# American Institute of Mathematical Sciences

January  2019, 15(1): 293-318. doi: 10.3934/jimo.2018044

## Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales

 1 Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 2 Department of Actuarial Studies and Business Analytics, Faculty of Business and Economics, Macquarie University, Sydney, Australia

Received  May 2017 Revised  October 2017 Published  April 2018

In this paper, we consider the valuation of vulnerable options under a Markov-modulated jump-diffusion model, where the option writer's asset value is subject to price pressure from other financial institutions due to distressed selling. A change of numéraire technique, proposed by Geman et al. [14], is employed to obtain a semi-analytical pricing formula for an vulnerable European option in the presence of regime switching effect. The method is numerically implemented using the multinomial approach in Costabile et al. [6]. We study the impacts of distressed selling and regime switching on the European option prices via numerical experiments.

Citation: Qing-Qing Yang, Wai-Ki Ching, Wanhua He, Tak-Kuen Siu. Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales. Journal of Industrial & Management Optimization, 2019, 15 (1) : 293-318. doi: 10.3934/jimo.2018044
##### References:
 [1] M. Anton and C. Polk, Connected stocks, The Journal of Finance, 69 (2014), 1099-1127.   Google Scholar [2] F. Black and M. Scholes, The valuation of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar [3] G. Cheang and G. Teh, Change of num$\acute{e}$raire and a jump-diffusion option pricing formula, Springer International Publishing, (2014), 371-389.   Google Scholar [4] R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar [5] R. Cont and L. Wagalath, Fire sales forensics: Measuring endogenous risk, Mathematical Finance, 26 (2016), 835-866.  doi: 10.1111/mafi.12071.  Google Scholar [6] M. Costabile, A. Leccadito, I. Massabó and E. Russo, Option Pricing under Regime-switching Jump-diffusion Models, Journal of Computational and Applied Mathematics, 256 (2014), 152-167.  doi: 10.1016/j.cam.2013.07.046.  Google Scholar [7] J. Coval and E. Stafford, Asset fire sales (and purchases) in equity markets, Journal of Financial Economics, 86 (2007), 479-512.   Google Scholar [8] D. Duffie and K. Singleton, Modeling Term Structures of Defaultable Bonds, Review of Financial Studies, 12 (1999), 197-226.   Google Scholar [9] R. Elliott, T. Siu, L. Chan and J. Lau, Pricing options under a generalized Markov-modulated jump-diffusion model, Stochastic Analysis and Applications, 25 (2007), 821-843.  doi: 10.1080/07362990701420118.  Google Scholar [10] R. Elliott and T. Siu, Option pricing and filtering with hidden Markov-modulated pure-jump processes, Applied Mathematical Finance, 20 (2013), 1-25.  doi: 10.1080/1350486X.2012.655929.  Google Scholar [11] R. Elliott and T. Siu, Asset pricing using trading volumes in a hidden regime-switching environment, Asia-Pacific Financial Market, 22 (2015), 133-149.   Google Scholar [12] F. A. Fard, Analytical pricing of vulnerable options under a generalized jump-diffusion model, Insurance: Mathematics and Economics, 60 (2015), 19-28.  doi: 10.1016/j.insmatheco.2014.10.007.  Google Scholar [13] I. Florescu, R. Liu and M. Mariani, Solutions to a partial integro-differential parabolic system arising in the pricing of financial options in regime-switching jump diffusion models, Electronic Journal of Differential Equations, 2012 (2012), 1-12.   Google Scholar [14] H. Geman, N. Karoui and J. Rochet, Change of numéaire, changes of probability measure and option pricing, Journal of Applied Probability, 32 (1995), 443-458.  doi: 10.2307/3215299.  Google Scholar [15] R. Greenwood and D. Thesmar, Stock price fragility, Journal of Financial Economics, 102 (2011), 471-490.   