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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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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. |



Parameter | Value | Parameter | Value |
Initial stock price | Correlation( |
||
Strike price | Correlation( |
||
Initial asset price | Correlation( |
||
Total liability | Intensity of Poisson process |
||
Default boundary | Intensity of Poisson process |
||
Deadweight of bankruptcy | Inverse mean-reverting speed | ||
Asset volatility | Total risk premium | ||
Risk-free rate | Standard deviation of |
||
Time to maturity |
Parameter | Value | Parameter | Value |
Initial stock price | Correlation( |
||
Strike price | Correlation( |
||
Initial asset price | Correlation( |
||
Total liability | Intensity of Poisson process |
||
Default boundary | Intensity of Poisson process |
||
Deadweight of bankruptcy | Inverse mean-reverting speed | ||
Asset volatility | Total risk premium | ||
Risk-free rate | Standard deviation of |
||
Time to maturity |
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 |
Strike price | |||||
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 | |||||
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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