# Pricing Vulnerable European Options under Lévy Process with Stochastic Volatility

Authors: Chaoqun Ma; Shengjie Yue; Yishuai Ren
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

This paper considers the pricing issue of vulnerable European option when the dynamics of the underlying asset value and counterparty’s asset value follow two correlated exponential Lévy processes with stochastic volatility, and the stochastic volatility is divided into the long-term and short-term volatility. A mean-reverting process is introduced to describe the common long-term volatility risk in underlying asset price and counterparty’s asset value. The short-term fluctuation of stochastic volatility is governed by a mean-reverting process. Based on the proposed model, the joint moment generating function of underlying log-asset price and counterparty’s log-asset value is explicitly derived. We derive a closed-form solution for the vulnerable European option price by using the Fourier inversion formula for distribution functions. Finally, numerical simulations are provided to illustrate the effects of stochastic volatility, jump risk, and counterparty credit risk on the vulnerable option price.


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## Body

## 1. Introduction

In general, options traded in the over-the-counter (OTC) market are associated with counterparty credit risk. Since the global financial crisis of 2007-2008, regulators and investors have widely recognized the effect of counterparty credit risk on OTC options (see, e.g., Crépey [1], Sayah [2], and Breton and Marzouk [3]). Unlike the options traded in regulated exchanges, there is no organized exchange to ensure that option writers will fulfill the contractual obligations in the OTC markets. Once the option writer defaults, the option holder will get only a fraction of necessary payments required by the contract. The holders of OTC options are likely to suffer a large loss if they ignore the counterparty credit risk. Thus, the participants in the OTC markets need to take into account counterparty credit risk when pricing OTC options. Those options with counterparty credit risk are referred to as vulnerable options.The valuation of vulnerable options has gained substantial attention in the finance literature. For modeling credit risk, the structural approach has been used to study vulnerable options by a lot of scholars, in which default events happen when the firm’s asset value is lower than a specific boundary. Based on this approach, the vulnerable European option pricing model is first presented by Johnson and Stulz [4]. Klein [5] extends the work of Johnson and Stulz [4] by assuming option writer to have other liabilities, in which the prices of underlying asset and the asset of counterparty are correlated. Subsequently, many extensions of the results in Klein [5] have been considered. For example, Hui et al. [6] and Hui et al. [7] discuss the pricing of vulnerable European option with stochastic default barrier. In contrast to constant interest rate in Klein [5], Klein and Inglis [8] incorporate stochastic interest rate into the vulnerable option pricing model. Liao and Huang [9] consider the possible counterparty default prior to the maturity of option; they derive explicit formulas for vulnerable options with stochastic interest rates. Taking jump into account, Tian et al. [10] investigate the vulnerable European option pricing where the dynamics of asset prices are governed by two correlated jump-diffusion processes. More broadly, Niu and Wang [11] discuss the pricing of vulnerable European option under regime switching setting.However, the above mentioned literature generally assume that the volatilities of the underlying asset and counterparty’s asset are constant, which is contrary to the actual situation, because it cannot explain many well-known empirical evidences such as the volatility clustering and the volatility smile. It is obvious that models with stochastic volatility will be more realistic. Some scholars have suggested alternative models for the price of vulnerable option. Lee et al. [12] derive the explicit expression for the vulnerable option price where the price processes of underlying asset and counterparty’s asset satisfy the Heston stochastic volatility model. Wang et al. [13] extend the model of Lee et al. [12] and assume that the price processes of underlying asset and counterparty’s asset follow two correlated two-factor stochastic volatility diffusion processes; they obtain a pricing formula of vulnerable option in a special case. However, Lee et al. [12] and Wang et al. [13] do not consider the effects of jump risk of underlying asset and counterparty’s asset on vulnerable option price. A lot of empirical evidences show that the arrival of new information and sudden disastrous events may lead to discontinuous changes in asset prices and the jump components of models are understood as the effects of market shocks on the asset prices. Therefore, many researchers have proposed pricing models that incorporate both stochastic volatility and jumps components. For example, Bates [14] studies pricing problem of option where the dynamics of underlying asset follows a jump-diffusion process with stochastic volatility, in which the jump components of asset price follow the compound Poisson process. However, Carr and Wu [15] and Huang and Wu [16] find the fact that the compound Poisson process cannot describe the many small jumps of financial assets. Moreover, many researches have considered more general jump structures. A more general process is Lévy process, which allows the jump components to have infinite activity and admits nearly an arbitrary distribution. Li et al. [17] apply Markov chain Monte Carlo (MCMC) methods to study infinite-activity Lévy processes with stochastic volatility; their empirical results show that infinite-activity jumps are essential for modeling the S&P 500 index returns. Subsequently, Zaevski et al. [18] derive the general formula for a European call option under stochastic volatility and tempered stable Lévy jumps. Liang and Li [19] study pricing of the European option under Normal Tempered Stable process with stochastic volatility. Li et al. [20] discuss the equity premium and option pricing under the general equilibrium under Lévy process and stochastic volatility.Along with the above research lines, based on the structural approach for default, we study the valuation of vulnerable European option when the price processes of the underlying asset and counterparty’s asset follow two correlated exponential Lévy processes with stochastic volatility. The volatilities of underlying asset and counterparty’s asset are both decomposed into long-term and short-term volatility. The short-term variance of asset price follows a mean-reverting process. A mean-reverting process is introduced to allow for the common long-term volatility risk in the underlying asset price and counterparty’s asset value, which differs from assumption that long-term volatility is constant in Wang et al. [13]. This assumption is consistent with the models in Christoffersen et al. [21] and Wong and Zhao [22]. The market is fluctuating in reality, and the market factor affects the long-term volatilities of underlying asset price and counterparty’s asset value. Thus, the long-term volatility of asset price is also changing over time. Based on the proposed model, the joint moment generating function of underlying log-asset price and counter’s log-asset value is derived. Moreover, we obtain an explicit formula for vulnerable European call option price by applying the Fourier inversion transform. Finally, sensitivity analysis and numerical simulation are provided for our results. By the numerical analysis, we discover that (i) the long-run mean of common long-term variance of underlying asset and counterparty’s asset exerts positive effect on the vulnerable option price; (ii) vulnerable option price is an increasing function with respect to the long-run mean of short-term variance of underlying asset price; (iii) a higher long-run mean of short-term variance of counterparty’s asset value induces a lower vulnerable option price; (iv) when the jump components of underlying asset price and counterparty’ asset value follow the CGMY model, the aggregate activity rate of jumps of underlying asset price has positive effect on the value of vulnerable option price. The higher aggregate activity rate of jumps of counterparty’s asset value, the lower vulnerable option price.Our contributions to the literature are summarized below: First, we extend the model of Wang et al. [13] by considering a more general financial market. In particular, compared to the constant long-term volatilities of underlying asset price and counterparty’s asset value in Wang et al. [13], we assume that the common long-term volatility risk in the underlying asset price and counterparty’s asset value is governed by a mean-reverting process. Besides, we simultaneously incorporate infinite-activity Lévy jumps and stochastic volatility into the dynamics of underlying asset and counterparty’s asset, which differs from the jump-free model in Wang [13]. Second, we derive an explicit expression for vulnerable option price by the Fourier inversion formula for distribution functions, which is different from the methods in Lee et al. [12] and Wang et al. [13].The rest of this paper is organized as follows. Section2 introduces the basic model setup and formulates the pricing problem of vulnerable European call option problem. In Section 3, the joint characteristic function of the underlying log-asset price and counterparty’s log-asset value is derived. By applying Fourier inversion transform, we derive an analytic solution for vulnerable European option price. Section 4 demonstrates our results with numerical examples, and Section 5 concludes this paper.