Google Scholar [16] X. Guo and Q. Zhang, Closed-form solutions for perpetual American put options with regime switching, SIAM Journal on Applied Mathematics, 64 (2004), 2034-2049.  doi: 10.1137/S0036139903426083.  Google Scholar [17] T. Hida, J. Potthoff and L. Streit, Dirichlet forms and white noise analysis, Communications in Mathematical Physics, 116 (1988), 235-245.  doi: 10.1007/BF01225257.  Google Scholar [18] M. Hung and Y. Liu, Pricing vulnerable options in incomplete markets, Journal of Futures Markets, 25 (2005), 135-170.   Google Scholar [19] R. Jarrow and S. Turnbull, Credit risk: Drawing the analogy, Risk Magazine, 5 (1992), 63-70.   Google Scholar [20] R. Jarrow and S. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), 53-85.   Google Scholar [21] H. Johnson and R. Stulz, The pricing of options with default risk, Journal of Finance, 42 (1987), 267-280.   Google Scholar [22] P. Klein, Pricing black-scholes options with correlated credit risk, Journal of Banking and Finance, 20 (1996), 1211-1229.   Google Scholar [23] S. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.   Google Scholar [24] L. Liew and T. Siu, A hidden markov regime-switching model for option valuation, Insurance: Mathematics and Economics, 47 (2010), 374-384.  doi: 10.1016/j.insmatheco.2010.08.003.  Google Scholar [25] R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.   Google Scholar [26] V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, The Journal of Finance, 48 (1993), 1969-1984.   Google Scholar [27] 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.  Google Scholar [28] A. Pascucci, PDE and Martingale Methods in Option Pricing, Bocconi & Springer Series, Springer-Verlag, New York, 2011.  Google Scholar [29] P. Pedler, Occupation time for two state markov chains, Journal of Applied Probability, 8 (1971), 381-390.  doi: 10.2307/3211908.  Google Scholar [30] W. J. Runggaldier, Jump diffusion models, In S. T. Rachev (Ed.), Handbook of heavy tailed distributions in finance, (2003), 169-209. Google Scholar [31] B. Sericola, Occupation times in Markov processes, Stochastic Models, 16 (2000), 479-510.  doi: 10.1080/15326340008807601.  Google Scholar [32] Y. Shen and T. Siu, Pricing variance swaps under a stochastic interest rate and volatility model with regime-switching, Operations Research Letters, 41 (2013), 180-187.  doi: 10.1016/j.orl.2012.12.008.  Google Scholar [33] A. Shleifer and R. Vishny, Liquidation values and debt capacity: A market equilibrium approach, Journal of Finance, 47 (1992), 1343-1366.   Google Scholar [34] A. Shleifer and R. Vishny, Fire sales in finance and macroeconomics, Journal of Economic Perspectives, 25 (2011), 29-48.   Google Scholar [35] S. Shreve, Stochastic calculus for finance II: Continuous-time models, Springer Finance Series, (2003), 404-459.   Google Scholar [36] T. Siu, J. Lau and H. Yang, Pricing participating products under a generalized jump-diffusion, Journal of Applied Mathematics and Stochastic Analysis, (2008), Article ID 474623, 30 Pages. doi: 10.1155/2008/474623.  Google Scholar [37] T. Siu, A BSDE approach to optimal investment of an insurer with hidden regime switching, Stochastic Analysis and Applications, 31 (2013), 1-18.  doi: 10.1080/07362994.2012.727144.  Google Scholar [38] T. Siu, A stochastic flows approach for asset allocation with hidden economic environment, International Journal of Stochastic Analysis, (2015), Article ID 462524, 11 pages. doi: 10.1155/2015/462524.  Google Scholar [39] R. Wiggins, T. Piontek and A. Metrick, The Lehman Brothers Bankruptcy A: Overview, Yale Program on Financial Stability Case Study, 2014. Google Scholar [40] S. Yang, M. Lee and J. Kim, Pricing Vulnerable Options under a Stochastic Volatility Model, Applied Mathematics Letters, 34 (2014), 7-12.  doi: 10.1016/j.aml.2014.03.007.  Google Scholar [41] Q. Yang, W. Ching, J. Gu and T. Siu, Optimal liquidation strategy across multiple exchanges under a jump-diffusion fast mean-reverting model, (2016), available at arXiv: 1607.04553. Google Scholar

