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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

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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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

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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. Google Scholar

[16]

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

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

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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. Google Scholar

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

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[31]

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

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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

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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. Google Scholar

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

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

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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. Google Scholar

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

[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. Google Scholar

[50]

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[52]

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

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[31]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[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. Google Scholar

[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. Google Scholar

[40]

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

[41]

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

[42]

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

[43]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[47]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[52]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[57]

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