# American Institute of Mathematical Sciences

May  2022, 18(3): 2077-2094. doi: 10.3934/jimo.2021057

## 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

Received  June 2020 Revised  February 2021 Published  May 2022 Early access  March 2021

Fund Project: 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.

Citation: Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2077-2094. doi: 10.3934/jimo.2021057
##### References:
 [1] B. Eraker, Do stock prices and volatility jump? Reconciling evidence from spot and option prices, The Journal of Finance, 59 (2004), 1367-1403.  doi: 10.1111/j.1540-6261.2004.00666.x. [2] J. P. Fouque, G. Papanicolaou and K. R. Sircar, Asymptotics of a two-scale stochastic volatility model, Equations aux Derivees Partielles et Applications, in honour of Jacques-Louis Lions, (1998), 517–525. [3] J. P. Fouque, G. Papanicolaou and K. R. Sircar, Mean-reverting stochastic volatility, International Journal of Theoretical and Applied Finance, 3 (2000), 101-142.  doi: 10.1142/S0219024900000061. [4] J. P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge: Cambridge University Press, 2011. doi: 10.1017/CBO9781139020534. [5] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [6] M. W. Hung and Y. H. Liu, Pricing vulnerable options in incomplete markets, The Journal of Future Markets, 25 (2005), 135-170.  doi: 10.1002/fut.20136. [7] J. Hull and A. White, The impact of default risk on the prices of options and other derivative securities, Journal of Banking and Finance, 19 (1995), 299-322.  doi: 10.1016/0378-4266(94)00050-D. [8] R. A. Jarrow and S. M. Turnbull, Pricing derivatives on financial securities subject to credit risk, The Journal of Finance, 50 (1995), 53-85.  doi: 10.1111/j.1540-6261.1995.tb05167.x. [9] H. Johnson and R. Stulz, The pricing of options with default risk, The Journal of Finance, 42 (1987), 267-280.  doi: 10.1111/j.1540-6261.1987.tb02567.x. [10] P. Klein, Pricing black-scholes options with correlated credit risk, Journal of Banking and Finance, 20 (1996), 1211-1229.  doi: 10.1016/0378-4266(95)00052-6. [11] P. Klein and M. Inglis, Valuation of European options subject to financial distress and interest rate risk, The Journal of Derivatives, 6 (1999), 44-56.  doi: 10.3905/jod.1999.319118. [12] P. Klein and M. Inglis, Pricing vulnerable European options when the option's payoff can increase the risk of financial distress, Journal of Banking & Finance, 25 (2001), 993-1012.  doi: 10.1016/S0378-4266(00)00109-6. [13] P. Klein and J. Yang, Vulnearable American options, Managerial Finance, 36 (2010), 414-430. [14] S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.  doi: 10.1287/mnsc.48.8.1086.166. [15] S. I. Liu and Y. C. Liu, Pricing vulnerable options by binomial trees, Available at SSRN 1365864 (2009). doi: 10.2139/ssrn.1365864. [16] C. Ma, S. Yue and Y. Ren, Pricing vulnerable European options under lévy process with stochastic volatility, Discrete Dynamics in Nature and Society, (2018), 3402703, 16 pp. doi: 10.1155/2018/3402703. [17] A. Melino and S. M. Turnbull, Pricing foreign currency options with stochastic volatility, Jounral of Econometrics, 45 (1990), 239-265.  doi: 10.1016/0304-4076(90)90100-8. [18] R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2. [19] N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081.  doi: 10.1287/mnsc.1110.1393. [20] H. Niu and D. Wang, Pricing vulnerable European options under a two-sided jump model via Laplace transform, Scientia Sinica Mathematica, 45 (2015), 195-212.  doi: 10.1360/012015-3. [21] H. Niu, and D. Wang, Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy, Quantitative Finance, 16 (2016), 1129-1145. doi: 10.1080/14697688.2015.1090623. [22] G. Petrella, An extension of the Euler Laplace transform inversion algorithm with applications in option pricing, Operations Research Letters, 32 (2004), 380-389.  doi: 10.1016/j.orl.2003.06.004. [23] A. G. Ramm, A simple proof of the Fredholm alternative and a characterization of the Fredholm operator, The American Mathematical Monthly, 108 (2001), 855-860.  doi: 10.1080/00029890.2001.11919820. [24] S. E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Vol. 11, Springer Science & Business Media, 2004. [25] L. Tian, G. Wang, X. Wang and Y. Wang, Pricing vulnerable options with correlated credit risk under jump-diffusion processes, The Journal of Futures Markets, 34 (2014), 957-979.  doi: 10.1002/fut.21629. [26] G. Wang, X. Wang and K. Zhou, Pricing vulnerable options with stochastic volatility, Physica A: Statistical Mechanics and its Applications, 485 (2017), 91-103.  doi: 10.1016/j.physa.2017.04.146. [27] W. Xu, W. Xu, H. Li and W. Xiao, A jump-diffusion approach to modelling vulnerable option pricing, Finance Research Letters, 9 (2012), 48-56.  doi: 10.1016/j.frl.2011.07.001. [28] S. J. Yang, M. K. Lee and J. H. Kim, Pricing vulnerable options under a stochastic volatility model, Applied Mathematics Letters, 34 (2014), 7-12.  doi: 10.1016/j.aml.2014.03.007.

