# Pricing Vulnerable Options in a Mixed Fractional Brownian Motion with Jumps

Authors: Panhong Cheng; Zhihong Xu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

A new framework for pricing European vulnerable options is developed in the case where the underlying stock price and firm value follow the mixed fractional Brownian motion with jumps, respectively. This research uses the actuarial approach to study the pricing problem of European vulnerable options. An analytic closed-form pricing formula for vulnerable options with jumps is obtained. For the purpose of understanding the pricing model, some properties of this pricing model are discussed in the paper. Finally, we compare and analyze the pricing results of different pricing models and discuss the influences of basic parameters on the pricing results of our proposed model by using numerical simulations, and the corresponding economic analyses about these influences are given.


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## Body

## 1. Introduction

Options are popular financial derivatives that play essential roles in financial markets. An option is an agreement that the holder has the right to buy or sell a certain amount of underlying assets at an exercise price during a fixed period. In 1973, a famous option pricing model was introduced by Black and Scholes [1], which paved the way for the wide application of options in financial markets. With the sustainable development of financial markets, there are more and more over-the-counter (in short, OTC) options, and the transaction scales of OTC options are also growing. In the OTC market, both sides of the transaction negotiate in private and make one-to-one transaction, and this market does not guarantee that the counterparty can realize the promise of payments to the option holder, so credit risk, also known as default risk, exists in OTC options trading. When the option holder is subject to credit risk, then the option with default is defined as a vulnerable option. Based on two kinds of credit risk models, structural model and reduced-form model, there is enormous literature studying the pricing problem of vulnerable options. Johnson and Stulz [2] first proposed the pricing model of vulnerable options. Since then, the study of vulnerable options has received more attention. For instance, Jarrow and Turnbull [3] studied the pricing of vulnerable options which had the assumption of independence between the options’ underlying assets and the default risk of the counterparty by adopting the reduced-form model. Klein [4] obtained an analytic pricing formula of vulnerable options by relaxing the independence assumption of the pricing model in Johnson and Stulz [2] and Jarrow and Turnbull [3] and applying the structural model. Hung and Liu [5] extended the model of Klein [4] to price vulnerable options when the market is incomplete.Motivated by the aforementioned works, some researchers further consider the construction of pricing model on vulnerable options under the conditions of stochastic interest rate, stochastic volatility, and jump risks. For instance, Liao and Huang [6] extended the framework of Klein and Inglis [7] to obtain closed-form pricing formulae for vulnerable options under stochastic interest rate by the risk-neutral pricing approach. Yoon and Kim [8] investigated the pricing problem of vulnerable options under the Hull–White interest rate and derived an analytic pricing formula for European vulnerable options by using double Mellin transforms. Lee et al. [9] incorporated Heston stochastic volatility model to describe the dynamics of underlying asset and the firm value of the counterparty and provided a closed-form analytic formula of vulnerable options price by using the Green function and Fourier transforms. Wang et al. [10] improved the model in Lee et al. [9]. Taking the differences between the volatility of the underlying asset and that of the asset of the counterparty into consideration, the authors in [10] used two-factor stochastic volatility diffusion processes to capture the changing characteristics of the underlying asset and that of the asset of the counterparty and derived a pricing formula for vulnerable options by applying equivalent martingale measure transformation method. All studies are devoted to constructing a pricing model for vulnerable options by diffusion processes in the above papers. However, owing to jump risk caused by rare events which occurs in the underlying asset and the counterparty asset, some literature proposed jump-diffusion models to consider vulnerable options pricing. Under the jump-diffusion assumptions about the underlying asset and the counterparty asset, Xu et al. [11] presented an improved method of vulnerable options pricing. Wang [12] carried out research into the pricing problem of vulnerable options by assuming that the underlying asset price and the value of counterparty asset both followed jump-diffusion processes and the default barrier was stochastic. Han et al. [13] investigated vulnerable options pricing considering the market prices of common systematic jump risks under regime-switching jump-diffusion models and derived explicit analytic pricing formulae for vulnerable options by risk-neutral pricing theory. Under the reduced-form framework, Niu et al. [14] incorporated jump risks and dynamical correlation between the underlying asset and the counterparty asset in vulnerable options to present jump-diffusion pricing models with stochastic correction. More studies about vulnerable options pricing could be seen in Ma et al. [15], Lee and Kim [16], Ma et al. [17], Han [18], Niu and Wang [19], Yang et al. [20], Pasricha and Goel [21], and the references therein.All the discussions mentioned above demonstrated that the logarithmic returns of the underlying asset were independent and identically distributed normal random variables, as well as the logarithmic returns of the counterparty asset. However, a series of financial empirical studies showed that the distribution of return on financial assets has the features of high peaks, heavy tails, and long-range dependence (see [22–26]). For example, Mandelbrot and Van Ness [22] first found that stock returns have long-range dependence and gave the definition of fractional Brownian motion (in short, FBM). Since then, FBM that exhibits self-similarity and long memory becomes a useful tool for capturing long memory behavior of the financial asset. However, the fractional Brownian motion is neither a Markov process nor a semimartingale (except H=1/2), so the usual stochastic calculus cannot be used to analyze it. Although Lin [27] developed a pathwise integral theory for FBM, Rogers [28] proved that the corresponding market has arbitrage. Duncan et al. [29] defined a new kind of integral based on Wick product that is called fractional Itô integral and derived the fractional Itô integral rule corresponding to the FBM. Hu and Øksendal [30] further studied the integral theory of FBM based on Wick product and showed that the corresponding Itô type fractional Black–Scholes market has no arbitrage under the fractional Itô integral rule, contrary to the situation when the pathwise integral was used in [28]. Moreover, Hu and Øksendal [30] proved that their Itô fractional Black–Scholes market is complete and derived the pricing formula for a European option at t=0. Necula [31] studied the pricing of European options under fractional Brownian motion environment by utilizing fractal geometry theory. Elliott and van der Hoek [32] analyzed the properties of fractional Brownian motion and its application in finance. Nualart [33] proposed a fractional Black–Scholes options pricing model as an improvement of the classical Black–Scholes model. Xiao et al. [34] constructed a new pricing model for European currency option in a fractional Brownian motion with jumps and gave the estimation method of parameters in the pricing model. Under the assumptions of the stock price obeying the fractional jump-diffusion model and the interest rate and default intensity obeying fractional Vasicek model, Wang et al. [35] derived the pricing formulae for European vulnerable options using the equivalent martingale measure method. Although some research has constructed no arbitrage fractional Black–Scholes model under Wick self-financing strategy, Björk and Hult [36] showed that the fractional market based on Wick product has no arbitrage but the definition of the corresponding self-financing trading strategy is too restrictive. Furthermore, the valuation of options may be negative prices when applying FBM. Thus, FBM based on Wick products is with limited applicability in finance. In order to overcome those problems and take the long-range dependence into account, it is reasonable to use the mixed fractional Brownian motion (in short, MFBM) to characterize the stochastic fluctuations of financial asset price [37, 38]. The mixed fractional Brownian motion, known as a generalization of FBM, is a family of centered Gaussian processes that is a linear combination of Brownian motion and independent fractional Brownian motion. Cheridito [39] first applied the MFBM to economics. It was proved that MFBM is equivalent to Brownian motion when H∈3/4,1, and hence the mixed fractional market has no arbitrage in [39]. Xiao et al. [40] studied pricing equity warrants in the MFBM environment and illustrated the validity of their proposed algorithm through numerical examples. Shokrollahi and Klllçman [41] developed a new framework for pricing European currency option, where the spot exchange rate obeyed a MFBM with jumps, and made numerical simulations to illustrate the flexibility of their model. Zhang et al. [42] employed stochastic analysis and fuzzy set theory to propose a fuzzy mixed fractional Brownian motion model with jumps. The closed-form solutions for the prices of European options under the fuzzy MFBM environment were derived through arbitrage-free pricing theory. Moreover, the empirical studies were made to illustrate the reasonableness of the proposed pricing model. Considering the long-range dependence of the underlying asset returns, Li and Wang [43] studied the valuation of the bid and ask prices of European options under the MFBM environment and obtained the explicit formulae for the bid and ask prices by using WANG-transform as a distortion function. Zhang et al. [44] assumed that the price of the underlying stock followed a MFBM, got the analytic pricing formulae for the fixed and floating strike geometric Asian power options through a partial differential equation approach, and presented the lower and upper bounds of the prices of fixed and floating strike geometric Asian power options by utilizing interval numbers. In addition, more studies into the pricing problems of options could be seen in [45–47] and the references therein.The analytic pricing formula of options has been derived usually by utilizing the risk-neutral pricing theory and equivalent martingale measure transformation method as most of the research quoted above has done. Nevertheless, an actuarial approach, turning the pricing problem of vulnerable options into a fair premium determination, might be a better choice which was put forward by Bladt and Rydberg [48] since it would not require the complexity of calculation of the probabilistic techniques. Yan et al. [49] obtained the pricing formulae of European options in the case where the underlying asset price followed Ornstein–Uhlenbeck process by actuarial approach. Shokrollahi and Klllçman [46] investigated the strategy of fair insurance premium actuarial approach for pricing foreign currency option in the MFBM environment with jumps.Further, to capture jumps in the underlying asset price and the value of counterparty asset and to take the long-range dependence of those financial assets, in this paper, we use the jump process, that is, Poisson process (see Xiao et al. [34]), to characterize the sudden changes and the diffusion process, that is, MFBM, to characterize the normal continuous fluctuations in financial assets. Closed-form formulae for vulnerable options are derived in MFBM environment with jumps by actuarial approach where we adopt the typical structural approach to describe default risk. Then, we illustrate some properties of our proposed pricing model. The comparative results of our model and other available models indicate that our model is closer to the actual market. In addition, vulnerable options values against different parameters changes in our model are investigated.The remainder of this article is structured as follows. In Section2, a pricing model for vulnerable options in the mixed fractional Brownian with jumps is constructed, and closed-form formulae are deduced by actuarial approach. Section 3 presents some properties of the pricing formulae. Section 4 gives numerical simulations according to our proposed pricing model. The concluding remarks are drawn in Section 5.


