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

Wan-Hua He; Chufang Wu; Jia-Wen Gu; Wai-Ki Ching; Chi-Wing Wong

doi:10.3934/jimo.2021057

Article Contents

2022, Volume 18, Issue 3: 2077-2094. Doi: 10.3934/jimo.2021057

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

1.
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
2.
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
3.
Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

^* Corresponding author: Chufang Wu
^* Corresponding author: Chufang Wu

Received: May 31, 2020

Revised: January 31, 2021

Published: May 2022

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

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Abstract

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.

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

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

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

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

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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 $

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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

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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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