Abstract
In this paper we develop a novel market model where asset variances–covariances evolve stochastically. In addition shocks on asset return dynamics are assumed to be linearly correlated with shocks driving the variance–covariance matrix. Analytical tractability is preserved since the model is linear-affine and the conditional characteristic function can be determined explicitly. Quite remarkably, the model provides prices for vanilla options consistent with observed smile and skew effects, while making it possible to detect and quantify the correlation risk in multiple-asset derivatives like basket options. In particular, it can reproduce and quantify the asymmetric conditional correlations observed on historical data for equity markets. As an illustrative example, we provide explicit pricing formulas for rainbow “Best-of” options.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ang A. and Chen J. (2002). Asymmetric correlations on equity portfolios. Journal of Financial Economics 63: 443–494
Bakshi G.S. and Madan D.B. (2000). Spanning and derivative-security valuation. Journal of Financial Economics 55: 205–238
Bru M.F. (1991). Wishart processes. Journal of Theoretical Probability 4: 725–743
Buraschi, A., Porchia, P., & Trojani, F. (2006). Correlation hedging. Preprint. Now available at SSRN: http://www.ssrn.com/abstract=908664 as “Correlation Risk and Optimal Portfolio Choice”.
Carr P. and Madan D.B. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance 2: 61–73
Da Fonseca, J., Grasselli, M., & Ielpo, F. (2007). Estimating the Wishart Affine Stochastic Correlation model using the empirical characteristic function. Working paper ESILV, RR-35.
Da Fonseca, J., Grasselli, M., & Tebaldi, C. (2005). Wishart multi-dimensional stochastic volatility. Working paper ESILV, RR-31. To appear in Quantitative Finance as “A Multifactor Volatility Heston Model”.
Dai Q. and Singleton K. (2000). Specification analysis of affine term structure models. The Journal of Finance 55: 1943–1978
Driessen, J., Maenhout, P. J., & Vilkov, G. (2005). Option implied correlations and the price of correlation risk. EFA 2005 Moscow Meetings. Available at SSRN: http://www.ssrn.com/abstract=67342.
Duffee G.R. (2002). Term premia and interest rate forecasts in affine models. Journal of Finance 57: 405–443
Duffie D., Filipovic D. and Schachermayer W. (2003). Affine processes and applications in finance. Annals of Applied Probability 13: 984–1053
Duffie D. and Kan R. (1996). A yield-factor model of interest rates. Mathematical Finance 6: 379–406
Duffie D., Pan J. and Singleton K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68: 1343–1376
Eraker B. (2004). Do stock prices and volatility jumps? Reconciling evidence from spot and option prices. The Journal of Finance 59: 1367–1403
Eraker B., Johannes M. and Polson N. (2007). The impact of jumps in volatility and return. The Journal of Finance 58: 1269–1300
Feller W. (1951). Two singular diffusion problems. Annals of Mathematics 54: 173–182
Fengler, M. R., & Schwendner, P. (2004). Quoting multiasset equity options in the presence of errors from estimating correlations. Journal of Derivatives, Summer, 43–54.
Gourieroux, C., & Sufana, R. (2004a). Wishart quadratic term structure models. CREF 03-10, HEC Montreal.
Gourieroux, C., & Sufana, R. (2004b). Derivative pricing with multivariate stochastic volatility: Application to credit risk. Working paper CREST.
Grasselli M. and Tebaldi C. (2008). Solvable affine term structure models. Mathematical Finance 18: 135–153
Heston S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies 6: 327–343
Heston S.L. and Nandi S. (2000). A closed-form GARCH option valuation model. The Review of Financial Studies 13: 585–625
Lewis, A. L. (2000). Option valuation under stochastic volatility with mathematica code. Finance Press.
Roll, R. (1988). The international crash of October, 1987. Financial Analysts Journal, September–October, 19–35.
Ross S. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory 13: 341–360
Sharpe W.F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19: 425–442
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fonseca, J.D., Grasselli, M. & Tebaldi, C. Option pricing when correlations are stochastic: an analytical framework. Rev Deriv Res 10, 151–180 (2007). https://doi.org/10.1007/s11147-008-9018-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11147-008-9018-x