American Institute of Mathematical Sciences

January  2018, 14(1): 199-229. doi: 10.3934/jimo.2017043

Neutral and indifference pricing with stochastic correlation and volatility

 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Nan-jing Huang

Received  January 2016 Revised  February 2017 Published  January 2018 Early access  April 2017

Fund Project: This work was supported by the National Natural Science Foundation of China (11471230,11671282).

In this paper, we consider a Wishart Affine Stochastic Correlation (WASC) model which accounts for the stochastic volatilities of the assets and for the stochastic correlations not only between the underlying assets' returns but also between their volatilities. Under the assumptions of the model, we derive the neutral and indifference pricing for general European-style financial contracts. The paper shows that comparing to risk-neutral pricing, the utility-based pricing methods are generally feasible and avoid factitiously dealing with some risk premia corresponding to the volatilities-correlations as a consequence of the incompleteness of the market.

Citation: Jia Yue, Nan-Jing Huang. Neutral and indifference pricing with stochastic correlation and volatility. Journal of Industrial and Management Optimization, 2018, 14 (1) : 199-229. doi: 10.3934/jimo.2017043
References:
 [1] L. Anderson and J. Andresen, Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Rev. Deriv. Res., 4 (2000), 231-262.  doi: 10.2139/ssrn.171438. [2] N. Bäuerle and Z. J. Li, Optimal portfolios for financial markets with Wishart volatility, J. Appl. Probab., 50 (2013), 1025-1043.  doi: 10.1017/S0021900200013772. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062. [4] M. Bru, Wishart process, J. Theoret. Probab., 4 (1991), 725-751.  doi: 10.1007/BF01259552. [5] R. Carmona, Indifference Pricing: Theory and Applications, Princeton University Press, Princeton, 2009. [6] R. Cont and J. D. Fonseca, Dynamics of implied volatility surfaces, Quant. Finance, 2 (2002), 45-60.  doi: 10.1088/1469-7688/2/1/304. [7] M. H. A. Davis, V. G. Panas and T. Zariphopoulou, European option pricing with transaction costs, SIAM J. Control Optim., 31 (1993), 470-493.  doi: 10.1137/0331022. [8] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [9] J. D. Fonseca, M. Grasselli and C. Tebaldi, Option pricing when correlations are stochastic: An analytical framework, Rev. Deriv. Res., 10 (2007), 151-180.  doi: 10.1007/s11147-008-9018-x. [10] A. Friedman, Stochastic differential equations, Stochastic Differential Equations and Applications, 1 (1975), 98-127.  doi: 10.1016/B978-0-12-268201-8.50010-4. [11] C. Gourieroux and R. Sufana, Derivative Pricing with Multivariate Stochastic Volatility: Application to Credit Risk, Working Paper, Les Cahiers du CREF, 2005. doi: 10.2139/ssrn.757312. [12] M. Grasselli and C. Tebaldi, Solvable affine term structure models, Math. Finance, 18 (2008), 135-153.  doi: 10.1111/j.1467-9965.2007.00325.x. [13] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [14] S. D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs, Rev. Futures Markets, 8 (1989), 222-239. [15] J. Kallsen, Utility-based derivative pricing in incomplete markets, Mathematical Finance-Bachelier Congress 2000, Springer, Berlin, (2002), 313-338. [16] M. Loretan and W. English, Evaluating Correlation Breakdowns During Periods of Market Volatility, Working paper, Board of Governors of the Federal Reserve System International Finance, 2000. doi: 10.2139/ssrn.231857. [17] O. Pfaffel, Wishart processes, available at: arXiv: 1201.3256, 2012. [18] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8. [19] J. M. Romo, A closed-form solution for outperformance options with stochastic correlation and stochastic volatility, J. Ind. Manag. Optim., 11 (2015), 1185-1209.  doi: 10.3934/jimo.2015.11.1185. [20] B. Solnik, C. Boucrelle and Y. L. Fur, International market correlation and volatility, Financ. Analysts J., 52 (1996), 17-34.  doi: 10.2469/faj.v52.n5.2021. [21] S. D. Stojanovic, Optimal momentum hedging via hypoelliptic reduced Monge-Ampère PDEs, SIAM J. Control Optim., 43 (2004), 1151-1173.  doi: 10.1137/S0363012903421170. [22] S. D. Stojanovic, Risk premium and fair option prices under stochastic volatility: The HARA solution, C. R. Math., 340 (2005), 551-556.  doi: 10.1016/j.crma.2004.11.002. [23] S. D. Stojanovic, Stochastic Volatility and Risk Premium, Lecture Notes, CARP, New York, 2005. [24] S. D. Stojanovic, Pricing and hedging of multi type contracts under multidimensional risks in incomplete markets modeled by general Itô SDE systems, Asia Pac. Financ. Mark., 13 (2006), 345-372. [25] S. D. Stojanovic, Neutral and Indifference Portfolio Pricing, Hedging and Investing, Springer, New York, 2011. [26] Y. Z. Zhu and M. Avellaneda, E-arch model for implied volatility term structure of fx options, Quantitative Analysis in Financial Markets, (1999), 277-291.  doi: 10.1142/9789812812599_0011.

