# Pricing Vulnerable Options with Market Prices of Common Jump Risks under Regime-Switching Models

Authors: Miao Han; Xuefeng Song; Huawei Niu; Shengwu Zhou
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

This paper investigates the valuation of vulnerable European options considering the market prices of common systematic jump risks under regime-switching jump-diffusion models. The way of regime-switching Esscher transform is adopted to identify an equivalent martingale measure for pricing vulnerable European options. Explicit analytical pricing formulae for vulnerable European options are derived by risk-neutral pricing theory. For comparison, the other two cases are also considered separately. The first case considers all jump risks as unsystematic risks while the second one assumes all jumps risks to be systematic risks. Numerical examples for the valuation of vulnerable European options are provided to illustrate our results and indicate the influence of the market prices of jump risks on the valuation of vulnerable European options.


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## Body

## 1. Introduction

Along with the development of the OTC market, people have recognized the influence of credit risk on financial derivatives pricing and attempted to establish all kinds of credit risk models. Johnson and Stulz [1] first incorporate credit risk with option pricing models and call it vulnerable options, which are vulnerable to counterparty default. They assume that if the counterparty writing an option is unable to make a promised payment, the holder of a derivative security would receive all assets of the counterparty. Klein [2] extends the work of Johnson and Stulz [1] by allowing the option writer to have other liabilities, which rank equally with payments under the option. It also extends the work of Hull and White [3] by relaxing the assumption of independence between the assets of the counterparty and the asset underlying the option. Jarrow and Turnbull [4] provide a new methodology for pricing and hedging derivative securities involving two types of credit risks. Klein and Inglis [5] also expand on the results of pricing vulnerable European options when the payoff of the option can increase the risk of financial distress from Johnson and Stulz [1] and Klein [2]. Hung and Liu [6] extend the framework of Klein [2] to price vulnerable option pricing when the market is incomplete. Chang and Huang [7] and Klein and Yang [8] study the valuation of vulnerable American options with correlated credit risk.However, the above literatures on vulnerable options all assume that the dynamics of the assets are modelled by geometric Brownian motions with constant drift and volatility. Indeed, over the past decade or two, dozens of empirical evidences have revealed that risky asset prices not only present sudden shocks due to the arrival of important new information in financial markets but also exhibit different behaviors in different time periods due to the time-inhomogeneity generated by the financial market. For the former case, Merton [9] introduces the jump-diffusion models with compound Poisson processes into option pricing (see Kou [10], Xu et al. [11], Tian et al. [12], etc.). For the latter case, Markov regime-switching models have provided us with a natural and convenient way to describe structural changes in market interest rate, exchange rate, stock returns, and so forth, since Hamilton [13] introduced this class of models into economics and finance.To better describe both the time-inhomogeneity and sudden shocks in the processes of asset prices, there have been lots of papers studying options pricing through incorporating jump-diffusion models with Markov regime-switching models. Elliott and Osakwe [14] consider option pricing for pure jump processes with Markov switching compensators. Bo et al. [15] study the pricing of some currency options based on the Markov-modulated jump-diffusion models for the spot foreign exchange rate. Wang and Wang [16] and Niu and Wang [17] study the pricing problem of vulnerable European options under the Markov regime switching jump-diffusion models. However, they regard all jump risks as systematic risks. Indeed, the jumps in an actual process may be caused by the market or may occur on its own. Tian et al. [12] divide the jumps into individual jumps for each asset price and common jumps that affect the prices of all assets. In this paper, considering the market prices of common systematic jump risks regardless of individual jump risks, we develop an equivalent martingale measure for two regime-switching jump-diffusion processes with correlated jumps via regime-switching Esscher transform and consider the differences between the physical jump-diffusion processes and the risk-neutral jump-diffusion processes. The dynamics of the risk assets prices in this paper are different from those obtained by considering the market prices of all jump risks in Niu and Wang [17] under the risk-neutral measure. In order to consider the influence of the market prices of jump risks on vulnerable option values, we also study vulnerable options pricing formulae with or without market prices of all jump risks. Finally, numerical results are also presented to illustrate our results by the Monte Carlo simulations.The rest of the paper is organized as follows. Section2 presents the basic setting and asset price model. In Section 3, we employ the regime-switching Esscher transform to determine an equivalent martingale measure considering the market prices of common systematic jump risks and derive the closed-form pricing formulae for vulnerable European options under a Markov regime switching jump-diffusion model. In Section 4, for comparison, the pricing formulae for vulnerable European options are obtained taking the market prices of all jump risks into consideration. Section 5 presents some numerical results to illustrate the valuation of vulnerable options for different parameters and different cases. Section 6 concludes the paper.


## 2. The Model Description

Consider a continuous-time financial market with a finite time horizonT≔0,T, where T<∞. We consider a complete probability space Ω,F,P, where P is a physical probability measure, under which all stochastic processes are defined. We equip the probability space Ω,F,P with a filtration F≔Ft∣t∈T and assume that there are three primary securities, namely, a risk-free bond B, underlying asset S, and option writer’s asset V, which are traded continuously over the time horizon T≔0,T. U={Ut}t∈T is a continuous-time finite-state Markov chain on Ω,F,P with a finite-state space S≔(s1,s2,…,sN). We use the states of U to indicate the states of the economy. We adopt the assumptions of Elliott et al. [18] that the state space of U is a limited collection of vectors e1,e2,…,eN, where ei=0,…,1,…,0′∈RN with “1” in theith component. Suppose that the time-invariant generator matrix of U is A with aiji,j=1,2,…,N. Then, following Elliott et al. [18], the semimartingale decomposition of U is given by(1)dUt=AUsds+dMt,where M={Mt}t∈T is an RN-valued martingale with respect to the filtration generated by {Ut}t∈T under P.LetBt denote the value of the bank currency account at time t. If one initially saves B0=1, then he can gain Bt=exp⁡(∫0t‍rsds) at time t. We assume that the instantaneous market interest rates rt depend on the economic states of U. Then, rt is given by (2)rt=r,Ut,where r=r1,r2,…,rN∈RN with ri>0, for each i=1,2,…,N; ·,· denotes the inner product in RN. For i≠j, ei,ej=0, else, ei,ei=1.Let the dynamics of the underlying asset valueSt of the option follow a regime-switching jump-diffusion process under the physical measure P,(3)dStSt-=αt-k1λSt∗dt+σStdWt1+eZSt-1-1dMt1,where Wt1 is a standard Brownian motion on Ω,F,P. The expected return rate αtt∈T and the volatility σStt∈T of the underlying asset St depend on {Ut}t∈T. They are defined as (4)αt=α,Ut,σSt=σS,Ut,where α=α1,α2,…,αN∈RN and σS=σS1,σS2,…,σSN∈RN with σSi>0 for each i=1,2,…,N. Mt1 is a Poisson process with intensity λSt∗. Following Tian et al. [12], both the jump term Mt1 and the intensity λSt∗ consist of two parts, (5)Mt1=Nt1+Nt,λSt∗=λSt+λt,where, specifically, shocks to the underlying asset price are also composed of two parts: individual shocks Nt1 for each asset price and common shocks Nt affecting the prices of all assets. Here, we take the common shocks as market factors, which are considered as systematic risks and consider the individual shocks as unsystematic risks. We assume Nt1 and Nt are independent Poisson processes with intensities λSt and λt, which depend on the states of {Ut}t∈T and are described by (6)λSt∗=λS∗,Ut,λSt=λS,Ut,λt=λ,Ut,where λS∗=λS1∗,λS2∗,…,λSN∗∈RN, λS=λS1,λS2,…,λSN∈RN, and λ=λ1,λ2,…,λN∈RN. If the jump occurs at time t, the jump amplitude of the underlying asset is controlled by ZSt1. For any time t≠s, we assume that ZSt1 and ZSs1 are independently and identically distributed and (7)ZSt1~Nμ1,σ12.The mean percentage jump size of the price is given by (8)k1=EeZSt1-1=EeZSt1-1=eμ1+1/2σ12-1.We assume the dynamics of the counterparty’s asset valueVt are also driven by the following regime-switching jump-diffusion process.(9)dVtVt-=βt-k2λVt∗dt+σVtdWt2+eZVt-2-1dMt2,where Wt2 is a standard Brownian motion on Ω,F,P. We assume the expected return rate βtt∈T and the volatility σVtt∈T of the counterparty’s asset value Vt depend on {Ut}t∈T and are defined by (10)βt=β,Ut,σVt=σV,Ut,where β=β1,β2,…,βN∈RN and σV=σV1,σV2,…,σVN∈RN with σVi>0 for each i=1,2,…,N. Mt2 is also a Poisson process with corresponding intensity λVt∗. In addition, shocks to Vt also include the individual component and the common component. Both the jump term Mt2 and the intensity λVt∗ consist of two parts, (11)Mt2=Nt2+Nt,λVt∗=λVt+λt,where Nt2 is a Poisson process with intensity λVt, which is independent of Nt and Nt1. The intensities λVt∗, λVt, and λt also depend on the states of {Ut}t∈T; they are given by (12)λVt∗=λV∗,Ut,λVt=λV,Ut,where λV∗=λV1∗,λV2∗,…,λVN∗∈RN and λV=λV1,λV2,…,λVN∈RN. If the jump occurs at time t, the jump amplitude of Vt is controlled by ZVt2. For t≠s, we assume that ZVt2 and ZVs2 are independently and identically distributed with distribution (13)ZVt2~Nμ2,σ22.The mean percentage jump size is given by (14)k2=EeZVt2-1=EeZVt2-1=eμ2+1/2σ22-1;Wt1 and Wt2 have the correlation coefficient ρ. Moreover, we assume that Wt1,Wt2, Nt, Nt1, Nt2, ZSt1, and ZVt2 are mutually independent.


