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

M. Jandacka and D. Sevcovic

Abstract
In this paper 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 synthetised 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 explain 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 data sets. We discuss how they can be used in qualitative and quantitative analysis of option market data.
AMS MOS 65N40; Secondary 60G40

Paper (Journal of Applied Mathematics, 3, 2005, 235-258)
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Lecture slides
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Analysis of the free boundary for the pricing of an American call option

D. Sevcovic

Abstract
The purpose of this paper is to analyze the free boundary problem for the Black-Scholes equation for pricing the American call option on stocks paying a continuous dividend. Using the Fourier integral transformation method we derive and analyze a nonlinear singular integral equation determining the shape of the free boundary. Numerical experiments based on this integral equation are also presented.

Keywords: Black-Scholes equation, American call option, early exercise free boundary, optimal stopping time, nonlinear integral equation
AMS MOS 65N40; Secondary 60G40

Paper (Euro. Journal on Applied Mathematics, 12 (2001), 25--37.)
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The early exercise boundary for the American put near expiry: numerical approximation

R.Stamicar, D. Sevcovic and J.Chadam

Abstract
It is well-known that the early exercise boundary for the American put approaches the strike price at expiry with infinite velocity. This causes difficulties in developing efficient and accurate numerical procedures and consequently trading strategies, during the volatile period near expiry. Based on the work of D.Sevcovic for the American call with dividend, an integral equation is derived for the free boundary for the American put which leads to an accurate numerical procedure and an interesting, and accurate, asymptotic solution for the early exercise boundary near expiry.

Keywords: American put near expiry, analytical and numerical approximation of early exercise boundary

Paper (Canad. Appl. Math. Quarterly, 7, No.4, (1999), 427-444)
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On some mathematical problems in the theory of pricing financial derivatives

(in Slovak: O niektorých matematických problémoch oceňovania finančných derivátov)

Daniel Sevcovic

Lecture slides