## 2. Pricing Model for Vulnerable Options in a Mixed Jump Fractional Brownian Environment

Considering the long-range dependence and abnormal fluctuations of financial assets, a mixed jump fractional Brownian motion pricing model for vulnerable options combining the MFBM and jump process is constructed. Closed-form pricing formulae are obtained by actuarial approach.


### 2.1. Basic Setting of the Pricing Model

Consider a complete probability spaceΩ,ℱ,ℱtt≥0,P, in which all the random variables and processes below are defined, and the information filtration ℱtt≥0 satisfies the usual conditions, such as being monotonic increasing and right continuous. A mixed fractional Brownian motion MtH is a linear combination of Brownian motion Bt and fractional Brownian motion BtH with Hurst parameter H∈0,1, namely,(1)MtH=αBt+βBtH,where Bt and BtH are independent, and α and β are two real constants such that α,β≠0,0.The MFBM is a centered Gaussian process with mean zero and the covariance(2)CovMtH,MsH=α2t∧s+β22t2H+s2H−t−s2H,s,t>0.In this paper, we will only considerH∈3/4,1, in order to get rid of arbitrage in the financial markets. Now, to derive the vulnerable options pricing formulae in a mixed jump fractional market, we shall make the following assumptions:(i)
All securities are perfectly divisible, and there are no transaction costs or taxes.(ii)
Security trading is continuous.(iii)
The short-term interest rate is constant during the lifetime of the derivative securities.(iv)
Dividends are not paid during the lifetime of the underlying asset.(v)
There are no riskless arbitrage opportunities.(vi)
The underlying asset priceSt and the firm value of the counterparty Vt both follow a MFBM with random jumps under the probability measure P, respectively. Thus,(3)dSt=Stμst−λSθSdt+StσSdBSt+σSHdBSHt+StJSt−1dNSt,(4)dVt=Vtμvt−λVθVdt+VtσVdBVt+σVHdBVHt+VtJVt−1dNVt,whereNit denotes a Poisson process with rate λi, which is ℱt-adapted. Jit−1 is a jump size (Ji0=1) with the mean EJi−1=θi. Jit is jump size percent at time t which is a sequence of lognormal, independent, and identically distributed variables with mean ln1+θi−σJi2/2 and variance σJi2. The drift μit is supposed to be nonrandom function of time t. The volatility (σi,σiH) is assumed to be a positive constant. Bit is a standard Brownian motion, and BiHt is a fractional Brownian motion with Hurst parameter H∈3/4,1. Bit and BiHt are both ℱt-adapted. Moreover, it is supposed that the Poisson processes NSt,NVt; the jump size percent terms JSt,JVt; the standard Brownian motion Bit; and the fractional Brownian motion BiHt are independent. i∈S,V is satisfied throughout the paper. The covariance of BSt and BVt is ρSVt, and the covariance of BSHt and BVHt is ρSVHt2H.The payoff of a European vulnerable call option and put option at maturity dateT are expressed in this paper and are consistent with Klein [4] as follows, respectively:(5)CTd=EST−K+IVT≥D∗+1−αDVTIVT<D∗,PTd=EK−ST+IVT≥D∗+1−αDVTIVT<D∗,where K is the strike price of the options. D∗ is a constant default boundary such that no credit loss occurs if VT is greater than D∗ and the option holder receives full payment. If VT is less than D∗, a credit loss occurs. In the event of the credit loss, the option holder only obtains the proportion 1−α/DVT of the nominal claim by the option writer, where α represents the deadweight costs associated with bankruptcy expressed as a percentage of the firm value of the counterparty and D is the value of total liabilities given by D∗ and an additional liability.


