American Institute of Mathematical Sciences

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September  2017, 22(7): 2595-2626. doi: 10.3934/dcdsb.2017100

Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model

Tak Kuen Siu 1,, and  Yang Shen 2,

1. 

Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia
2. 

Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

* Corresponding author

Received  July 2016 Revised  August 2016 Published  March 2017

Fund Project: Tak Kuen Siu would like to acknowledge a Discovery Grant from the Australian Research Council (ARC), (Project No.: DP130103517). Yang Shen would like to acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), (Project No.: RGPIN-2016-05677).

A risk-minimizing approach to pricing contingent claims in a general non-Markovian, regime-switching, jump-diffusion model is discussed, where a convex risk measure is used to describe risk. The pricing problem is formulated as a two-person, zero-sum, stochastic differential game between the seller of a contingent claim and the market, where the latter may be interpreted as a ''fictitious'' player. A backward stochastic differential equation (BSDE) approach is applied to discuss the game problem. Attention is given to the entropic risk measure, which is a particular type of convex risk measures. In this situation, a pricing kernel selected by an equilibrium state of the game problem is related to the one selected by the Esscher transform, which was introduced to the option-pricing world in the seminal work by [38].

Keywords: Game theory, option valuation, non-Markovian regime-switching jump diffusion, convex risk measures, backward stochastic differential equations, Esscher transforms.
Mathematics Subject Classification: Primary:58F15, 58F17;Secondary:53C35.
Citation: Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100
References:
[1]

P. Artzner, F. Delbaen, J. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.

[2]

A. Badescu, R. J. Elliott and T. K. Siu, Esscher transforms and consumption-based models, Insurance: Mathematics and Economics, 45 (2009), 337-347.  doi: 10.1016/j.insmatheco.2009.08.001.

[3]

P. Barrieu and N. El Karoui, Inf-convolution of risk measures and optimal risk transfer, Finance and Stochastics, 9 (2005), 269-298.  doi: 10.1007/s00780-005-0152-0.

[4]

P. Barrieu and N. El Karoui, Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: R. Carmona, (Eds. ), Volume on indifference Pricing, Princeton: Princeton University Press, 2009.

[5]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.

[6]

O. Bobrovnytska and M. Schweizer, Mean-variance hedging and stochastic control: Beyond the Brownian setting, IEEE Transactions on Automatic Control, 49 (2004), 396-408.  doi: 10.1109/TAC.2004.824468.

[7]

H. Bülhmann, F. Delbaen, P. Embrechts and A. N. Shiryaev, No-arbitrage, change of measure and conditional Esscher transforms, CWI Quarterly, 9 (1996), 291-317. 

[8]

S. N. Cohen, R. J. Elliott and C. E. M. Pearce, A general comparison theorem for backward stochastic differential equations, Advances in Applied Probability, 42 (2010), 878-898. 

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R. Cont and P. Tankov, Financial Modelling with Jump Processes London: Chapman & Hall / CRC Press, 2004.

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J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.

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X. De~Scheemaekere, Risk indifference pricing and backward stochastic differential equation, CEB Working Paper No. 08/027. September 2008, Solvay Business School, Brussels, Belgium, 2008.

[12]

F. Delbaen, S. Peng and R. Rosazza-Gianin, Representation of the penalty term of dynamic concave utilities, Finance and Stochastics, 14 (2010), 449-472.  doi: 10.1007/s00780-009-0119-7.

[13]

O. Deprez and H. U. Gerber, On convex principles of premium calculation, Insurance: Mathematics and Economics, 4 (1985), 179-189.  doi: 10.1016/0167-6687(85)90014-9.

[14]

B. Dupire, Functional Ità calculus, Preprint, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS, Bloomberg L. P. , 2009.

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N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

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R. J. Elliott, Double martingales, Probability Theory and Related Fields, 34 (1976), 17-28.  doi: 10.1007/BF00532686.

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R. J. Elliott, Stochastic Calculus and Applications New York: Springer Verlag, 1982.

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R. J. Elliott, A partially observed control problem for Markov chains, Applied Mathematics and Optimization, 2 (1992), 151-169.  doi: 10.1007/BF01182478.

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R. J. Elliott, L. Aggoun and J. Moore, Hidden Markov Models: Estimation and Control New York: Springer-Verlag, 1995.

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R. J. Elliott, L. L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432.  doi: 10.1007/s10436-005-0013-z.

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R. J. Elliott, T. K. Siu, L. L. Chan and J. W. Lau, Pricing options under a generalized Markov modulated jump diffusion model, Stochastic Analysis and Applications, 25 (2007), 821-843.  doi: 10.1080/07362990701420118.

