discussion atthe end of this work shows that, at least in the case of the CIR intensity model, even thesimple first-order approximation has a remarkable computational efficiency.
Keywords : Credit Value Adjustment; Defaultable Claims; Counterparty Credit Risk; WrongWay Risk; XVA; Affine Processes.
Many financial institutions trade contracts in over-the-counter (OTC) markets, their counter-parties being other financial institutions or corporate clients. However, many of those contractsare subject, to some extent, to counterparty risk, or in other words, they are subject to somedefault event concerning the solvency of either one of the parties, that might take place duringthe lifetime of the contract. These are called defaultable. Initially, the evaluation regarded Euro-pean options, named vulnerable, when the seller’s default was the only risk and two approachesemerged over the years: the structural approach and the reduced form approach.Historically, the structural approach came first introduced by Johnson and Stulz in [29] whenthey considered the option as the sole liability of the counterparty. In the same framework, in [31] ∗ University of L’Aquila, [email protected] † Dept. Economics and Finance, University of Roma - Tor Vergata, [email protected] ‡ Dept. Enterprise Engineering, University of Roma - Tor Vergata, [email protected] a r X i v : . [ q -f i n . P R ] J u l lein discussed more general liability structures, in [32] he included interest rate risk, and in [33]he considered a (stochastic) default barrier depending on the value of the option. More recently,[36] extended this approach to jump-diffusion models, [27] considered multiple correlations, [18]treated it by using copulas.Then researchers developed the alternative reduced-form approach. For a comprehensivepresentation of the topic, we refer the reader to [34]. In [19], and the references therein, one canfind a general overview of the approach for defaultable bonds. Later, the approach’s mathemat-ical framework was carefully formalized in [5] and [6], and recently [17] and [21] extended it todefaultable claims in Levy market models.In the last decade, after the financial crisis of 2008-09, the interest in Counterparty CreditRisk increased remarkably, and attention focused on building a general framework to define andevaluate the premium to compensate the risk connected to defaultable products (in particularof Interest Rate Swaps). This premium took the name of Credit Value Adjustment (CVA) inthe seminal paper by Zhu and Pykhtin [37], and it defines the appropriate reduction of thedefault-free value of a portfolio, to compensate for the default risk. This discount became thecrucial quantity to take into account when trading derivatives in OTC markets, spurring muchresearch in the field: see, for instance, [4], [10], [25].Over the years, other value adjustments were introduced in the contract’s evaluation, leadingto the acronym (X)VA. Here, X stands for D= debt, L= liquidity, F=funding, to include alsothe risks due to the default of both parties, funding investment strategies, lack of liquidity. Werefer the reader to [26] for a comprehensive exposition on the matter. In [24], one might find anupdated overview of the recent research directions under investigation. We point out that therethe characterization of the adjusted value as the solution of a BSDE is very well explained. Ina Markovian setting, the connection between bilateral CVA and Partial Differential Equations(PDEs) is also thoroughly investigated in [15] and further developed in [16].isIn this work, we treat a European claim, whose price is influenced by the default probabilitiesof either party as well by liquidity, financing, and collateralization risks when exploiting theintensity approach for the default times of both parties.In a remarkable series of papers, ([12], [13], [14]), Brigo et al. describe in detail how intro-ducing all the value adjustments implies the loss of an explicit expression for the adjusted value.Indeed the BSDE characterizing the contract’s value is generally nonlinear and hence hardlysolvable. It depends on the asset’s price and many other, possibly correlated, factors such asdefault intensities, interest rate, stochastic volatility, so that even in a Markovian setting, theexpectation representing the solution of the associated PDE becomes extremely difficult to eval-uate. Hence to provide a numerical approximation, one may resort only to the discretization ofthe PDE characterizing the solution of the BSDE (see [30]) or to Monte Carlo simulations (as in[13]). Either approach, on average computational resources, results to be computationally veryexpensive.We are interested in devising an approximation procedure simple and computationally effi-cient even in the presence of many stochastic factors, provided we make some modeling choices.Indeed, we suggest to view the evaluation expectation as a smooth function of the correlationparameters and to approximate it by its Taylor polynomial expansion around the zero vector(the independent case), in the hope that the first or second-order are enough to provide an accu-rate approximation. We apply our method to estimate the price contribution that comes from2onsidering stochastic default intensities correlated with the underlying’s price. We remark,though, that we can straightforward extend the same technique to include further stochasticfactors.To evaluate Taylor polynomial’s coefficients, we follow a two-step procedure to exploit, when-ever possible, explicit formulae from option and bond’s pricing theory. First, we condition theunderlying’s price with respect to the stochastic factors, retrieving a conditional Black & Sc-holes formula. Then, assuming the intensities to be described by affine models, we represent thesingle terms of the expansion using a change of Numeraire technique (similar to the one in [9])to disentangle the correlation among the asset’s price and the default intensities. The affinityof the processes makes it possible to use a “bond-like” expression for the default component.To carry out the calculations in detail and to perform the numerical analysis of the method,we represent the intensities by two Cox Ingersoll Ross (CIR) processes. The final section showsthe method’s efficiency using Monte Carlo simulations as a benchmark.A strong point of this approach is that it provides a relatively simple method that one canuse with many correlated processes. Correlation often destroys any affine property the dynami-cal system might have, making the Riccati equations/Fourier transform framework inapplicable,and one can resort only to Monte Carlo or PDE’s approximations. The latter are both com-putationally expensive in several dimensions, hence the construction of an alternative with aremarkable gain in computational time, without loss in accuracy, becomes very important.Our method becomes particularly convenient when the correlation structure (as Monte Carlosimulations point out for the CIR model) seems to follow a linear pattern. In this case, a first-order Taylor’s polynomial is enough to produce an accurate approximation, providing a ratherhandy evaluation formula. We finally remark that the conditioning and change of numerairetechniques allow us to keep the coefficients’ approximations to a minimum. The expansion’szeroth term corresponds to the independent case, and we need to have a semi-explicit formulato evaluate it. This fact forced us to restrict our model choices.The paper is structured as follows. In the next section, we describe the general problemleading to the BSDE characterization under the reduced-form approach. We specify the modeland the two-step evaluation procedure to compute Taylor’s approximation in Section 3, while inSection 4, we specialize the calculations when the default intensities are CIR processes. Section5 concerns the numerical analysis of our results. We consider a finite time interval [0 , T ] and a complete probability space (Ω , F , P ), endowedwith a filtration {F t } t ∈ [0 ,T ] , augmented with the P − null sets and made right continuous. Weassume that all processes have a c´adl´ag version.The market is described by the interest rate process r t determining the money market accountand by an adapted process X t representing an asset log-price (we will specify its dynamics later),which may also depend on additional stochastic factors. We assume • that the filtration {F t } t ∈ [0 ,T ] is rich enough (and possibly more) to support all the stochas-tic processes that describe the market; 3 to be in absence of arbitrage; • that the given probability P is a risk-neutral measure, already selected by some criterion.In this market model (as in [13]) we consider two parties ( I = investor, C = counterparty)exchanging some European claim with default-free payoff f ( X T ), where f is a function (notnecessarily nonnegative) as regular as needed. We take for granted that the market processesfulfill the necessary integrability hypotheses to guarantee a good definition of all the expectationswe are going to write.Both parties might default, due to some critical credit state, with respective random times τ (Counterparty) and τ (Investor), which are not stopping times with respect to the filtration F t . In this context we define the filtration G t = F t ∨ H t ∨ H t , where H it = σ ( { τ i ≤ s } , s ≤ t ), i = 1 ,
