# The European Vulnerable Option Pricing with Jumps Based on a Mixed Model

Authors: Chao Wang; Jianmin He; Shouwei Li
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

In this paper, we combine the reduced-form model with the structural model to discuss the European vulnerable option pricing. We define that the default occurs when the default process jumps or the corporate goes bankrupt. Assuming that the underlying asset follows the jump-diffusion process and the default follows the Vasicek model, we can have the expression of European vulnerable option. Then we use the measure transformation and martingale method to derive the explicit solution of it.


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## Body

## 1. Instruction

Vulnerable option is a kind of option with credit risk. The pricing of credit risk has been concerned by scholars for a long time. Merton (1974) [1] firstly introduced option pricing into zero coupon bond with credit risk. He assumed that the capital structure of corporate consisted of two parts which are assets and liabilities; the default occurred when the corporate was insolvent at maturity. This is the basic of structural model. Black and Cox (1976) [2] researched the corporate bond pricing with subordinated debt capital structure. They improved the definition of default and took the interest and dividend into consideration to obtain the corporate bond pricing. Johnson and Stulz (1987) [3] discussed the option pricing with credit risk based on the structural model and put forward the conception of vulnerable option firstly. Longstaff and Schwartz (1995) [4] assumed that the default boundary was a constant and the recovery rate was an exogenous ratio. Then they derived the zero coupon bond pricing. Hull and White (1995) [5] assumed that the underlying asset and the counterparty asset were independent of each other and derived the vulnerable option pricing. Jarrow and Turnbull (2000) [6] presented the reduced-form model. They considered that the occurrence of default was a kind of jumps and they used the Poisson process with constant intensity to describe the density of default. Rich (1996) [7] presented that the default could occur before maturity and the equity was exercised immediately if default occurred. Klein (1996) [8] supposed that the underlying assets and credit risk were correlated and deduced the option pricing with martingale method. Madan and Unal (1998) [9] put forward a model with stochastic default intensity. According to Lando (1998) [10], the default-free bond pricing method was used for default bond pricing by adjusting the short-term interest rate. Assuming that the default occurs at any time with specific probability, Duffie and Singleton (1999) [11] deduced the default bond pricing at initial time. Klein and Inglis (2001) [12] took the stochastic default boundary which depends on options and counterparty debts into consideration to discuss option pricing. Ammann (2002) [13] deduced the explicit solution of vulnerable option by using structural method on the assumptions of stochastic interest and boundary. The unexpected risks were introduced into default by Zhou (2001) [14]; he assumed that the corporate assets were composed of continuous process and jump process and presented the corporate assets value model based on the jump-diffusion process firstly. Hui et al. (2003) [15] discussed the pricing model with dynamic default boundary and derived the explicit solution of the vulnerable option by using the method of partial differential equation. Lakner and Liang (2008) [16] studied the credit default bonds pricing based on the structure model and the reduced-form model by using martingale method. Wang and Wang (2010) [17] assumed that the underlying asset followed the jump-diffusion process and derived the expression of the European vulnerable option pricing under the Markov regime switching model. Tchuindjo (2011) [18] studied the pricing of bond and bond option under the condition of stochastic default intensity and obtained the explicit solution. Su and Wang (2012) [19] assumed that the default intensity followed the stochastic model with jumps; then the vulnerable option pricing was given based on the reduced-form model by the martingale method. Wang et al. (2015) [20] deduced the explicit solution of the European vulnerable option which was derived by the fractional Brownian motion with jumps. Yoon and Kim (2015) [21] used double Mellin transforms to study European vulnerable options under constant as well as stochastic interest rates and obtained an analytic closed-form pricing formula in each interest rate case. Fard (2015) [22] obtained a closed-form price for the vulnerable option by using the Esscher transform under a completely random generalized jump-diffusion model. Wang (2016) [23] presented a pricing model which allows for the correlation between the intensity of default and the variance of the underlying asset and derived a closed-form solution for the vulnerable option. Lee et al. (2016) [24] studied the pricing of European-type vulnerable options when the underlying asset follows the Heston dynamics and obtained a closed-form analytic formula of the option price as a stochastic volatility extension of the classical Heston formula. Jeon et al. (2017) [25] studied the pricing of vulnerable path-dependent options using double Mellin transforms and obtained an explicit form pricing formula or semianalytic formula in each path-dependent option.


