Figure 1.
Solutions to $F_\gamma$ and F (n = 2)
-
Previous Article
Optimal control of switched systems with multiple time-delays and a cost on changing control
- JIMO Home
- This Issue
-
Next Article
Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims
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 |
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.
Keywords:
Wishart affine stochastic correlation,
utility-based pricing,
neutral and indifference pricing,
stochastic correlation and volatility.
Mathematics Subject Classification:
Primary: 91G20; Secondary: 91G80.
Citation:
Jia Yue, Nan-Jing Huang. Neutral and indifference pricing with stochastic correlation and volatility. Journal of Industrial & 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. |
show all references
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. |
Figure 2.
Solutions to $F_\gamma$ and F (n = 1)
Figure 3.
Price under different delivery times and positions($\gamma = 0.5$ )
Figure 4.
Price under different delivery times and positions($\gamma = 2$ )
Figure 5.
Price under different risk-aversion parameters($\kappa_0 = -5,T = 1,\gamma\neq 1$ )
Figure 6.
Price under different risk-aversion parameters($\kappa_0 = 5,T = 1,\gamma\neq 1$ )
Figure 7.
Price under different risk-aversion parameters($\kappa_0 = -0.01,T = 1,\gamma\neq 1$ )
Figure 8.
Price under different risk-aversion parameters($\kappa_0 = 0,T = 1,\gamma\neq 1$ )
Figure 9.
Price under different risk-aversion parameters and positions($T = 1$ )
| [1] |
Jacinto Marabel Romo. A closed-form solution for outperformance options with stochastic correlation and stochastic volatility. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1185-1209. doi: 10.3934/jimo.2015.11.1185 |
| [2] |
Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli. Pricing realized variance options using integrated stochastic variance options in the Heston stochastic volatility model. Conference Publications, 2007, 2007 (Special) : 354-363. doi: 10.3934/proc.2007.2007.354 |
| [3] |
Xiao-Qian Jiang, Lun-Chuan Zhang. A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-978. doi: 10.3934/dcdss.2019065 |
| [4] |
Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521 |
| [5] |
María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics & Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004 |
| [6] |
Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296 |
| [7] |
Lixin Wu, Fan Zhang. LIBOR market model with stochastic volatility. Journal of Industrial & Management Optimization, 2006, 2 (2) : 199-227. doi: 10.3934/jimo.2006.2.199 |
| [8] |
Lassi Roininen, Markku S. Lehtinen, Sari Lasanen, Mikko Orispää, Markku Markkanen. Correlation priors. Inverse Problems & Imaging, 2011, 5 (1) : 167-184. doi: 10.3934/ipi.2011.5.167 |
| [9] |
Mathias Fink, Josselin Garnier. Ambient noise correlation-based imaging with moving sensors. Inverse Problems & Imaging, 2017, 11 (3) : 477-500. doi: 10.3934/ipi.2017022 |
| [10] |
Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435 |
| [11] |
Zhiping Zhou, Xinbao Liu, Jun Pei, Panos M. Pardalos, Hao Cheng. Competition of pricing and service investment between iot-based and traditional manufacturers. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1203-1218. doi: 10.3934/jimo.2018006 |
| [12] |
Ming Su, Arne Winterhof. Hamming correlation of higher order. Advances in Mathematics of Communications, 2018, 12 (3) : 505-513. doi: 10.3934/amc.2018029 |
| [13] |
Vladimír Špitalský. Local correlation entropy. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5711-5733. doi: 10.3934/dcds.2018249 |
| [14] |
Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965 |
| [15] |
Laurent Devineau, Pierre-Edouard Arrouy, Paul Bonnefoy, Alexandre Boumezoued. Fast calibration of the Libor market model with stochastic volatility and displaced diffusion. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-31. doi: 10.3934/jimo.2019025 |
| [16] |
Nian Li, Xiaohu Tang, Tor Helleseth. A class of quaternary sequences with low correlation. Advances in Mathematics of Communications, 2015, 9 (2) : 199-210. doi: 10.3934/amc.2015.9.199 |
| [17] |
Xin-Guo Liu, Kun Wang. A multigrid method for the maximal correlation problem. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 785-796. doi: 10.3934/naco.2012.2.785 |
| [18] |
Kaitlyn (Voccola) Muller. SAR correlation imaging and anisotropic scattering. Inverse Problems & Imaging, 2018, 12 (3) : 697-731. doi: 10.3934/ipi.2018030 |
| [19] |
Yan Zhang, Yonghong Wu, Benchawan Wiwatanapataphee, Francisca Angkola. Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-31. doi: 10.3934/jimo.2018141 |
| [20] |
Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295 |
2017 Impact Factor: 0.994
Tools
Metrics
Other articles
by authors
[Back to Top]