## 2. Basic Setting of the Model

In this section, we describe theoretical framework for valuing of vulnerable European call option. The underlying asset price and counterparty’s asset value follow exponential Lévy processes with stochastic volatility. The volatility risk is divided into short-term risk and long-term risk. We incorporate a mean-reverting process to describe the common long-term volatility risk in underlying asset price and counterparty’s asset value. The short-term fluctuation of stochastic volatility also follows a mean-reverting process.Time is indexed byt∈[0,T], where T is a positive finite constant representing the time horizon. To model uncertainties, suppose that we have a given complete probability space with filtration (Ω,F,{Ft}0≤t≤T,Q), and the filtration {Ft} satisfies the usual condition (increase, right-continuous and augmented). It represents the flow of information available to the investor. Q is a risk-neutral probability. Suppose W1S, W2S, WjZ(j=1,2,3), W1V, and W3V are all standard Brownian motions which are Ft-adapted. Moreover, cov(dW1S(t),dW1Z(t))=ρ1sdt, cov(dW2S(t),dW2Z(t))=ρ2sdt, cov(dW1V(t),dW1Z(t))=ρ1vdt, cov(dW3V(t),dW3Z(t))=ρ3vdt, cov(dW1S(t),dW1V(t))=ρsvdt, and any other Brownian motions are pairwise independent. Under a risk neutral measure Q, the underlying asset price S(t) is modeled as the exponential of the lévy process with stochastic volatility:(1)dStSt-=rdt+η1Z1tdW1St+Z2tdW2St+∫Rey-1N~ydt,dy,(2)dZ1t=κ1θ1-Z1tdt+σ1Z1tdW1Zt,(3)dZ2t=κ2θ2-Z2tdt+σ2Z2tdW2Zt,where S(t-) stands for the value of S(t) before a possible jump occurs, y∈R=R∖{0}; r is the interest rate, and the parameters κj, θj, and σj(j=1,2) are the mean-reverting rate, the long-term mean; and volatility of the variance process Zj(t), respectively. We require 2κjθj≥σj2 to ensure that the process Zj(t) remains strictly positive. η12Z1(t) represents the long-term variance of underlying asset price, which reflect the effect of corporate performance and market states on the underlying asset price. Here we assume that Z1(t) follow a mean-reverting process, which represents the common systematic risk caused by market factors. Z2(t) is short-term variance of underlying asset value, which depends on investors’ trading and reflects company’s idiosyncratic risk. In addition, N~y(dt,dy)=Ny(dt,dy)-νy(dy)dt is a compensated jump measure, where Ny(dt,dy) is the jump measure and Lévy kernel (or density) νy(dy) satisfies ∫Rmin⁡(1,y2)νy(dy)<∞; the underlying asset price in (1) contains two orthogonal martingales: two purely continuous martingale and a purely discontinuous (jump) martingale.Next, letV(t) denote the market value of counterparty’s asset, and the dynamic of V(t) is given by the SDE under risk neutral measure Q,(4)dVtVt-=rdt+η2Z1tdW1Vt+Z3tdW3Vt+∫Reξ-1N~ξdt,dξ,(5)dZ3t=κ3θ3-Z3tdt+σ3Z3tdW3Zt,where V(t-) stands for the value of V(t) before a possible jump occurs, ξ∈R=R∖{0}; η22Z1(t) represents the long-term variance of counterparty’s asset value, and Z1(t) describes the systematic risk caused by market shocks. Z3(t) is short-term variance of counterparty’s asset price, which depends on the investor’s trading activities. The parameters κ3, θ3, and σ3 are the mean-reverting rate, the long-term mean, and volatility of the variance process Z3(t), respectively. We require 2κ3θ3≥σ32 to ensure that the process Z3(t) remains strictly positive. In addition, N~ξ(dt,dξ)=Nξ(dt,dξ)-νξ(dξ)dt is a compensated jump measure, where Nξ(dt,dξ) is the jump measure and Lévy kernel (or density) νξ(dξ) satisfies ∫Rmin(1,ξ2)νξ(dξ)<∞.In the proposed framework, we consider stochastic volatility and jumps in the dynamics of both the underlying asset and the counterparty’s asset, and the long-term stochastic volatility and the short-term stochastic volatility are not constants. The underlying asset price and the counterparty’s asset value are correlated, and the correlation coefficient isρSV. Furthermore, correlation between stochastic volatility and asset prices is also considered in the proposed model. The short-term stochastic volatilities of the underlying asset and the counterparty reflect company’s idiosyncratic risk while the long-term stochastic volatilities of the underlying asset and the counterparty describe systematic risk caused by common market shocks. In addition, we also focus on the discontinuous changes in the underlying asset and counterparty’s asset; the jump components in (1) and (4) reflect the abnormal vibrations in the asset price due to the arrival of important new information and sudden disastrous events.


