Abstract
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.
1. 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 quasi-martingale 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.
2. Market Environment
Let the uncertainty in the economy be described by the filtered probability space . is the short-term interest rate which is consistently positive and -measurable in this space. Assume that is a risk neutral martingale measure under which the discounted asset price processes are martingales.
Suppose the stock price is given bywhere is positive constant and is a fractional Brownian motion whose parameter is in the space . is a composite Poisson process, , and represents the th jump range of (using a convention that if there are no jumps). Consider . is a sequence of independent identically distributed random variables with the finite expected value. and are mutually independent and and are mutually independent. is a Poisson process whose intensity is . is a nonnegative adapted stochastic process which is integrable on any finite time interval. , , and are mutually independent.
Suppose that follows log-normal distribution so that ; then we have
Suppose that the interest rate and default intensity follow Vasicek model under the risk neutral measurewhere , and are all positive constants. The covariance matrix of is
In order to prove the theorem, we introduce two lemmas firstly.
Lemma 1 (see [8]). We denote by the quasi-conditional expectation with respect to the risk neutral measure. Then the price at every of a bounded -measurable claim is given by
Lemma 2 (see [12]). If , where , , thenwhere is a standard normal distribution.
3. 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 of an European vulnerable call option with strike price and maturity is given by
Note that ; we have
Since is a filtration, then . 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
For convenience, let , where and ; and are given as follows.
Theorem 3. Considerwhere
Proof. By means of Lemma 1 we can see that the discounted asset price process whose numeraire is is a quasi-martingale. So we can define different equivalent quasi-martingale measures for different numeraires.
Suppose there is a bank account and a coupon with maturity . The price at every of the coupon is . Then the forward quasi-martingale measure equivalent to by the Radon-Nikodym derivative is given as SincehenceUsing (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 obtainLet , using multidimensional fractional Itô’s lemma [16]So the Radon-Nikodym derivative isUsing fractional Girsanov’s theorem [14] and are fractional Brownian motions under measure .
So, we can calculate under measure .
Sincewe define and for convenience. We will calculate firstly.
Using (1) we have the expression for We substitute into it; thenUsing fractional Girsanov’s theorem, we have the expression for under measure whereSince , we substitute into ; thensince When ,According to the nature of normal distribution, when , we havewhere SoThen we will calculate .
Since , we substitute into ; then
Using Lemma 2, when ,where so Hence, when ,
Theorem 4. Considerwhere
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 haveLet ; thenUsing multidimensional fractional Itô lemma, let ; thenSoUsing fractional Girsanov’s theoremwhere Under the probability measure , the processes , and are fractional Brownian motions. The covariance matrix of is the same as .
Then we can calculate under .
Sincewe define and for convenience. We will calculate firstly.
Using fractional Girsanov’s theorem, we have the expression for under measure where We substitute into ; when , thenwhereSoThen we will calculate .
Since , we substitute into Using Lemma 2, when , we haveLet and thenwhere So Hence
Theorem 5. The price at time of the European vulnerable call option is
Note 1. When there is no jump process, , and recovery rate is 1, (65) can be simplified as
4. Numerical Experiments
In this section, we mainly discuss the influence of different parameters on option prices. Figures are about the relationship between option prices and strike prices with different parameters.
The values of different parameters are as follows:
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.
Figure 2 is 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.
Figures 3, 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.
5. 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the Fundamental Research Funds for the Central Universities (2013XK03).