2, which is the smallest filtration extension that makes both random variables stoppingtimes. Moreover, we assume there exists a unique extension of the risk-neutral probability to G t , that we keep denoting by P .In general, the following fundamental assumption, known as the H-hypothesis (see e.g. [23]and [22] and the references therein), ensures price coherence:(H) Every F t − martingale remains a G t − martingale.By Lemma 7.3.5.1 in [28], (H) is automatically satisfied, under square integrability of the payoff,by the default-free price of any European contingent claim, whence we may affirm thate (cid:82) t r u du e X t = E (e (cid:82) T r u du e X T |F t ) = E (e (cid:82) T r u du e X T |G t )e (cid:82) t r u du c ( t, T ) := E (e (cid:82) T r u du f ( X T ) |F t ) = E (e (cid:82) T r u du f ( X T ) |G t )remain G t − martingales under P , for all t ∈ [0 , T ].In what follows, to stress the significance of the term “adjustment”, we will point the correc-tions out step by step, with their signs determined by the fact that we are taking the investor?sviewpoint.We start assuming full knowledge that is we are in the G t − filtration. The contract makessense only if the default of either party has not occurred yet at the evaluation time t . Denotingby τ = min( τ , τ ), this fact is represented by the indicator function { τ>t } to be placed in frontof the price.Either party may default, so a bilateral adjustment is needed. For the moment we assumenothing is recovered at default. Denoting by CVA ( t, T ) the Credit Value Adjustment due tothe counterparty’s default, this quantity has to act as a discount to the default-free price tobalance the investor’s risk assumption. On the other hand, the Debt Value Adjustment dueto the investor’s default, DVA ( t, T ), has to act as an accrual of the default-free price as itcompensates the counterparty’s risk assumption. So, for the G t − adapted adjusted value of theEuropean claim c G ( t, T ), we may write { τ>t } c G ( t, T ) = { τ>t } (cid:104) c ( t, T ) − CVA ( t, T ) + DVA ( t, T ) (cid:105) , (1)where CVA ( t, T ) and DVA ( t, T ) ≥
0. 4ow, let us admit the defaulting party might partially compensate for the loss due tohis/her default. In this case, we have to include other two nonnegative terms, CVA rec ( t, T )and DVA rec ( t, T ) (respectively for the counterparty and the investor), and we can rewrite theabove as { τ>t } c G ( t, T ) = { τ>t } (cid:104) c ( t, T ) − CVA ( t, T ) + DVA ( t, T ) + CVA rec ( t, T ) − DVA rec ( t, T ) (cid:105) . Moreover, as explained in [14], the two parties might be asked to collateralize their participationto the contract, they might need to borrow money to finance this participation and/or the riskyasset(s) from a repo market to rea,lize their hedging strategies. All this leads to funding andliquidity risks that, again, have to be included for the correct contract’s evaluation. Thus, weshould write { τ>t } c G ( t, T ) = { τ>t } (cid:104) c ( t, T ) − CVA ( t, T ) + DVA ( t, T )+ CVA rec ( t, T ) − DVA rec ( t, T ) + FVA( t, T ) + LVA( t, T ) (cid:105) , (2)with FVA( t, T ), LVA( t, T ) ∈ R .The first represents the Funding Value Adjustment, the sec-ond the Liquidity Value Adjustment, and they are both determined by strategy financing andcollateralization.It is then necessary to model these terms to get to a manageable formula. The range ofpossible choices of mechanisms to include in the formation of prices is quite broad, and we referthe reader again to [12], [13] and [14] for a detailed discussion. Of course, there is an interplayamong the different cash flows. For instance, collateralization changes the parties? exposures,the amount of cash borrowed at rate r increases its value at a rate r s .Here we use the following set of assumptions.1. The claim pays no dividends.2. The adjustment processes all depend on a close-out value, (cid:15) t , determined by a contractualagreement. It is natural to consider it F t − adapted since it is established on the basis ofthe information before default. Usually, it is taken as the default-free price or as the priceof the defaultable claim itself.3. We denote the collateralization process by C s and it is a, possibly time-varying, percentageof the close-out value C s = (cid:40) α s (cid:15) + s , when due by the counterparty α s (cid:15) − s , when due by the investor 0 < α s < , ∀ s ∈ [0 , T ] . (3)Thus the net exposure is ( (cid:15) s − C s ) + = (1 − α s ) (cid:15) + s for the investor and ( (cid:15) s − C s ) − = (1 − α s ) (cid:15) − s for the counterparty.Moreover, we assume that collateralizing happens at rate r cs .4. We denote by R ( s ) the recovery percentage of the close-out value in case of counterparty’s default and by R ( s ), when investor’s default occurs. Mirror-like we define the LossGiven Default as L i ( s ) = (1 − R i ( s )), i = 1 , r t Risk-free rate τ Default time Counterparty r φt Funding rate τ Default time Investor r ct Collateral rate (cid:15) t Close-out value h t Hedging rate λ it Default intensities α t collateralization level f ( · ) Option payoff R i ( t ) Recovery rates i = 1 , v t (cid:82) Tt v s ds ˜ r t r φt − h t ˆ r t r φt − r ct Table 1: Summary of notations.5. To build investing strategies, the parties may invest in the riskless asset at a rate r φ andthe risky asset(s) at a rate h t , the latter happening in a parallel repo market. We denote by φ u the quantity of riskless asset the contract globally requires (either positive or negative)and by H t the value of the portion of the risky asset(s) (either positive or negative) tradedon the repo market.Since at the same time the investor’s purchase generates wealth at a rate r s , and as wellthe borrow/sale of the risky asset generates wealth at a rate r φ , also this aspect will haveto be taken into account.As we said, the recovery and the collateral agreements are usually a fraction of the close-outvalue, and therefore they should be F t − adapted. On the contrary, the funding and hedgingprocesses ( φ, H ) might incorporate the contribution of the default events, and therefore theycould be a priori G t − adapted.Finally, the price should be given by the three components c G ( t, T ) = φ t + H t + C t . (4)Following the crystal clear exposition in [13] (but also in [12] and [14] ), keeping in mind hy-pothesis (H) and (4), one can obtain the following BSDE in the G− filtration { τ>t } c G ( t, T ) = { τ>t } (cid:40) E (cid:104) e − (cid:82) Tt r u du f ( X T ) { τ>T } (cid:12)(cid:12) G t (cid:105) + E (cid:104) e − (cid:82) τt r u du { τ ≤ T } (cid:16) (cid:15) τ − (1 − α τ ) (cid:2) L ( τ ) (cid:15) + τ { τ = τ } − L ( τ ) (cid:15) − τ { τ = τ } (cid:3)(cid:17)(cid:12)(cid:12)(cid:12) G t (cid:105) + (cid:20)(cid:90) τ ∧ Tt e − (cid:82) st r u du (cid:110) [ r s − r φs ] c G ( s, T ) ds + [ r φs − r cs ] C s + [ h s − r s ] H s (cid:111) ds (cid:12)(cid:12)(cid:12) G t (cid:21) (cid:41) . (5)The random variables τ i , i = 1 ,
2, are not F t − stopping times, hence the traders can observe onlywhether the default events happened or not, conditioned to the available information. Thus, anyrisk-neutral evaluation that would naturally take place in the G− filtration, needs translating interms of {F t } . For that, we have the following well known Key Lemma, to be found in [6] or[4], just to quote some references. 6 emma 2.1 Given a G t − stopping time τ , for any integrable G T − measurable r.v. Y , the follow-ing equality holds E (cid:104) { τ>t } Y |G t (cid:105) = { τ>t } E (cid:104) { τ>t } Y |F t (cid:105) P ( τ > t |F t ) . (6)This Lemma calls for the conditional distributions of the default times that we are going totreat within the (Cox) reduced-form framework. We denote the conditional distribution of therandom times as F it = P ( τ i ≤ t |F t ) , i = 1 , ∀ t ≥ , (7)and we assume that they both verify F it <
1. Hence we can define the corresponding F - hazardprocesses of the τ i ’s asΓ it := − ln(1 − F it ) ⇒ F it = 1 − e − Γ it ∀ t > , Γ = 0 , (8)which we assume to be differentiable, defining the so-called F t − adapted intensity processes λ i by Γ it = (cid:90) t λ iu du ⇒ F it = 1 − e − (cid:82) t λ iu du . As in the classical framework of [20], we assume conditional independence for the default times,i.e. for any t > t , t ∈ [0 , t ] P ( τ > t , τ > t |F t ) = P ( τ > t |F t ) P ( τ > t |F t ) , so that we may conclude that λ t := λ t + λ t is the intensity process of τ = inf { τ , τ } . Remark 2.2
It is worth noting that the independence assumption certainly simplifies compu-tations, but it does not take into consideration default contagion effects. Within the intensityframework, more realistic models allowing default dependence were recently proposed (see [7], [8]and the references therein), and we remark that we could extend our method to the correlatedcase, provided we introduce an additional parameter.