## 3. Vulnerable European Option Pricing Considering the Market Prices of Common Jump Risks

### 3.1. Equivalent Martingale Measure via Esscher Transform

The financial market described by a regime-switching model and the jump components is incomplete; hence the risk-neutral measure is not uniquely determined. In order to obtain the valuation of vulnerable European options, we need to choose a special martingale measure. Esscher transform, which was first introduced by Gerber and Shiu [19], is a method of selecting the equivalent martingale measure and has extensive applications in the fields of finance and insurance. In this subsection, we adopt the random Esscher transform of Elliott et al. [20] to identify an equivalent martingale measure.Here, we divide jump risks into two parts and price common jump risks caused by market factors in the model while neglecting the jump risks caused by the assets themselves.First, we define two processesY1t and Y2t as follows:(15)Y1t=ln⁡StS0=∫0tαs-k1λSs∗-12σSs2ds+∫0tσSsdWs1+∫0tZSs-1dNs1+∫0tZSs-1dNs,Y2t=ln⁡VtV0=∫0tβs-k2λVs∗-12σVs2ds+∫0tσVsdWs2+∫0tZVs-2dNs2+∫0tZVs-2dNs.For simplicity, let (16)Y1t=C1t+J1t+Jt,Y2t=C2t+J2t+Jt′,where Cit,Jiti=1,2 and JtJt′ are the continuous diffusive part, common jump part, and individual jump part, respectively, of Yit,i=1,2. They have the following forms: (17)C1t=∫0tαs-k1λSs∗-12σSs2ds+∫0tσSsdWs1,Jt=∫0tZSs-1dNs1,J1t=∫vtZSs-1dNs,C2t=∫0tβs-k2λVs∗-12σVs2ds+∫0tσVsdWs2,Jt′=∫0tZVs-2dNs2,J2t=∫0tZVs-2dNs.Let{FtU}t∈T and {Fti}t∈T denote the P-augmentations of natural filtrations generated by U and Yii=1,2, respectively. For each i=1,2, set Gti=FtU∨Fti. For any t∈0,T, let Ht=Gt1∨Gt2. Then, the random Esscher transform Qθ~P on HT with regime-switching parameters θ1t=θ1tc1,θ1t-J1t∈0,T and θ2t=θ2tc2,θ2t-J2t∈[0,T] is given as follows:(18)dQθdPHT=exp⁡∫0Tθ1tc1dC1t+∫0Tθ1t-J1dJ1t+∫0Tθ2tc2dC2t+∫0Tθ2t-J2dJ2tEexp⁡∫0Tθ1tc1dC1t+∫0Tθ1t-J1dJ1t+∫0Tθ2tc2dC2t+∫0Tθ2t-J2dJ2t∣FTU,where E· denote the expectation under measure P. For m∈c1,J1, θ1tm=θ1m,Ut, and θ1m=θ11m,θ12m,…,θ1Nm∈RN, and for m∈c2,J2, θ2tm=θ2m,Ut and θ2m=θ21m,θ22m,…,θ2Nm∈RN. Since (19)Eexp⁡∫0Tθ1tc1dC1t∫0Tθ1t-J1dJ1t+∫0Tθ2tc2dC2t+∫0Tθ2t-J2dJ2t∣FTU=exp⁡∫0Tθ1tc1αt-k1λSt∗-12σSt2dt+∫0Tθ2tc2βt-k2λVt∗-12σVt2dt+12∫0Tθ1tc12σSt2+θ2tc22σVt2+2ρθ1tc1σStθ2tc2σVtdt+∫0Tλteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22-1dt,then(20)dQθdPHT=exp⁡∫0Tθ1tc1σStdWt1+∫0Tθ2tc2σVtdWt2-12∫0Tθ1tc12σSt2+θ2tc22σVt2+2ρθ1tc1σStθ2tc2σVtdt×exp⁡∫0Tθ1t-J1ZSt-1+θ2t-J2ZVt-2dNt-∫0Tλteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22-1dt.By the fundamental theorem of asset pricing, the absence of arbitrage opportunities is “essentially” equivalent to the existence of an equivalent martingale measure under which the discounted stock price process is a martingale. Following Elliott et al. [20], due to the presence of the uncertainty generated by the Markov chain process, the martingale condition is obtained by considering an enlarged filtration H. The following theorem presents the result.Theorem 1.
The martingale conditions are satisfied if and only if(21)αt-rt-k1λt+θ1tc1σSt2+ρσStθ2tc2σVt+λ~tkStθ1t-J1=0,βt-rt-k2λt+θ2tc2σVt2+ρσStθ1tc1σVt+λ~tkVtθ2t-J2=0,where(22)λ~t=λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22,kStθ1t-J1=k1+1eθ1t-J1σ12-1,kVtθ2t-J2=k2+1eθ2t-J2σ22-1.Proof.
Let(23)MStC=∫ut1+θ1sc1σSsdWs1+∫utθ2sc2σVsdWs2-12∫ut1+θ1sc12σSs2+θ2sc22σVs2+2ρ1+θ1sc1σSsθ2sc2σVsds,MStJ1=∫ut1+θ1s-J1ZSs-1+θ2s-J2ZVs-2dNs-∫utλse1+θ1s-J1μ1+θ2s-J2μ2+1/21+θ1s-J12σ12+θ2s-J2σ22-1ds.Then, by Bayes’ rule, we have (24)EQθe-∫0trsdsSt∣Hu=Ee-∫0trsdsSt×dQθdP∣HuEdQθdP∣Hu=e-∫0ursdsSuEexp⁡∫utαs-rs-k1λSs∗+θ1sc1σSs2+ρσSsθ2sc2σVsds+MStc×exp⁡-∫utλseθ1s-J1μ1+θ2s-J2μ2+1/2θ1s-J1σ12+θ2s-J2σ22-1ds+∫utλse1+θ1s-J1μ1+θ2s-J2μ2+1/21+θ1s-J12σ12+θ2s-J2σ22-1ds+MStJ1×exp⁡∫utZSs-1dNs1=e-∫0ursdsSuexp⁡∫utαs-rs-k1λSs∗+θ1sc1σSs2+ρσSsθ2sc2σVsds×exp⁡-∫utλseθ1s-J1μ1+θ2s-J2μ2+1/2θ1s-J1σ12+θ2s-J2σ22-1ds+∫utλse1+θ1s-J1μ1+θ2s-J2μ2+1/21+θ1s-J12σ12+θ2s-J2σ22-1ds×exp⁡∫utλSseμ1+1/2σ12-1ds.The martingale condition (25)EQθe-∫0trsdsSt∣Hu=e-∫0ursdsSuis satisfied if and only if (26)αt-rt-k1λt+θ1tc1σSt2+ρσStθ2tc2σVt+λ~tkStθ1t-J1=0,where(27)λ~t=λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22,kStθ1t-J1=k1+1eθ1t-J1σ12-1.Similarly, since (28)EQθe-∫0trsdsVt∣Hu=e-∫0ursdsVu,it follows that (29)βt-rt-k2λt+θ2tc2σVt2+ρσStθ1tc1σVt+λ~tkVtθ2t-J2=0,where λ~t=λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22 and kVtθ2t-J2=k2+1eθ2t-J2σ22-1.Proposition 2.
Conditional onFTU and under the equivalent martingale measure Qθ, St and Vt follow stochastic differential equations:(30)dStSt-=rt-k1λSt-λ~tkStθ1t-J1dt+σStdW~t1+eZSt-1-1dNt1+eZ~St-1-1dN~t,dVtVt-=rt-k2λVt-λ~tkVtθ2t-J2dt+σVtdW~t2+eZVt-2-1dNt2+eZ~Vt-2-1dN~t,where θ1tc1, θ1t-J1, θ2tc2, and θ2t-J2 are given by(21). Moreover,(31)W~t1=Wt1-∫0tσSsθ1sc1ds-ρ∫0tσVsθ2sc2ds,W~t2=Wt2-∫0tσVsθ2sc2ds-ρ∫0tσSsθ1sc1dsare two standard Qθ Brownian motions. N~t is still a Poisson process under the measure Qθ with the intensity rate λ~t. λ~t is given as follows:(32)λ~t=λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22,and the mean percentage jump sizes of Z~St-1 and Z~Vt-2 are(33)kStθ1t-J1=k1+1eθ1t-J1σ12-1,kVtθ2t-J2=k2+1eθ2t-J2σ22-1.Proof.
Under the conditions ofFTU, we use Girsanov’s theorem and (20) to obtain W~t1,W~t2 which are two standard Qθ Brownian motions with the correlation coefficient ρ. Note that the parameters describing the common jumps in the two risky assets values have been changed. Now, we prove the formulae of (32) and (33). (34)EQθexp⁡iu∫0TZSt-1dNt∣FTU=Eexp⁡iu∫0TZSt-1dNt×dQθdP∣FTU=exp⁡∫0tλteiu+θ1t-J1μ1+θ2t-J2μ2+1/2iu+θ1t-J12σ12+θ2t-J22σ22-1λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J12σ12+θ2t-J22σ22-1dt-λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J12σ12+θ2t-J22σ22-1dt=exp⁡∫0Tλ~t∫-∞+∞eiuz-1f~USzdt.
Hence, under the conditionFTU and the probability measure Qθ, the intensity rate of Nt is λ~t=λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22. The density function of ZSt-1 is (35)f~USz=12πσ1e-z-μ1+θ1t-J1σ122/2σ12,z∈-∞,∞.Similarly, we can also prove that the density function of ZVt-2 is (36)f~UVz=12πσ2e-z-μ2+θ2t-J2σ222/2σ22,z∈-∞,∞.
Note that the risk-neutral probability measureQθ is selected by the Esscher transform, and the probability law of the Markov chain U remains the same after the measure change; that is, under Qθ, U still has the same semimartingale dynamics. For each 0≤t≤T, we find that the solutions of (21) are not unique. Hence, a special case needs to be considered to obtain an equivalent martingale measure. We suppose (37)αt-rt-k1λt+θ1tc1σSt2+ρσStθ2tc2σVt=0,λ~teμ1+1/2σ12+θ1t-J1σ12-1=0,βt-rt-k2λt+θ2tc2σVt2+ρσStθ1tc1σVt=0,λ~teμ2+1/2σ22+θ2t-J2σ22-1=0.Then the following corollary gives a pair of solutions for the martingale condition.Corollary 3.
The martingale conditions (21) are satisfied if the parameters are given as follows:(38)θ1tc1∗=βt-rt-k2λtρσSt-αt-rt-k1λtσVt1-ρ2σSt2σVt,θ2tc2∗=αt-rt-k1λtρσVt-βt-rt-k2λtσSt1-ρ2σVt2σSt,θ1t-J1∗=-μ1+1/2σ12σ12,θ2t-J2∗=-μ2+1/2σ22σ22,k~Stθ1t-J1=k1+1eθ1t-J1σ12-1=0,k~Vtθ2t-J2=k2+1eθ2t-J2σ22-1=0,λ~t∗=λtexp⁡-μ122σ12+σ128×exp⁡-μ222σ22+σ228.The explicit Esscher parameters in Corollary 3 will be used to the Monte Carlo simulations in Section 5.


