Neutral and indifference pricing with stochastic correlation and volatility

Jia Yue; Nan-Jing Huang

doi:10.3934/jimo.2017043

Article Contents

2018, Volume 14, Issue 1: 199-229. Doi: 10.3934/jimo.2017043

This issue Previous Article Optimal control of switched systems with multiple time-delays and a cost on changing control Next Article Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims

Neutral and indifference pricing with stochastic correlation and volatility

Jia Yue and
Nan-Jing Huang^,

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

* Corresponding author: Nan-jing Huang
* Corresponding author: Nan-jing Huang

Received: December 31, 2015

Revised: January 31, 2017

Early access: March 2017

Published: January 2018

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

Abstract Full Text(HTML) Figure(9) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 91G20; Secondary: 91G80.

Citation:

Full Text(HTML)

Figure 1. Solutions to $F_\gamma$ and F (n = 2)

Download: Full-size image PowerPoint slide

Figure 2. Solutions to $F_\gamma$ and F (n = 1)

Download: Full-size image PowerPoint slide

Figure 3. Price under different delivery times and positions($\gamma = 0.5$)

Download: Full-size image PowerPoint slide

Figure 4. Price under different delivery times and positions($\gamma = 2$)

Download: Full-size image PowerPoint slide

Figure 5. Price under different risk-aversion parameters($\kappa_0 = -5,T = 1,\gamma\neq 1$)

Download: Full-size image PowerPoint slide

Figure 6. Price under different risk-aversion parameters($\kappa_0 = 5,T = 1,\gamma\neq 1$)

Download: Full-size image PowerPoint slide

Figure 7. Price under different risk-aversion parameters($\kappa_0 = -0.01,T = 1,\gamma\neq 1$)

Download: Full-size image PowerPoint slide

Figure 8. Price under different risk-aversion parameters($\kappa_0 = 0,T = 1,\gamma\neq 1$)

Download: Full-size image PowerPoint slide

Figure 9. Price under different risk-aversion parameters and positions($T = 1$)

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.