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On some difference equations with exponential nonlinearity
Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model
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 |
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 [
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. |
[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.
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[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. |
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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.
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T. K. Siu and Y. Shen, Risk-based asset allocation under stochastic volatility with jumps, Working Paper, 2016. |
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Hybrid Switching Diffusions: Properties and Applications New York: Springer, 2010.
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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,
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