### 3.2. Vulnerable European Option Pricing

As in Klein [2], we assume that default will only occur if the counterparty’s asset value at maturity VT is less than some amount D∗. Additionally, this amount is not set to the value of the option but corresponds to the amount of the counterparty’s claim D at exercise time T. Once a credit loss occurs at exercise time T, only the proportion 1-αVT/D of the value of the option at T is paid out by the counterparty. Then, the payoff of a vulnerable European call option is given by (39)ST-K+1VT≥D∗+1-αVTST-K+D1VT<D∗,where K is the strike price of the option and α is the deadweight cost associated with bankruptcy, expressed as a percentage of the value of the counterparty’s assets. By the risk-neutral arguments, under the risk-neutral measure Qθ, the price of a vulnerable European call option at time 0 is given as(40)CS,V,U=EQθe-∫0TrsdsST-K+1VT≥D∗+1-αVTD1VT<D∗.rt is the risk-free rate. CS,V,U depends on U for the Markov regime-switching in our pricing model.Under the conditionsGTn1,n2,n3≔N~T=n,NT1=n1,NT2=n2 and FTU, for m1=n+n1,m2=n+n2, the value of a vulnerable European call option at time 0 is given as(41)Cm1,m2=EQθe-∫0TrsdsST,m1-K+1VT,m2≥D∗+1-αVT,m2D1VT,m2<D∗.Clearly, Cm1,m2 represents the value of the vulnerable option given FTU, conditional on m1 and m2 jumps on the underlying asset and the assets of the counterparty, respectively.Therefore, the conditional vulnerable European call option price givenFTU is(42)C∗=EQθe-∫0TrsdsST-K+1VT≥D∗+1-αVTD1VT<D∗∣FTU=∑n=0∞∑n1=0∞∑n2=0∞QN~T=n,NT1=n1,NT2=n2Cm1,m2.Let Ji denote the occupation time of U≔{Ut}t∈T in state i over the time duration 0,T, where t∈T. Then (43)R¯T=1T∫0Trtdt=1T∑i=1NriJi,U¯1T=1T∫0TσSt2dt=1T∑i=1NσSi2Ji,U¯2T=1T∫0TσVt2dt=1T∑i=1NσVi2Ji,U¯3T=1T∫0TσStσVtdt=1T∑i=1NσSiσViJi,λ¯ST=1T∫0TλStdt=1T∑i=1NλSiJi,λ¯VT=1T∫0TλVtdt=1T∑i=1NλViJi,λ~¯T=1T∫0Tλ~tdt=1T∑i=1Nλ~iJi.From Buffington and Elliott [21], the price of a vulnerable European call option is given by(44)CS,V,U=∫0,TNC∗×ψJ1,J2,…,JNdJ1dJ2⋯dJN,where ψJ1,J2,…,JN is the joint distribution of the occupation time J1,J2,…,JN under the condition U0=U and the probability measure Qθ. Write J≔J1,J2,…,JN for the vector of occupation times. Let G denote a diagonal matrix with the elements in the vector ζ=ζ1,ζ2,…,ζN as its diagonal. Then, for any ζ∈RN, the characteristic function of J under the condition U0=U and the martingale measure Qθ is given by (45)EQθeiζ,J∣U0=U=UeA+iGT,I,where i=-1 and I=1,1,…,1∈RN. Note that ψJ1,J2,…,JN can be completely determined by the characteristic function. As for the proof, refer to Buffington and Elliott [21].At this point, the key problem that needs to be solved is to determineCm1,m2. According to theItô formula, the following equalities hold: (46)ln⁡ST=ln⁡S0+∫0Trt-k1λSt-λ~tkStθ1t-J1-12σSt2dt+∫0TσStdW~t1+∑i=1NT1ZSi1+∑i=1N~TZ~Si1,ln⁡VT=ln⁡V0+∫0Trt-k2λVt-λ~tkVtθ2t-J2-12σVt2dt+∫0TσVtdW~t2+∑i=1NT2ZVi2+∑i=1N~TZ~Vi2,where cov⁡dW~t1,dW~t2=ρdt, ln⁡ST,m1/ln⁡S0,ln⁡VT,m2/ln⁡V0 is bivariate normally distributed under the conditions GTn,n1,n2≔N~T=n,NT1=n1,NT2=n2 and FTU with the following properties:(47)MSm1=Eln⁡ST,m1ln⁡S0=∫0Trt-k1λSt-λ~tkStθ1t-J1-12σSt2dt+m1μ1+σ12∑i=1nθ1ti-J1,MVm2=Eln⁡VT,m2ln⁡V0=∫0Trt-k2λVt-λ~tkVtθ2t-J2-12σVt2dt+m2μ2+σ22∑i=1nθ2ti-J2,cov⁡ln⁡ST,m1ln⁡S0,ln⁡VT,m2ln⁡V0=∫0TρσStσVtdt.Let (48)ln⁡ST,m1ln⁡S0=MSm1+∫0TσSt2dt+m1σ12ξ1,ln⁡VT,m2ln⁡V0=MVm2+∫0TσVt2dt+m2σ22ξ2,where ξ1 and ξ2 are standard normal random variables and the correlation coefficient is (49)ρ¯m1,m2=U¯3TTU¯1TT+m1σ12U¯2TT+m2σ22ρ.Now we need to calculate Cm1,m2, and the derivation process can be referred to Tian et al. [12].(50)Cm1,m2=e-R¯TTEQθST,m1-K+1VT,m2≥D∗+1-αDVT,m21VT,m2<D∗=e-R¯TTA1m1,m2+A2m1,m2+A3m1,m2+A4m1,m2,where(51)A1m1,m2=EQθST,m11ST,m1≥K,VT,m2≥D∗,A2m1,m2=-KEQθ1ST,m1≥K,VT,m2≥D∗,A3m1,m2=1-αDEQθST,m1VT,m21ST,m1≥K,VT,m2<D∗,A4m1,m2=-1-αDKEQθVT,m21ST,m1≥K,VT,m2<D∗.Then, we can get A1m1,m2, A2m1,m2, A3m1,m2, and A4m1,m2 in closed form: (52)A1m1,m2=S0eMSm1+1/2U¯1TT+m1σ12N2a1m1,a2m2,ρ¯m1,m2,A2m1,m2=-KN2b1m1,b2m2,ρ¯m1,m2,A3m1,m2=1-αDS0V0eMSm1+1/2U¯1TT+m1σ12+MVm2+1/2U¯2TT+m2σ22+ρU¯3TT×N2c1m1,m2,-ρ¯m1,m2,A4m1,m2=-1-αDKV0eMVm2+1/2U¯2TT+m2σ22N2d1m1,d2m2,-ρ¯m1,m2,where(53)N2x1,x2,ρ=12π1-ρ2∫-∞x1∫-∞x2e-1/21-ρ2p2-2ρpq+q2dpdq.The parameters are expressed as follows:(54)b1m1=ln⁡S0/K+MSm1U¯1TT+m1σ12,b2m2=ln⁡V0/D∗+MVm2U¯2TT+m2σ22,a1m1=b1m1+U¯1TT+m1σ12,a2m2=b2m2+ρ¯U¯1TT+m1σ12,c1m1=b1m1+U¯1TT+m1σ12+ρ¯U¯2TT+m2σ22,c2m2=-b2m2-U¯2TT+m2σ22+ρ¯U¯1TT+m1σ12,d1m1=b1m1+ρ¯U¯2TT+m2σ22,d2m2=-b2m2-U¯2TT+m2σ22.Therefore, the conditional vulnerable European call option price given FTU is(55)C∗=∑n=0∞∑n1=0∞∑n2=0∞QN~T=n,NT1=n1,NT2=n2Cm1,m2=∑n=0∞∑n1=0∞∑n2=0∞λ~¯TTnn!λ¯STTm1-nm1-n!λ¯VTTm2-nm2-n!e-λ~¯TT-λ¯STT-λ¯VTTCm1,m2.Note the valuation of vulnerable European call options depends on the parametersθ1tc1, θ2tc2, θ1t-J1, and θ2t-J2. For the numerical analysis, we use the special parameters that are given by Corollary 3:(56)MSm1=Eln⁡ST,m1ln⁡S0=∫0Trt-k1λSt-12σSt2dt+n1μ1-12nσ12,MVm2=Eln⁡VT,m2ln⁡V0=∫0Trt-k2λVt-12σVt2dt+n2μ2-12nσ22.Note. If we adopt the hypothesis of Merton [9] and regard the jump risks as unsystematic risks which should not be priced, thus, the parameters describing the jumps in the two risky asset price processes will not be changed after application of the measure transform from the physical measure P to risk-neutral measure. That is, the parameters θ1t-J1 and θ2t-J2 in (18) are equal to 0. From Theorem 1 and Proposition 2, we can obtain the results of the following two propositions specifically.Proposition 4.
The martingale conditions are satisfied if and only if(57)θ1tc1=βt-rtρσSt-αt-rtσVt1-ρ2σSt2σVt,θ2tc2=αt-rtρσVt-βt-rtσSt1-ρ2σVt2σSt.Proposition 5.
Conditional onFTU and under the equivalent martingale measure, the dynamics of St and Vt are given by(58)dStSt-=rt-k1λSt∗dt+σStdW~t1+eZSt-1-1dMt1,dVtVt-=rt-k2λVt∗dt+σVtdW~t2+eZVt-1-1dMt2,where W~t(1) and W~t(2) are two standard Brownian motions which is defined by (31).Under the conditionsGTn,n1,n2≔NT=n,NT1=n1,NT2=n2 and FTU, for m1=n+n1,m2=n+n2, ln⁡ST,m1/ln⁡S0,ln⁡VT,m2/ln⁡V0 is also bivariate normally distributed with the following properties:(59)MS′m1=Eln⁡ST,m1ln⁡S0=∫0Trt-k1λSt∗-12σSt2dt+m1μ1,MV′m2=Eln⁡VT,m2ln⁡V0=∫0Trt-k2λVt∗-12σVt2dt+m2μ2,cov⁡ln⁡ST,m1ln⁡S0,ln⁡VT,m2ln⁡V0=∫0TρσStσVtdt.Similar to the previous results, we can obtain(60)C∗=∑n=0∞∑n1=0∞∑n2=0∞QNT=n,NT1=n1,NT2=n2Cm1,m2′=∑n=0∞∑n1=0∞∑n2=0∞λ¯TTnn!λ¯STTm1-nm1-n!λ¯VTTm2-nm2-n!e-λ¯TT-λ¯STT-λ¯VTTCm1,m2′,where λ¯T=1/T∫0T‍λtdt=1/T∑i=1N‍λiJi. So(61)Cm1,m2′=e-R¯TTA1′m1,m2+A2′m1,m2+A3′m1,m2+A4′m1,m2.The forms of A1′m1,m2, A2′m1,m2, A3′m1,m2, and A4′m1,m2 are consistent with those given in (50), as long as MSm1 is replaced with MS′m1 and MVm2 is replaced with MV′m2.


