Journal of Applied Mathematics
Volume 2005 (2005), Issue 3, Pages 235-258
doi:10.1155/JAM.2005.235
On the risk-adjusted pricing-methodology-based valuation
of vanilla options and explanation of the volatility smile
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.
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.