## 2. The European Vulnerable Option Pricing Model

Suppose that the uncertainty in the economy is described by the probability spaceΩ,F,Q,Ft0≤t≤T where Q is a risk neutral martingale measure in which the discounted assets price is a martingale. The underlying assets and corporate assets are given by(1)dSt=r-θλ-Stdt+σ1StdW1t+StdJt,(2)dVt=rVtdt+σ2VtdW2t,where σ1, σ2, and r are constants and W1t and W2t are Brownian motions in the probability space. Jt=∑i=1NtYi is a composite Poisson process and Yi (set Y0=0) is the jump range of it. Yi is a sequence of independent identically distributed random variables with the finite expected value and EYi=θ. Yi, W1t, W2t, W3t, and Nt are independent of each other where Nt is a Poisson process with parameter λ-. Suppose that Yi+1 follows log-normal distribution, then we have ln⁡Yi+1~NμJ,σJ2.Suppose that default intensityλt is Ft-measurable in the space and it follows the Vasicek model in the risk neutral measure(3)dλt=αβ-λtdt+σ3dW3t,where α, β, and σ3 are all constants. The covariance matrix of W1t, W2t, and W3t is(4)1ρ12ρ13ρ121ρ23ρ13ρ231t.We will combine the reduced-form model with the structural model to discuss the European vulnerable option pricing. We define that the default occurs when the default process jumps or the corporate goes bankrupt. Suppose that the underlying asset follows the jump-diffusion process and the corporate asset follows the Brownian motion. Then we can have the European vulnerable option pricing at initial time in the risk neutral measure. We assume that the maturity isT, the strike price is K, the default time is τ, the proportion of bankruptcy costs in writer’s assets is w, and the default boundary is a constant D; then the European vulnerable option pricing is(5)C0,T=Ee-∫0TruduST-K+1-wDVT1-Iτ>T,VT>D+Iτ>T,VT>D∣F0=Ee-∫0Trudu1-wDVTST-K+∣F0+Ee-∫0TruduST-K+Iτ>T,VT>D∣F0-Ee-∫0Trudu1-wDVTST-K+Iτ>T,VT>D∣F0.Since F0 is a filter space and T≥0, we have F0⊆FT. Suppose that there is no default at initial time. According to the law of iterated expectations and Fubini theorem we have (6)EIτ>T∣F0=EEIτ>T∣F0∣FT=EEIτ>T∣FT∣F0.Since the path of λt:0≤t≤T is known at time T, then(7)EIτ>T∣FT=Pτ>T=PNT-N0=0=exp⁡-∫0Tλudu.So we have(8)C0,T=Ee-∫0Trudu1-wDVTST-K+∣F0+Ee-∫0Tru+λuduST-K+IVT>D∣F0-Ee-∫0Tru+λudu1-wDVTST-K+IVT>D∣F0.