Exploiting the key Lemma and the intensity processes as in [3], the above equation getsprojected on the smaller filtration, obtaining { τ>t } c G ( t, T ) = { τ>t } E (cid:104) e − (cid:82) Tt ( r u + λ u ) du f ( X T )+ (cid:90) Tt e − (cid:82) st ( r u + λ u ) du (cid:2) λ s (cid:15) s − (1 − α s ) (cid:0) λ s L ( s ) (cid:15) + s − λ s L ( s ) (cid:15) − s (cid:1)(cid:3) ds + (cid:90) Tt e − (cid:82) st ( r u + λ u ) du (cid:2)(cid:0) r s − r φs (cid:1) c G ( s, T ) + (cid:0) r φs − r cs (cid:1) α s (cid:15) s + ( h s − r s ) H s (cid:3) ds (cid:12)(cid:12)(cid:12) F t (cid:105) . (9)Applying the Key Lemma and Lemma 2 in [13] (extension of the key lemma) to (9), we mayconclude that there exists an F t − adapted adjusted price of the European claim, c a ( t, T ) and anadapted hedging strategy (the part hedging the default-free risks) ˜ H such that c a ( t, T ) { τ>t } = c G ( t, T ) { τ>t } , ˜ H t { τ>t } = H t { τ>t } , { τ > t } { τ>t } c a ( t, T ) = { τ>t } E (cid:104) e − (cid:82) Tt ( r u + λ u ) du f ( X T )+ (cid:90) Tt e − (cid:82) st ( r u + λ u ) du (cid:2) λ s (cid:15) s − (1 − α s ) (cid:0) λ s L ( s ) (cid:15) + s − λ s L ( s ) (cid:15) − s (cid:1)(cid:3) ds + (cid:90) Tt e − (cid:82) st ( r u + λ u ) du (cid:2)(cid:0) r s − r φs (cid:1) c a ( s, T ) + (cid:0) r φs − r cs (cid:17) α s (cid:15) s + ( h s − r s ) ˜ H s (cid:3) ds (cid:12)(cid:12) F t (cid:105) (10) Remark 2.3
Following [14], a few issues about the above BSDE need to be addressed.1. We remark that this equation has a unique strong solution as long as we take square inte-grable close-out value and intensities and, for instance, we assume the processes r, r c , r φ , h to be bounded. This is going to be our standing assumption.2. The process ˜ H t is linked to the solution of the BSDE. If we restrict to a diffusion settingwith deterministic coefficients, the theory of BSDE’s gives an explicit representation forthe process ˜ H . To deal with this, we extend the observation made in [14] when they assumedeterministic intensities.More precisely, we assume that the stock price, S u = e X u , and the intensities processes,under the given risk-neutral probability, verify dS u = r u S u du + σ ( t, S u ) dY u , and dλ iu = a i ( u, λ iu ) du + b i ( u, λ iu ) dB iu , i = 1 , , for correlated Brownian motions Y, B , B and deterministic coefficients σ ( u, x ) , a i ( u, λ ) , b i ( u, λ ) chosen to ensure the existence and uniqueness of strong solutions. Then (10) can be equiv-alently written on { τ > t } as e − (cid:82) t ( r u + λ u ) du c a ( t, T ) = c a (0 , T ) + (cid:90) t Z s dY s + M t − (cid:90) t e − (cid:82) s ( r u + λ u ) du (cid:2) λ s (cid:15) s − (1 − α s ) (cid:16) λ s L ( s ) (cid:15) + s − λ s L ( s ) (cid:15) − s (cid:1)(cid:105) ds − (cid:90) t e − (cid:82) s ( r u + λ u ) du (cid:104)(cid:0) r s − r φs (cid:1) c a ( s, T ) + (cid:0) r φs − r cs (cid:17) α s (cid:15) s + ( h s − r s ) ˜ H s (cid:105) ds, (11) where Z is the component of the solution of the BSDE coming from the martingale repre-sentation theorem, while M is a martingale depending on the intensities and possibly onsome other stochastic factors (again represented by diffusions). In this context, c a ( t, T ) is a deterministic function of the state variables, and assuming enough regularity of thisfunction, ˜ H should represent the δ − hedging of the contract ˜ H u = ∂c a ( u, T ) ∂S S u . n the other hand, the Markovian setting gives also that Z is given by Z u = σ ( u, S u ) ∂c a ( u, T ) ∂S ⇒ ˜ H u = S u σ ( u, S u ) Z u , provided that σ ( u, x ) > for all u, x . From now on, in addition to the hypotheses stated in the first of the previous remarks, weassume that 0 < σ x ≤ σ ( u, x ) ≤ σ x, ∀ u, x for some constants σ and σ .This implies, as in [13] or [14], that we may apply Girsanov’s theorem to change the Brownianmotion driving the above BSDE to include the term ˜ H . Indeed, B t = Y t + (cid:90) t ( r u − h u ) S u σ ( u, S u ) du is a new Brownian motion with respect to the probability defined by the Radon-Nykodim deriva-tive d Q d P = e − (cid:82) T ( r u − h u ) Suσ ( u,Su ) dY u + (cid:82) T ( r u − h u ) S uσ u,Su ) du which verifies the Novikov condition. Consequently, under Q the asset price equation and (11)become dS t = S t h t dt + σ ( t, S t ) dB t e − (cid:82) t ( r u + λ u ) du c a ( t, T ) = c a (0 , T ) + (cid:90) t Z s dB s + M t − (cid:90) t e − (cid:82) s ( r u + λ u ) du (cid:2) λ s (cid:15) s − (1 − α s ) (cid:0) λ s L ( s ) (cid:15) + s − λ s L ( s ) (cid:15) − s (cid:1)(cid:3) ds − (cid:90) t e − (cid:82) s ( r u + λ u ) du (cid:2)(cid:0) r s − r φs (cid:1) c a ( s, T ) + (cid:0) r φs − r cs (cid:17) α s (cid:15) s (cid:3) ds. (12)Passing again to the conditional expectation and multiplying both sides by e (cid:82) t ( r u + λ u ) du , weobtain { τ>t } c a ( t, T ) = { τ>t } E Q (cid:104) e − (cid:82) Tt ( r u + λ u ) du f ( X T )+ (cid:90) Tt e − (cid:82) st ( r u + λ u ) du (cid:2) λ s (cid:15) s − (1 − α s ) (cid:0) λ s L ( s ) (cid:15) + s − λ s L ( s ) (cid:15) − s (cid:1)(cid:3) ds + (cid:90) Tt e − (cid:82) st ( r u + λ u ) du (cid:2)(cid:0) r s − r φs (cid:1) c a ( s, T ) + (cid:0) r φs − r cs (cid:17) α s (cid:15) s (cid:3) ds (cid:12)(cid:12) F t (cid:105) . (13)The latter equation is linear or nonlinear depending on the choice of (cid:15) s . In the literature thereare fundamentally two possible choices: either (cid:15) s = c ( s, T ) (the default-free value of the claim)or (cid:15) s = c a ( s, T ).The first choice will always give a solvable linear BSDE. With the second choice, we mightobtain a solvable linear BSDE if the adjusted value stays always nonnegative (or nonpositive),otherwise the negative and positive parts generate a nonlinear, not explicitly solvable, BSDE.9o exploit explicit formulas, when possible, we decide to choose always (cid:15) s = c ( s, T ) (thatcorresponds to asking collateralization proportional to the default-free price rather than to thecurrent price), to guarantee the solvability of the BSDE for all European claims.With this choice (13) becomes on { τ > t } c a ( t, T ) = E Q (cid:104) e − (cid:82) Tt ( r u + λ u ) du f ( X T ) + (cid:90) Tt e − (cid:82) st ( r u + λ u ) du (cid:2) Ψ s + ( r s − r φs (cid:1) c a ( s, T ) (cid:3) ds (cid:12)(cid:12) F t (cid:105) where Ψ s = (cid:2) λ s + ( r φs − r cs ) α s (cid:3) c ( s, T ) − (1 − α ) (cid:2) λ s L ( s ) c ( s, T ) + − λ s L ( s ) c ( s, T ) − (cid:3) , which can be solved obtaining { τ>t } c a ( t, T ) = { τ>t } E Q (cid:104) e − (cid:82) Tt ( r φu + λ u ) du f ( X T ) + (cid:90) Tt e − (cid:82) st ( r φu + λ u ) du Ψ s ds (cid:12)(cid:12) F t (cid:105) . (14)We remark we could have proposed a more general situation, considering different collateralrates and recovery processes and close-out values for the two parties. All these generalizationswould have led to a more articulate, but not mathematically more difficult, equation. Indeed,the main nonlinearity is due to the recovery terms, once one decides to consider as close-outvalue the adjusted price of the contract.In the next section, we introduce the market model and in the following two, we describe ourevaluation procedure by steps, leading to approximations handier than Monte Carlo simulations. Remark 2.4
We remark that if we are in absence of default of either part, λ = λ = 0 ,funding, collateralization, rehypothecation are considered and the close-out value is taken equalto the contract’s current value, then the solution of (13) becomes c a ( t, T ) = E Q (cid:104) e − (cid:82) Tt [(1 − α u ) r φu + α u r cu ] du f ( X T ) (cid:12)(cid:12) F t (cid:105) , which reduces to the usual Black & Scholes setting, only if the collateralization, funding, reporates all coincide with the risk-free rate. From now on we omit the probability Q in the notation of the expectation and we will alwaysbe referring to (14). In what follows we specify the market model, where the asset price is represented as a stochasticexponential, and the default intensities are assumed to be affine processes. Then we illustrate aconditioning procedure that helps to exploit explicit expressions for the default-free price, as ithappens in the Black & Scholes model when considering European Vanilla Options or Futures.Finally, we apply a change of Numeraire that allows using the well-known expression for Zero-Coupon Bonds when interest rates are affine processes. This last step helps to disentangle thecontribution due to the intensities and the one coming from the derivative.10n section 4 we specialize this procedure to the case when the intensities are CIR processes.We will be able to derive semi-explicit formulas, that we approximate by means of a Taylor’sexpansion with respect to the correlation parameters, up to the first or second order. We donot consider the other very popular affine Vasicek model since it is well known explicit formulascan be derived in this case.