## 3. Vulnerable European Call Option Pricing

In this section, we derive the joint moment generating function of the underlying log-asset price and counterparty’ log-asset value. In this framework, by using the Fourier inversion transform, we obtain the explicit expression for vulnerable European option price.LetT denote the option maturity; for vulnerable European call options, the boundary condition is determined by a payoff function depending upon the financial distress circumstance at the time t=T. As in Klein (1996) and its subsequent results, then the payoff of a vulnerable European call option is given by(6)CT=ST-K+1VT≥D∗+1-γVTST-K+D1VT<D∗,where K is the strike price of the option. γ is the deadweight cost related with the bankruptcy or reorganization process of the firm expressed as a percentage of the counterparty’ asset value. D is the total amount of claims of the counterparty, and D∗ is an exogenous default barrier. D∗ may be less than D due to the possibility of a counterparty continuing its operation. A credit loss happens if the counterparty’s asset value V(T) is less than the amount D∗. If V(T) is higher than or equal to the default barrier D∗, the claim is paid out in full. Otherwise, default event occurs and only the proportion (1-γ)V(T)/D of the nominal claim at T is paid out by the option writer.By the standard risk-neutral arguments, under a risk-neutral measureQ, the price of a vulnerable European call option at time 0 is given as(7)CS,V,t,T=EQe-rT-tST-K+1VT≥D∗+1-γVTST-K+D1VT<D∗∣Ft,where EQ denotes risk neutral expectations. e-r(T-t) denotes the stochastic discount factor. Expression (7) shows that the value of a vulnerable European call option contains two parts which are conditional on whether V(T) is higher or less than the default barrier D∗ under risk-neutral measure Q. If V(T) is higher than D∗, the option writer can afford to pay all the claim. If V(T) is lower than D∗, only a proportion of the nominal claim is paid.In the following, we derive the explicit expression for the value of vulnerable option price by using Fourier inversion formula for the probability distribution functions. In the process of calculation, we need to use the joint moment generating function ofln⁡S(T) and ln⁡V(T). Thus, it is necessary for us to give this definition of the joint moment generating function. Given the dynamics of the underlying asset price and the counterparty’s asset value under the probability measure Q, it is possible to obtain the joint moment generating function of ln⁡S(T) and ln⁡V(T). Let ln⁡S(t)=x1, ln⁡V(t)=x2, and the joint moment generating function of ln⁡S(T) and ln⁡V(T) under probability measure Q is defined as(8)fx1,x2,τ;φ1,φ2=EQeφ1ln⁡ST+φ2ln⁡VT∣Ft,where T≥t, τ=T-t. Then, the following lemma holds.Lemma 1.
Suppose that underlying asset price and counterpart’s asset value follow the dynamics in (1) and (4), respectively. Let ln⁡S(t)=x1, ln⁡V(t)=x2, Z1(t)=z1, Z2(t)=z2, Z3(t)=z3, and the joint moment generating function of ln⁡S(T) and ln⁡V(T) in (8) is given by(9)fx1,x2,τ;φ1,φ2=eφ1x1+φ2x2+Aτz1+Bτz2+Dτz3+Eτ,where(10)Aτ=-a11-e-Δ1τ2Δ1-Δ1+b11-e-Δ1τ,(11)Bτ=-a21-e-Δ2τ2Δ2-Δ2+b21-e-Δ2τ,(12)Dτ=-a31-e-Δ3τ2Δ3-Δ3+b31-e-Δ3τ,(13)Eτ=rφ1+φ2+∫Reφ1y-1-φ1ey-1νydy+∫Reφ2ξ-1-φ2eξ-1νξdξτ-κ1θ1σ12Δ1+b1τ+2ln⁡1-Δ1+b12Δ11-e-Δ1τ-κ2θ2σ22Δ2+b2τ+2ln⁡1-Δ2+b22Δ11-e-Δ2τ-κ3θ3σ32Δ3+b3τ+2ln⁡1-Δ3+b32Δ31-e-Δ3τ,where(14)Δ1=b12+σ12a1,a1=η12φ1+η22φ2-η12φ12-η22φ22-2η1η2ρsvφ1φ2,b1=η1ρ1sσ1φ1+η2ρ1vσ1φ2-κ1,Δ2=b22+σ22a2,a2=φ1-φ12,b2=φ1σ2ρ2s-κ2,Δ3=b32+σ32a3,a3=φ2-φ22,b3=φ2σ3ρ3v-κ3.Proof.
See AppendixA.Now, we turn to derive the pricing formula for vulnerable option pricing. We need to rewrite pricing formula (7) of vulnerable European call option as follows:(15)CS,V,t,T=EQe-rT-tST-K+1VT≥D∗+1-γVTST-K+D1VT<D∗∣Ft=e-rT-tC1t,T-KC2t,T+e-rT-t1-γDC3t,T-KC4t,T,where C1(t,T)-C4(t,T) are given by(16)C1t,T=EQST1lnST≥lnK,lnVT≥lnD∗∣Ft,(17)C2t,T=EQ1lnST≥lnK,lnVT≥lnD∗∣Ft,(18)C3t,T=EQSTVT1ln⁡ST≥ln⁡K,ln⁡VT<ln⁡D∗∣Ft,(19)C4t,T=EQVT1ln⁡ST≥ln⁡K,ln⁡VT<ln⁡D∗∣Ft.In the following, from the standard probability theory, we can obtain the probability distribution function by using the joint moment generating function. In Theorem2, we can derive the explicit expression of vulnerable European option by applying Fourier inversion transform.Theorem 2.
The price of vulnerable option pricing at timet is formally given by(20)Ct,T=e-rT-tΠ1t,T+14fx1,x2,τ,1,0-KΠ2t,T+e-rT-t1-γDΠ3t,T+14fx1,x2,τ,1,1-KΠ4t,T-14Kfx1,x2,τ,0,1,where f(x1,x2,τ;φ1,φ2) is the moment generating function in (8), and Π1(t,T), Π2(t,T), Π3(t,T), and Π4(t,T) are given in (B.11), (B.12), (B.17), and (B.22), respectively.Proof.
See AppendixB.