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R. J. Elliott and T. K. Siu, On risk minimizing portfolios under a Markovian regime-switching Black-Scholes economy, Annals of Operations Research, 176 (2010), 271-291.  doi: 10.1007/s10479-008-0448-5.

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R. J. Elliott and T. K. Siu, A risk-based approach for pricing American options under a generalized Markov regime-switching model, Quantitative Finance, 11 (2011), 1633-1646.  doi: 10.1080/14697688.2011.615215.

[25]

R. J. Elliott and T. K. Siu, A BSDE approach to a risk-based optimal investment of an insurer, Automatica J. IFAC, 47 (2011), 253-261.  doi: 10.1016/j.automatica.2010.10.032.

[26]

R. J. Elliott, T. K. Siu and A. Badescu, On pricing and hedging options in regime-switching models with feedback effect, Journal of Economic Dynamics and Control, 35 (2011), 694-713.  doi: 10.1016/j.jedc.2010.12.014.

[27]

R. J. Elliott and T. K. Siu, A BSDE approach to convex risk measures for derivative securities, Stochastic Analysis and Applications, 30 (2012), 1083-1101.  doi: 10.1080/07362994.2012.727141.

[28]

R. J. Elliott and T. K. Siu, Reflected backward stochastic differential equations, convex risk measures and American options, Stochastic Analysis and Applications, 31 (2013), 1077-1096.  doi: 10.1080/07362994.2013.830459.

[29]

R. J. Elliott, T. K. Siu and S. N. Cohen, Backward stochastic difference equations for dynamic convex risk measures on a binomial tree, Journal of Applied Probability, 52 (2015), 771-785.  doi: 10.1017/S0021900200113427.

[30]

F. Esscher, On the probability function in the collective theory of risk, Skandinavisk Aktuarietidskrift, 15 (1932), 175-195. 

[31]

H. Föllmer and A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics, 6 (2002), 429-447.  doi: 10.1007/s007800200072.

[32]

H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time (2nd Edition) Berlin-New York: Walter de Gruyter, 2004. doi: 10.1515/9783110212075.

[33]

H. Föllmer and T. Knispel, Entropic risk measures: coherence v.s. convexity, model ambiguity, and robust large deviations, Stochastics and Dynamics, 11 (2011), 333-351.  doi: 10.1142/S0219493711003334.

[34]

M. Frittelli, Introduction to a theory of value coherent to the no arbitrage principle, Finance and Stochastics, 4 (2000), 275-297.  doi: 10.1007/s007800050074.

[35]

M. Frittelli and E. Rosazza-Gianin, Putting order in risk measures, Journal of Banking and Finance, 26 (2002), 1473-1486.  doi: 10.1016/S0378-4266(02)00270-4.

[36]

J. Fu and H. Yang, Equilibrium approach of asset pricing under Lévy process, European Journal of Operational Research, 223 (2012), 701-708.  doi: 10.1016/j.ejor.2012.06.037.

[37]

H. U. Gerber, An Introduction to Mathematical Risk Theory Huebner, 1979.

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H. U. Gerber and E. S. W. Shiu, Option pricing by Esscher transforms (with discussions), Transactions of the Society of Actuaries, 46 (1994), 99-191. 

[39]

M. J. Goovaerts, F. E. C. De Vylder and J. Haezendonck, Insurance Premiums Amsterdam: North-Holland Publishing, 1984. doi: 10.1007/978-94-009-6354-2.

[40]

X. Guo, Information and option pricings, Quantitative Finance, 1 (2001), 38-44.  doi: 10.1080/713665550.

[41]

L. P. Hansen and T. J. Sargent, Robustness Princeton: Princeton University Press, 2008.

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

[43]

S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Management Science, 50 (2004), 1178-1192. 

[44]

D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Annals of Applied Probability, 9 (1999), 904-950.  doi: 10.1214/aoap/1029962818.

[45]

A. L. Lewis, A simple option formula for general jump-diffusion and other exponential Lévy processes, Preprint, Envision Financial Systems and OptionCity. net, United States, 2001. doi: 10.2139/ssrn. 282110.

[46]

J. Liu, J. Pan and T. Wang, An equilibrium model of rare-event premia and its implication for option smirks, Review of Financial Studies, 18 (2005), 131-164.  doi: 10.1093/rfs/hhi011.

[47]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching London: Imperial College Press, 2006. doi: 10.1142/p473.

[48]

S. Mataramvura and B. ∅ksendal, Risk minimizing portfolios and HJB equations for stochastic differential games, Stochastics, 80 (2007), 317-337.  doi: 10.1080/17442500701655408.