## 3. The Explicit Solution of European Vulnerable Option Pricing

Suppose that the assets price follows the jump-diffusion process, then the measure transformation of the continuous diffusion process can be derived by Girsanov’s theorem and the measure transformation of the jump-diffusion process is given in Theorem1.Theorem 1.
In the probability spaceΩ,F,Q, if one has Radon-Nikodym derivative Zt(9)Zt∣Ft=e∑i=0Ntln⁡Yi+1-θλt,where Nt,t≥0 is a Poisson process with intensity λ and Yi is a sequence of independent identically distributed random variables that the mean is θ and ln⁡Yi+1~NμJ,σJ2, then Nt can be transformed into measure Q~ by Zt where the intensity λ~=1+θλ.Proof.
According to the definition of moment generating function, we have(10)E~eu∑i=0Ntln⁡Yi+1∣Ft=Eeu∑i=0Ntln⁡Yi+1Zt∣Ft=Eeu+1∑i=0Ntln⁡Yi+1-θλt∣Ft=exp⁡λtφJu+1-1-θλt=exp⁡λtEeu+1lnYi+1-λt-θλt=exp⁡λtEeu+1ln⁡Yi+1-1+θ=exp⁡λt1+θEeu+1ln⁡Yi+11+θ-1.Since (11)Eeln⁡Yi+1=1+θ,we have(12)E~eu∑i=0Ntln⁡Yi+1∣Ft=exp⁡λt1+θEeuln⁡Yi+1-1.Let g=ln⁡Yi+1; then(13)E~eu∑i=0Ntln⁡Yi+1∣Ft=exp⁡λ~t∫-∞∞eugfgdg-1.So in condition Ft, the intensity of Nt is λ~=1+θλ.Theorem 2.
The European vulnerable option pricing at initial time in the risk neutral measure is(14)C0,T=∑n=0∞λ-~Tnn!e-λ-~T1-wDS0V0erT+ρ12σ1σ2TNd1n+e-rTA0,TX0,TNd3n,d4n,ρ-e-rT1-wDA0,TZ0,TNd7n,d8n,ρ-∑n=0∞λ-Tnn!e-λ-T1-wDKV0Nd2n+e-rTKA0,TNd5n,d6n,ρ-e-rT1-wDKA0,TY0,TNd9n,d10n,ρ,where(15)λ-~=1+θλ-,ρ=ρ12σ1σ2Tσ12T+σ22T,Mu,T,α=1α1-e-αT-u,A0,T=exp⁡-βT-λ0-βM0,T,α+12σ32∫0TM2u,T,αdu,X0,T=S0exp⁡rT-ρ13σ1σ3∫0TMu,T,αdu,Y0,T=V0exp⁡rT-ρ23σ2σ3∫0TMu,T,αdu,Z0,T=S0V0exp⁡2rT-ρ13σ1+ρ23σ2σ3∫0TMu,T,αdu+ρ12σ1σ2,d1n=ln⁡S0/K+rT+1/2σ12T+ρ12σ1σ2T-θλ-~T+nμJσ12T+nσJ2,d2n=ln⁡S0/K+rT-1/2σ12T+ρ12σ1σ2T-θλ-T+nμJσ12T+nσJ2,d3n=ln⁡S0/K+rT+1/2σ12T-ρ13σ1σ3∫0TMu,T,αdu-θλ-~T+nμJσ12T+nσJ2,d4n=ln⁡V0/D+rT-1/2σ22T+ρ12σ1σ2T-ρ23σ2σ3∫0TMu,T,αduσ22T,d5n=ln⁡S0/K+rT-1/2σ12T-ρ13σ1σ3∫0TMu,T,αdu-θλ-T+nμJσ12T+nσJ2,d6n=ln⁡V0/D+rT-1/2σ22T-ρ23σ2σ3∫0TMu,T,αduσ22T,d7n=ln⁡S0/K+rT+1/2σ12T-ρ13σ1σ3∫0TMu,T,αdu+ρ12σ1σ2T-θλ-~T+nμJσ12T+nσJ2,d8n=ln⁡V0/D+rT+1/2σ22T+ρ12σ1σ2T-ρ23σ2σ3∫0TMu,T,αduσ22T=d4n+σ22T,d9n=ln⁡S0/K+rT-1/2σ12T-ρ13σ1σ3∫0TMu,T,αdu+ρ12σ1σ2T-θλ-T+nμJσ12T+nσJ2,d10n=ln⁡V0/D+rT+1/2σ22T-ρ23σ2σ3∫0TMu,T,αduσ22T=d6n+σ22T.Proof.
For convenience, we defineC0,T=I+II-III, where(16)I=Ee-rT1-wDVTST-K+∣F0,II=Ee-rT-∫0TλuduST-K+IVT>D∣F0,III=Ee-rT-∫0Tλudu1-wDVTST-K+IVT>D∣F0.Then(17)I=1-wDS0V0erT+ρ12σ1σ2T∑n=0∞λ-~Tnn!e-λ-~TNd1n-1-wDKV0∑n=0∞λ-Tnn!e-λ-TNd2n,II=e-rTA0,TX0,T∑n=0∞λ-~Tnn!e-λ-~TNd3n,d4n,ρ-e-rTKA0,T∑n=0∞λ-Tnn!e-λ-TNd5n,d6n,ρ,III=-e-rT1-wDA0,TZ0,T∑n=0∞λ-~Tnn!e-λ-~TNd7n,d8n,ρ-e-rT1-wDKA0,TY0,T∑n=0∞λ-Tnn!e-λ-TNd9n,d10n,ρ.
