American Institute of Mathematical Sciences

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

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

Qing-Qing Yang 1, , Wai-Ki Ching 1, , Wanhua He 1, and  Tak-Kuen Siu 2,

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  January 2019 Early access  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.

Keywords: Vulnerable option, regime-switching, change of numéraire, distressed selling, multinomial approach.
Mathematics Subject Classification: Primary: 91G80; Secondary: 90B50.
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 and 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. 

[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.

[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. 

[4]

R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004.

[5]

R. Cont and L. Wagalath, Fire sales forensics: Measuring endogenous risk, Mathematical Finance, 26 (2016), 835-866.  doi: 10.1111/mafi.12071.

[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.

[7]

J. Coval and E. Stafford, Asset fire sales (and purchases) in equity markets, Journal of Financial Economics, 86 (2007), 479-512. 

[8]

D. Duffie and K. Singleton, Modeling Term Structures of Defaultable Bonds, Review of Financial Studies, 12 (1999), 197-226. 

[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.

[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.

[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. 

[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.

[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. 

[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.

[15]

R. Greenwood and D. Thesmar, Stock price fragility, Journal of Financial Economics, 102 (2011), 471-490. 

[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.

[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.

[18]

M. Hung and Y. Liu, Pricing vulnerable options in incomplete markets, Journal of Futures Markets, 25 (2005), 135-170. 

[19]

R. Jarrow and S. Turnbull, Credit risk: Drawing the analogy, Risk Magazine, 5 (1992), 63-70. 

[20]

R. Jarrow and S. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), 53-85. 

[21]

H. Johnson and R. Stulz, The pricing of options with default risk, Journal of Finance, 42 (1987), 267-280. 

[22]

P. Klein, Pricing black-scholes options with correlated credit risk, Journal of Banking and Finance, 20 (1996), 1211-1229. 

[23]

S. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101. 

[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.

[25]

R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. 

[26]

V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, The Journal of Finance, 48 (1993), 1969-1984. 

[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.

[28]

A. Pascucci, PDE and Martingale Methods in Option Pricing, Bocconi & Springer Series, Springer-Verlag, New York, 2011.

[29]

P. Pedler, Occupation time for two state markov chains, Journal of Applied Probability, 8 (1971), 381-390.  doi: 10.2307/3211908.

[30]

W. J. Runggaldier, Jump diffusion models, In S. T. Rachev (Ed.), Handbook of heavy tailed distributions in finance, (2003), 169-209.

[31]

B. Sericola, Occupation times in Markov processes, Stochastic Models, 16 (2000), 479-510.  doi: 10.1080/15326340008807601.

[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.

[33]

A. Shleifer and R. Vishny, Liquidation values and debt capacity: A market equilibrium approach, Journal of Finance, 47 (1992), 1343-1366. 

[34]

A. Shleifer and R. Vishny, Fire sales in finance and macroeconomics, Journal of Economic Perspectives, 25 (2011), 29-48. 

[35]

S. Shreve, Stochastic calculus for finance II: Continuous-time models, Springer Finance Series, (2003), 404-459. 

[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.

[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.

[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.

[39]

R. Wiggins, T. Piontek and A. Metrick, The Lehman Brothers Bankruptcy A: Overview, Yale Program on Financial Stability Case Study, 2014.

[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.

[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.

show all references

References:
[1]

M. Anton and C. Polk, Connected stocks, The Journal of Finance, 69 (2014), 1099-1127. 

[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.

[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. 

[4]

R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004.

[5]

R. Cont and L. Wagalath, Fire sales forensics: Measuring endogenous risk, Mathematical Finance, 26 (2016), 835-866.  doi: 10.1111/mafi.12071.

[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.

[7]

J. Coval and E. Stafford, Asset fire sales (and purchases) in equity markets, Journal of Financial Economics, 86 (2007), 479-512. 

[8]

D. Duffie and K. Singleton, Modeling Term Structures of Defaultable Bonds, Review of Financial Studies, 12 (1999), 197-226. 

[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.

[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.

[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. 

[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.

[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. 

[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.

[15]

R. Greenwood and D. Thesmar, Stock price fragility, Journal of Financial Economics, 102 (2011), 471-490. 

[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.

[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.

[18]

M. Hung and Y. Liu, Pricing vulnerable options in incomplete markets, Journal of Futures Markets, 25 (2005), 135-170. 

[19]

R. Jarrow and S. Turnbull, Credit risk: Drawing the analogy, Risk Magazine, 5 (1992), 63-70. 

[20]

R. Jarrow and S. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), 53-85. 

[21]

H. Johnson and R. Stulz, The pricing of options with default risk, Journal of Finance, 42 (1987), 267-280. 

[22]

P. Klein, Pricing black-scholes options with correlated credit risk, Journal of Banking and Finance, 20 (1996), 1211-1229. 

[23]

S. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101. 

[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.

[25]

R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. 

[26]

V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, The Journal of Finance, 48 (1993), 1969-1984. 

[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.

[28]

A. Pascucci, PDE and Martingale Methods in Option Pricing, Bocconi & Springer Series, Springer-Verlag, New York, 2011.

[29]

P. Pedler, Occupation time for two state markov chains, Journal of Applied Probability, 8 (1971), 381-390.  doi: 10.2307/3211908.

[30]

W. J. Runggaldier, Jump diffusion models, In S. T. Rachev (Ed.), Handbook of heavy tailed distributions in finance, (2003), 169-209.

[31]

B. Sericola, Occupation times in Markov processes, Stochastic Models, 16 (2000), 479-510.  doi: 10.1080/15326340008807601.

[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.

[33]

A. Shleifer and R. Vishny, Liquidation values and debt capacity: A market equilibrium approach, Journal of Finance, 47 (1992), 1343-1366. 

[34]

A. Shleifer and R. Vishny, Fire sales in finance and macroeconomics, Journal of Economic Perspectives, 25 (2011), 29-48. 

[35]

S. Shreve, Stochastic calculus for finance II: Continuous-time models, Springer Finance Series, (2003), 404-459. 

[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.

[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.

[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.

[39]

R. Wiggins, T. Piontek and A. Metrick, The Lehman Brothers Bankruptcy A: Overview, Yale Program on Financial Stability Case Study, 2014.

[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.

[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.

Figure 1.  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.
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Table 1.  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 $N=100$
Table Options

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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 $N=100$
Table 2.  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
Table Options

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$\mathcal{Q}=\left[ \begin{matrix} -5&5 \\ 3&-3 \\ \end{matrix} \right]$ $\mathcal{Q}=\left[ \begin{matrix} -3&3 \\ 5&-5 \\ \end{matrix} \right]$ $\mathcal{Q}=\left[ \begin{matrix} -5&5 \\ 5&-5 \\ \end{matrix} \right]$ $\mathcal{Q}=\left[ \begin{matrix} -3&3 \\ 3&-3 \\ \end{matrix} \right]$
$\frac{S(0)}{K}$ $\chi_0$ 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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