The Pricing of Vulnerable Options in a Fractional Brownian Motion Environment

Under the assumption of the stock price, interest rate, and default intensity obeying the stochastic differential equation driven by fractional Brownian motion, the jump-diffusion model is established for the financial market in fractional Brownian motion setting. With the changes of measures, the traditional pricing method is simplified and the general pricing formula is obtained for the European vulnerable option with stochastic interest rate. At the same time, the explicit expression for it comes into being.


Introduction
Vulnerable option is a kind of option with credit risk that refers to a risk, a borrower that will default on any type of debt by failing to make required payments.Johnson and Stulz [1] firstly substituted default risk into option pricing and advanced a new definition called vulnerable option.Klein [2] obtained the pricing formula for vulnerable option with martingale method.Ammann [3] developed Klein's model on the basis of structural approach.He finally obtained the explicit expression for vulnerable option under the assuming of interest rate and default intensity obeying the stochastic differential equation.What is more, other academics such as Chang and Hung [4] also discussed this problem, while all the discussions stated above are in the environment of geometric Brownian motion.Because of the inadequacies of geometric Brownian motion in describing the self-similarity and long-term dependence of stock prices, fractional Brownian motion is widely used into asset pricing.Hu and Øksendal [5] developed the structural approach in the condition that the stock prices followed a fractional Brownian motion and they proved that the correspondence to fractional Black-Scholes market had no arbitrage.For more literature on fractional Brownian motion, we can refer to Øksendal [6].But there is another problem that fractional Black-Scholes market does not have equivalent martingale measure according to Sottinen and Valkeila [7].Necula [8] applied quasimartingale method to the risk neutral measure.Huang et al. [9] obtained the explicit expression for the European option price under the assuming of fractional Black-Scholes market.Su and Wang [10] and Li and Ma [11] derived the closed form formula for the price of the vulnerable European option by the method of changing measures.
In this paper, we will use quasi-martingale method to change measures, so we can derive the general pricing formula for the European vulnerable option under the assuming of the stock price obeying the jump-diffusion model, the interest rate and default intensity obeying Vasicek model which are driven by fractional Brownian motion.

Market Environment
Let the uncertainty in the economy be described by the filtered probability space (Ω, F, , (F  ) 0≤≤ ).() is the short-term interest rate which is consistently positive and F measurable in this space.Assume that  is a risk neutral martingale measure under which the discounted asset price processes are martingales.
Suppose that   +1 follows log-normal distribution ln Suppose that the interest rate and default intensity follow Vasicek model under the risk neutral measure where , , , , In order to prove the theorem, we introduce two lemmas firstly.

Pricing Options
In this section, we intend to discuss pricing vulnerable options in a Fractional Brownian Motion Environment.
We define that the default time is  and  represents the recovery rate due to the bankruptcy or reorganization, where  is a constant.When the writer of the option defaults, the payoff is given by  times the payoff of the default-free option at maturity.The price at every  ∈ [0, ] of an European vulnerable call option with strike price  and maturity  is given by Note that  {≤} +  {>} = 1; we have Since (F  ) 0≤≤ is a filtration, then F  ⊆ F  .Suppose there is no default at present time, by the law of iterated conditional expectations; therefore, Obviously  − ∫   () (  − ) +  {>} is bonded and absolutely integrable, so we can interchange the two expectations by Fubini's theorem Since the full path of () is known at time , we have [13] Then ( 9) can be written as Theorem 3. Consider where Proof.By means of Lemma 1 we can see that the discounted asset price process whose numeraire is () is a quasimartingale.So we can define different equivalent quasimartingale measures for different numeraires.Suppose there is a bank account   = exp{∫  0 ()} and a coupon with maturity .The price at every  ∈ [0, ] of the coupon is (, ).Then the forward quasi-martingale measure   equivalent to  by the Radon-Nikodym derivative is given as Since hence Using (3) we can have the expression for () [14] We integrate two sides from [, ] exclusively: Using Fubini's theorem [15], we exchange the last item's integral sequence and obtain Let (, , ) = (1/)(1 −  −(−) ), using multidimensional fractional Itô's lemma [16]  (, ) =  [ − ∫   () So the Radon-Nikodym derivative is Using fractional Girsanov's theorem [14] We substitute ∫   () into it; then where Since since When   = , According to the nature of normal distribution, when   = , we have where So where Hence, when (43) Proof.Using Lemma 1, the quasi-martingale measure   equivalent to  by the Radon-Nikodym derivative is given as Using ( 4) we can have the expression for () Then we have So Using fractional Girsanov's theorem where where We substitute   into  {  ≥} ; when   = , then where Using Lemma 2, when   = , we have Let and then where Hence

Numerical Experiments
In this section, we mainly discuss the influence of different parameters on option prices.( For all figures, the horizontal axis shows strike price and the vertical axis shows option value.
Figure 1 is about the influence of  parameter  on option value.As we can see from the figure, the influence will become larger with the increasing of strike prices within limits.

Conclusion
The method of changing measures is widely used for pricing options.In this paper, we develop this method and prove its feasibility in pricing options under the assumption of fractional Brownian motion.What is more, we also take jump process into consideration and obtain the general pricing formula for the European vulnerable option.Finally, we verify its accuracy through the numerical experiments.

Figure 2
Figure2is about the influence of recovery rate  on option value.The default risk will decline with the increasing of recovery rate.So the option prices will fall down.Figures3, 4, and 5 are about the influences of different covariance.According to these figures, we can see that different covariance has different influences on option value.