[49]

H. Meng and T. K. Siu, Risk-based asset allocation under Markov-modulated pure jump processes, Stochastic Analysis and Applications, 32 (2014), 191-206.  doi: 10.1080/07362994.2014.858551.

[50]

R. C. Merton, The theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.

[51]

R. C. Merton, Option pricing when the underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2.

[52]

Y. Miyahara, Geometric Lévy processes and MEMM: pricing model and related estimation problems, Asia-Pacific Financial Markets, 8 (2001), 45-60. 

[53]

V. Naik, Option valuation and hedging strategies with jumps in volatility of asset returns, Journal of Finance, 48 (1993), 1969-1984.  doi: 10.1111/j.1540-6261.1993.tb05137.x.

[54]

B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions Berlin, Heidelberg, New York: Springer Verlag, 2007. doi: 10.1007/978-3-540-69826-5.

[55]

B. Oksendal and A. Sulem, A game theoretic approach to martingale measures in incomplete markets, Surveys of Applied and Industrial Mathematics, 15 (2008), 18-24. 

[56]

B. Oksendal and A. Sulem, Risk indifference pricing in jump diffusion markets, Mathematical Finance, 19 (2009), 619-637.  doi: 10.1111/j.1467-9965.2009.00382.x.

[57]

B. Oksendal and A. Sulem, Portfolio optimization under model uncertainty and BSDE games, Quantitative Finance, 11 (2011), 1665-1674.  doi: 10.1080/14697688.2011.615219.

[58]

V. Piterbarg, Markovian projection method for volatility calibration SSRN (2006), 906473, 22pp. doi: 10.2139/ssrn. 906473.

[59]

Y. Shen and T. K. Siu, Stochastic differential game, Esscher transform and general equilibrium under a Markovian regime-switching Lévy model, Insurance: Mathematics and Economics, 53 (2013), 757-768.  doi: 10.1016/j.insmatheco.2013.09.016.

[60]

Y. Shen, K. Fan and T. K. Siu, Option valuation under a double regime-switching model, Journal of Futures Markets, 34 (2014), 451-478.  doi: 10.1002/fut.21613.

[61]

T. K. Siu, A game theoretic approach to option valuation under Markovian regime-switching models, Insurance: Mathematics and Economics, 42 (2008), 1146-1158.  doi: 10.1016/j.insmatheco.2008.03.003.

[62]

T. K. Siu, J. W. Lau and H. Yang, Pricing participating products under a generalized jump-diffusion Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008), Article ID 474623, 30 Pages. doi: 10.1155/2008/474623.

[63]

T. K. Siu, A BSDE approach to risk-based asset allocation of pension funds with regime switching, Annals of Operations Research, 201 (2012), 449-473.  doi: 10.1007/s10479-012-1211-5.

[64]

T. K. Siu, Functional Ità's calculus and dynamic convex risk measures for derivative securities, Communications on Stochastic Analysis, 6 (2012), 339-358. 

[65]

T. K. 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.

[66]

T. K. Siu, A functional Ità's calculus approach to convex risk measures with jump diffusion, European Journal of Operational Research, 250 (2016), 874-883.  doi: 10.1016/j.ejor.2015.10.032.

[67]

T. K. Siu and Y. Shen, Risk-based asset allocation under stochastic volatility with jumps, Working Paper, 2016.

[68]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications New York: Springer, 2010. doi: 10.1007/978-1-4419-1105-6.

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F. L. Yuen and H. Yang, Option pricing in a jump-diffusion model with regime switching, ASTIN Bulletin, 39 (2009), 515-539.  doi: 10.2143/AST.39.2.2044646.

show all references

References:
[1]

P. Artzner, F. Delbaen, J. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.

[2]

A. Badescu, R. J. Elliott and T. K. Siu, Esscher transforms and consumption-based models, Insurance: Mathematics and Economics, 45 (2009), 337-347.  doi: 10.1016/j.insmatheco.2009.08.001.

[3]

P. Barrieu and N. El Karoui, Inf-convolution of risk measures and optimal risk transfer, Finance and Stochastics, 9 (2005), 269-298.  doi: 10.1007/s00780-005-0152-0.

[4]

P. Barrieu and N. El Karoui, Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: R. Carmona, (Eds. ), Volume on indifference Pricing, Princeton: Princeton University Press, 2009.

[5]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.

[6]

O. Bobrovnytska and M. Schweizer, Mean-variance hedging and stochastic control: Beyond the Brownian setting, IEEE Transactions on Automatic Control, 49 (2004), 396-408.  doi: 10.1109/TAC.2004.824468.