### 2.2. Actuarial Approach for Pricing Vulnerable Options

In this section, we deal with the pricing problem for European vulnerable options by actuarial approach, when the underlying stock price and firm value obey the SDEs (3) and (4), respectively. The pricing model can be applied not only to the arbitrage-free, equilibrium, and complete markets, but also to the arbitrage, nonequilibrium, and incomplete markets.Definition 1.
Assume thatβSw and βVw (w∈0,T) are the expectation return rates of the stochastics processes St,t≥0 and Vt,t≥0 on t∈0,T, respectively, which are defined as follows:(6)ESTS0=exp∫0TβSwdw,(7)EVTV0=exp∫0TβVwdw.Definition 2.
Suppose thatC0d and P0d are the values of the European vulnerable call and put option at time t=0, respectively, whose stock price is St, firm value of the counterparty is Vt, strike price is K, and maturity date is T. Then, the values of the European vulnerable options by actuarial approach are presented as follows:(8)C0d=Ee−∫0TβSwdwST−e−rTKIA1,B1+1−αe−rTDe−∫0TβVwdwVTIA1,B2,(9)P0d=Ee−rTK−e−∫0TβSwdwSTIA2,B1+1−αe−rTDe−∫0TβVwdwVTIA2,B2,where A1=e−∫0TβSwdwST≥e−rTK and A2=e−∫0TβSwdwST<e−rTK are the essential conditions for performing the European vulnerable call and put option on the maturity date T, respectively. B1=e−∫0TβVwdwVT≥e−rTD∗ and B2=e−∫0TβVwdwVT<e−rTD∗ indicate whether a credit loss occurs or not. I• is an indicator function.Lemma 1.
The expectation return ratesβSw and βVw, defined by (6) and (7), respectively, satisfy the following equalities:(10)∫0TβSwdw=∫0TμSwdw,(11)∫0TβVwdw=∫0TμVwdw.Proof.
By applying mixed fractional Itô formula, it is easy to get the solution of SDE (3):(12)ST=S0e∫0Tμswdw−λSθST−1/2σS2T−1/2σSH2T2H+σSBST+σSHBSHT+∑i=0NSTlnJSti.
Thus,(13)ESTS0=Ee∫0Tμswdw−λSθST−1/2σS2T−1/2σSH2T2H+σSBST+σSHBSHT+∑i=0NSTlnJSti=e∫0Tμswdw−λSθST−1/2σS2T−1/2σSH2T2H×EexpσSBST+σSHBSHTEexp∑i=0NSTlnJSti.
SinceBST∼N0,T,BSHT∼N0,T2H, BST, and BSHT are independent, (14)EeσSBST+σSHBSHT=e1/2σs2T+1/2σSH2T2H.
Using the independence ofNST and JSti and the theory of Poisson distribution with intensity λST, we have(15)Eexp∑i=0NSTlnJSti=EEexp∑i=0NSTlnJSti|NST=∑m=0∞PNST=mEexp∑i=0mlnJSti=∑m=0∞λSTmm!e−λSTEexplnJStim=∑m=0∞λSTmm!e−λST1+θSm=eλSθST.
Therefore,(16)ESTS0=e∫0TμSwdw.
Combining (6) and (16), we have the conclusion (10) of Lemma 1. Similarly, we can prove that equality (11) is satisfied.Theorem 1.
The prices of European vulnerable options, defined by (8) and (9), are represented as(17)C0d=∑m=0∞∑n=0∞PmλSTPnλVTS0e−λSθST1+θSmN2a1m,a2n,ρ−Ke−rTN2b1m,b2n,ρ+1−αDS0V0er−λSθS−λVθVT+ρδSδV1+θSm1+θVnN2c1m,−c2n,−ρ−1−αDKV0e−λVθVT1+θVnN2d1m,−d2n,−ρ,(18)P0d=∑m=0∞∑n=0∞PmλSTPnλVTKe−rTN2−b1m,b2n,−ρ−S0e−λSθST1+θSmN2−a1m,a2n,−ρ+1−αDKV0e−λVθVT1+θVnN2−d1m,−d2n,ρ−1−αDS0V0er−λSθS−λVθVT+ρδSδVN2−c1m,−c2n,ρ,where PmλST and PnλVT are the probability functions of NST and NVT throughout the paper, respectively, with PmλST=λSTm/m!e−λST,PnλVT=λVTn/n!e−λVT. N2 denotes the bivariate normal cumulative distribution function given by(19)N2x,y,ρ=12π1−ρ2∫−∞x∫−∞ye−1/21−ρ2u2−2ρuv+v2dudv,a1m=b1m+δS,a2n=b2n+ρδS,b1m=lnS0/K+r−λSθST+mln1+θS−δS2/2δS,b2n=lnV0/D∗+r−λVθVT+nln1+θv−δV2/2δV,c1m=b1m+δS+ρδV,c2n=b2n+δV+ρδS,d1m=b1m+ρδV,d2n=b2n+δV,δS2=σS2T+σSH2T2H+mσJS2,δV2=σV2T+σVH2T2H+nσJV2,ρ=ρSVσSσVT+ρSVHσSHσVHT2HδSδV.Proof.
For the convenience, (8) can be rewritten as(20)C0d=E1−E2+E3−E4,where(21)E1=Ee−∫0TβSwdwSTIA1,B1,E2=e−rTKEIA1,B1,E3=1−αe−rTDEe−∫0TβSwdwSTe−∫0TβVwdwVTIA1,B2,E4=1−αKDEe−∫0TβVwdwVTIA1,B2.
Equality (21) illustrates that the value of a European vulnerable call option consists of four terms E1, E2, E3, and E4. Let(22)STm=S0e∫0TμSwdw−λSθST−1/2σS2T−1/2σSH2T2H+σSBST+σSHBSHT+∑i=0mlnJSti,VTn=V0e∫0TμVwdw−λVθVT−1/2σV2T−1/2σVH2T2H+σVBVT+σVHBVHT+∑i=0nlnJVti,z1=σSBST+σSHBSHT+∑i=0mlnJSti−mln1+θS+m/2σJS2δS,δS2=σS2T+σSH2T2H+mσJS2,z2=σVBVT+σVHBVHT+∑i=0nlnJVti−nln1+θV+n/2σJV2δV,δV2=σV2T+σVH2T2H+nσJV2,b1m=lnS0/K+r−λSθST+mln1+θS−δS2/2δS,b2n=lnV0/D∗+r−λVθVT+nln1+θv−δV2/2δV.
Now, we treat each of these terms separately.
Firstly, we calculate the termE2.(23)E2=e−rTKEIA1,B1=e−rTKEEIA1,B1|NST,NVT=e−rTK∑m=0∞∑n=0∞PmλSTPnλVT×EIe−∫0TβSwdwSTm≥e−rTK,e−∫0TβVwdwVTn≥e−rTD∗=e−rTK∑m=0∞∑n=0∞PmλSTPnλVTEIz1≥−b1m,z2≥−b2n,where equality (23) comes from Lemma 1 and the independence of BiT,BiHT, JSt,JVt,NST,NVT,i∈S,V.
Note thatcovBST,BVT=ρSVT,covBSHT,BVHT=ρSVHT2H; thus,(24)z1,z2∼N0,1;0,1;ρ,ρ=ρSVσSσVT+ρSVHσSHσVHT2HδSδV.
Therefore, we obtain(25)E2=e−rTK∑m=0∞∑n=0∞PmλSTPnλVTN2b1m,b2n,ρ.
Now, we are in a position to calculate the termE1. From Lemma 1 and equality (12), we can get(26)E1=Ee−∫0TβSwdwSTIA1,B1=EEe−∫0TβSwdwSTIA1,B1|NST,NVT=∑m=0∞∑n=0∞PmλSTPnλVT×Ee−∫0TβSwdwSTmIe−∫0TβSwdwSTm≥e−rTK,e−∫0TβVwdwVTn≥e−rTD∗=S0e−λSθST∑m=0∞∑n=0∞PmλSTPnλVT1+θSmEeδSz1−δS2/2Iz1≥−b1m,z2≥−b2n=S0e−λSθST∑m=0∞∑n=0∞PmλSTPnλVT1+θSm∫−b1m+∞∫−b2n+∞eδSz1−δS2/2φz1,z2,ρdz1dz2,where φz1,z2,ρ, the standard bivariate normal probability density function, is as follows:(27)φz1,z2,ρ=12π1−ρ2e−1/21−ρ2z12+z22−2ρz1z2.
By using the method of undetermined coefficients, the exponent of the integrand in (26) is of the form(28)−121−ρ2z1+a2+z2+b2−2ρz1+az2+b+M,where a=−δS,b=−ρδS,M=0. By setting a1m=b1m+δS,a2n=b2n+ρδS, E1 is given by the formula(29)E1=S0e−λSθST∑m=0∞∑n=0∞PmλSTPnλVT1+θSm∫−b1m+∞∫−b2n+∞φz1−δS,z2−ρδS,ρdz1dz2=S0e−λSθST∑m=0∞∑n=0∞PmλSTPnλVT1+θSmN2b1m+δS,b2n+ρδS=S0e−λSθST∑m=0∞∑n=0∞PmλSTPnλVT1+θSmN2a1m,a2n,ρ.
Next, we calculate the termE3. Noting that e−∫0TβVwdwVTn<e−rTD∗ is equivalent to z2<−b2n, we deduce that(30)E3=1−αe−rTDEe−∫0TβSwdwSTe−∫0TβVwdwVTIA1,B2=1−αS0V0erT−λSθS+λVθVTD∑m=0∞∑n=0∞PmλSTPnλVT1+θSm1+θVn×∫−b1m+∞∫−∞−b2neδSz1+δVz2−δS2+δV2/2φz1,z2,ρdz1dz2=1−αS0V0erT−λSθS+λVθVT+ρδSδVD∑m=0∞∑n=0∞PmλSTPnλVT1+θSm1+θVn×∫−b1m+∞∫−∞−b2nφz1−δS−ρδV,z2−δV−ρδS,ρdz1dz2.
Settingc1m=b1m+δS+ρδV and c2n=b2n+δV+ρδS, we have(31)E3=1−αS0V0erT−λSθS+λVθVT+ρδSδVD∑m=0∞∑n=0∞PmλSTPnλVT×1+θSm1+θVnN2c1m,−c2n,−ρ.
Finally, we deal with the termE4. Similar to calculating E1, it is easy to obtain(32)E4=1−αKDEe−∫0TβVwdwVTIA1,B2=1−αKV0e−λVθVTD∑m=0∞∑n=0∞PmλSTPnλVT1+θVn×∫−b1m+∞∫−∞−b2nφz1−ρδV,z2−δV,ρdz1dz2.
Settingd1m=b1m+ρδV and d2n=b2n+δV, we have(33)E4=1−αKV0e−λVθVTD∑m=0∞∑n=0∞PmλSTPnλVT1+θVnN2d1m,−d2n,−ρ.
Combining everything together, (17) is satisfied. The proof of (18) is the same.