The derivation ofI, II, and III is separately shown in Proof I, Proof II, and Proof III.
Proof I. We define I=Ee-rT(1-w/D)VTST-K+∣F0=I1-I2, where(18)I1=Ee-rT1-wDSTVTIST>K∣F0,I2=Ee-rT1-wDKVTIST>K∣F0.
We deduceI2 firstly. We introduce a new measure (19)dQVdQ∣F0=VTEVT∣F0=exp⁡∫0Tσ2dW2u-12∫0Tσ22du.According to (2)(20)VT=V0exp⁡rT-12σ22T+∫0Tσ2dW2u;then(21)I2=1-wDKV0Eexp⁡∫0Tσ2dW2u-12σ22TIST>K∣F0=1-wDKV0EVIST>K∣F0.Using (19) and Girsanov’s theorem, we have(22)W1VT=W1T-ρ12σ2T,W2VT=W2T-σ2T.According to (1)(23)ST=S0exp⁡rT-12σ12T+∫0Tσ1dW1u-θλ-T+∑i=0NTln⁡Yi+1.So in the measure QV, we have(24)ST=S0exp⁡rT-12σ12T+∫0Tσ1dW1Vu+ρ12σ1σ2T-θλ-T+∑i=0NTln⁡Yi+1.Since (25)∫0Tσ1dW1Vu~N0,σ12Tand when NT=n, (26)∑i=0Ntln⁡Yi+1~NnμJ,nσJ2,then we have(27)EVIST>K∣F0=∑n=0∞λ-Tnn!e-λ-TNd2n,where(28)d2n=ln⁡S0/K+rT-1/2σ12T+ρ12σ1σ2T-θλ-T+nμJσ12T+nσJ2.
Then we will deduceI1. We introduce a new measure(29)dQSVdQ∣F0=STVTESTVT∣F0=exp⁡∫0Tσ1dW1u+∫0Tσ2dW2u-12σ12+σ22+2ρ12σ1σ2T-θλ-T+∑i=0NTln⁡Yi+1.
According to Girsanov’s theorem and Theorem1(30)W1SVT=W1T-σ1T-ρ12σ2T.So in measure QSV(31)ST=S0exp⁡rT+12σ12T+∫0Tσ1dW1SVu+ρ12σ1σ2T-θλ-~T+∑i=0NTln⁡Yi+1.Since (32)ESTVT∣F0=S0V0e2rT+ρ12σ1σ2T,we have(33)I1=1-wDS0V0erT+ρ12σ1σ2TESVIST>K∣F0.When NT=n(34)ESVIST>K∣F0=∑n=0∞λ-~Tnn!e-λ-~TNd1n,where(35)d1n=ln⁡S0/K+rT+1/2σ12T+ρ12σ1σ2T-θλ-~T+nμJσ12T+nσJ2.
Proof II. We define II=Ee-rT-∫0TλuduST-K+IVT>D∣F0=I3-I4, where(36)I3=Ee-rT-∫0TλuduSTIST>K,VT>D∣F0,I4=Ee-rT-∫0TλuduKIST>K,VT>D∣F0.
We deduceI4 firstly. We define an equivalent martingale measure Qλ according to Radon-Nikodym derivative as follows:(37)dQλdQ∣F0=e-∫0TλuduEe-∫0Tλudu∣F0.
According to (3)(38)-∫0Tλudu=-βT-λ0-βM0,T,α-σ3∫0TMu,T,αdW3u.
LetA0,T=Ee-∫0Tλudu∣F0; usingItô lemma we have(39)A0,T=exp⁡-βT-λ0-βM0,T,α+12σ32∫0TM2u,T,αdu.
So(40)dQλdQ∣F0=e-∫0TλuduEe-∫0Tλudu∣F0=exp⁡-σ3∫0TMu,T,αdW3u-12σ32∫0TM2u,T,αdu.Using Girsanov’s theorem, (41)W1λT=W1T+ρ13σ3∫0TMu,T,αdu,W2λT=W2T+ρ23σ3∫0TMu,T,αdu,W3λT=W3T+σ3∫0TMu,T,αdu.Then the solutions of ST and VT in measure Qλ are (42)ST=S0exp⁡rT-12σ12T+∫0Tσ1dW1u-θλ-T+∑i=0NTln⁡Yi+1=S0exp⁡rT-12σ12T+∫0Tσ1dW1λu-ρ13σ1σ3∫0TMu,T,αdu-θλ-T+∑i=0NTln⁡Yi+1,(43)VT=V0exp⁡rT-12σ22T+∫0Tσ2dW2u=V0exp⁡rT-12σ22T+∫0Tσ2dW2λu-ρ23σ2σ3∫0TMu,T,αdu.Since (44)I4=Ee-rT-∫0TλuduKIST>K,VT>D∣F0=e-rTKEe-∫0TλuduIST>K,VT>D∣F0=e-rTKA0,TEλIST>K,VT>D∣F0,when NT=n, substituting ST and VT into IST>K,VT>D, we have(45)EλIST>K,VT>D∣F0=∑n=0∞λ-Tnn!e-λ-TNd5n,d6n,ρ,where(46)d5n=ln⁡S0/K+rT-1/2σ12T-ρ13σ1σ3∫0TMu,T,αdu-θλ-T+nμJσ12T+nσJ2,d6n=ln⁡V0/D+rT-1/2σ22T-ρ23σ2σ3∫0TMu,T,αduσ22T.