## 4. Numerical Analysis

In this section, we present numerical results of the vulnerable European call option price under Lévy process with stochastic volatility. Numerical examples are presented to investigate the effects of the parameters on the vulnerable European call option. The pricing models of vulnerable option in Klein [5] and Wang et al. [13] and pricing models of nonvulnerable option in Sun [23] and Li et. al [20] are chosen as reference models. Our choice of the four reference models is made to study the effects of stochastic volatility, jump risks, and credit risk on vulnerable option prices. In the following numerical analysis, we change one of the parameter values to illustrate its effect on the price with other parameters taking on the values in Tables 1–3. Our model allows a general distribution for jump components of underlying asset price and counterparty’s asset value and thus can be easily deduced to the simple cases such that the jump components follow compound Poisson process in Merton [24] and Kou [25], the CGMY model of Carr et al. [26], etc. By assuming different types of Lévy kernel, our model has different forms. In the following, we will first study a simple case with jump components following compound Poisson processes and then a more general case with the jump components following a CGMY model.Table 1
parameter Values in the Base Case.
Parameter Value Parameter Value Time to maturity T=1 Mean-reverting rate of volatility k 3 = 2 Time t=0 Long-term mean of volatility θ3=0.05 Initial stock price S(0)=10 Volatility of volatility σ 3 = 0.4 Initial volatility value Z1(0)=0.05 Volatility of the asset η1=1 Mean-reverting rate of volatility k 1 = 1 Volatility of the asset η2=0.5 Long-term mean of volatility θ1=0.05 Outstanding claims D = 30 Volatility of volatility σ 1 = 0.3 Default barrier D ∗ = 30 Initial volatility value Z 2 ( 0 ) = 0.06 Deadweight cost γ = 0.4 Mean-reverting rate of volatility k 2 = 2 Strike price K=10 Long-term mean of volatility θ 2 = 0.06 Initial asset value V ( 0 ) = 30 Volatility of volatility σ 2 = 0.5 Correlation coefficient ρ 1 s = - 0.5, ρ2s=-0.5 Initial volatility value Z 3 ( 0 ) = 0.05 Correlation coefficient ρsv=0.5 Correlation coefficient ρ s v = 0.5 Correlation coefficient ρ1v=-0.5 Interest rate r=0.03 Correlation coefficient ρ3v=-0.5Table 2
parameter values.
λ 1 μ 1 δ 1 λ 2 μ 2 δ2  1 0 0.1 1 0 0.1Table 3
parameter values.
λ 1 p 1 β 11 β 12 λ 2 p 2 β 21 β22  1 0.5 5 5 1 0.4 10 10In the numerical analysis, we use the functionquadgk available in MATLAB to deal with the single integrals for the Fourier inversion transform. The double integrals for the Fourier inversion transform are computed by using the functionintegral2 available in MATLAB.