## 2.1. Basic Setting of the Pricing Model

Consider a complete probability spaceΩ,ℱ,ℱtt≥0,P, in which all the random variables and processes below are defined, and the information filtration ℱtt≥0 satisfies the usual conditions, such as being monotonic increasing and right continuous. A mixed fractional Brownian motion MtH is a linear combination of Brownian motion Bt and fractional Brownian motion BtH with Hurst parameter H∈0,1, namely,(1)MtH=αBt+βBtH,where Bt and BtH are independent, and α and β are two real constants such that α,β≠0,0.The MFBM is a centered Gaussian process with mean zero and the covariance(2)CovMtH,MsH=α2t∧s+β22t2H+s2H−t−s2H,s,t>0.In this paper, we will only considerH∈3/4,1, in order to get rid of arbitrage in the financial markets. Now, to derive the vulnerable options pricing formulae in a mixed jump fractional market, we shall make the following assumptions:(i)
All securities are perfectly divisible, and there are no transaction costs or taxes.(ii)
Security trading is continuous.(iii)
The short-term interest rate is constant during the lifetime of the derivative securities.(iv)
Dividends are not paid during the lifetime of the underlying asset.(v)
There are no riskless arbitrage opportunities.(vi)
The underlying asset priceSt and the firm value of the counterparty Vt both follow a MFBM with random jumps under the probability measure P, respectively. Thus,(3)dSt=Stμst−λSθSdt+StσSdBSt+σSHdBSHt+StJSt−1dNSt,(4)dVt=Vtμvt−λVθVdt+VtσVdBVt+σVHdBVHt+VtJVt−1dNVt,whereNit denotes a Poisson process with rate λi, which is ℱt-adapted. Jit−1 is a jump size (Ji0=1) with the mean EJi−1=θi. Jit is jump size percent at time t which is a sequence of lognormal, independent, and identically distributed variables with mean ln1+θi−σJi2/2 and variance σJi2. The drift μit is supposed to be nonrandom function of time t. The volatility (σi,σiH) is assumed to be a positive constant. Bit is a standard Brownian motion, and BiHt is a fractional Brownian motion with Hurst parameter H∈3/4,1. Bit and BiHt are both ℱt-adapted. Moreover, it is supposed that the Poisson processes NSt,NVt; the jump size percent terms JSt,JVt; the standard Brownian motion Bit; and the fractional Brownian motion BiHt are independent. i∈S,V is satisfied throughout the paper. The covariance of BSt and BVt is ρSVt, and the covariance of BSHt and BVHt is ρSVHt2H.The payoff of a European vulnerable call option and put option at maturity dateT are expressed in this paper and are consistent with Klein [4] as follows, respectively:(5)CTd=EST−K+IVT≥D∗+1−αDVTIVT<D∗,PTd=EK−ST+IVT≥D∗+1−αDVTIVT<D∗,where K is the strike price of the options. D∗ is a constant default boundary such that no credit loss occurs if VT is greater than D∗ and the option holder receives full payment. If VT is less than D∗, a credit loss occurs. In the event of the credit loss, the option holder only obtains the proportion 1−α/DVT of the nominal claim by the option writer, where α represents the deadweight costs associated with bankruptcy expressed as a percentage of the firm value of the counterparty and D is the value of total liabilities given by D∗ and an additional liability.