## 3.1. Equivalent Martingale Measure via Esscher Transform

The financial market described by a regime-switching model and the jump components is incomplete; hence the risk-neutral measure is not uniquely determined. In order to obtain the valuation of vulnerable European options, we need to choose a special martingale measure. Esscher transform, which was first introduced by Gerber and Shiu [19], is a method of selecting the equivalent martingale measure and has extensive applications in the fields of finance and insurance. In this subsection, we adopt the random Esscher transform of Elliott et al. [20] to identify an equivalent martingale measure.Here, we divide jump risks into two parts and price common jump risks caused by market factors in the model while neglecting the jump risks caused by the assets themselves.First, we define two processesY1t and Y2t as follows:(15)Y1t=ln⁡StS0=∫0tαs-k1λSs∗-12σSs2ds+∫0tσSsdWs1+∫0tZSs-1dNs1+∫0tZSs-1dNs,Y2t=ln⁡VtV0=∫0tβs-k2λVs∗-12σVs2ds+∫0tσVsdWs2+∫0tZVs-2dNs2+∫0tZVs-2dNs.For simplicity, let (16)Y1t=C1t+J1t+Jt,Y2t=C2t+J2t+Jt′,where Cit,Jiti=1,2 and JtJt′ are the continuous diffusive part, common jump part, and individual jump part, respectively, of Yit,i=1,2. They have the following forms: (17)C1t=∫0tαs-k1λSs∗-12σSs2ds+∫0tσSsdWs1,Jt=∫0tZSs-1dNs1,J1t=∫vtZSs-1dNs,C2t=∫0tβs-k2λVs∗-12σVs2ds+∫0tσVsdWs2,Jt′=∫0tZVs-2dNs2,J2t=∫0tZVs-2dNs.Let{FtU}t∈T and {Fti}t∈T denote the P-augmentations of natural filtrations generated by U and Yii=1,2, respectively. For each i=1,2, set Gti=FtU∨Fti. For any t∈0,T, let Ht=Gt1∨Gt2. Then, the random Esscher transform Qθ~P on HT with regime-switching parameters θ1t=θ1tc1,θ1t-J1t∈0,T and θ2t=θ2tc2,θ2t-J2t∈[0,T] is given as follows:(18)dQθdPHT=exp⁡∫0Tθ1tc1dC1t+∫0Tθ1t-J1dJ1t+∫0Tθ2tc2dC2t+∫0Tθ2t-J2dJ2tEexp⁡∫0Tθ1tc1dC1t+∫0Tθ1t-J1dJ1t+∫0Tθ2tc2dC2t+∫0Tθ2t-J2dJ2t∣FTU,where E· denote the expectation under measure P. For m∈c1,J1, θ1tm=θ1m,Ut, and θ1m=θ11m,θ12m,…,θ1Nm∈RN, and for m∈c2,J2, θ2tm=θ2m,Ut and θ2m=θ21m,θ22m,…,θ2Nm∈RN. Since (19)Eexp⁡∫0Tθ1tc1dC1t∫0Tθ1t-J1dJ1t+∫0Tθ2tc2dC2t+∫0Tθ2t-J2dJ2t∣FTU=exp⁡∫0Tθ1tc1αt-k1λSt∗-12σSt2dt+∫0Tθ2tc2βt-k2λVt∗-12σVt2dt+12∫0Tθ1tc12σSt2+θ2tc22σVt2+2ρθ1tc1σStθ2tc2σVtdt+∫0Tλteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22-1dt,then(20)dQθdPHT=exp⁡∫0Tθ1tc1σStdWt1+∫0Tθ2tc2σVtdWt2-12∫0Tθ1tc12σSt2+θ2tc22σVt2+2ρθ1tc1σStθ2tc2σVtdt×exp⁡∫0Tθ1t-J1ZSt-1+θ2t-J2ZVt-2dNt-∫0Tλteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22-1dt.By the fundamental theorem of asset pricing, the absence of arbitrage opportunities is “essentially” equivalent to the existence of an equivalent martingale measure under which the discounted stock price process is a martingale. Following Elliott et al. [20], due to the presence of the uncertainty generated by the Markov chain process, the martingale condition is obtained by considering an enlarged filtration H. The following theorem presents the result.Theorem 1.
The martingale conditions are satisfied if and only if(21)αt-rt-k1λt+θ1tc1σSt2+ρσStθ2tc2σVt+λ~tkStθ1t-J1=0,βt-rt-k2λt+θ2tc2σVt2+ρσStθ1tc1σVt+λ~tkVtθ2t-J2=0,where(22)λ~t=λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22,kStθ1t-J1=k1+1eθ1t-J1σ12-1,kVtθ2t-J2=k2+1eθ2t-J2σ22-1.Proof.