[7]

H. Bülhmann, F. Delbaen, P. Embrechts and A. N. Shiryaev, No-arbitrage, change of measure and conditional Esscher transforms, CWI Quarterly, 9 (1996), 291-317. 

[8]

S. N. Cohen, R. J. Elliott and C. E. M. Pearce, A general comparison theorem for backward stochastic differential equations, Advances in Applied Probability, 42 (2010), 878-898. 

[9]

R. Cont and P. Tankov, Financial Modelling with Jump Processes London: Chapman & Hall / CRC Press, 2004.

[10]

J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.

[11]

X. De~Scheemaekere, Risk indifference pricing and backward stochastic differential equation, CEB Working Paper No. 08/027. September 2008, Solvay Business School, Brussels, Belgium, 2008.

[12]

F. Delbaen, S. Peng and R. Rosazza-Gianin, Representation of the penalty term of dynamic concave utilities, Finance and Stochastics, 14 (2010), 449-472.  doi: 10.1007/s00780-009-0119-7.

[13]

O. Deprez and H. U. Gerber, On convex principles of premium calculation, Insurance: Mathematics and Economics, 4 (1985), 179-189.  doi: 10.1016/0167-6687(85)90014-9.

[14]

B. Dupire, Functional Ità calculus, Preprint, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS, Bloomberg L. P. , 2009.

[15]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[16]

R. J. Elliott, Double martingales, Probability Theory and Related Fields, 34 (1976), 17-28.  doi: 10.1007/BF00532686.

[17]

R. J. Elliott, Stochastic Calculus and Applications New York: Springer Verlag, 1982.

[18]

R. J. Elliott, A partially observed control problem for Markov chains, Applied Mathematics and Optimization, 2 (1992), 151-169.  doi: 10.1007/BF01182478.

[19]

R. J. Elliott, L. Aggoun and J. Moore, Hidden Markov Models: Estimation and Control New York: Springer-Verlag, 1995.

[20]

R. J. Elliott, L. L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432.  doi: 10.1007/s10436-005-0013-z.

[21]

R. J. Elliott, T. K. Siu, L. L. Chan and J. W. Lau, Pricing options under a generalized Markov modulated jump diffusion model, Stochastic Analysis and Applications, 25 (2007), 821-843.  doi: 10.1080/07362990701420118.

[22]

R. J. Elliott and T. K. Siu, Risk-based indifference pricing under a stochastic volatility model, Communications on Stochastic Analysis: Special Issue for Professor G. Kallianpur, 4 (2010), 51-73. 

[23]

R. J. Elliott and T. K. Siu, On risk minimizing portfolios under a Markovian regime-switching Black-Scholes economy, Annals of Operations Research, 176 (2010), 271-291.  doi: 10.1007/s10479-008-0448-5.

[24]

R. J. Elliott and T. K. Siu, A risk-based approach for pricing American options under a generalized Markov regime-switching model, Quantitative Finance, 11 (2011), 1633-1646.  doi: 10.1080/14697688.2011.615215.

[25]

R. J. Elliott and T. K. Siu, A BSDE approach to a risk-based optimal investment of an insurer, Automatica J. IFAC, 47 (2011), 253-261.  doi: 10.1016/j.automatica.2010.10.032.

[26]

R. J. Elliott, T. K. Siu and A. Badescu, On pricing and hedging options in regime-switching models with feedback effect, Journal of Economic Dynamics and Control, 35 (2011), 694-713.  doi: 10.1016/j.jedc.2010.12.014.

[27]

R. J. Elliott and T. K. Siu, A BSDE approach to convex risk measures for derivative securities, Stochastic Analysis and Applications, 30 (2012), 1083-1101.  doi: 10.1080/07362994.2012.727141.

[28]

R. J. Elliott and T. K. Siu, Reflected backward stochastic differential equations, convex risk measures and American options, Stochastic Analysis and Applications, 31 (2013), 1077-1096.  doi: 10.1080/07362994.2013.830459.

[29]

R. J. Elliott, T. K. Siu and S. N. Cohen, Backward stochastic difference equations for dynamic convex risk measures on a binomial tree, Journal of Applied Probability, 52 (2015), 771-785.  doi: 10.1017/S0021900200113427.

[30]

F. Esscher, On the probability function in the collective theory of risk, Skandinavisk Aktuarietidskrift, 15 (1932), 175-195. 

[31]

H. Föllmer and A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics, 6 (2002), 429-447.  doi: 10.1007/s007800200072.

[32]

H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time (2nd Edition) Berlin-New York: Walter de Gruyter, 2004. doi: 10.1515/9783110212075.