## 2.2. Actuarial Approach for Pricing Vulnerable Options

In this section, we deal with the pricing problem for European vulnerable options by actuarial approach, when the underlying stock price and firm value obey the SDEs (3) and (4), respectively. The pricing model can be applied not only to the arbitrage-free, equilibrium, and complete markets, but also to the arbitrage, nonequilibrium, and incomplete markets.Definition 1.
Assume thatβSw and βVw (w∈0,T) are the expectation return rates of the stochastics processes St,t≥0 and Vt,t≥0 on t∈0,T, respectively, which are defined as follows:(6)ESTS0=exp∫0TβSwdw,(7)EVTV0=exp∫0TβVwdw.Definition 2.
Suppose thatC0d and P0d are the values of the European vulnerable call and put option at time t=0, respectively, whose stock price is St, firm value of the counterparty is Vt, strike price is K, and maturity date is T. Then, the values of the European vulnerable options by actuarial approach are presented as follows:(8)C0d=Ee−∫0TβSwdwST−e−rTKIA1,B1+1−αe−rTDe−∫0TβVwdwVTIA1,B2,(9)P0d=Ee−rTK−e−∫0TβSwdwSTIA2,B1+1−αe−rTDe−∫0TβVwdwVTIA2,B2,where A1=e−∫0TβSwdwST≥e−rTK and A2=e−∫0TβSwdwST<e−rTK are the essential conditions for performing the European vulnerable call and put option on the maturity date T, respectively. B1=e−∫0TβVwdwVT≥e−rTD∗ and B2=e−∫0TβVwdwVT<e−rTD∗ indicate whether a credit loss occurs or not. I• is an indicator function.Lemma 1.
The expectation return ratesβSw and βVw, defined by (6) and (7), respectively, satisfy the following equalities:(10)∫0TβSwdw=∫0TμSwdw,(11)∫0TβVwdw=∫0TμVwdw.Proof.
By applying mixed fractional Itô formula, it is easy to get the solution of SDE (3):(12)ST=S0e∫0Tμswdw−λSθST−1/2σS2T−1/2σSH2T2H+σSBST+σSHBSHT+∑i=0NSTlnJSti.
Thus,(13)ESTS0=Ee∫0Tμswdw−λSθST−1/2σS2T−1/2σSH2T2H+σSBST+σSHBSHT+∑i=0NSTlnJSti=e∫0Tμswdw−λSθST−1/2σS2T−1/2σSH2T2H×EexpσSBST+σSHBSHTEexp∑i=0NSTlnJSti.
SinceBST∼N0,T,BSHT∼N0,T2H, BST, and BSHT are independent, (14)EeσSBST+σSHBSHT=e1/2σs2T+1/2σSH2T2H.
Using the independence ofNST and JSti and the theory of Poisson distribution with intensity λST, we have(15)Eexp∑i=0NSTlnJSti=EEexp∑i=0NSTlnJSti|NST=∑m=0∞PNST=mEexp∑i=0mlnJSti=∑m=0∞λSTmm!e−λSTEexplnJStim=∑m=0∞λSTmm!e−λST1+θSm=eλSθST.
Therefore,(16)ESTS0=e∫0TμSwdw.
Combining (6) and (16), we have the conclusion (10) of Lemma 1. Similarly, we can prove that equality (11) is satisfied.Theorem 1.
The prices of European vulnerable options, defined by (8) and (9), are represented as(17)C0d=∑m=0∞∑n=0∞PmλSTPnλVTS0e−λSθST1+θSmN2a1m,a2n,ρ−Ke−rTN2b1m,b2n,ρ+1−αDS0V0er−λSθS−λVθVT+ρδSδV1+θSm1+θVnN2c1m,−c2n,−ρ−1−αDKV0e−λVθVT1+θVnN2d1m,−d2n,−ρ,(18)P0d=∑m=0∞∑n=0∞PmλSTPnλVTKe−rTN2−b1m,b2n,−ρ−S0e−λSθST1+θSmN2−a1m,a2n,−ρ+1−αDKV0e−λVθVT1+θVnN2−d1m,−d2n,ρ−1−αDS0V0er−λSθS−λVθVT+ρδSδVN2−c1m,−c2n,ρ,where PmλST and PnλVT are the probability functions of NST and NVT throughout the paper, respectively, with PmλST=λSTm/m!e−λST,PnλVT=λVTn/n!e−λVT. N2 denotes the bivariate normal cumulative distribution function given by(19)N2x,y,ρ=12π1−ρ2∫−∞x∫−∞ye−1/21−ρ2u2−2ρuv+v2dudv,a1m=b1m+δS,a2n=b2n+ρδS,b1m=lnS0/K+r−λSθST+mln1+θS−δS2/2δS,b2n=lnV0/D∗+r−λVθVT+nln1+θv−δV2/2δV,c1m=b1m+δS+ρδV,c2n=b2n+δV+ρδS,d1m=b1m+ρδV,d2n=b2n+δV,δS2=σS2T+σSH2T2H+mσJS2,δV2=σV2T+σVH2T2H+nσJV2,ρ=ρSVσSσVT+ρSVHσSHσVHT2HδSδV.Proof.