We keep denoting by t ∈ [0 , T ] the initial time and we make the following simplifying hypothesesfor (14):1. all the rates, r, r c , r φ , h are deterministic;2. for i = 1 ,
2, (1 − α ) L i are constant and we will keep denoting them simply by L i .So we have Ψ s = (cid:2) λ s + ( r φs − r cs ) α (cid:3) c ( s, T ) − (cid:2) λ s L c ( s, T ) + − λ s L c ( s, T ) − (cid:3) . We also choose thefollowing model for our state variables for fixed initial conditions ( t, x, λ , λ ) ∈ R + × R × R + × R + , ∀ s ∈ [ t, T ] X s = x + (cid:90) st ( h u − σ du + σ ( B s − B t ) x ∈ R (15) λ is = λ i + (cid:90) st [ γ iu λ iu + β iu ] du + (cid:90) st [ η iu λ iu + δ iu ] dB iu , λ i > , i = 1 , σ > r, γ i , β i , η i , δ i , i = 1 , B , B , B ) is a 3-dimensional Brownian motion, with B s = ρ B s + ρ B s + (cid:113) − ρ − ρ B s , ρ + ρ ≤ . The processes X s , λ s , λ s are Markovian, therefore c ( s, T ) and c a ( s, T ) are deterministic functionsrespectively of the state variables X and ( X, λ , λ ), and depending also on the correlationparameters ρ = ( ρ , ρ ).For any t ≤ s ≤ T , we define the processes N i ( u, s ) := E (e − (cid:82) st λ iv dv |F u ) , i = 1 , , (17)which are martingales for t ≤ u ≤ s and that, having chosen the intensities as affine processes,by Fourier transform have an explicit expression for their initial values N i ( t, s ) = e A i ( t,s ) λ i + B i ( t,s ) ⇒ N i ( u, s ) = e A i ( u,s ) λ i + B i ( u,s ) − (cid:82) ut λ iv dv , (18)where λ i is the initial condition of the intensity and A i and B i are deterministic functionsverifying a set of Riccati equations. We remark that by independence of the intensities we alsohave N ( u, s ) := E (e − (cid:82) st λ v dv |F u ) = E (e − (cid:82) st ( λ v + λ v ) dv |F u ) = N ( u, s ) N ( u, s ) , dN i ( u, s ) = N i ( u, s ) A i ( u, s )( η iu λ iu + δ iu ) dB iu dN ( u, s ) = N ( u, s ) (cid:104) A ( u, s )( η u λ u + δ u ) dB u + A ( u, T )( η u λ u + δ u ) dB u (cid:105) . (19)In some classical specifications of the affine modeling framework: • γ iu = − γ i , β i ( λ ) = γ i θ i , δ iu = δ i , η iu = 0 (Vasicek) • γ iu = − γ i , β i ( λ ) = γ i θ i , δ iu = 0, η iu = η i (CIR),for γ i , θ i , i = 1 , A i ( t, s ) and B i ( t, s ) in closed form. In this subsection, we express an alternative formulation for the expectations in (14), whichmay be useful to write (conditionally) whenever possible, the explicit formula for the default-free price. To simplify notation, from now on we denote by E t the conditional expectation withrespect to F t .Since the interest rate r φ is deterministic, we rewrite (14) as { τ>t } c a ( t, T ) = { τ>t } (cid:110) e − (cid:82) Tt r φu du E t (cid:16) e − (cid:82) Tt λ u du f ( X T ) (cid:17) + { τ>t } (cid:90) Tt e − (cid:82) st r φu du E t (cid:16) e − (cid:82) st λ u du Ψ s (cid:17) ds (20)and we focus on the inner expectations. Proposition 3.1
Let A ts = F B ,B s ∨ F t = σ ( { B u , B u , u ≤ s } ) ∨ F t , t ≤ s ≤ T. Then E t (cid:104) e − (cid:82) Tt λ u du f ( X T ) (cid:105) = e (cid:82) Tt h u du E t (cid:104) e − (cid:82) Tt λ u du E (cid:16) e − (cid:82) Tt h u du f ( X T ) (cid:12)(cid:12)(cid:12) A tT (cid:17)(cid:105) , where X T (cid:12)(cid:12)(cid:12) A tT ∼ N (cid:16) ζ T ( ρ ) + (cid:90) Tt (cid:0) h u du − Σ ( ρ )2 (cid:1) du ; Σ ( ρ )( T − t ) (cid:17) and ζ T ( ρ ) = x + σ ( B T − B t ) ρ + σ ( B T − B t ) ρ − σ | ρ | T − t ) , Σ( ρ ) = σ (cid:112) − | ρ | . Proof:
From (15) the log-price at time T is X T = ζ T ( ρ ) + (cid:90) Tt h u du + Σ( ρ )( B T − B t ) − Σ ( ρ )2 ( T − t ) , and a simple application of the conditional expectation’s tower-property gives E t (cid:104) e − (cid:82) Tt λ u du f ( X T ) (cid:105) = E t (cid:104) E (cid:16) e − (cid:82) Tt λ u du f ( X T ) (cid:12)(cid:12)(cid:12) A tT (cid:17)(cid:105) = E t (cid:104) e − (cid:82) Tt λ u du E (cid:16) f ( X T ) (cid:12)(cid:12)(cid:12) A tT (cid:17)(cid:105) = e (cid:82) Tt h u du E t (cid:104) e − (cid:82) Tt λ u du E (cid:16) e − (cid:82) Tt h u du f ( X T ) (cid:12)(cid:12)(cid:12) A tT (cid:17)(cid:105) . (cid:50) .3 Changing Numeraires As a final step to evaluate the expectations E t in the previous expression, we apply the followingfamily of changes of probability d Q s d Q (cid:12)(cid:12)(cid:12) F s = N ( s, s ) N ( t, s ) , (21)defining the s − forward measures, for any t ≤ s ≤ T . Recalling (19), by Girsanov’s theorem,under