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.

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 [

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.

Let the uncertainty in the economy be described by the filtered probability space

Suppose the stock price is given by

Suppose that

Suppose that the interest rate and default intensity follow Vasicek model under the risk neutral measure

In order to prove the theorem, we introduce two lemmas firstly.

We denote by

If

In this section, we intend to discuss pricing vulnerable options in a Fractional Brownian Motion Environment.

We define that the default time is

Note that

Since

Obviously

Since the full path of

For convenience, let

Consider

By means of Lemma

Suppose there is a bank account

Using (

Using fractional

So, we can calculate

Since

Using (

Since

According to the nature of normal distribution, when

Then we will calculate

Since

Using Lemma

Consider

Using Lemma

Using fractional

Then we can calculate

Since

we define

Using fractional

Then we will calculate

Since

The price at time

When there is no jump process,

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

The influence of

Figure

The influence of recovery rate

Figures

The influence of default covariance

The influence of covariance

The influence of covariance

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.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Fundamental Research Funds for the Central Universities (2013XK03).