For the convenience, (8) can be rewritten as(20)C0d=E1−E2+E3−E4,where(21)E1=Ee−∫0TβSwdwSTIA1,B1,E2=e−rTKEIA1,B1,E3=1−αe−rTDEe−∫0TβSwdwSTe−∫0TβVwdwVTIA1,B2,E4=1−αKDEe−∫0TβVwdwVTIA1,B2.
Equality (21) illustrates that the value of a European vulnerable call option consists of four terms E1, E2, E3, and E4. Let(22)STm=S0e∫0TμSwdw−λSθST−1/2σS2T−1/2σSH2T2H+σSBST+σSHBSHT+∑i=0mlnJSti,VTn=V0e∫0TμVwdw−λVθVT−1/2σV2T−1/2σVH2T2H+σVBVT+σVHBVHT+∑i=0nlnJVti,z1=σSBST+σSHBSHT+∑i=0mlnJSti−mln1+θS+m/2σJS2δS,δS2=σS2T+σSH2T2H+mσJS2,z2=σVBVT+σVHBVHT+∑i=0nlnJVti−nln1+θV+n/2σJV2δV,δV2=σV2T+σVH2T2H+nσJV2,b1m=lnS0/K+r−λSθST+mln1+θS−δS2/2δS,b2n=lnV0/D∗+r−λVθVT+nln1+θv−δV2/2δV.
Now, we treat each of these terms separately.
Firstly, we calculate the termE2.(23)E2=e−rTKEIA1,B1=e−rTKEEIA1,B1|NST,NVT=e−rTK∑m=0∞∑n=0∞PmλSTPnλVT×EIe−∫0TβSwdwSTm≥e−rTK,e−∫0TβVwdwVTn≥e−rTD∗=e−rTK∑m=0∞∑n=0∞PmλSTPnλVTEIz1≥−b1m,z2≥−b2n,where equality (23) comes from Lemma 1 and the independence of BiT,BiHT, JSt,JVt,NST,NVT,i∈S,V.
Note thatcovBST,BVT=ρSVT,covBSHT,BVHT=ρSVHT2H; thus,(24)z1,z2∼N0,1;0,1;ρ,ρ=ρSVσSσVT+ρSVHσSHσVHT2HδSδV.
Therefore, we obtain(25)E2=e−rTK∑m=0∞∑n=0∞PmλSTPnλVTN2b1m,b2n,ρ.
Now, we are in a position to calculate the termE1. From Lemma 1 and equality (12), we can get(26)E1=Ee−∫0TβSwdwSTIA1,B1=EEe−∫0TβSwdwSTIA1,B1|NST,NVT=∑m=0∞∑n=0∞PmλSTPnλVT×Ee−∫0TβSwdwSTmIe−∫0TβSwdwSTm≥e−rTK,e−∫0TβVwdwVTn≥e−rTD∗=S0e−λSθST∑m=0∞∑n=0∞PmλSTPnλVT1+θSmEeδSz1−δS2/2Iz1≥−b1m,z2≥−b2n=S0e−λSθST∑m=0∞∑n=0∞PmλSTPnλVT1+θSm∫−b1m+∞∫−b2n+∞eδSz1−δS2/2φz1,z2,ρdz1dz2,where φz1,z2,ρ, the standard bivariate normal probability density function, is as follows:(27)φz1,z2,ρ=12π1−ρ2e−1/21−ρ2z12+z22−2ρz1z2.
By using the method of undetermined coefficients, the exponent of the integrand in (26) is of the form(28)−121−ρ2z1+a2+z2+b2−2ρz1+az2+b+M,where a=−δS,b=−ρδS,M=0. By setting a1m=b1m+δS,a2n=b2n+ρδS, E1 is given by the formula(29)E1=S0e−λSθST∑m=0∞∑n=0∞PmλSTPnλVT1+θSm∫−b1m+∞∫−b2n+∞φz1−δS,z2−ρδS,ρdz1dz2=S0e−λSθST∑m=0∞∑n=0∞PmλSTPnλVT1+θSmN2b1m+δS,b2n+ρδS=S0e−λSθST∑m=0∞∑n=0∞PmλSTPnλVT1+θSmN2a1m,a2n,ρ.
Next, we calculate the termE3. Noting that e−∫0TβVwdwVTn<e−rTD∗ is equivalent to z2<−b2n, we deduce that(30)E3=1−αe−rTDEe−∫0TβSwdwSTe−∫0TβVwdwVTIA1,B2=1−αS0V0erT−λSθS+λVθVTD∑m=0∞∑n=0∞PmλSTPnλVT1+θSm1+θVn×∫−b1m+∞∫−∞−b2neδSz1+δVz2−δS2+δV2/2φz1,z2,ρdz1dz2=1−αS0V0erT−λSθS+λVθVT+ρδSδVD∑m=0∞∑n=0∞PmλSTPnλVT1+θSm1+θVn×∫−b1m+∞∫−∞−b2nφz1−δS−ρδV,z2−δV−ρδS,ρdz1dz2.
Settingc1m=b1m+δS+ρδV and c2n=b2n+δV+ρδS, we have(31)E3=1−αS0V0erT−λSθS+λVθVT+ρδSδVD∑m=0∞∑n=0∞PmλSTPnλVT×1+θSm1+θVnN2c1m,−c2n,−ρ.
Finally, we deal with the termE4. Similar to calculating E1, it is easy to obtain(32)E4=1−αKDEe−∫0TβVwdwVTIA1,B2=1−αKV0e−λVθVTD∑m=0∞∑n=0∞PmλSTPnλVT1+θVn×∫−b1m+∞∫−∞−b2nφz1−ρδV,z2−δV,ρdz1dz2.
Settingd1m=b1m+ρδV and d2n=b2n+δV, we have(33)E4=1−αKV0e−λVθVTD∑m=0∞∑n=0∞PmλSTPnλVT1+θVnN2d1m,−d2n,−ρ.
Combining everything together, (17) is satisfied. The proof of (18) is the same.