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References:
 [1] L. Anderson and J. Andresen, Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Rev. Deriv. Res., 4 (2000), 231-262.  doi: 10.2139/ssrn.171438. [2] N. Bäuerle and Z. J. Li, Optimal portfolios for financial markets with Wishart volatility, J. Appl. Probab., 50 (2013), 1025-1043.  doi: 10.1017/S0021900200013772. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062. [4] M. Bru, Wishart process, J. Theoret. Probab., 4 (1991), 725-751.  doi: 10.1007/BF01259552. [5] R. Carmona, Indifference Pricing: Theory and Applications, Princeton University Press, Princeton, 2009. [6] R. Cont and J. D. Fonseca, Dynamics of implied volatility surfaces, Quant. Finance, 2 (2002), 45-60.  doi: 10.1088/1469-7688/2/1/304. [7] M. H. A. Davis, V. G. Panas and T. Zariphopoulou, European option pricing with transaction costs, SIAM J. Control Optim., 31 (1993), 470-493.  doi: 10.1137/0331022. [8] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [9] J. D. Fonseca, M. Grasselli and C. Tebaldi, Option pricing when correlations are stochastic: An analytical framework, Rev. Deriv. Res., 10 (2007), 151-180.  doi: 10.1007/s11147-008-9018-x. [10] A. Friedman, Stochastic differential equations, Stochastic Differential Equations and Applications, 1 (1975), 98-127.  doi: 10.1016/B978-0-12-268201-8.50010-4. [11] C. Gourieroux and R. Sufana, Derivative Pricing with Multivariate Stochastic Volatility: Application to Credit Risk, Working Paper, Les Cahiers du CREF, 2005. doi: 10.2139/ssrn.757312. [12] M. Grasselli and C. Tebaldi, Solvable affine term structure models, Math. Finance, 18 (2008), 135-153.  doi: 10.1111/j.1467-9965.2007.00325.x. [13] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [14] S. D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs, Rev. Futures Markets, 8 (1989), 222-239. [15] J. Kallsen, Utility-based derivative pricing in incomplete markets, Mathematical Finance-Bachelier Congress 2000, Springer, Berlin, (2002), 313-338. [16] M. Loretan and W. English, Evaluating Correlation Breakdowns During Periods of Market Volatility, Working paper, Board of Governors of the Federal Reserve System International Finance, 2000. doi: 10.2139/ssrn.231857. [17] O. Pfaffel, Wishart processes, available at: arXiv: 1201.3256, 2012. [18] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8. [19] J. M. Romo, A closed-form solution for outperformance options with stochastic correlation and stochastic volatility, J. Ind. Manag. Optim., 11 (2015), 1185-1209.  doi: 10.3934/jimo.2015.11.1185. [20] B. Solnik, C. Boucrelle and Y. L. Fur, International market correlation and volatility, Financ. Analysts J., 52 (1996), 17-34.  doi: 10.2469/faj.v52.n5.2021. [21] S. D. Stojanovic, Optimal momentum hedging via hypoelliptic reduced Monge-Ampère PDEs, SIAM J. Control Optim., 43 (2004), 1151-1173.  doi: 10.1137/S0363012903421170. [22] S. D. Stojanovic, Risk premium and fair option prices under stochastic volatility: The HARA solution, C. R. Math., 340 (2005), 551-556.  doi: 10.1016/j.crma.2004.11.002. [23] S. D. Stojanovic, Stochastic Volatility and Risk Premium, Lecture Notes, CARP, New York, 2005. [24] S. D. Stojanovic, Pricing and hedging of multi type contracts under multidimensional risks in incomplete markets modeled by general Itô SDE systems, Asia Pac. Financ. Mark., 13 (2006), 345-372. [25] S. D. Stojanovic, Neutral and Indifference Portfolio Pricing, Hedging and Investing, Springer, New York, 2011. [26] Y. Z. Zhu and M. Avellaneda, E-arch model for implied volatility term structure of fx options, Quantitative Analysis in Financial Markets, (1999), 277-291.  doi: 10.1142/9789812812599_0011.
Solutions to $F_\gamma$ and F (n = 2)
Solutions to $F_\gamma$ and F (n = 1)
Price under different delivery times and positions($\gamma = 0.5$)
Price under different delivery times and positions($\gamma = 2$)
Price under different risk-aversion parameters($\kappa_0 = -5,T = 1,\gamma\neq 1$)
Price under different risk-aversion parameters($\kappa_0 = 5,T = 1,\gamma\neq 1$)
Price under different risk-aversion parameters($\kappa_0 = -0.01,T = 1,\gamma\neq 1$)
Price under different risk-aversion parameters($\kappa_0 = 0,T = 1,\gamma\neq 1$)
Price under different risk-aversion parameters and positions($T = 1$)
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