Let(23)MStC=∫ut1+θ1sc1σSsdWs1+∫utθ2sc2σVsdWs2-12∫ut1+θ1sc12σSs2+θ2sc22σVs2+2ρ1+θ1sc1σSsθ2sc2σVsds,MStJ1=∫ut1+θ1s-J1ZSs-1+θ2s-J2ZVs-2dNs-∫utλse1+θ1s-J1μ1+θ2s-J2μ2+1/21+θ1s-J12σ12+θ2s-J2σ22-1ds.Then, by Bayes’ rule, we have (24)EQθe-∫0trsdsSt∣Hu=Ee-∫0trsdsSt×dQθdP∣HuEdQθdP∣Hu=e-∫0ursdsSuEexp⁡∫utαs-rs-k1λSs∗+θ1sc1σSs2+ρσSsθ2sc2σVsds+MStc×exp⁡-∫utλseθ1s-J1μ1+θ2s-J2μ2+1/2θ1s-J1σ12+θ2s-J2σ22-1ds+∫utλse1+θ1s-J1μ1+θ2s-J2μ2+1/21+θ1s-J12σ12+θ2s-J2σ22-1ds+MStJ1×exp⁡∫utZSs-1dNs1=e-∫0ursdsSuexp⁡∫utαs-rs-k1λSs∗+θ1sc1σSs2+ρσSsθ2sc2σVsds×exp⁡-∫utλseθ1s-J1μ1+θ2s-J2μ2+1/2θ1s-J1σ12+θ2s-J2σ22-1ds+∫utλse1+θ1s-J1μ1+θ2s-J2μ2+1/21+θ1s-J12σ12+θ2s-J2σ22-1ds×exp⁡∫utλSseμ1+1/2σ12-1ds.The martingale condition (25)EQθe-∫0trsdsSt∣Hu=e-∫0ursdsSuis satisfied if and only if (26)αt-rt-k1λt+θ1tc1σSt2+ρσStθ2tc2σVt+λ~tkStθ1t-J1=0,where(27)λ~t=λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22,kStθ1t-J1=k1+1eθ1t-J1σ12-1.Similarly, since (28)EQθe-∫0trsdsVt∣Hu=e-∫0ursdsVu,it follows that (29)βt-rt-k2λt+θ2tc2σVt2+ρσStθ1tc1σVt+λ~tkVtθ2t-J2=0,where λ~t=λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22 and kVtθ2t-J2=k2+1eθ2t-J2σ22-1.Proposition 2.
Conditional onFTU and under the equivalent martingale measure Qθ, St and Vt follow stochastic differential equations:(30)dStSt-=rt-k1λSt-λ~tkStθ1t-J1dt+σStdW~t1+eZSt-1-1dNt1+eZ~St-1-1dN~t,dVtVt-=rt-k2λVt-λ~tkVtθ2t-J2dt+σVtdW~t2+eZVt-2-1dNt2+eZ~Vt-2-1dN~t,where θ1tc1, θ1t-J1, θ2tc2, and θ2t-J2 are given by(21). Moreover,(31)W~t1=Wt1-∫0tσSsθ1sc1ds-ρ∫0tσVsθ2sc2ds,W~t2=Wt2-∫0tσVsθ2sc2ds-ρ∫0tσSsθ1sc1dsare two standard Qθ Brownian motions. N~t is still a Poisson process under the measure Qθ with the intensity rate λ~t. λ~t is given as follows:(32)λ~t=λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22,and the mean percentage jump sizes of Z~St-1 and Z~Vt-2 are(33)kStθ1t-J1=k1+1eθ1t-J1σ12-1,kVtθ2t-J2=k2+1eθ2t-J2σ22-1.Proof.
Under the conditions ofFTU, we use Girsanov’s theorem and (20) to obtain W~t1,W~t2 which are two standard Qθ Brownian motions with the correlation coefficient ρ. Note that the parameters describing the common jumps in the two risky assets values have been changed. Now, we prove the formulae of (32) and (33). (34)EQθexp⁡iu∫0TZSt-1dNt∣FTU=Eexp⁡iu∫0TZSt-1dNt×dQθdP∣FTU=exp⁡∫0tλteiu+θ1t-J1μ1+θ2t-J2μ2+1/2iu+θ1t-J12σ12+θ2t-J22σ22-1λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J12σ12+θ2t-J22σ22-1dt-λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J12σ12+θ2t-J22σ22-1dt=exp⁡∫0Tλ~t∫-∞+∞eiuz-1f~USzdt.
Hence, under the conditionFTU and the probability measure Qθ, the intensity rate of Nt is λ~t=λteθ1t-J1μ1+θ2t-J2μ2+1/2θ1t-J1σ12+θ2t-J2σ22. The density function of ZSt-1 is (35)f~USz=12πσ1e-z-μ1+θ1t-J1σ122/2σ12,z∈-∞,∞.Similarly, we can also prove that the density function of ZVt-2 is (36)f~UVz=12πσ2e-z-μ2+θ2t-J2σ222/2σ22,z∈-∞,∞.
Note that the risk-neutral probability measureQθ is selected by the Esscher transform, and the probability law of the Markov chain U remains the same after the measure change; that is, under Qθ, U still has the same semimartingale dynamics. For each 0≤t≤T, we find that the solutions of (21) are not unique. Hence, a special case needs to be considered to obtain an equivalent martingale measure. We suppose (37)αt-rt-k1λt+θ1tc1σSt2+ρσStθ2tc2σVt=0,λ~teμ1+1/2σ12+θ1t-J1σ12-1=0,βt-rt-k2λt+θ2tc2σVt2+ρσStθ1tc1σVt=0,λ~teμ2+1/2σ22+θ2t-J2σ22-1=0.Then the following corollary gives a pair of solutions for the martingale condition.Corollary 3.
The martingale conditions (21) are satisfied if the parameters are given as follows:(38)θ1tc1∗=βt-rt-k2λtρσSt-αt-rt-k1λtσVt1-ρ2σSt2σVt,θ2tc2∗=αt-rt-k1λtρσVt-βt-rt-k2λtσSt1-ρ2σVt2σSt,θ1t-J1∗=-μ1+1/2σ12σ12,θ2t-J2∗=-μ2+1/2σ22σ22,k~Stθ1t-J1=k1+1eθ1t-J1σ12-1=0,k~Vtθ2t-J2=k2+1eθ2t-J2σ22-1=0,λ~t∗=λtexp⁡-μ122σ12+σ128×exp⁡-μ222σ22+σ228.The explicit Esscher parameters in Corollary 3 will be used to the Monte Carlo simulations in Section 5.


