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# Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility

• * Corresponding author: Chufang Wu

The first author is supported by Research Grants Council of Hong Kong under Grant Number 17301519

• In this paper, we propose a model to price vulnerable European options where the dynamics of the underlying asset value and the counter-party's asset value follow two jump-diffusion processes with fast mean-reverting stochastic volatility. First, we derive an equivalent risk-neutral measure and transfer the pricing problem into solving a partial differential equation (PDE) by the Feynman-Kac formula. We then approximate the solution of the PDE by pricing formulas with constant volatility via multi-scale asymptotic method. The pricing formula for vulnerable European options is obtained by applying a two-dimensional Laplace transform when the dynamics of the underlying asset value and the counter-party's asset value follow two correlated jump-diffusion processes with constant volatilities. Thus, an analytic approximation formula for the vulnerable European options is derived in our setting. Numerical experiments are given to demonstrate our method by using Laplace inversion.

Mathematics Subject Classification: Primary: 91G20, 35R60; Secondary: 35C20.

 Citation: • • Figure 1.  The relationship between initial stock price $S_0$ and the correction term $P_1$ under difference values of strike price $K$

Figure 2.  The relationship between initial asset value $V_0$ and the correction term $P_1$ under different values of total liability $D$

Figure 3.  The relationship between correlation parameters $\rho_{sy}$ (rho0), $\rho_{sv}$ (rho1), $\rho_{vy}$ (rho2) and the correction term $P_1$

Table 1.  Preference parameters

 Parameter Value Parameter Value Initial stock price $S_0=40$ Correlation($S$ $\&$ $Y$) $\rho_{sy}=-0.1$ Strike price $K=40$ Correlation($S$ $\&$ $V$) $\rho_{sv}=0.2$ Initial asset price $V_0=100$ Correlation($Y$ $\&$ $V$) $\rho_{vy}=0.1$ Total liability $D=100$ Intensity of Poisson process $N_t^S$ $\lambda^S=1$ Default boundary $\tilde{D}=100$ Intensity of Poisson process $N_t^V$ $\lambda^V=0$ Deadweight of bankruptcy $\gamma=0.6$ Inverse mean-reverting speed $\epsilon=0.001$ Asset volatility $\sigma=0.2$ Total risk premium $\Lambda=2$ Risk-free rate $r=0.05$ Standard deviation of $Y$ $u=\frac{1}{\sqrt{2}}$ Time to maturity $T=1$

Table 2.  Numerical approximation for the option price $P^\epsilon$

 Strike price T=0.5 T=1 T=1.5 T=2 30 7.9925 8.6168 9.2444 9.8764 35 4.7952 5.4263 6.0566 6.6868 40 2.2740 2.8658 3.4625 4.0619 45 0.8294 1.2551 1.7232 2.2222 50 0.2386 0.4569 0.7373 1.0737

Table 3.  Numerical approximation for the option price $P^\epsilon$

 Strike price $\lambda^S=0.5$ $\lambda^S=1$ $\lambda^S=1.5$ $\lambda^S=2$ $\lambda^S=2.5$ 30 8.6090 8.6168 8.6257 8.6355 8.6462 35 5.3874 5.4263 5.4653 5.5045 5.5438 40 2.7907 2.8658 2.9387 3.0095 3.0785 45 1.1744 1.2551 1.3336 1.4098 1.4839 50 0.3964 0.4569 0.5167 0.5756 0.6338
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