Journal of Applied Mathematics
Volume 2005 (2005), Issue 3, Pages 235-258

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

Martin Jandačka1 and Daniel Ševčovič2

1Department of Economic and Financial Models, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava 842 48, Slovakia
2Institute of Applied Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava 842 48, Slovakia

Received 16 June 2004; Revised 28 January 2005

Copyright © 2005 Martin Jandačka and Daniel Ševčovič. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.