## 3.2. Vulnerable European Option Pricing

As in Klein [2], we assume that default will only occur if the counterparty’s asset value at maturity VT is less than some amount D∗. Additionally, this amount is not set to the value of the option but corresponds to the amount of the counterparty’s claim D at exercise time T. Once a credit loss occurs at exercise time T, only the proportion 1-αVT/D of the value of the option at T is paid out by the counterparty. Then, the payoff of a vulnerable European call option is given by (39)ST-K+1VT≥D∗+1-αVTST-K+D1VT<D∗,where K is the strike price of the option and α is the deadweight cost associated with bankruptcy, expressed as a percentage of the value of the counterparty’s assets. By the risk-neutral arguments, under the risk-neutral measure Qθ, the price of a vulnerable European call option at time 0 is given as(40)CS,V,U=EQθe-∫0TrsdsST-K+1VT≥D∗+1-αVTD1VT<D∗.rt is the risk-free rate. CS,V,U depends on U for the Markov regime-switching in our pricing model.Under the conditionsGTn1,n2,n3≔N~T=n,NT1=n1,NT2=n2 and FTU, for m1=n+n1,m2=n+n2, the value of a vulnerable European call option at time 0 is given as(41)Cm1,m2=EQθe-∫0TrsdsST,m1-K+1VT,m2≥D∗+1-αVT,m2D1VT,m2<D∗.Clearly, Cm1,m2 represents the value of the vulnerable option given FTU, conditional on m1 and m2 jumps on the underlying asset and the assets of the counterparty, respectively.Therefore, the conditional vulnerable European call option price givenFTU is(42)C∗=EQθe-∫0TrsdsST-K+1VT≥D∗+1-αVTD1VT<D∗∣FTU=∑n=0∞∑n1=0∞∑n2=0∞QN~T=n,NT1=n1,NT2=n2Cm1,m2.Let Ji denote the occupation time of U≔{Ut}t∈T in state i over the time duration 0,T, where t∈T. Then (43)R¯T=1T∫0Trtdt=1T∑i=1NriJi,U¯1T=1T∫0TσSt2dt=1T∑i=1NσSi2Ji,U¯2T=1T∫0TσVt2dt=1T∑i=1NσVi2Ji,U¯3T=1T∫0TσStσVtdt=1T∑i=1NσSiσViJi,λ¯ST=1T∫0TλStdt=1T∑i=1NλSiJi,λ¯VT=1T∫0TλVtdt=1T∑i=1NλViJi,λ~¯T=1T∫0Tλ~tdt=1T∑i=1Nλ~iJi.From Buffington and Elliott [21], the price of a vulnerable European call option is given by(44)CS,V,U=∫0,TNC∗×ψJ1,J2,…,JNdJ1dJ2⋯dJN,where ψJ1,J2,…,JN is the joint distribution of the occupation time J1,J2,…,JN under the condition U0=U and the probability measure Qθ. Write J≔J1,J2,…,JN for the vector of occupation times. Let G denote a diagonal matrix with the elements in the vector ζ=ζ1,ζ2,…,ζN as its diagonal. Then, for any ζ∈RN, the characteristic function of J under the condition U0=U and the martingale measure Qθ is given by (45)EQθeiζ,J∣U0=U=UeA+iGT,I,where i=-1 and I=1,1,…,1∈RN. Note that ψJ1,J2,…,JN can be completely determined by the characteristic function. As for the proof, refer to Buffington and Elliott [21].At this point, the key problem that needs to be solved is to determineCm1,m2. According to theItô formula, the following equalities hold: (46)ln⁡ST=ln⁡S0+∫0Trt-k1λSt-λ~tkStθ1t-J1-12σSt2dt+∫0TσStdW~t1+∑i=1NT1ZSi1+∑i=1N~TZ~Si1,ln⁡VT=ln⁡V0+∫0Trt-k2λVt-λ~tkVtθ2t-J2-12σVt2dt+∫0TσVtdW~t2+∑i=1NT2ZVi2+∑i=1N~TZ~Vi2,where cov⁡dW~t1,dW~t2=ρdt, ln⁡ST,m1/ln⁡S0,ln⁡VT,m2/ln⁡V0 is bivariate normally distributed under the conditions GTn,n1,n2≔N~T=n,NT1=n1,NT2=n2 and FTU with the following properties:(47)MSm1=Eln⁡ST,m1ln⁡S0=∫0Trt-k1λSt-λ~tkStθ1t-J1-12σSt2dt+m1μ1+σ12∑i=1nθ1ti-J1,MVm2=Eln⁡VT,m2ln⁡V0=∫0Trt-k2λVt-λ~tkVtθ2t-J2-12σVt2dt+m2μ2+σ22∑i=1nθ2ti-J2,cov⁡ln⁡ST,m1ln⁡S0,ln⁡VT,m2ln⁡V0=∫0TρσStσVtdt.Let (48)ln⁡ST,m1ln⁡S0=MSm1+∫0TσSt2dt+m1σ12ξ1,ln⁡VT,m2ln⁡V0=MVm2+∫0TσVt2dt+m2σ22ξ2,where ξ1 and ξ2 are standard normal random variables and the correlation coefficient is (49)ρ¯m1,m2=U¯3TTU¯1TT+m1σ12U¯2TT+m2σ22ρ.Now we need to calculate Cm1,m2, and the derivation process can be referred to Tian et al. [12].(50)Cm1,m2=e-R¯TTEQθST,m1-K+1VT,m2≥D∗+1-αDVT,m21VT,m2<D∗=e-R¯TTA1m1,m2+A2m1,m2+A3m1,m2+A4m1,m2,where(51)A1m1,m2=EQθST,m11ST,m1≥K,VT,m2≥D∗,A2m1,m2=-KEQθ1ST,m1≥K,VT,m2≥D∗,A3m1,m2=1-αDEQθST,m1VT,m21ST,m1≥K,VT,m2<D∗,A4m1,m2=-1-αDKEQθVT,m21ST,m1≥K,VT,m2<D∗.Then, we can get A1m1,m2, A2m1,m2, A3m1,m2, and A4m1,m2 in closed form: (52)A1m1,m2=S0eMSm1+1/2U¯1TT+m1σ12N2a1m1,a2m2,ρ¯m1,m2,A2m1,m2=-KN2b1m1,b2m2,ρ¯m1,m2,A3m1,m2=1-αDS0V0eMSm1+1/2U¯1TT+m1σ12+MVm2+1/2U¯2TT+m2σ22+ρU¯3TT×N2c1m1,m2,-ρ¯m1,m2,A4m1,m2=-1-αDKV0eMVm2+1/2U¯2TT+m2σ22N2d1m1,d2m2,-ρ¯m1,m2,where(53)N2x1,x2,ρ=12π1-ρ2∫-∞x1∫-∞x2e-1/21-ρ2p2-2ρpq+q2dpdq.The parameters are expressed as follows:(54)b1m1=ln⁡S0/K+MSm1U¯1TT+m1σ12,b2m2=ln⁡V0/D∗+MVm2U¯2TT+m2σ22,a1m1=b1m1+U¯1TT+m1σ12,a2m2=b2m2+ρ¯U¯1TT+m1σ12,c1m1=b1m1+U¯1TT+m1σ12+ρ¯U¯2TT+m2σ22,c2m2=-b2m2-U¯2TT+m2σ22+ρ¯U¯1TT+m1σ12,d1m1=b1m1+ρ¯U¯2TT+m2σ22,d2m2=-b2m2-U¯2TT+m2σ22.Therefore, the conditional vulnerable European call option price given FTU is(55)C∗=∑n=0∞∑n1=0∞∑n2=0∞QN~T=n,NT1=n1,NT2=n2Cm1,m2=∑n=0∞∑n1=0∞∑n2=0∞λ~¯TTnn!λ¯STTm1-nm1-n!λ¯VTTm2-nm2-n!e-λ~¯TT-λ¯STT-λ¯VTTCm1,m2.Note the valuation of vulnerable European call options depends on the parametersθ1tc1, θ2tc2, θ1t-J1, and θ2t-J2. For the numerical analysis, we use the special parameters that are given by Corollary 3:(56)MSm1=Eln⁡ST,m1ln⁡S0=∫0Trt-k1λSt-12σSt2dt+n1μ1-12nσ12,MVm2=Eln⁡VT,m2ln⁡V0=∫0Trt-k2λVt-12σVt2dt+n2μ2-12nσ22.Note. If we adopt the hypothesis of Merton [9] and regard the jump risks as unsystematic risks which should not be priced, thus, the parameters describing the jumps in the two risky asset price processes will not be changed after application of the measure transform from the physical measure P to risk-neutral measure. That is, the parameters θ1t-J1 and θ2t-J2 in (18) are equal to 0. From Theorem 1 and Proposition 2, we can obtain the results of the following two propositions specifically.Proposition 4.
The martingale conditions are satisfied if and only if(57)θ1tc1=βt-rtρσSt-αt-rtσVt1-ρ2σSt2σVt,θ2tc2=αt-rtρσVt-βt-rtσSt1-ρ2σVt2σSt.Proposition 5.
Conditional onFTU and under the equivalent martingale measure, the dynamics of St and Vt are given by(58)dStSt-=rt-k1λSt∗dt+σStdW~t1+eZSt-1-1dMt1,dVtVt-=rt-k2λVt∗dt+σVtdW~t2+eZVt-1-1dMt2,where W~t(1) and W~t(2) are two standard Brownian motions which is defined by (31).Under the conditionsGTn,n1,n2≔NT=n,NT1=n1,NT2=n2 and FTU, for m1=n+n1,m2=n+n2, ln⁡ST,m1/ln⁡S0,ln⁡VT,m2/ln⁡V0 is also bivariate normally distributed with the following properties:(59)MS′m1=Eln⁡ST,m1ln⁡S0=∫0Trt-k1λSt∗-12σSt2dt+m1μ1,MV′m2=Eln⁡VT,m2ln⁡V0=∫0Trt-k2λVt∗-12σVt2dt+m2μ2,cov⁡ln⁡ST,m1ln⁡S0,ln⁡VT,m2ln⁡V0=∫0TρσStσVtdt.Similar to the previous results, we can obtain(60)C∗=∑n=0∞∑n1=0∞∑n2=0∞QNT=n,NT1=n1,NT2=n2Cm1,m2′=∑n=0∞∑n1=0∞∑n2=0∞λ¯TTnn!λ¯STTm1-nm1-n!λ¯VTTm2-nm2-n!e-λ¯TT-λ¯STT-λ¯VTTCm1,m2′,where λ¯T=1/T∫0T‍λtdt=1/T∑i=1N‍λiJi. So(61)Cm1,m2′=e-R¯TTA1′m1,m2+A2′m1,m2+A3′m1,m2+A4′m1,m2.The forms of A1′m1,m2, A2′m1,m2, A3′m1,m2, and A4′m1,m2 are consistent with those given in (50), as long as MSm1 is replaced with MS′m1 and MVm2 is replaced with MV′m2.