### 4.1. The Merton Model

When jump components of underlying asset and counterparty’s asset follow compound Poisson processes, the jump magnitude of asset price is assumed to be drawn from a log-normal distribution. Furthermore, the Lévy kernels are given by(21)νydy=λ112πδ12exp⁡-y-μ122δ12dy,(22)νξdξ=λ212πδ22exp⁡-ξ-μ222δ22dξ,where λj(j=1,2) is the jump intensity of the asset process, μj is the mean of jump size, and δj2 is the variance of jump size. Following the Merton [24] and Sun [23], the two integrals in (13) are given by(23)∫Reφ1y-1νydy=λ1eφ1μ1+1/2φ12δ12-1,(24)∫Reφ2ξ-1νξdξ=λ2eφ2μ2+1/2φ22δ22-1.In Table2, we will give the specimen values for the model parameters of assets’ jump components. Figures 1–5 illustrate the effects of basic parameters on vulnerable option prices where the parameters include the default barrier, long-run mean of stochastic volatility, and jump intensity. Denote prices of nonvulnerable European option in Sun [23] by CNE, and let COU be values of vulnerable option in proposed model. We denote the values of vulnerable option in Wang et al. [13] and Klein [5] prices by CSV and CKL, respectively.Figure 1
The effect ofD∗ on the vulnerable option prices.Figure 2
The effect of long-run mean of stochastic volatilityθ1 on the option prices.Figure 3
The effects of long-run means of stochastic volatilityθ2 and θ3 on the vulnerable option prices.Figure 4
The effect of the jump intensityλ1 on the option prices.Figure 5
The effect of the jump intensityλ2 on the vulnerable option prices.Figure1 illustrates that the vulnerable option price is a decreasing function of default barrier D∗. When the default barrier is relatively small, the value of vulnerable European option decreases slowly with D∗; while the default barrier is close to the initial value of counterparty’s asset, the vulnerable option price decreases more dramatically with D∗. Large default barrier D∗ will increase the default probability; the counterparty credit risk has a more significant impact on the vulnerable option price if the default barrier increases. In addition, the value in proposed model is higher than those of models proposed by Klein [5] and Wang et al. [13]. The phenomenon is consistent with the results presented in existing literature, because the jump parts of underlying asset increase the chance that the underlying asset price will become very low or very high at maturity; the option holder will gain benefit from the increase of underlying asset price and suffers limited downside risk when underlying asset price decreases. Therefore jump risk enhances option value. Moreover, the value of nonvulnerable option price is higher than those of vulnerable European options. Once the credit events happen, the option holder will only get a fraction of promised payoff; thus, the vulnerable option is less valuable.Figure2 displays that the value of option price with stochastic volatility increases with the long-run mean of the process Z1(t). The reason for this is that a higher value θ1 yields a higher value of Z1(t). Large Z1(t) will increase probability that the underlying asset price and counterparty’s asset value will become very low or very high at maturity; when the underlying asset price and counterparty’s asset value increase, the option holder will gain benefit; when underlying asset price and counterparty’s asset value decrease, the option holder faces a limited loss. Therefore the option becomes valuable if θ1 increases.Figure3 plots the effect of the long-term mean of stochastic volatility on vulnerable option price. The solid and dotted lines correspond to the vulnerable option prices obtained by the proposed model against long-run means of short-term variances θ2 and θ3, respectively. For parameter θ2, we can observe that θ2 exerts positive effect on the vulnerable option price. The reason is the same as parameter θ1. Large long-run mean of variance θ2 will increase the probability that underlying asset price turns into very high or very low at maturity. Consequently, the vulnerable option price is valuable under large θ2. For parameter θ3, the large long-run mean of variance θ3 will lead to counterparty’s asset value being very high or very low at maturity. Consequently, the default probability will increase. Thus, the vulnerable option price decreases with θ3.As shown in Figure4, the value of option with jump risk increases with the jump intensity λ1. The reason is the same as that of θ2. The jump intensity will result in underlying asset price doing very poorly and very well at maturity, but the option holder has a limited loss. Thus, the option with jump risk is valuable under large jump intensity λ1. Figure 5 shows that the jump intensity λ2 has a negative effect on the value of vulnerable option with jump risk. Counterparty credit risk is related to the jump intensity of the counterparty’s asset. A stronger jump intensity λ2 of counterparty’ asset value corresponds to a larger default probability. Accordingly, the vulnerable option price decreases with jump intensity λ2. In addition, the vulnerable option price is a decreasing function of γ, the deadweight cost coefficient γ affects the recovery rate when default event happens, and a higher value of γ gives rise to a smaller recovery rate. Thus, the option holder suffer a greater loss under large γ.