Q s W iv = B iv − (cid:90) vt A i ( u, s )( η iu λ iu + δ iu ) du, i = 1 , , t ≤ v ≤ s define independent Brownian motions and the market dynamics, for t ≤ v ≤ s ≤ T , become X v = x + (cid:90) vt (cid:16) h u − σ σ (cid:88) i =1 , ρ i A i ( u, s )( η iu λ iu + δ iu ) (cid:17) du + σ ( W s − W t ) (22) λ iv = λ i + (cid:90) vt (cid:104) ( γ iu + A i ( u, s ) η iu ) λ iu + ( β iu + A i ( u, s ) δ iu ) (cid:105) du + (cid:90) vt [ η iu λ iu + δ iu ] dW iu , (23)where ( W , W , B ) is a 3-dimensional Brownian motion, on [ t, s ] with W v = ρ W v + ρ W v + (cid:113) − ρ − ρ B v , ρ + ρ ≤ s , different Brownian motions are generated. We keep denoting them in the samemanner, as they all have the same distributional properties.Hence, for any t ≤ s ≤ T and any F s − measurable random variable Y , we have E t (cid:16) e (cid:82) st λ u du Y (cid:17) = N ( t, s ) E st (cid:0) Y (cid:1) , (24)where E st , denotes expectations under Q s . In this section, we restrict to considering a European call with strike price e κ and maturity T ,for which we may exploit the Black & Scholes formula, at least in a conditional fashion. Weremark that in this case, by exploiting the put-call parity, it is possible to extend the evaluationmethod also to forward contracts.We treat the case when the intensities are both described by a CIR process. We do notconsider here the Vasicek model, since not appropriate for intensities, as it does not guaranteethe positivity of the process, even though it has been previously considered in credit risk modeling(see for instance [21]) as it allows to write very computable explicit formulas.13 .1 The CIR specification In this case, the dynamics of the market, for any t ≤ s ≤ T , are given by X s = x + (cid:90) st (cid:16) h u − σ (cid:17) du + σ ( B s − B t ) (25) λ is = λ i + (cid:90) st γ i ( θ i − λ iu ) du + η i (cid:90) st (cid:113) λ iu dB iu , i = 1 , . (26)We denote by ˜ r u = r φu − h u and , ˆ r u = r φu − r cu we have to compute { τ>t } c a ( t, T ) = { τ>t } (cid:110) e − (cid:82) Tt ˜ r u du E t (cid:104) e − (cid:82) Tt λ u du e − (cid:82) Tt h u du f ( X T ) (cid:105) + (cid:90) Tt e − (cid:82) st r φu du E t (cid:104) e − (cid:82) st λ u du Λ s c ( s, T ) (cid:105) ds (cid:27) (27)where Λ s = λ s + α ˆ r s − L λ s . Proposition 4.1
Let f ( x ) = (e x − e κ ) + and c ( s, T ) ≡ c ( s, T ) + = c BS ( X s , s, ¯ v s , σ ) c BS ( x, s, ¯ v s , σ ) = e x N (cid:0) d ( x, s, ¯ v s , σ ) (cid:1) − e κ − ¯ v s N (cid:0) d ( x, s, ¯ v s , σ ) (cid:1) d , ( x, s, ¯ v s , σ ) = x − κ + ¯ v s ± σ ( T − s ) σ (cid:112) ( T − s ) , where we denoted by ¯ v s = (cid:90) Ts v u du , for any v : [0 , T ] −→ R . Then we have { τ>t } c a ( t, T ; ρ ) = { τ>t } (cid:110) e − (cid:82) Tt ˜ r u du E t (cid:104) e − (cid:82) Tt λ u du c BS (cid:0) ζ T ( ρ ) , t, ¯ h t , Σ( ρ ) (cid:1)(cid:105) + (cid:90) Tt e − (cid:82) st r φu du E t (cid:104) e − (cid:82) st λ u du Λ s c BS (cid:0) X s ( ρ ) , s, ¯ r s , σ (cid:1)(cid:105) ds (cid:27) . (28) Proof:
Applying inside the first expectation the conditioning with respect to A tT , we obtain E t (cid:104) e − (cid:82) Tt λ u du e − (cid:82) Tt h u du f ( X T ) (cid:105) = E t (cid:104) E t (cid:16) e − (cid:82) Tt λ u du e − (cid:82) Tt h u du f ( X T ) |A tT (cid:17)(cid:105) = E t (cid:104) e − (cid:82) Tt λ u du E t (cid:16) e − (cid:82) Tt h u du f ( X T ) (cid:12)(cid:12)(cid:12) A tT (cid:17)(cid:105) = E t (cid:104) e − (cid:82) Tt λ u du c BS (cid:0) ζ tT ( ρ ) , t, ¯ h t , Σ( ρ ) (cid:1)(cid:105) and we may view the second expectation in (27) as E t (cid:104) e − (cid:82) st λ u du Λ s c BS ( X s ( ρ ) , s, ¯ r s , σ ) (cid:105) where,for t ≤ s ≤ T , setting M is = B is − B it , for i = 1 ,
2, we have X s ( ρ ) = x + (cid:90) st (cid:16) h u − σ (cid:17) du + σ (cid:0) M s ρ + M s ρ + M s (cid:112) − | ρ | (cid:1) . d , ( ζ T ( ρ ) , t, ¯ h t , Σ( ρ )) = (cid:104) d ( x, t, ¯ h t , σ ) + M T σ √ T − t ρ + M T σ √ T − t ρ − σ √ T − t | ρ | (cid:105) √ −| ρ | (cid:104) d ( x, s, ¯ h t , σ ) + M T σ √ T − t ρ + M T σ √ T − t ρ (cid:105) √ −| ρ | . Pointing out the dependence on ρ of c a ( t, T ), we get (28). (cid:50) We want to approximate (28) by a Taylor expansion with respect to the correlation param-eters ρ = ( ρ , ρ ) around = (0 ,
0) on { τ > t } . The first-order approximation would hencebe c a ( t, T ; ρ ) ≈ c a ( t, T ; ) + ∂c a ( t, T ; ) ∂ρ ρ + ∂c a ( t, T ; ) ∂ρ ρ . Remark 4.2