## 4. Vulnerable European Option Pricing Considering the Market Prices of All Jump Risks

For comparison, we can follow Bo et al. [15] and assume that all jump risks are systematic risks which should be priced. Therefore, the parameters describing the jumps in the two risky asset value processes should be changed. As in (15), we still define two processes Y1t and Y2t. For simplicity, let (62)Y1t=C1t+J1t+Jt,Y2t=C2t+J2t+Jt′,and they admit the following forms: (63)C1t=∫0tαs-k1λSs∗-12σSs2ds+∫0tσSsdWs1,J1t=∫0tZSs-1dNs1,Jt=∫0tZSs-1dNs,C2t=∫0tβs-k2λVs∗-12σVs2ds+∫0tσVsdWs2,J2t=∫0tZVs-2dNs2,Jt′=∫0tZVs-2dNs.Because the Brownian motions and jump processes are independent, the regime-switching Esscher transforms for diffusion processes and jump processes are defined separately as follows.We adopt the Esscher transform for the diffusion parts to determine an equivalent martingale measure.(64)dQθcdPHT=exp⁡∫0Tθ1tc1dC1t+∫0Tθ2tc2dC2tEexp⁡∫0Tθ1tc1dC1t+∫0Tθ2tc2dC2t∣FTU,where E[·] is the expectation under measure P. θ1tc1=θ1c1,Ut, θ1c1=θ11c1,θ12c1,…,θ1Nc1∈RN; θ2tc2=θ2c2,Ut, θ2c2=θ21c2,θ22c2,…,θ2Nc2∈RN. It is easy to obtain(65)dQθcdPHT=exp⁡∫0Tθ1tc1σStdWt1+∫0Tθ2tc2σVtdWt2-12∫0Tθ1tc12σSt2+θ2tc22σVt2+2ρθ1tc1σStθ2tc2σVtdt.The Esscher transform for the jump processes to determine an equivalent martingale measure is given below:(66)dQθJdPHT=exp⁡∫0Tθ1tJ1dJ1t+∫0TθtJdJt+∫0Tθ2tJ2dJ2t+∫0TθtJ′dJt′Eexp⁡∫0Tθ1tJ1dJ1t+∫0TθtJdJt+∫0Tθ2tJ2dJ2t+∫0TθtJ′dJt′∣FTU,from which we can obtain(67)dQθJdPHT=exp⁡∫0Tθ1tJ1ZSt-1dNt1-∫0TλSteθ1tJ1μ1+1/2θ1tJ1σ12-1dt×exp⁡∫0Tθ2tJ2ZVt2dNt2-∫0TλVteθ2tJ2μ2+1/2θ2tJ2σ22-1dt×exp⁡∫0TθtJZSt1+θtJ′ZVt2dNt-∫0TλteθtJμ1+θtJ′μ2+1/2θtJσ12+θtJ′σ22-1dt.Based on the above analysis, we now define a new probability measureQθ′ by setting (68)dQθ′dPHT=dQθJdPHT×dQθcdPHT.Theorem 6.
The martingale conditions are satisfied if and only if(69)αt-rt-k1λSt∗+θ1tc1σSt2+ρσStθ2tc2σVt+λ~StkStθ1tJ1+λ~tktθtJ=0,βt-rt-k2λVt∗+θ2tc2σVt2+ρσStθ1tc1σVt+λ~VtkVtθ2tJ2+λ~tktθtJ′=0,where(70)λ~t=λteθtJμ1+θtJ′μ2+1/2θtJσ12+θtJ′σ22,λ~St=λSteθ1tJ1μ1+1/2θ1tJ1σ12,λ~Vt=λVteθ2tJ2μ2+1/2θ2tJ2σ22,ktθtJ=k1+1eθtJσ12-1,ktθtJ′=k2+1eθtJ′σ22-1,kStθt1J1=k1+1eθ1tJ1σ12-1,kVtθt2J2=k2+1eθ2tJ2σ22-1.The proof is the same as for Theorem1 in Section 3, and we omit it here.Proposition 7.
Conditional onFTU and under the equivalent martingale measure Qθ′, one has (71)ln⁡ST=ln⁡S0+∫0Trt-λ~StkStθ1tJ1-λ~tktθtJ-12σSt2dt+∫0TσStdW~t1+∑i=1N~T1Z~Si1+∑i=1N~TZ^Si1,ln⁡VT=ln⁡V0+∫0Trt-λ~VtkVtθ2tJ2-λ~tktθtJ′-12σVt2dt+∫0TσVtdW~t2+∑i=1N~T2Z~Vi2+∑i=1N~TZ^Vi2,where W~t1 and W~t2 are also two standard Brownian motions following (31).Then, all the parameters describing the jumps in the two risky asset values are changed according to (70). The probability law of the chain U remains the same after the measure change; that is, under Qθ′, U still has the semimartingale dynamics.Moreover, because the solutions of (69) are not unique for each 0≤t≤T, we obtain a special pair of solutions by making some assumptions. Let(72)αt-rt-k1λSt∗+θ1tc1σSt2+ρσStθ2tc2σVt=0,λ~Steμ1+1/2σ12+θ1tJ1σ12-1=0,λ~teμ1+1/2σ12+θtJσ12-1=0,βt-rt-k2λVt∗+θt2c2σVt2+ρσStθ1tc1σVt=0,λ~Vteμ2+1/2σ22+θ2tJ2σ22-1=0,λ~teμ2+1/2σ22+θtJ′σ22-1=0,then we can have the following result.Corollary 8.
Under the assumptions of (72), the martingale conditions can be satisfied if the parameters are given as follows:(73)θ1tc1∗=βt-rt-k2λVt∗ρσSt-αt-rt-k1λSt∗σVt1-ρ2σSt2σVt,θ2tc2∗=αt-rt-k1λSt∗ρσVt-βt-rt-k2λVt∗σSt1-ρ2σVt2σSt,θ1tJ1∗=θtJ∗=-μ1+1/2σ12σ12,θ2tJ2∗=θtJ′∗=-μ2+1/2σ22σ22,k~Stθ1tJ1=k~tθtJ=k1+1eθ1tJ1∗σ12-1=0,k~Vtθ2tJ2=k~tθtJ′=k2+1eθ2tJ2∗σ22-1=0,λ~t∗=λtexp⁡-μ122σ12+σ128×exp⁡-μ222σ22+σ228,λ~St∗=λStexp⁡-μ122σ12+σ128,λ~Vt∗=λVtexp⁡-μ222σ22+σ228.
The explicit Esscher parameters in Corollary8 are used into numerical analysis. Consequently, we have the dynamics of the asset value processes under the equivalent martingale measure Qθ′ with the Esscher parameters in Corollary 8. Under the conditions GTn,n1,n2≔N~T=n,N~T1=n1,N~T2=n2 and FTU, for m1=n+n1,m2=n+n2, ln⁡ST,m1/ln⁡S0,ln⁡VT,m2/ln⁡V0 is also bivariate normally distributed with the following properties:(74)MS′′m1=Eln⁡ST,m1ln⁡S0=∫0Trt-12σSt2dt-12m1σ12,MV′′m2=Eln⁡VT,m2ln⁡V0=∫0Trt-12σVt2dt-12m2σ22,cov⁡ln⁡ST,m1ln⁡S0,ln⁡VT,m2ln⁡V0=∫0TρσStσVtdt.Similar to the previous results, we can obtain(75)C∗=∑n=0∞∑n1=0∞∑n2=0∞QN~T=n,N~T1=n1,N~T2=n2Cm1,m2″=∑n=0∞∑n1=0∞∑n2=0∞λ~¯TTnn!λ~¯STTm1-nm1-n!λ~¯VTTm2-nm2-n!e-λ~¯TT-λ~¯STT-λ~¯VTTCm1,m2″,where(76)λ~¯ST=1T∫0Tλ~Stdt=1T∑i=1Nλ~StiJi,λ~¯VT=1T∫0Tλ~Vtdt=1T∑i=1Nλ~VtiJi.So(77)Cm1,m2′′=e-R¯TTA1″m1,m2+A2″m1,m2+A3″m1,m2+A4″m1,m2,and the forms of A1″m1,m2, A2″m1,m2, A3″m1,m2, and A4″m1,m2 are consistent with those given in Section 3, as long as MSm1 is replaced with MS″m1 and MVm2 is replaced with MV″m2.