show all references

##### References:
 [1] B. Eraker, Do stock prices and volatility jump? Reconciling evidence from spot and option prices, The Journal of Finance, 59 (2004), 1367-1403.  doi: 10.1111/j.1540-6261.2004.00666.x. [2] J. P. Fouque, G. Papanicolaou and K. R. Sircar, Asymptotics of a two-scale stochastic volatility model, Equations aux Derivees Partielles et Applications, in honour of Jacques-Louis Lions, (1998), 517–525. [3] J. P. Fouque, G. Papanicolaou and K. R. Sircar, Mean-reverting stochastic volatility, International Journal of Theoretical and Applied Finance, 3 (2000), 101-142.  doi: 10.1142/S0219024900000061. [4] J. P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge: Cambridge University Press, 2011. doi: 10.1017/CBO9781139020534. [5] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [6] M. W. Hung and Y. H. Liu, Pricing vulnerable options in incomplete markets, The Journal of Future Markets, 25 (2005), 135-170.  doi: 10.1002/fut.20136. [7] J. Hull and A. White, The impact of default risk on the prices of options and other derivative securities, Journal of Banking and Finance, 19 (1995), 299-322.  doi: 10.1016/0378-4266(94)00050-D. [8] R. A. Jarrow and S. M. Turnbull, Pricing derivatives on financial securities subject to credit risk, The Journal of Finance, 50 (1995), 53-85.  doi: 10.1111/j.1540-6261.1995.tb05167.x. [9] H. Johnson and R. Stulz, The pricing of options with default risk, The Journal of Finance, 42 (1987), 267-280.  doi: 10.1111/j.1540-6261.1987.tb02567.x. [10] P. Klein, Pricing black-scholes options with correlated credit risk, Journal of Banking and Finance, 20 (1996), 1211-1229.  doi: 10.1016/0378-4266(95)00052-6. [11] P. Klein and M. Inglis, Valuation of European options subject to financial distress and interest rate risk, The Journal of Derivatives, 6 (1999), 44-56.  doi: 10.3905/jod.1999.319118. [12] P. Klein and M. Inglis, Pricing vulnerable European options when the option's payoff can increase the risk of financial distress, Journal of Banking & Finance, 25 (2001), 993-1012.  doi: 10.1016/S0378-4266(00)00109-6. [13] P. Klein and J. Yang, Vulnearable American options, Managerial Finance, 36 (2010), 414-430. [14] S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.  doi: 10.1287/mnsc.48.8.1086.166. [15] S. I. Liu and Y. C. Liu, Pricing vulnerable options by binomial trees, Available at SSRN 1365864 (2009). doi: 10.2139/ssrn.1365864. [16] C. Ma, S. Yue and Y. Ren, Pricing vulnerable European options under lévy process with stochastic volatility, Discrete Dynamics in Nature and Society, (2018), 3402703, 16 pp. doi: 10.1155/2018/3402703. [17] A. Melino and S. M. Turnbull, Pricing foreign currency options with stochastic volatility, Jounral of Econometrics, 45 (1990), 239-265.  doi: 10.1016/0304-4076(90)90100-8. [18] R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2. [19] N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081.  doi: 10.1287/mnsc.1110.1393. [20] H. Niu and D. Wang, Pricing vulnerable European options under a two-sided jump model via Laplace transform, Scientia Sinica Mathematica, 45 (2015), 195-212.  doi: 10.1360/012015-3. [21] H. Niu, and D. Wang, Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy, Quantitative Finance, 16 (2016), 1129-1145. doi: 10.1080/14697688.2015.1090623. [22] G. Petrella, An extension of the Euler Laplace transform inversion algorithm with applications in option pricing, Operations Research Letters, 32 (2004), 380-389.  doi: 10.1016/j.orl.2003.06.004. [23] A. G. Ramm, A simple proof of the Fredholm alternative and a characterization of the Fredholm operator, The American Mathematical Monthly, 108 (2001), 855-860.  doi: 10.1080/00029890.2001.11919820. [24] S. E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Vol. 11, Springer Science & Business Media, 2004. [25] L. Tian, G. Wang, X. Wang and Y. Wang, Pricing vulnerable options with correlated credit risk under jump-diffusion processes, The Journal of Futures Markets, 34 (2014), 957-979.  doi: 10.1002/fut.21629. [26] G. Wang, X. Wang and K. Zhou, Pricing vulnerable options with stochastic volatility, Physica A: Statistical Mechanics and its Applications, 485 (2017), 91-103.  doi: 10.1016/j.physa.2017.04.146. [27] W. Xu, W. Xu, H. Li and W. Xiao, A jump-diffusion approach to modelling vulnerable option pricing, Finance Research Letters, 9 (2012), 48-56.  doi: 10.1016/j.frl.2011.07.001. [28] S. J. Yang, M. K. Lee and J. H. Kim, Pricing vulnerable options under a stochastic volatility model, Applied Mathematics Letters, 34 (2014), 7-12.  doi: 10.1016/j.aml.2014.03.007.
The relationship between initial stock price $S_0$ and the correction term $P_1$ under difference values of strike price $K$
The relationship between initial asset value $V_0$ and the correction term $P_1$ under different values of total liability $D$
The relationship between correlation parameters $\rho_{sy}$ (rho0), $\rho_{sv}$ (rho1), $\rho_{vy}$ (rho2) and the correction term $P_1$
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 $T=1$
 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 $T=1$
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
 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
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
 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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