For the sake of exposition, we decided to restrict our discussion to the first orderapproximation, which may turn to be extremely satisfying when the model seems to exhibit aroughly linear dependence upon the correlation parameters. This was highlighted by the MonteCarlo simulations for the CIR intensity setting (section 5) and the accuracy of our methodturned out to be very good. If the dependence on the correlation parameters is more markedlynonlinear, one may develop Taylor’s polynomial to a higher order to capture this behavior. Weexplicitly wrote also a second-order formula: it is computationally longer, but it does not presentany additional theoretical complexity. We did not report it here to keep the exposition lighter.
Since the integrability conditions are satisfied, the derivatives pass under the integral and ex-pectation signs and the problem is reduced to computing the derivatives with respect to thecorrelation parameters of c BS (cid:0) ζ T ( ρ ) , t, T, Σ( ρ ) (cid:1) and of c BS (cid:0) X s ( ρ ) , s, T, σ (cid:1) and evaluating themat . After some calculations, one arrives at the following expressions c BS (cid:0) ζ T ( ρ ) , t, ¯ h t , Σ( ρ ) (cid:1) ≈ c BS (cid:0) x, t, ¯ h t , σ (cid:1) + σ e x N (cid:0) d ( x, t, ¯ h t , σ ) (cid:1)(cid:2) M T ρ + M T ρ (cid:3) and c BS (cid:0) X s ( ρ ) , s, ¯ r s , σ (cid:1) ≈ c BS (cid:0) X s ( ) , s, ¯ r s , σ (cid:1) + σ e X s ( ) N (cid:0) d ( X s ( ) , s, ¯ r s , σ ) (cid:1)(cid:2) M s ρ + M s ρ (cid:3) to be plugged into (27), with each term to be computed following the procedure outlined in theprevious sections. Thus, exploiting the independence between X s ( ) and B , B we have c a ( t, T ; ρ ) ≈ e − (cid:82) Tt ˜ r u du (cid:40) N ( t, T ) c BS (cid:0) x, t, ¯ h t , σ (cid:1) + σ e x N (cid:0) d ( x, t, ¯ h t , σ ) (cid:1) E t (cid:104) e − (cid:82) Tt λ u du ( M T ρ + M T ρ ) (cid:105)(cid:41) + (cid:90) Tt e − (cid:82) st r φu du (cid:40) E t (cid:104) e (cid:82) st λ u du Λ s (cid:105) E t (cid:104) c BS (cid:0) X s ( ) , s, ¯ r s , σ (cid:1)(cid:105) + σ E t (cid:104) e X s ( ) N (cid:0) d ( X s ( ) , s, ¯ r s , σ ) (cid:1)(cid:105) (cid:88) i =1 E t (cid:16) e − (cid:82) st λ u du Λ s M is (cid:17) ρ i (cid:41) ds and we have to compute every single expectation. We proceed by steps, showing that we mayreduce to computing some basic cases. 15. Noticing that M s ∼ N (0; σ ( s − t )) X s ( ) = x + (cid:90) st ( h u − σ du + M s ∼ N (cid:16) x + (cid:90) st ( h u − σ du ; σ ( s − t ) (cid:17) ,d i ( X s ( ) , s, ¯ r s , σ ) = X s ( ) − k + ¯ r s ± σ ( T − s ) σ √ T − s = M s √ T − s + d i ( x, s, ¯ r s , σ ) + 1 σ √ T − s (cid:90) st ( h u − σ du ∼ N (cid:16) d i ( x, s, ¯ r s , σ ) + 1 σ √ T − s (cid:90) st ( h u − σ du, s − tT − s (cid:17) , i = 1 , E t (cid:104) c BS (cid:0) X s ( ) , s, ¯ r s , σ (cid:1)(cid:105) = E t (cid:104) e X s ( ) N (cid:0) d ( X s ( ) , s, ¯ r s , σ ) (cid:1)(cid:105) − e κ − ¯ r s E t (cid:104) N (cid:0) d ( X s ( ) , s, ¯ r s , σ ) (cid:1)(cid:105) , the Gaussian integrals can be computed explicitly E t (cid:104) e X s ( ) N (cid:0) d ( X s ( ) , s, ¯ r s , σ ) (cid:1)(cid:105) =e x + (cid:82) st h u du N (cid:16) d ( x + (¯ r s − ¯ h s ) , t, ¯ h t , σ ) (cid:17) E t (cid:104) N (cid:0) d ( X s ( ) , s, ¯ r s , σ ) (cid:1)(cid:105) = N (cid:16) d ( x + (¯ r s − ¯ h s ) , t, ¯ h t , σ ) (cid:17) , by applying the following Lemma 4.3
Let p ∈ R and X ∼ N ( µ, ν ) , then E (e pX N ( X )) = e pµ + ( pν )22 N (cid:18) µ + pν √ ν (cid:19) where by N we denote the standard Normal distribution function.Proof: see Zacks (1981) for p = 0, the general case follows by a “completing the squares”argument. (cid:3) Therefore we may conclude that E t (cid:104) c BS (cid:0) X s ( ) , s, ¯ r s , σ (cid:1)(cid:105) = e − (¯ r s − ¯ h s )+ (cid:82) st h u du c BS (cid:0) x + (¯ r s − ¯ h s ) , t, ¯ h t , σ (cid:1) (29)2. It remains to evaluate the expectations E t (cid:16) e − (cid:82) st λ u du Λ s (cid:17) , E t (cid:16) e − (cid:82) st λ u du ( B is − B it ) (cid:17) , E t (cid:16) e − (cid:82) st λ u du Λ s ( B is − B it ) (cid:17) i = 1 , s = λ s + α ˆ r s − L λ s the above expressions reduce to computing E t (cid:104) e − (cid:82) st λ u du ( λ is ) α ( B js − B jt ) k (cid:105) for i, j = 1 ,
2, and α, k = 0 , E t (cid:104) e − (cid:82) st λ u du ( λ is ) α ( B js − B jt ) k (cid:105) .