### 4.2. The Double Exponential Jump Model

When jump components of underlying asset and counterparty’s asset follow compound Poisson processes, we assume that the jump magnitude of asset price has asymmetric double exponential distribution. The Lévy kernels are given by(25)νydy=λ1p1β11e-β11y1y≥0+q1β12eβ12y1y<0dy,β11>1,β12>0,p1+q1=1,(26)νξdξ=λ2p2β21e-β21ξ1ξ≥0+q2β22eβ22ξ1ξ<0dξ,β11>1,β12>0,p2+q2=1,where λj(j=1,2) is the jump intensity of the asset process and pj and qj are probability of up-move jump and probability of down-move jump, respectively. βj1 and βj2 are mean of positive jumps and mean of negative jumps, respectively. Following Kou [25], the two integrals in (13) are given by(27)∫Reφ1y-1νydy=λ1p1β11β11-φ1+q1β12β12+φ1-1,∫Reφ2ξ-1νξdξ=λ2p2β21β21-φ2+q2β22β22+φ2-1.In Table 3, we will give the specimen values for the model parameters of assets’ jump components. Figures 6 and 7 illustrate the effects of jump intensity λ1 and jump intensity λ2 on vulnerable option price.Figure 6
The effect of the jump intensityλ1 on the vulnerable option prices.Figure 7
The effect of the jump intensityλ2 on the vulnerable option prices.Figures6 and 7 illustrate the sensitivities of the vulnerable option price with respect to the jump intensity λ1 and the jump intensity λ2. From Figure 6, we observe that high jump intensity λ1 leads to a high vulnerable option. Large jump intensity λ1 will increase the chance that the underlying asset price will become very low or very high; the option holder will benefit from the price increase. Figure 7 shows that the vulnerable option price is a decreasing function of λ2. Large jump intensity λ2 will increase default probability. Therefore the vulnerable option price becomes less valuable if λ2 increases.


### 4.3. The CGMY Model

A more general case can be considered if jump components of underlying asset and counterparty’s asset follow CGMY model developed by Carr et al. [26], and the Lévy kernels are given by(28)νyy=C1e-G1∣y∣y1+Y1,y<0,C1e-M1∣y∣y1+Y1,y>0,(29)νξξ=C2e-G2∣ξ∣ξ1+Y2,ξ<0,C2e-M2∣ξ∣ξ1+Y2,ξ>0,where Cj>0, Gj>0, Mj>0, and Yj<2(j=1,2). Cj determines the aggregate activity rate of the CGMY model while the parameters Gj and Mj determine the height of the upward and downward jumps, respectively. Yj controls both tails of the spot return distribution. Based on this, following Carr et al. [26], the two integrals in (13) are given by(30)∫Reφ1y-1νydy=C1Γ-Y1M1-φ1Y1-M1Y1+G1+φ1Y1-G1Y1,∫Reφ2ξ-1νξdξ=C2Γ-Y2M2-φ2Y2-M2Y2+G2+φ2Y2-G2Y2.In Table4, we will give the specimen values for the model parameters of jump parts of underlying asset price and counterparty’s asset value. Figures 8–11 illustrate the effects of basic parameters on vulnerable option prices. We denote prices of nonvulnerable European option in Li et al. [20] by CNV.Table 4
parameter values of CGMY model.
C 1 G 1 M 1 Y 1 C 2 G 2 M 2 Y2 1.5 12 25 0.25 1 13 22 0.2Figure 8
The effect ofC1 on the option prices.Figure 9
The effect ofY1 on the option prices.Figure 10
The effect ofC2 on the vulnerable option prices.Figure 11
The effect ofY2 on the vulnerable option prices.From Figure8, we can see that the value of option with jumps increases with parameter C1. The parameter C1 determines the aggregate activity rate of jumps of the underlying asset price; the large jump frequency will lead to the underlying asset price doing either very well or very poorly at maturity. While the option holder has limited downside risk, the vulnerable option becomes valuable if C1 increases. As shown in Figure 9, the price of option with jumps is a monotonic increasing function of Y1. The parameter Y1 controls both tails of the underlying return distribution. As Y1 increases, both tails of underlying return distribution will become fatter, and the underlying asset price becomes very high at maturity; it is not surprising that the option with jumps would become more valuable if Y1 increases.Figure10 denotes vulnerable option price is a decreasing function of parameter C2. The parameter C2 controls the aggregate activity rate of jumps of the counterparty’s asset value. Large C2 will increase the probability that counterparty’s asset value becomes very low at maturity, which implies that the default probability will increase. Thus, the vulnerable option price will decrease with C2. Figure 11 shows that effect of parameter Y2 on the vulnerable option price; the parameter Y2 controls both tails of the spot return distribution of counterparty’s asset. As Y2 increases, default probability will increase. Accordingly, the vulnerable option price is likely to decrease with Y2.


