Journal of Applied Mathematics

On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile

Martin Jandačka and Daniel Ševčovič

Full-text: Open access

Abstract

We analyse a model for pricing derivative securities in the presence of both transaction costs as well as the risk from a volatile portfolio. The model is based on the Black-Scholes parabolic PDE in which transaction costs are described following the Hoggard, Whalley, and Wilmott approach. The risk from a volatile portfolio is described by the variance of the synthesized portfolio. Transaction costs as well as the volatile portfolio risk depend on the time lag between two consecutive transactions. Minimizing their sum yields the optimal length of the hedge interval. In this model, prices of vanilla options can be computed from a solution to a fully nonlinear parabolic equation in which a diffusion coefficient representing volatility nonlinearly depends on the solution itself giving rise to explaining the volatility smile analytically. We derive a robust numerical scheme for solving the governing equation and perform extensive numerical testing of the model and compare the results to real option market data. Implied risk and volatility are introduced and computed for large option datasets. We discuss how they can be used in qualitative and quantitative analysis of option market data.

Article information

Source
J. Appl. Math., Volume 2005, Number 3 (2005), 235-258.

Dates
First available in Project Euclid: 25 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.jam/1122298273

Digital Object Identifier
doi:10.1155/JAM.2005.235

Mathematical Reviews number (MathSciNet)
MR2201973

Zentralblatt MATH identifier
1128.91025

Citation

Jandačka, Martin; Ševčovič, Daniel. On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile. J. Appl. Math. 2005 (2005), no. 3, 235--258. doi:10.1155/JAM.2005.235. https://projecteuclid.org/euclid.jam/1122298273


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