[33]

H. Föllmer and T. Knispel, Entropic risk measures: coherence v.s. convexity, model ambiguity, and robust large deviations, Stochastics and Dynamics, 11 (2011), 333-351.  doi: 10.1142/S0219493711003334.

[34]

M. Frittelli, Introduction to a theory of value coherent to the no arbitrage principle, Finance and Stochastics, 4 (2000), 275-297.  doi: 10.1007/s007800050074.

[35]

M. Frittelli and E. Rosazza-Gianin, Putting order in risk measures, Journal of Banking and Finance, 26 (2002), 1473-1486.  doi: 10.1016/S0378-4266(02)00270-4.

[36]

J. Fu and H. Yang, Equilibrium approach of asset pricing under Lévy process, European Journal of Operational Research, 223 (2012), 701-708.  doi: 10.1016/j.ejor.2012.06.037.

[37]

H. U. Gerber, An Introduction to Mathematical Risk Theory Huebner, 1979.

[38]

H. U. Gerber and E. S. W. Shiu, Option pricing by Esscher transforms (with discussions), Transactions of the Society of Actuaries, 46 (1994), 99-191. 

[39]

M. J. Goovaerts, F. E. C. De Vylder and J. Haezendonck, Insurance Premiums Amsterdam: North-Holland Publishing, 1984. doi: 10.1007/978-94-009-6354-2.

[40]

X. Guo, Information and option pricings, Quantitative Finance, 1 (2001), 38-44.  doi: 10.1080/713665550.

[41]

L. P. Hansen and T. J. Sargent, Robustness Princeton: Princeton University Press, 2008.

[42]

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

[43]

S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Management Science, 50 (2004), 1178-1192. 

[44]

D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Annals of Applied Probability, 9 (1999), 904-950.  doi: 10.1214/aoap/1029962818.

[45]

A. L. Lewis, A simple option formula for general jump-diffusion and other exponential Lévy processes, Preprint, Envision Financial Systems and OptionCity. net, United States, 2001. doi: 10.2139/ssrn. 282110.

[46]

J. Liu, J. Pan and T. Wang, An equilibrium model of rare-event premia and its implication for option smirks, Review of Financial Studies, 18 (2005), 131-164.  doi: 10.1093/rfs/hhi011.

[47]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching London: Imperial College Press, 2006. doi: 10.1142/p473.

[48]

S. Mataramvura and B. ∅ksendal, Risk minimizing portfolios and HJB equations for stochastic differential games, Stochastics, 80 (2007), 317-337.  doi: 10.1080/17442500701655408.

[49]

H. Meng and T. K. Siu, Risk-based asset allocation under Markov-modulated pure jump processes, Stochastic Analysis and Applications, 32 (2014), 191-206.  doi: 10.1080/07362994.2014.858551.

[50]

R. C. Merton, The theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.

[51]

R. C. Merton, Option pricing when the underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2.

[52]

Y. Miyahara, Geometric Lévy processes and MEMM: pricing model and related estimation problems, Asia-Pacific Financial Markets, 8 (2001), 45-60. 

[53]

V. Naik, Option valuation and hedging strategies with jumps in volatility of asset returns, Journal of Finance, 48 (1993), 1969-1984.  doi: 10.1111/j.1540-6261.1993.tb05137.x.

[54]

B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions Berlin, Heidelberg, New York: Springer Verlag, 2007. doi: 10.1007/978-3-540-69826-5.

[55]

B. Oksendal and A. Sulem, A game theoretic approach to martingale measures in incomplete markets, Surveys of Applied and Industrial Mathematics, 15 (2008), 18-24. 

[56]

B. Oksendal and A. Sulem, Risk indifference pricing in jump diffusion markets, Mathematical Finance, 19 (2009), 619-637.  doi: 10.1111/j.1467-9965.2009.00382.x.

[57]

B. Oksendal and A. Sulem, Portfolio optimization under model uncertainty and BSDE games, Quantitative Finance, 11 (2011), 1665-1674.  doi: 10.1080/14697688.2011.615219.

[58]

V. Piterbarg, Markovian projection method for volatility calibration SSRN (2006), 906473, 22pp. doi: 10.2139/ssrn. 906473.

[59]

Y. Shen and T. K. Siu, Stochastic differential game, Esscher transform and general equilibrium under a Markovian regime-switching Lévy model, Insurance: Mathematics and Economics, 53 (2013), 757-768.  doi: 10.1016/j.insmatheco.2013.09.016.

[60]

Y. Shen, K. Fan and T. K. Siu, Option valuation under a double regime-switching model, Journal of Futures Markets, 34 (2014), 451-478.  doi: 10.1002/fut.21613.

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