## 4.1. The Merton Model

When jump components of underlying asset and counterparty’s asset follow compound Poisson processes, the jump magnitude of asset price is assumed to be drawn from a log-normal distribution. Furthermore, the Lévy kernels are given by(21)νydy=λ112πδ12exp⁡-y-μ122δ12dy,(22)νξdξ=λ212πδ22exp⁡-ξ-μ222δ22dξ,where λj(j=1,2) is the jump intensity of the asset process, μj is the mean of jump size, and δj2 is the variance of jump size. Following the Merton [24] and Sun [23], the two integrals in (13) are given by(23)∫Reφ1y-1νydy=λ1eφ1μ1+1/2φ12δ12-1,(24)∫Reφ2ξ-1νξdξ=λ2eφ2μ2+1/2φ22δ22-1.In Table2, we will give the specimen values for the model parameters of assets’ jump components. Figures 1–5 illustrate the effects of basic parameters on vulnerable option prices where the parameters include the default barrier, long-run mean of stochastic volatility, and jump intensity. Denote prices of nonvulnerable European option in Sun [23] by CNE, and let COU be values of vulnerable option in proposed model. We denote the values of vulnerable option in Wang et al. [13] and Klein [5] prices by CSV and CKL, respectively.Figure 1
The effect ofD∗ on the vulnerable option prices.Figure 2
The effect of long-run mean of stochastic volatilityθ1 on the option prices.Figure 3
The effects of long-run means of stochastic volatilityθ2 and θ3 on the vulnerable option prices.Figure 4
The effect of the jump intensityλ1 on the option prices.Figure 5
The effect of the jump intensityλ2 on the vulnerable option prices.Figure1 illustrates that the vulnerable option price is a decreasing function of default barrier D∗. When the default barrier is relatively small, the value of vulnerable European option decreases slowly with D∗; while the default barrier is close to the initial value of counterparty’s asset, the vulnerable option price decreases more dramatically with D∗. Large default barrier D∗ will increase the default probability; the counterparty credit risk has a more significant impact on the vulnerable option price if the default barrier increases. In addition, the value in proposed model is higher than those of models proposed by Klein [5] and Wang et al. [13]. The phenomenon is consistent with the results presented in existing literature, because the jump parts of underlying asset increase the chance that the underlying asset price will become very low or very high at maturity; the option holder will gain benefit from the increase of underlying asset price and suffers limited downside risk when underlying asset price decreases. Therefore jump risk enhances option value. Moreover, the value of nonvulnerable option price is higher than those of vulnerable European options. Once the credit events happen, the option holder will only get a fraction of promised payoff; thus, the vulnerable option is less valuable.Figure2 displays that the value of option price with stochastic volatility increases with the long-run mean of the process Z1(t). The reason for this is that a higher value θ1 yields a higher value of Z1(t). Large Z1(t) will increase probability that the underlying asset price and counterparty’s asset value will become very low or very high at maturity; when the underlying asset price and counterparty’s asset value increase, the option holder will gain benefit; when underlying asset price and counterparty’s asset value decrease, the option holder faces a limited loss. Therefore the option becomes valuable if θ1 increases.Figure3 plots the effect of the long-term mean of stochastic volatility on vulnerable option price. The solid and dotted lines correspond to the vulnerable option prices obtained by the proposed model against long-run means of short-term variances θ2 and θ3, respectively. For parameter θ2, we can observe that θ2 exerts positive effect on the vulnerable option price. The reason is the same as parameter θ1. Large long-run mean of variance θ2 will increase the probability that underlying asset price turns into very high or very low at maturity. Consequently, the vulnerable option price is valuable under large θ2. For parameter θ3, the large long-run mean of variance θ3 will lead to counterparty’s asset value being very high or very low at maturity. Consequently, the default probability will increase. Thus, the vulnerable option price decreases with θ3.As shown in Figure4, the value of option with jump risk increases with the jump intensity λ1. The reason is the same as that of θ2. The jump intensity will result in underlying asset price doing very poorly and very well at maturity, but the option holder has a limited loss. Thus, the option with jump risk is valuable under large jump intensity λ1. Figure 5 shows that the jump intensity λ2 has a negative effect on the value of vulnerable option with jump risk. Counterparty credit risk is related to the jump intensity of the counterparty’s asset. A stronger jump intensity λ2 of counterparty’ asset value corresponds to a larger default probability. Accordingly, the vulnerable option price decreases with jump intensity λ2. In addition, the vulnerable option price is a decreasing function of γ, the deadweight cost coefficient γ affects the recovery rate when default event happens, and a higher value of γ gives rise to a smaller recovery rate. Thus, the option holder suffer a greater loss under large γ.


## 4.2. The Double Exponential Jump Model

When jump components of underlying asset and counterparty’s asset follow compound Poisson processes, we assume that the jump magnitude of asset price has asymmetric double exponential distribution. The Lévy kernels are given by(25)νydy=λ1p1β11e-β11y1y≥0+q1β12eβ12y1y<0dy,β11>1,β12>0,p1+q1=1,(26)νξdξ=λ2p2β21e-β21ξ1ξ≥0+q2β22eβ22ξ1ξ<0dξ,β11>1,β12>0,p2+q2=1,where λj(j=1,2) is the jump intensity of the asset process and pj and qj are probability of up-move jump and probability of down-move jump, respectively. βj1 and βj2 are mean of positive jumps and mean of negative jumps, respectively. Following Kou [25], the two integrals in (13) are given by(27)∫Reφ1y-1νydy=λ1p1β11β11-φ1+q1β12β12+φ1-1,∫Reφ2ξ-1νξdξ=λ2p2β21β21-φ2+q2β22β22+φ2-1.In Table 3, we will give the specimen values for the model parameters of assets’ jump components. Figures 6 and 7 illustrate the effects of jump intensity λ1 and jump intensity λ2 on vulnerable option price.Figure 6
The effect of the jump intensityλ1 on the vulnerable option prices.Figure 7
The effect of the jump intensityλ2 on the vulnerable option prices.Figures6 and 7 illustrate the sensitivities of the vulnerable option price with respect to the jump intensity λ1 and the jump intensity λ2. From Figure 6, we observe that high jump intensity λ1 leads to a high vulnerable option. Large jump intensity λ1 will increase the chance that the underlying asset price will become very low or very high; the option holder will benefit from the price increase. Figure 7 shows that the vulnerable option price is a decreasing function of λ2. Large jump intensity λ2 will increase default probability. Therefore the vulnerable option price becomes less valuable if λ2 increases.