Then we will deduceI3. We define a new measure(47)dQSdQλ∣F0=STEλST∣F0.Let X0,T=EλST∣F0; according to (42)(48)X0,T=S0exp⁡rT-ρ13σ1σ3∫0TMu,T,αdu.Then (49)dQSdQλ∣F0=exp⁡-12σ12T+∫0Tσ1dW1λu-θλ-T+∑i=0NTln⁡Yi+1.According to Girsanov’s theorem and Theorem 1(50)W1ST=W1λT-σ1T,W2ST=W2λT-ρ12σ1T.So in measure, we have(51)ST=S0exp⁡rT+12σ12T+∫0Tσ1dW1Su-ρ13σ1σ3∫0TMu,T,αdu-θλ-~T+∑i=0NTln⁡Yi+1,VT=V0exp⁡rT-12σ22T+∫0Tσ2dW2Su+ρ12σ1σ2T-ρ23σ2σ3∫0TMu,T,αdu.Since (52)I3=Ee-rT-∫0TλuduSTIST>K,VT>D∣F0=e-rTEe-∫0TλuduSTIST>K,VT>D∣F0=e-rTA0,TEλSTIST>K,VT>D∣F0=e-rTA0,TX0,TESIST>K,VT>D∣F0,when N0=n,(53)ESIST>K,VT>D∣F0=∑n=0∞λ-~Tnn!e-λ-~TNd3n,d4n,ρ,where (54)d3n=ln⁡S0/K+rT+1/2σ12T-ρ13σ1σ3∫0TMu,T,αdu-θλ-~T+nμJσ12T+nσJ2,d4n=ln⁡V0/D+rT-1/2σ22T+ρ12σ1σ2T-ρ23σ2σ3∫0TMu,T,αduσ22T.
Proof III. We define III=Ee-rT-∫0Tλudu((1-w)/D)VTST-K+IVT>D∣F0=I5-I6, where(55)I5=Ee-rT-∫0Tλudu1-wDSTVTIST>K,VT>D∣F0,I6=Ee-rT-∫0Tλudu1-wDKVTIST>K,VT>D∣F0.
We deduceI6 firstly. We introduce a new measure(56)dQVdQλ∣F0=VTEλVT∣F0.Let Y0,T=EλVT∣F0; according to (43)(57)Y0,T=V0exp⁡rT-ρ23σ2σ3∫0TMu,T,αdu.So(58)dQVdQλ∣F0=exp⁡-12σ22T+∫0Tσ2dW2λu.According to Girsanov’s theorem, (59)W1VT=W1λT-ρ12σ2T,W2VT=W2λT-σ2T.So, in measure QV, we have(60)ST=S0exp⁡rT-12σ12T+∫0Tσ1dW1Vu-ρ13σ1σ3∫0TMu,T,αdu+ρ12σ1σ2T-θλ-T+∑i=0NTln⁡Yi+1,VT=V0exp⁡rT+12σ22T+∫0Tσ2dW2Vu-ρ23σ2σ3∫0TMu,T,αdu.Since (61)I6=Ee-rT-∫0Tλudu1-wDKVTIST>K,VT>D∣F0=e-rT1-wDKEe-∫0TλuduVTIST>K,VT>D∣F0=e-rT1-wDKA0,TEλVTIST>K,VT>D∣F0=e-rT1-wDKA0,TY0,TEVIST>K,VT>D∣F0,when N0=n, we have(62)EVIST>K,VT>D∣F0=∑n=0∞λ-Tnn!e-λ-TNd9n,d10n,ρ,where (63)d9n=ln⁡S0/K+rT-1/2σ12T-ρ13σ1σ3∫0TMu,T,αdu+ρ12σ1σ2T-θλ-T+nμJσ12T+nσJ2,d10n=ln⁡V0/D+rT+1/2σ22T-ρ23σ2σ3∫0TMu,T,αduσ22T=d6n+σ22T.
Then we will deduceI5. We introduce the measure as follows:(64)dQSVdQλ∣F0=STVTEλSTVT∣F0.Let Z0,T=EλSTVT∣F0; according to (42) and (43)(65)Z0,T=S0V0exp⁡2rT-ρ13σ1+ρ23σ2σ3∫0TMu,T,αdu+ρ12σ1σ2.So(66)dQSVdQλ∣F0=exp⁡-12σ12+σ22+ρ12σ1σ2T+∫0Tσ1dW1λu+∫0Tσ2dW2λu-θλ-T+∑i=0NTln⁡Yi+1.According to Girsanov’s theorem and Theorem 1(67)W1SVT=W1λT-σ1T-ρ12σ2T,W2SVT=W2λT-σ2T-ρ12σ1T.So in measure QSV(68)ST=S0exp⁡rT+12σ12T+∫0Tσ1dW1SVu-ρ13σ1σ3∫0TMu,T,αdu+ρ12σ1σ2T-θλ-~T+∑i=0NTln⁡Yi+1,VT=V0exp⁡rT+12σ22T+∫0Tσ2dW2SVu-ρ23σ2σ3∫0TMu,T,αdu+ρ12σ1σ2T.Since(69)I5=Ee-rT-∫0Tλudu1-wDSTVTIST>K,VT>D∣F0=e-rT1-wDEe-∫0TλuduSTVTIST>K,VT>D∣F0=e-rT1-wDA0,TEλSTVTIST>K,VT>D∣F0=e-rT1-wDA0,TZ0,TESVIST>K,VT>D∣F0,when N0=n, we have(70)ESVIST>K,VT>D∣F0=∑n=0∞λ-~Tnn!e-λ-~TNd7n,d8n,ρ,where (71)d7n=ln⁡S0/K+rT+1/2σ12T-ρ13σ1σ3∫0TMu,T,αdu+ρ12σ1σ2T-θλ-~T+nμJσ12T+nσJ2,d8n=ln⁡V0/D+rT+1/2σ22T+ρ12σ1σ2T-ρ23σ2σ3∫0TMu,T,αduσ22T=d4n+σ22T.