show all references

##### References:
 [1] M. Anton and C. Polk, Connected stocks, The Journal of Finance, 69 (2014), 1099-1127.   Google Scholar [2] F. Black and M. Scholes, The valuation of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar [3] G. Cheang and G. Teh, Change of num$\acute{e}$raire and a jump-diffusion option pricing formula, Springer International Publishing, (2014), 371-389.   Google Scholar [4] R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar [5] R. Cont and L. Wagalath, Fire sales forensics: Measuring endogenous risk, Mathematical Finance, 26 (2016), 835-866.  doi: 10.1111/mafi.12071.  Google Scholar [6] M. Costabile, A. Leccadito, I. Massabó and E. Russo, Option Pricing under Regime-switching Jump-diffusion Models, Journal of Computational and Applied Mathematics, 256 (2014), 152-167.  doi: 10.1016/j.cam.2013.07.046.  Google Scholar [7] J. Coval and E. Stafford, Asset fire sales (and purchases) in equity markets, Journal of Financial Economics, 86 (2007), 479-512.   Google Scholar [8] D. Duffie and K. Singleton, Modeling Term Structures of Defaultable Bonds, Review of Financial Studies, 12 (1999), 197-226.   Google Scholar [9] R. Elliott, T. Siu, L. Chan and J. Lau, Pricing options under a generalized Markov-modulated jump-diffusion model, Stochastic Analysis and Applications, 25 (2007), 821-843.  doi: 10.1080/07362990701420118.  Google Scholar [10] R. Elliott and T. Siu, Option pricing and filtering with hidden Markov-modulated pure-jump processes, Applied Mathematical Finance, 20 (2013), 1-25.  doi: 10.1080/1350486X.2012.655929.  Google Scholar [11] R. Elliott and T. Siu, Asset pricing using trading volumes in a hidden regime-switching environment, Asia-Pacific Financial Market, 22 (2015), 133-149.   Google Scholar [12] F. A. Fard, Analytical pricing of vulnerable options under a generalized jump-diffusion model, Insurance: Mathematics and Economics, 60 (2015), 19-28.  doi: 10.1016/j.insmatheco.2014.10.007.  Google Scholar [13] I. Florescu, R. Liu and M. Mariani, Solutions to a partial integro-differential parabolic system arising in the pricing of financial options in regime-switching jump diffusion models, Electronic Journal of Differential Equations, 2012 (2012), 1-12.   Google Scholar [14] H. Geman, N. Karoui and J. Rochet, Change of numéaire, changes of probability measure and option pricing, Journal of Applied Probability, 32 (1995), 443-458.  doi: 10.2307/3215299.  Google Scholar [15] R. Greenwood and D. Thesmar, Stock price fragility, Journal of Financial Economics, 102 (2011), 471-490.   Google Scholar [16] X. Guo and Q. Zhang, Closed-form solutions for perpetual American put options with regime switching, SIAM Journal on Applied Mathematics, 64 (2004), 2034-2049.  doi: 10.1137/S0036139903426083.  Google Scholar [17] T. Hida, J. Potthoff and L. Streit, Dirichlet forms and white noise analysis, Communications in Mathematical Physics, 116 (1988), 235-245.  doi: 10.1007/BF01225257.  Google Scholar [18] M. Hung and Y. Liu, Pricing vulnerable options in incomplete markets, Journal of Futures Markets, 25 (2005), 135-170.   Google Scholar [19] R. Jarrow and S. Turnbull, Credit risk: Drawing the analogy, Risk Magazine, 5 (1992), 63-70.   Google Scholar [20] R. Jarrow and S. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), 53-85.   Google Scholar [21] H. Johnson and R. Stulz, The pricing of options with default risk, Journal of Finance, 42 (1987), 267-280.   Google Scholar [22] P. Klein, Pricing black-scholes options with correlated credit risk, Journal of Banking and Finance, 20 (1996), 1211-1229.   Google Scholar [23] S. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.   