## 3. Properties of Pricing Formula

In the previous section, we obtain the pricing formulae of European vulnerable options. In this section, we will discuss the properties of the pricing formulae.Remark 1.
The pricing formulae (17) and (18) are both convergent.
It is clear that0≤N2x,y,ρ≤1. Then, in order to illustrate the convergence of (17), we only need to show the convergence of these series(34)∑m=0∞∑n=0∞PmλSTPnλVTS0e−λSθST1+θSm,∑m=0∞∑n=0∞PmλSTPnλVT1−αDKV0e−λVθVT1+θVn,∑m=0∞∑n=0∞PmλSTPnλVTKe−rT,∑m=0∞∑n=0∞PmλSTPnλVT1−αDS0V0er−λSθS−λVθVT+ρδSδV1+θSm1+θVn.
Here, we show that∑m=0∞∑n=0∞PmλSTPnλVTS0e−λSθST1+θSm is convergent.(35)∑m=0∞∑n=0∞PmλSTPnλVTS0e−λSθST1+θSm=∑m=0∞PmλSTS0e−λSθST1+θSm∑n=0∞PnλVT=∑m=0∞e−λS1+θSTS0m!λST1+θSm.
Settingum=e−λS1+θSTS0/m!λST1+θSm, we have(36)limm⟶∞um+1um=limm⟶∞λST1+θSm+1=0<1.
According to D’Alembert’s test,∑m=0∞∑n=0∞PmλSTPnλVTS0e−λSθST1+θSm is convergent.
The convergence of the other three series will be proved in the same way. Therefore, the pricing formula (17) is convergent.
Similarly, the convergence of (18) can be shown.Remark 2.
If there is no risk of bankruptcy, i.e.,D∗⟶0 and σSH=0, the option value is similar to the Merton jump-diffusion model in [50]. Here, the pricing formulae for European call and put options under Merton jump-diffusion model are as follows:(37)C0=∑m=0∞PmλSTS0e−λSθST1+θSmNa1^m−Ke−rTNb1^m,(38)P0=∑m=0∞PmλSTKe−rTN−b1^m−S0e−λSθST1+θSmN−a1^m,where a1^m=b1^m+δS^,b1^m=lnS0/K+r−λSθST+mln1+θS−δS^2/2/δS^, δS^2=σS2T+mσJS2, and N• denotes the standard normal cumulative distribution function.
It is easily seen thata2⟶+∞,b2⟶+∞,c2⟶+∞,d2⟶+∞ as D∗⟶0 in the pricing formulae (17) and (18). Note that N2z,−∞,ρ⟶0,N2z,+∞,ρ⟶Nz; then, (17) and (18) can be simplified to (37) and (38), respectively.
Moreover, if no jump occurs in the stock price, our model will reduce to the Black–Scholes model in [1]. Formulae (17) and (18) can be expressed as the following pricing formulae for European call and put options:(39)C0=S0Na−Ke−rTNb,P0=Ke−rTN−b−S0N−a,where a=b+σST,b=lnS0/K+r−σS2/2T/σST.Remark 3.
The basic diffusion model which was proposed by Klein [4] is a special case of our model in this article when σSH=σVH=0,λs=λv=0,m=n=0. Here, the pricing formulae (17) and (18) can be rewritten as follows:(40)C0d=S0N2a1,a2,ρSV−Ke−rTN2b1,b2,ρSV+1−αDS0V0erT+ρSVσSσVTN2c1,−c2,−ρSV−1−αDKV0N2d1,−d2,−ρSV,P0d=Ke−rTN2−b1,b2,−ρSV−S0N2−a1,a2,−ρSV+1−αDKV0N2−d1,−d2,ρSV−1−αDS0V0erT+ρSVσSσVTN2−c1,−c2,ρSV,where(41)a1=b1+σST,a2=b2+ρSVσST,b1=lnS0/K+r−σS2/2TσST,b2=lnV0/D∗+r−σV2/2TσVT,c1=b1+σST+ρSVσVT,c2=b2+σVT+ρSVσST,d1=b1+ρSVσVT,d2=b2+σVT.Remark 4.
When the volatilityσSH=σVH=0,λV=0,n=0, our jump mixed fractional motion model for pricing vulnerable options will reduce to the jump-diffusion model in the Brownian motion environment which was studied by Xu et al. [51].Remark 5.
Assume that the stock price pays a continuous dividend yieldq; then, the pricing formulae of vulnerable options can be given when S0 is replaced with S0e−qT in (17) and (18), and r is replaced with r − q in the first set of parameters to the bivariate normal distribution function. (the ak,bk,ck,dk term, k=1). With those adjustments, the equations are presented as follows:(42)C0d=∑m=0∞∑n=0∞PmλSTPnλVTS0e−q+λSθST1+θSmN2a1˜m,a2˜n,ρ−Ke−rTN2b1˜m,b2˜n,ρ+1−αDS0V0er−q−λSθS−λVθVT+ρδSδV1+θSm1+θVnN2c1˜m,−c2˜n,−ρ−1−αDKV0e−λVθVT1+θVnN2d1˜m,−d2˜n,−ρ,(43)P0d=∑m=0∞∑n=0∞PmλSTPnλVTKe−rTN2−b1˜m,b2˜n,−ρ−S0e−q+λSθST1+θSmN2−a1˜m,a2˜n,−ρ+1−αDKV0e−λVθVT1+θVnN2−d1˜m,−d2˜n,ρ−1−αDS0V0er−q−λSθS−λVθVT+ρδSδV1+θSm1+θVnN2−c1˜m,−c2˜n,ρ,where(44)a1˜m=b1˜m+δS,a2˜n=b2˜n+ρδS,b1˜m=lnS0/K+r−q−λSθST+mln1+θS−δS2/2δS,b2˜n=lnV0/D∗+r−λVθVT+nln1+θv−δV2/2δV,c1˜m=b1˜m+δS+ρδV,c2˜n=b2˜n+δV+ρδS,d1˜m=b1˜m+ρδV,d2˜n=b2˜n+δV,δS2=σS2T+σSH2T2H+mσJS2,δV2=σV2T+σVH2T2H+nσJV2,ρ=ρSVσSσVT+ρSVHσSHσVHT2HδSδV.
The above jump-diffusion model with dividend based on MFBM can be used to value currency options. A pricing model for currency options with jumps could be seen in references [41, 46]. It is a special case of the jump-diffusion model with dividend when q=rf,λv=0,n=0, and D∗⟶0. Here, the European call and put currency options at time t=0 are as follows, respectively:(45)C0=∑m=0∞PmλSTS0e−rf+λSθST1+θSmNa1˜m−Ke−rTNb1˜m,P0=∑m=0∞PmλSTKe−rTN−b1˜m−S0e−rf+λSθST1+θSmN−a1˜m,where a1˜m and b1˜m are given above. Moreover, when no jumps occur in stock price, the above pricing formulae (42) and (43) can be simplified to the following equations which are consistent with the conclusion obtained in [45].(46)C0=S0e−rfTNc−Ke−rTNd,P0=Ke−rTN−d−S0e−rfTN−c,where(47)c=d+σS2T+σSH2T2H,d=lnS0/K+r−rfT−1/2σS2T+σSH2T2HσS2T+σSH2T2H.


## 4. Numerical Analysis

This section aims to present numerical results of European vulnerable options with jumps given by (17) and (18). For these purposes, we carry out two sets of numerical experiments. The first set is the comparisons with theoretical prices under different models: Klein’s model (1996), Black–Scholes’ model (1973), and our mixed jump fractional Brownian motion model (hereafter MJFBM). The purpose of these three models we chose is to investigate the effect of jump risk on the values of vulnerable options. The second set is used to analyze the prices of vulnerable options for different parameters under MJFBM. To implement our analysis, the set of basic parameters are given in Table 1, unless otherwise stated. Specifically, the parameters S0,K,V0,σS,σV,ρSV,r,D,D∗,T, and α are based on the values reported by Klein [4].Table 1
Parameter values of European vulnerable call option in the base case.
ParameterValueParameterValueInitial value ofSS0=40Initial value ofVV0=5Strike priceK=40Interest rater=0.4833Value of total liabilitiesD=5Default boundaryD∗=5Ratio of bankruptcy costsα=0.5Time to maturityT=0.3333Volatility ofSσS=0.3Volatility ofVσV=0.3Volatility ofS under FBMσSH=0.3Volatility ofV under FBMσVH=0.3Correlation coefficient betweenBSt and BVtρSV=0.5Correlation coefficient betweenBSHt and BVHtρSVH=0.5Jump intensity ofSλS=5Jump intensity ofVλV=5Mean jump size ofSθS=−0.3Mean jump size ofVθV=−0.3Standard deviation oflnJStσJS=0.15Standard deviation oflnJVtσJV=0.15Jump frequency ofSm=1Jump frequency ofVn=1