## 5. Numerical Results

In this section, we adopt Monte Carlo simulation to preform numerical experiments for the vulnerable European call option prices under the regime-switching jump-diffusion models. We first present numerical results for the vulnerable European call option prices considering the market prices of common systematic jump risks. The effects of some basic variables on vulnerable option prices which are obtained from formula (55) with the Esscher parameters in Corollary 3 are illustrated in Figures 1–4. Suppose one year has 252 trading days. Then, the sample interval is 1/252. We parsimoniously assume that the Markov chain U has two states; that is, N=2. The first and second regimes, namely, e1=1,0′ and e2=0,1′, can be interpreted as good and bad economic states, respectively. We suppose that the transition probability matrix Q of the two-state Markov chain U is given by(78)p11p12p21p22=0.20.80.70.3.Figure 1
Vulnerable European call option price against spot-to-strike ratio.Figure 2
Vulnerable European call option price against the initial prices ofS and V.Figure 3
Vulnerable European call option price against mean jump sizes ofS and V.Figure 4
Vulnerable European call option price against standard deviation of the jump sizes ofS and V.We choose these parameters to be broadly consistent with those used in the literature such as Tian et al. [12] to calibrate the results. The preference parameters listed in Table 1 are independent of the Markov chain, but the parameters listed in Table 2 depend on the Markov chain. We suppose that the initial state of the economy be e1. In Figures 1–6, we change one of the parameter values to investigate its impact on the vulnerable option price, with the other variables taking on the values listed in Tables 1 and 2.Table 1
Parameters that are independent of the Markov chain.
Parameter name
Value
Mean jump size ofS
μ
1
=
0
Volatility of jump size ofS
σ
1
=
0.1
Initial value ofS
S
0
=
10
Correlation coefficientρ
ρ
=
0.5
Default boundary
D
∗
=
10
Time to maturity
T
=
1
Mean jump size ofV
μ
2
=
0
Volatility of jump size ofV
σ
2
=
0.1
Initial value ofV
V
0
=
10
Strike price
K
=
10
Outstanding claims
D
=
10
Deadweight cost of bankruptcy
α
=
0.5Table 2
Parameters that depend on the Markov chain.
Parameter name
Value in statee1
Value in statee2
Interest rate
r
1
=
0.02
r
2
=
0.04
Volatility ofS
σ
S
1
=
0.3
σ
S
2
=
0.4
Volatility ofV
σ
V
1
=
0.3
σ
V
2
=
0.4
Common jump intensity
λ
1
=
1
λ
2
=
2
Individual jump intensity ofS
λ
S
1
=
1
λ
S
2
=
2
Individual jump intensity ofV
λ
V
1
=
1
λ
V
1
=
2Figure 5
Vulnerable European call option price against spot-to-strike ratio for three cases with parametersμ1=0,μ2=0,σ1=0.1, and σ2=0.1.Figure 6
Vulnerable European call option price against spot-to-strike ratio for three cases with parametersμ1=0.01,μ2=0.09,σ1=0.1,σ2=0.3.Figure1 shows the vulnerable European call option price plotted against the spot-to-strike ratio for three different cases. The “+”, “∘”, and “∗” lines correspond to the cases of mixed states with transition probability matrix Q, pure state 1, and pure state 2, respectively. It can be seen that the option values when considering the mixed states of the economy are higher than the values in pure state 1 (good state) and lower than the values in pure state 2 (bad state). Mixed states of the economy can better consider the influence of a change in the economic system on the option value. If the model considers only state 1 and ignores the regime-switching and jump risk, the numerical results are the same as the result in Figure 3 in Tian et al. [12].Three-dimensional numerical analysis of the impact of the initial prices ofSS0 and VV0 on the vulnerable European call option price is performed. We can see that the option price increases with the increase in S0 and V0, but the impact of S0 on the option value is stronger than that of V0.Figure3 shows the effects of the mean jump sizes μ1 of St and μ2 of Vt on the option prices under mixed states. If μ2=0 remains unchanged, the vulnerable option price traces out a U-shaped curve as μ1 change from −1 to 1. If μ1=0 remains unchanged, the option value presents an inverse U-shaped curve when μ2 changes from −1 to 1. By comparing the curvature degrees, we can illustrate that the impact of μ1 on the option value is stronger than that of μ2. Compared to Figure 12 in Tian et al. [12], the option value curves are of similar shape, but the values are distinct.Figure4 shows the effects of the standard deviations of the jump sizes σ1 of S and σ2 of V on the option prices under mixed states. If σ2=0.1 remains unchanged, the option value presents an increasing trend as σ1 changes from 0 to 0.8, while if σ1=0.1 remains unchanged, the option value declines as σ2 changes from 0 to 0.8. We can see from the rates of change that the influence of σ1 is more prominent. The conclusion is similar to that of Figure 13 in Tian et al. [12]. Figures 3 and 4 indicate that the impacts of the mean jump size and the standard deviation of the jump size of the underlying asset on the option price are more significant than that of the assets of the counterparty.Below, we consider the option prices against the parameters for three cases of incorporating the market prices of jump risks in Figures5-6. The “∘”, “+”, and “∗” lines correspond to the cases of considering the market prices of common jump risks obtained from formula (55) with the Esscher parameters in Corollary 3, neglecting the market prices of all jump risks obtained from formula (60) and considering the market prices of all jump risks obtained from formula (75) with the Esscher parameters in Corollary 8, respectively. Figure 5 shows that the difference in vulnerable European call option prices is small against the spot-to-strike ratio for three cases with parameters μ1=0, μ2=0, σ1=0.1, σ2=0.1 and other parameters taking on the values listed in Tables 1 and 2. The numerical results are inconsistent with what I had expected that incorporating the market prices of jump risks would increase the option prices. One of the reasons may be the choice of special regime-switching parameters for the equivalent martingale measure. This problem needs to be researched. To investigate whether the choice of model parameters has an impact on the results, we change the parameters to μ1=0.01, μ2=0.09, σ1=0.1, σ2=0.3. From Figure 6, we can see clearly that, with gradual incorporation of the market prices of jump risks, the option prices increase against the spot-to-strike ratio. Moreover, with the increase in S0/K, the differences in the option prices between the three cases increase. Thus, we can see that the introduction of the market prices of jump risks affects the option value and that the option price when considering the market prices of common jump risks is between the option prices in the other two cases. Therefore, studying this problem has practical significance. Note that the effects of the other parameters of the models can also be discussed.


## 6. Conclusion

This paper proposes the pricing problem of vulnerable European options, in which the dynamics of the underlying asset of the option and the asset value of the counterparty are driven by two correlated Markov regime-switching jump-diffusion processes. Comparing with most of the existing regime-switching jump-diffusion models, the main advantage of our model is that we incorporate the market prices of common jump risks as systematic risks into the models because the jump processes are divided into individual jumps for each asset price and common jumps that affect the prices of all assets. We employ the Esscher transform to determine an equivalent martingale measure in the incomplete market and obtain the vulnerable options pricing formulae. To illustrate the influence of the market prices of jump risks on the price of vulnerable options, vulnerable options pricing formulae are also given in the case of taking and without taking the market prices of all jump risks into consideration. The numerical results indicate that jump risks have a more significant impact on the option prices. As potential future works, we might consider incorporating regime-switching risk into options pricing and discuss the effect of the choice of regime-switching Esscher parameters on the option prices.



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