Google Scholar [24] L. Liew and T. Siu, A hidden markov regime-switching model for option valuation, Insurance: Mathematics and Economics, 47 (2010), 374-384.  doi: 10.1016/j.insmatheco.2010.08.003.  Google Scholar [25] R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.   Google Scholar [26] V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, The Journal of Finance, 48 (1993), 1969-1984.   Google Scholar [27] 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.  Google Scholar [28] A. Pascucci, PDE and Martingale Methods in Option Pricing, Bocconi & Springer Series, Springer-Verlag, New York, 2011.  Google Scholar [29] P. Pedler, Occupation time for two state markov chains, Journal of Applied Probability, 8 (1971), 381-390.  doi: 10.2307/3211908.  Google Scholar [30] W. J. Runggaldier, Jump diffusion models, In S. T. Rachev (Ed.), Handbook of heavy tailed distributions in finance, (2003), 169-209. Google Scholar [31] B. Sericola, Occupation times in Markov processes, Stochastic Models, 16 (2000), 479-510.  doi: 10.1080/15326340008807601.  Google Scholar [32] Y. Shen and T. Siu, Pricing variance swaps under a stochastic interest rate and volatility model with regime-switching, Operations Research Letters, 41 (2013), 180-187.  doi: 10.1016/j.orl.2012.12.008.  Google Scholar [33] A. Shleifer and R. Vishny, Liquidation values and debt capacity: A market equilibrium approach, Journal of Finance, 47 (1992), 1343-1366.   Google Scholar [34] A. Shleifer and R. Vishny, Fire sales in finance and macroeconomics, Journal of Economic Perspectives, 25 (2011), 29-48.   Google Scholar [35] S. Shreve, Stochastic calculus for finance II: Continuous-time models, Springer Finance Series, (2003), 404-459.   Google Scholar [36] T. Siu, J. Lau and H. Yang, Pricing participating products under a generalized jump-diffusion, Journal of Applied Mathematics and Stochastic Analysis, (2008), Article ID 474623, 30 Pages. doi: 10.1155/2008/474623.  Google Scholar [37] T. Siu, A BSDE approach to optimal investment of an insurer with hidden regime switching, Stochastic Analysis and Applications, 31 (2013), 1-18.  doi: 10.1080/07362994.2012.727144.  Google Scholar [38] T. Siu, A stochastic flows approach for asset allocation with hidden economic environment, International Journal of Stochastic Analysis, (2015), Article ID 462524, 11 pages. doi: 10.1155/2015/462524.  Google Scholar [39] R. Wiggins, T. Piontek and A. Metrick, The Lehman Brothers Bankruptcy A: Overview, Yale Program on Financial Stability Case Study, 2014. Google Scholar [40] S. Yang, M. Lee and J. Kim, Pricing Vulnerable Options under a Stochastic Volatility Model, Applied Mathematics Letters, 34 (2014), 7-12.  doi: 10.1016/j.aml.2014.03.007.  Google Scholar [41] Q. Yang, W. Ching, J. Gu and T. Siu, Optimal liquidation strategy across multiple exchanges under a jump-diffusion fast mean-reverting model, (2016), available at arXiv: 1607.04553. Google Scholar
Vulnerable European call option price against spot-to-strike ratio. The figure above shows the trend of the time-zero call price conditional on the Markov chain being in the good state at the initial time, and the figure below illustrates the momentum of the time zero call price conditional on the Markov chain being in the bad state at the initial time.
The choices of the jump parameters and volatility parameters indicate that the risky asset values have larger jumps and larger volatilities in the bad economic state than in the good one.