16o do so, we apply the change of Numeraire described in subsection 3.3, obtaining E t (cid:104) e − (cid:82) st λ u du ( λ is ) α ( B js − B jt ) k (cid:105) = N ( t, s ) E st (cid:104) ( λ is ) α (cid:2) ( W js − W jt ) + η j (cid:90) st A j ( u, s ) (cid:113) λ ju du (cid:3) k (cid:105) . We can exploit the independence of W and W , so that the last expectation, for i (cid:54) = j becomes η j E st (cid:104) ( λ is ) α (cid:105)(cid:34) (cid:90) st A j ( u, s ) E st (cid:16)(cid:113) λ ju (cid:17) du (cid:35) k , where for t ≤ u ≤ sλ iu = λ i + (cid:90) ut (cid:104) γ i θ i − (cid:16) γ i − η i A i ( v, s ) (cid:17) λ iu (cid:105) dv + η i (cid:90) ut (cid:113) λ iv dW iv . When i = j , if k = 0, clearly we have only the first expectation, if α = 0 only the second,and for α = k = 1, we end up with E st (cid:104) λ is ( W is − W it ) (cid:105) + η j (cid:90) st A i ( u, s ) E st (cid:104) λ is (cid:113) λ iu (cid:105) du.
3. Thus we have reduced the problem to considering the expectations for u ≤ s E st (cid:0) λ is (cid:1) , E st (cid:0)(cid:113) λ iu (cid:1) , E st (cid:0) λ is (cid:113) λ iu (cid:1) , (30) E st (cid:0) λ is ( W is − W it ) (cid:1) , (31)The third of (30), again by the independence of the increments, can be written as E st (cid:0) λ is (cid:113) λ iu (cid:1) = E st (cid:0) ( λ is − λ iu ) (cid:113) λ iu (cid:1) + E st (cid:0) ( λ iu ) (cid:1) = E st (cid:0) λ is − λ iu (cid:1) E st (cid:0)(cid:113) λ iu (cid:1) + E st (cid:0) ( λ iu ) (cid:1) . By applying Itˆo’s formula and taking expectations, for t ≤ u ≤ s ≤ T we have E st (cid:0) λ iu (cid:1) = e − (cid:82) ut [ γ i − η i A i ( ξ,s )] dξ (cid:26) λ i + γ i θ i (cid:90) ut e (cid:82) vt [ γ i − η i A i ( ξ,s )] dξ dv (cid:27) , E st (cid:104)(cid:113) λ iu (cid:105) = e − (cid:82) ut [ γ i − η i A i ( ξ,s )] dξ (cid:34)(cid:112) λ i + 12 (cid:104) γ i θ i − η i (cid:105)(cid:90) ut e (cid:82) vt [ γ i − η i A i ( ξ,s )] dξ E st (cid:104) (cid:112) λ iv (cid:105) dv (cid:35) , E st (cid:104) ( λ iu ) (cid:105) = e − (cid:82) ut [ γ i − η i A i ( ξ,s )] dξ (cid:20) ( λ i ) + 32 (cid:104) γ i θ i + η i (cid:105)(cid:90) ut e (cid:82) vt [ γ i − η i A i ( ξ,s )] dξ E st (cid:104)(cid:113) λ iv (cid:105) dv (cid:21) , and we approximate √ λ iv by 1 √ λ i or 1 √ θ i , freezing the process either at the initial conditionor at the mean reversion parameter. This choice usually provides simple and numericallyquite accurate approximations of the powers of a CIR process. Finally, we may use inte-gration by parts for the expectation (31) and we may conclude E st (cid:0) λ is ( W is − W it ) (cid:1) = η i (cid:90) st e − (cid:82) su [ γ i − η i A i ( ξ,s )] dξ E st (cid:104)(cid:113) λ iu (cid:105) du In conclusion, all the pieces appearing in (4.1) can be computed explicitly, provided weperform the mentioned freezing for ( λ iu ) − .17ummarizing c a ( t, T ; ρ ) ≈ g ( t, T ; ) + g ( t, T ; ) ρ + g ( t, T ; ) ρ (32)where the zeroth term is (with R = 1 − L ) g ( t, T ; ) = e − (cid:82) Tt ˜ r u du N ( t, T ) c BS (cid:0) x, t, ¯ h t ,σ (cid:1) + (cid:90) Tt e − (cid:82) st ˜ r u du − (¯ r s − ¯ h s ) N ( t, s ) (cid:2) R E st ( λ s )+ E st ( λ s ) + α ˆ r s (cid:3) c BS (cid:0) x + (¯ r s − ¯ h s ) , t, ¯ h t , σ (cid:1) ds (33)and the first-order coefficients are g ( t, T ; ) = σ (cid:40) η e x − (cid:82) Tt ˜ r u du N ( t, T ) N (cid:0) d ( x, t, ¯ h t , σ ) (cid:1) (cid:90) Tt A ( s, T ) E Tt (cid:0)(cid:112) λ s (cid:1) ds + (cid:90) Tt e x − (cid:82) st ˜ r u du N ( t, s ) N (cid:16) d ( x + (¯ r s − ¯ h s ) , t, ¯ h t , σ ) (cid:17)(cid:104) R E st (cid:0) λ s ( W s − W t ) (cid:1) + η (cid:90) st A ( u, s ) (cid:2) E st ( λ s − λ u ) E st ( (cid:112) λ u ) + E st (cid:0) ( λ u ) (cid:1) + (cid:0) E st ( λ s ) + α ˆ r s (cid:1) E st ( (cid:112) λ u ) (cid:3) du (cid:105) ds (cid:41) (34) g ( t, T ; ) = σ (cid:40) η e x − (cid:82) Tt ˜ r u du N ( t, T ) N (cid:0) d ( x, t, ¯ h t , σ ) (cid:1) (cid:90) Tt A ( s, T ) E Tt (cid:0)(cid:112) λ s (cid:1) ds + (cid:90) Tt e x − (cid:82) st ˜ r u du N ( t, s ) N (cid:16) d ( x + (¯ r s − ¯ h s ) , t, ¯ h t , σ ) (cid:17)(cid:104) E st (cid:0) λ s ( W s − W t ) (cid:1) + η (cid:90) st A ( u, s ) (cid:2)(cid:0) R E st ( λ s ) + α ˆ r s (cid:1) E st ( (cid:112) λ u ) + E st ( λ s − λ u ) E st ( (cid:112) λ u ) + E st (cid:0) ( λ u ) (cid:1)(cid:3) du (cid:105) ds (cid:41) (35)where, for t ≤ s ≤ T and i = 1 ,
2, we have N i ( t, s ) = e A i ( t,s ) λ i + B i ( t,s ) , N ( t, s ) = N ( t, s ) N ( t, s )with h i = (cid:113) γ i + 2 η i , A i ( t, T ) = − h i ( T − t ) − h i − γ i + ( h i + γ i )e h i ( T − t ) B i ( t, T ) = 2 γ i θ i η i ln (cid:32) h i e γ i + h i ( T − t ) h i − γ i + ( h i + γ i )e h i ( T − t ) (cid:33) . In this section, we present some numerical results of our approximation method for the callprice. As a first step, we assess the performance of the first-order approximation (32) by usingthe Monte Carlo evaluations with control variates as a benchmark, employing the default-freeprice as control: in the considered cases, this reduces the length of the confidence interval by atleast one order of magnitude. For the simulations, we generated M = 10 sample paths with a18ime step equal to 