## 4. Numerical Experiments

In this section, we mainly discuss the influence of different parameters on option value. The parameters are as follows:(72)α=0.2;β=0.05;r0=0.05;θ=1;λ-=0.1;λ-~=0.2;K=100;D=60;T=1;σ1=0.5;σ2=0.5;σ3=0.5;σJ=0.5;μJ=0.2;ρ12=0.5;ρ23=0.5;ρ13=0.5;w=0.8.Figure1 shows the influences of underlying assets and corporate assets on option value. We can see that the option value increases gradually with the increase of underlying assets and corporate assets, but the corporate assets have a higher influence on option value than that of underlying assets.Figure 1
Option value with different underlying assets and corporate assets.From Figure2 we can see that the jump process has an obvious influence on option pricing which can make the option value more accurate. Because of the expansion of default, the default probability becomes higher. So the option value is lower than that in Black-Scholes model and this is coincident with the reality.Figure 2
Option value in different pricing models.Figure3 shows the changes of option value with different jump intensity. With the increase of jump intensity, the option value will decrease because of the rising default risk. We can choose the most suitable jump intensity according to the historical data to make the option value more accurate in reality.Figure 3
Option value with different jump intensity.From Figures4 and 5 we can see that the option value decreases obviously when the default intensity and the default boundary are less than certain critical value while the it changes slowly when the default intensity and the default boundary exceed the critical value. In real application, the default intensity and the default boundary can be set according to the corporate reputation, so that the option pricing will be more reasonable.Figure 4
Option value with different default intensity.Figure 5
Option value with different default boundary.Figure6 shows that the option value and the proportion of bankruptcy costs are negatively correlated. The higher the bankruptcy costs, the lower the option value.Figure 6
Option value with different proportion of bankruptcy costs.


## 5. Conclusion

The traditional definition of default probability is divided into two forms: the reduced-form model says that the corporate will default as long as the jump occurs and the default probability is determined by the default intensity while the structural model says that the default occurs when the corporate assets are less than the default boundary and the default probability is determined by the relationship between corporate assets and default boundary. In this paper, we combine reduced-form model with structural model to discuss the European vulnerable option pricing. We define that the default occurs when the default process jumps or the corporate goes bankrupt. Supposing that the underlying asset follows the jump-diffusion process and the default follows the Vasicek model, we deduce the expression of vulnerable option. Then we use the measure transformation and martingale method to derive the explicit solution of it. The results show that the model we put forward is more in line with the real financial market.



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