 Parameters Values Parameters Values Dimension of $W_t$ $n=4$ Tolerance level $\epsilon=0.01$ Transition rate $\theta_1=5$ Transition rate $\theta_2=5$ Jump in $S$ $u^{S,1}=1.015$ Jump in $S$ $d^{S,1}=0.98$ $u^{S,2}=1.25$ $d^{S,2}=0.75$ Jump in $B$ $u^{B,1}=1.0125$ Jump in $B$ $d^{B,1}=0.99$ $u^{B,2}=1.1250$ $d^{B,2}=0.85$ Jump in $X$ $u^{X,1}=1.04$ Jump in $X$ $d^{X,1}=0.97$ $u^{X,2}=1.15$ $d^{X,2}=0.85$ Jump in $V$ $u^{V, 1}=1.03$ Jump in $V$ $d^{V, 1}=0.96$ $u^{V, 2}=1.36$ $d^{V, 2}=0.75$ Probability $p_1=0.75$ Probability $p_2=0.5$ Market depth $L=5000$ MLR $\eta=10$ $\alpha^S(1)$ $(0.1,0.15,0.1,0.2)$ $\alpha^S(2)$ $(0.3,0.15,0.13,0.3)$ $\alpha^B(1)$ $(0.2,0.12,0.13,0.25)$ $\alpha^B(2)$ $(0.02,0.1,0.3,0.005)$ $\alpha^X(1)$ $(0.26,0.4,0.1,0.15)$ $\alpha^X(2)$ $(0.6,0.4,0.16,0.1)$ $\alpha^V(1)$ $(0.1,0.2,0.1,0.2)$ $\alpha^V(2)$ $(0.1,0.3,0.2,0.27)$ Intensity $\lambda_1=2$ Intensity $\lambda_2=3$ Default boundary $d^*=5$ Outstanding Claims $d=25$ Deadweight cost $\alpha=0.4$ Time to maturity $T=1$ Initial price $S_0=40$ Strike price $K=40$ Initial price $V_0=50$ Initial price $X_0=100$ Initial price $B(0,T)=0.05$ Time steps $N=100$
 Parameters Values Parameters Values Dimension of $W_t$ $n=4$ Tolerance level $\epsilon=0.01$ Transition rate $\theta_1=5$ Transition rate $\theta_2=5$ Jump in $S$ $u^{S,1}=1.015$ Jump in $S$ $d^{S,1}=0.98$ $u^{S,2}=1.25$ $d^{S,2}=0.75$ Jump in $B$ $u^{B,1}=1.0125$ Jump in $B$ $d^{B,1}=0.99$ $u^{B,2}=1.1250$ $d^{B,2}=0.85$ Jump in $X$ $u^{X,1}=1.04$ Jump in $X$ $d^{X,1}=0.97$ $u^{X,2}=1.15$ $d^{X,2}=0.85$ Jump in $V$ $u^{V, 1}=1.03$ Jump in $V$ $d^{V, 1}=0.96$ $u^{V, 2}=1.36$ $d^{V, 2}=0.75$ Probability $p_1=0.75$ Probability $p_2=0.5$ Market depth $L=5000$ MLR $\eta=10$ $\alpha^S(1)$ $(0.1,0.15,0.1,0.2)$ $\alpha^S(2)$ $(0.3,0.15,0.13,0.3)$ $\alpha^B(1)$ $(0.2,0.12,0.13,0.25)$ $\alpha^B(2)$ $(0.02,0.1,0.3,0.005)$ $\alpha^X(1)$ $(0.26,0.4,0.1,0.15)$ $\alpha^X(2)$ $(0.6,0.4,0.16,0.1)$ $\alpha^V(1)$ $(0.1,0.2,0.1,0.2)$ $\alpha^V(2)$ $(0.1,0.3,0.2,0.27)$ Intensity $\lambda_1=2$ Intensity $\lambda_2=3$ Default boundary $d^*=5$ Outstanding Claims $d=25$ Deadweight cost $\alpha=0.4$ Time to maturity $T=1$ Initial price $S_0=40$ Strike price $K=40$ Initial price $V_0=50$ Initial price $X_0=100$ Initial price $B(0,T)=0.05$ Time steps $N=100$
We use the multinomial recombining grid approximation method to calculate the vulnerable European call option prices with different strike prices, initial states and state persistence under different setting of market impact, without market impact (No) and subject to market impact (Impact). The default choices are given by the basic parameters in Table 1. We present, in this table, a comparison for the vulnerable European call option prices subject to different levels of persistence of the underlying economic state process in the good and bad economic states.
 $\mathcal{Q}=\left[ \begin{matrix} -5&5 \\ 3&-3 \\ \end{matrix} \right]$ $\mathcal{Q}=\left[ \begin{matrix} -3&3 \\ 5&-5 \\ \end{matrix} \right]$ $\mathcal{Q}=\left[ \begin{matrix} -5&5 \\ 5&-5 \\ \end{matrix} \right]$ $\mathcal{Q}=\left[ \begin{matrix} -3&3 \\ 3&-3 \\ \end{matrix} \right]$ $\frac{S(0)}{K}$ $\chi_0$ No Impact No Impact No Impact No Impact 0.8 1 0.5445 0.2324 0.0469 0.0200 0.0836 0.0357 0.3153 0.3146 2 1.7845 0.7617 0.2812 0.1200 0.4341 0.1853 1.1786 0.5030 1.0 1 0.5520 0.2354 0.0475 0.0203 0.0848 0.0362 0.3197 0.1363 2 1.8093 0.7716 0.2851 0.1216 0.4403 0.1877 1.1949 0.5096 1.25 1 0.5581 0.2378 0.0480 0.0205 0.0857 0.0365 0.3232 0.1377 2 1.8291 0.7796 0.2882 0.1228 0.4450 0.1897 1.2080 0.5148
 $\mathcal{Q}=\left[ \begin{matrix} -5&5 \\ 3&-3 \\ \end{matrix} \right]$ $\mathcal{Q}=\left[ \begin{matrix} -3&3 \\ 5&-5 \\ \end{matrix} \right]$ $\mathcal{Q}=\left[ \begin{matrix} -5&5 \\ 5&-5 \\ \end{matrix} \right]$ $\mathcal{Q}=\left[ \begin{matrix} -3&3 \\ 3&-3 \\ \end{matrix} \right]$ $\frac{S(0)}{K}$ $\chi_0$ No Impact No Impact No Impact No Impact 0.8 1 0.5445 0.2324 0.0469 0.0200 0.0836 0.0357 0.3153 0.3146 2 1.7845 0.7617 0.2812 0.1200 0.4341 0.1853 1.1786 0.5030 1.0 1 0.5520 0.2354 0.0475 0.0203 0.0848 0.0362 0.3197 0.1363 2 1.8093 0.7716 0.2851 0.1216 0.4403 0.1877 1.1949 0.5096 1.25 1 0.5581 0.2378 0.0480 0.0205 0.0857 0.0365 0.3232 0.1377 2 1.8291 0.7796 0.2882 0.1228 0.4450 0.1897 1.2080 0.5148
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