10 − for any considered maturity. The benchmark Monte Carlo method wasimplemented to approximate the call price (14) by using the Euler discretization scheme withfull truncation for the intensity processes λ t and λ t (see [35]) and with an exact simulation ofthe Brownian motion for the underlying X t . The running integrals appearing in the expectationswere evaluated by means of trapezoidal routine. All the algorithms were implemented in MatLab(R2019b).The evaluation of the zeroth and first-order terms of our approximation ((33), (34), (35))requires the computation of nested one-dimensional integrals of well-behaved functions once foreach set of chosen parameters and this step was implemented through the vectorized globaladaptive quadrature MatLab algorithm.The parameters of the intensity processes were set as in [11] and [3] (see Table 2) and theyagree with calibrated default intensities. The strike price was fixed to K = e κ = 100 and weconsidered two maturities, T = 0 . T = 2. Lastly, without loss of generality, we took t = 0,the log-asset’s initial value was set to 4 . σ = 40%. The remainingparameters were chosen as r = h = 0 . r φ = 0 . r c = 0 .
002 and α = 0 . T of thecontract. It is apparent how the approximation is highly satisfactory for short term maturitywhile it tends to deteriorate a little when the horizon increases.In Table (5) we highlight the separate contributions of the zeroth and first-order terms, g (0 , T ; ), g (0 , T ; ) and g (0 , T ; ) in (32), which are not significantly affected in relativemagnitude by changes in the values of the parameters. In particular, the contribution due tothe correlation between the underlying and the intensities is quite sizeable and it supports thechoice of stochastic processes versus deterministic functions to represent the intensities. Wenotice that the contribution of the term g is more significant compared to that of g whichappears to be always rather small. This is to be expected since we are considering a call optionand default of the Investor is bound to have a limited impact on the overall value; on thecontrary, the term g is more relevant being connected to the counterparty’s default and, asnatural, it decreases as the collateralization tends to one.The contribution coming from the stochastic nature of the intensities can be better appreci-ated by looking at the results of the further set of numerical experiments reported in Table (6).There, in order to compare with the results in [14], we considered the rates r = 0 . h = 0 . r φ = 0 . r c = 0 .
002 and we chose λ = 0 . λ = 0 .
02 and the other parameters as in (2).The losses given default were set to L = L = 60% and we took T = 0 .
5. The correction thatwe obtain with respect to the prices in [14] is of the order of 10 − , which can, of course, becomevery relevant as the volume of the transaction grows.As a final remark, we write explicitly our evaluation formula when constant intensities λ it ≡ λ i are taken. It is immediately seen by using (29) that the price (27) becomes c a ( t, T ) = e ( λ + λ − ( r φ − h ))( T − t ) c BS ( x, t, ¯ h, σ ) + ( λ + λ + ( r φ − r c ) α − λ L ) × (cid:90) Tt e − ( λ + λ +( r φ − h ))( s − t ) e − ( r − h )( T − s ) c BS ( x + ( r − h )( T − s ) , t, ¯ h, σ ) ds (36)which, as noticed in [13] and [14], shows that the interplay among all the rates in this framework19 γ θ η τ (counterparty) 0.03 0.02 0.161 0.08 0.9848 0.9371 τ (investor) 0.035 0.35 0.45 0.15 0.9660 0.7399Table 2: Parameter sets for the CIR default intensities.-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6 -7.478e-04 -5.850e-04 -3.852e-04 -1.951e-04 -4.362e-05 7.122e-05 1.881e-04-0.4 -5.338e-04 -3.423e-04 -1.508e-04 5.306e-05 1.955e-04 3.118e-04 3.636e-04-0.2 -3.104e-04 -9.415e-05 8.240e-05 2.456e-04 3.640e-04 4.693e-04 5.321e-040 -1.194e-04 8.440e-05 2.489e-04 4.203e-04 5.234e-04 6.252e-04 7.105e-040.2 5.723e-05 2.527e-04 4.217e-04 5.816e-04 7.102e-04 8.091e-04 9.161e-040.4 2.584e-04 4.708e-04 6.296e-04 7.458e-04 8.760e-04 9.736e-04 1.079e-030.6 4.854e-04 6.768e-04 8.431e-04 9.614e-04 1.074e-03 1.167e-03 1.241e-03Table 3: Approximation errors, Set 1 for τ , Set 2 for τ , T = 0 .
5. The average length of the95% confidence interval for the MC estimates is 5 . e − \ ρ -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6 -6.619e-02 -5.728e-02 -4.887e-02 -3.983e-02 -3.114e-02 -2.303e-02 -1.427e-02-0.4 -5.191e-02 -4.320e-02 -3.409e-02 -2.552e-02 -1.726e-02 -9.017e-03 -7.615e-04-0.2 -3.706e-02 -2.828e-02 -1.938e-02 -1.138e-02 -3.327e-03 4.780e-03 1.299e-020 -2.246e-02 -1.338e-02 -5.165e-03 2.822e-03 1.095e-02 1.877e-02 2.686e-020.2 -7.224e-03 1.505e-03 9.585e-03 1.776e-02 2.559e-02 3.352e-02 4.164e-020.4 8.800e-03 1.771e-02 2.568e-02 3.327e-02 4.091e-02 4.864e-02 5.639e-020.6 2.543e-02 3.414e-02 4.206e-02 4.961e-02 5.704e-02 6.453e-02 7.191e-02Table 4: Approximation errors, Set 1 for τ , Set 2 for τ , T = 2. The average length of the 95%confidence interval for the MC estimates is 0 . T g g g . . − . . . − . . τ and Set 2 for τ . The corresponding default-free prices according to the B&S formulaare c BS ( X , , ¯ r s , σ ) = 11 . T = 0 .
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