## 4.3. The CGMY Model

A more general case can be considered if jump components of underlying asset and counterparty’s asset follow CGMY model developed by Carr et al. [26], and the Lévy kernels are given by(28)νyy=C1e-G1∣y∣y1+Y1,y<0,C1e-M1∣y∣y1+Y1,y>0,(29)νξξ=C2e-G2∣ξ∣ξ1+Y2,ξ<0,C2e-M2∣ξ∣ξ1+Y2,ξ>0,where Cj>0, Gj>0, Mj>0, and Yj<2(j=1,2). Cj determines the aggregate activity rate of the CGMY model while the parameters Gj and Mj determine the height of the upward and downward jumps, respectively. Yj controls both tails of the spot return distribution. Based on this, following Carr et al. [26], the two integrals in (13) are given by(30)∫Reφ1y-1νydy=C1Γ-Y1M1-φ1Y1-M1Y1+G1+φ1Y1-G1Y1,∫Reφ2ξ-1νξdξ=C2Γ-Y2M2-φ2Y2-M2Y2+G2+φ2Y2-G2Y2.In Table4, we will give the specimen values for the model parameters of jump parts of underlying asset price and counterparty’s asset value. Figures 8–11 illustrate the effects of basic parameters on vulnerable option prices. We denote prices of nonvulnerable European option in Li et al. [20] by CNV.Table 4
parameter values of CGMY model.
C 1 G 1 M 1 Y 1 C 2 G 2 M 2 Y2 1.5 12 25 0.25 1 13 22 0.2Figure 8
The effect ofC1 on the option prices.Figure 9
The effect ofY1 on the option prices.Figure 10
The effect ofC2 on the vulnerable option prices.Figure 11
The effect ofY2 on the vulnerable option prices.From Figure8, we can see that the value of option with jumps increases with parameter C1. The parameter C1 determines the aggregate activity rate of jumps of the underlying asset price; the large jump frequency will lead to the underlying asset price doing either very well or very poorly at maturity. While the option holder has limited downside risk, the vulnerable option becomes valuable if C1 increases. As shown in Figure 9, the price of option with jumps is a monotonic increasing function of Y1. The parameter Y1 controls both tails of the underlying return distribution. As Y1 increases, both tails of underlying return distribution will become fatter, and the underlying asset price becomes very high at maturity; it is not surprising that the option with jumps would become more valuable if Y1 increases.Figure10 denotes vulnerable option price is a decreasing function of parameter C2. The parameter C2 controls the aggregate activity rate of jumps of the counterparty’s asset value. Large C2 will increase the probability that counterparty’s asset value becomes very low at maturity, which implies that the default probability will increase. Thus, the vulnerable option price will decrease with C2. Figure 11 shows that effect of parameter Y2 on the vulnerable option price; the parameter Y2 controls both tails of the spot return distribution of counterparty’s asset. As Y2 increases, default probability will increase. Accordingly, the vulnerable option price is likely to decrease with Y2.


## 5. Conclusion

This paper investigates the pricing problem of vulnerable European call options where the price processes of underlying asset and counterparty’s asset are governed by two correlated exponential Lévy processes with stochastic volatility. We divide volatility risk of underlying asset and counterparty’s asset into long-term and short-term risk. A mean-reverting process is introduced to allow for the common long-term volatility risk in the underlying asset price and counterparty’s asset value. The short-term fluctuation of stochastic volatility also follows a mean-reverting process. Based on the proposed model, we explicitly obtain the joint moment generating function of underlying log-asset price and counter’s log-asset value. By using the Fourier inversion transform, an analytic solution of vulnerable European call option is derived. Finally, sensitivity analysis and numerical simulation are provided to illustrate the effects of model parameters on the option price. The main findings are as follows: (i) the long-run means of common long-term variance of underlying asset and counterparty’s asset have positive effect on the vulnerable option price. A large long-run mean of common long-term variance will increase the possibility that underlying asset price and counterparty’s asset value will become very well or very poor at maturity. The option holder will benefit form increase of underlying asset price and faces a limited loss. Therefore the vulnerable option becomes valuable. (ii) The value of vulnerable option price increases with long-run mean of short-term variance of underlying asset price and decreases with long-run mean of short-term variance of counterparty’ asset value; (iii) the value in proposed model is higher than those of Wang et al. [13] and Klein [5]. The jump components of underlying asset price will increase the probability that the underlying asset price will turn to very low or very high at maturity, and the vulnerable option with jump risk becomes valuable; (iv) when the jump components of underlying asset price and counterparty’ asset value follow the CGMY model, the value of option price increases with the aggregate activity rate of jumps of underlying asset price and decreases with the aggregate activity rate of jumps of counterparty’s asset value.However, two further extensions of vulnerable European option pricing would be interesting. One possible extension is to incorporate stochastic interest into pricing problem of vulnerable option. Another extension would be to extend the study on vulnerable option pricing under Markov regime-switching setting.



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