### 4.1. Comparisons with Option Prices

Pricing results for call options under different pricing models are presented in Table2, where PJ−MF denotes the prices calculated according to the MJFBM, PK denotes the prices computed by Klein’s model, PBS denotes the prices computed by Black–Scholes’ model. Since negative jumps in underlying asset reduce the stock prices, and negative jumps in firm values increase the likelihood of default, the call option holder will face an unfavorable situation. From Table 2, we can see that the option prices of the proposed model are smaller than those of Klein’s model. This numerical analysis shows that a jump process has a significant impact on option pricing. Using jump-diffusion processes to characterize the fluctuations of the underlying asset prices and firm values is necessary, and the introduction of jump process can make the vulnerable option pricing more accurate. Moreover, due to the effect of default risk, the prices of the proposed model and Klein’s model are both lower than the prices of the classical Black–Scholes’ model.Table 2
European vulnerable call option pricing under different models.
Underlying asset pricesDifferent pricing resultsPJ−MFPKPBS300.369 20.737 20.748 7320.505 31.370 01.398 9341.659 72.261 62.323 7362.828 53.393 53.516 3384.010 24.754 64.943 2405.201 16.246 46.557 1426.395 97.860 68.308 9448.595 79.540 310.156 04610.800 611.247 712.065 34811.006 712.941 514.013 25012.213 814.575 815.984 2


### 4.2. Influence of Parameters

In order to further understand the performance of our pricing model, we carry out the sensitivity analysis in order to investigate the price changes of European vulnerable options with respect to different parameters including initial value of firm, default barrier, ratio of bankruptcy costs, and Hurst index in our model.Figure1(a) illustrates how the vulnerable call option prices change with the initial firm value of the counterparty. We can see that the option prices increase with the rising of the initial firm value V0. Figure 1(b) shows the option prices against the default boundary D∗. We observe that option prices are a decreasing function of D∗. The reason for this observation is that as the default boundary D∗ increases, the likelihood of default increases and then the option prices decrease. Figure 1(c) describes the relationship between the option prices and the ratio of bankruptcy costs α. We notice that the option prices obtained by the proposed model decrease sharply for the high ratio of bankruptcy costs α. This agrees with the financial truth that as the ratio of bankruptcy costs α increases, the recovery rate 1−α/DVT decreases, and then the expected payoff of the option holder at maturity time T decreases; hence, the option prices decrease. From Figure 1(d), we can observe that the option prices have increasing trends as Hurst parameter H increases. This shows that the Hurst parameter has an important impact on option pricing.Figure 1
Option price against different parameters: (a) option price against the initial value ofV; (b) option price against default boundary D∗; (c) option price against the ratio of bankruptcy cost α; (d) option price against Hurst index H.
(a)(b)(c)(d)


## 4.1. Comparisons with Option Prices

Pricing results for call options under different pricing models are presented in Table2, where PJ−MF denotes the prices calculated according to the MJFBM, PK denotes the prices computed by Klein’s model, PBS denotes the prices computed by Black–Scholes’ model. Since negative jumps in underlying asset reduce the stock prices, and negative jumps in firm values increase the likelihood of default, the call option holder will face an unfavorable situation. From Table 2, we can see that the option prices of the proposed model are smaller than those of Klein’s model. This numerical analysis shows that a jump process has a significant impact on option pricing. Using jump-diffusion processes to characterize the fluctuations of the underlying asset prices and firm values is necessary, and the introduction of jump process can make the vulnerable option pricing more accurate. Moreover, due to the effect of default risk, the prices of the proposed model and Klein’s model are both lower than the prices of the classical Black–Scholes’ model.Table 2
European vulnerable call option pricing under different models.
Underlying asset pricesDifferent pricing resultsPJ−MFPKPBS300.369 20.737 20.748 7320.505 31.370 01.398 9341.659 72.261 62.323 7362.828 53.393 53.516 3384.010 24.754 64.943 2405.201 16.246 46.557 1426.395 97.860 68.308 9448.595 79.540 310.156 04610.800 611.247 712.065 34811.006 712.941 514.013 25012.213 814.575 815.984 2


## 4.2. Influence of Parameters

In order to further understand the performance of our pricing model, we carry out the sensitivity analysis in order to investigate the price changes of European vulnerable options with respect to different parameters including initial value of firm, default barrier, ratio of bankruptcy costs, and Hurst index in our model.Figure1(a) illustrates how the vulnerable call option prices change with the initial firm value of the counterparty. We can see that the option prices increase with the rising of the initial firm value V0. Figure 1(b) shows the option prices against the default boundary D∗. We observe that option prices are a decreasing function of D∗. The reason for this observation is that as the default boundary D∗ increases, the likelihood of default increases and then the option prices decrease. Figure 1(c) describes the relationship between the option prices and the ratio of bankruptcy costs α. We notice that the option prices obtained by the proposed model decrease sharply for the high ratio of bankruptcy costs α. This agrees with the financial truth that as the ratio of bankruptcy costs α increases, the recovery rate 1−α/DVT decreases, and then the expected payoff of the option holder at maturity time T decreases; hence, the option prices decrease. From Figure 1(d), we can observe that the option prices have increasing trends as Hurst parameter H increases. This shows that the Hurst parameter has an important impact on option pricing.Figure 1
Option price against different parameters: (a) option price against the initial value ofV; (b) option price against default boundary D∗; (c) option price against the ratio of bankruptcy cost α; (d) option price against Hurst index H.
(a)(b)(c)(d)


## 5. Conclusions

Reasonable option pricing can not only improve the operation efficiency of the financial markets, but also help investors make effective decisions in the complex and changeable financial markets, so as to maximize their own returns. Since a mixed fractional Brownian motion is an important stochastic process to capture long-range dependence memory of financial phenomena, and jump risks caused by rare events occurring in the underlying stock price and firm value cannot be ignored in the financial markets, we use a mixed jump fractional Brownian motion to capture the behavior characteristics of the stock price and firm value. The actuarial approach turns the pricing problem of vulnerable options into a fair premium determination, which does not require the complexity of the calculation as in the probabilistic techniques. On the basis of actuarial approach, a closed-form pricing formula of European vulnerable options in the MJFBM environment is deduced. Furthermore, some properties of the vulnerable options pricing formulae are derived. Moreover, numerical results in Section4 show that our mixed jump fractional pricing model is an efficient model for pricing vulnerable options and that some parameters in our model have great influences on the pricing results.



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