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We analyse a model for pricing derivative securities in thepresence of both transaction costs as well as the risk from avolatile portfolio. The model is based on the Black-Scholesparabolic PDE in which transaction costs are described followingthe Hoggard, Whalley, and Wilmott approach. The risk from a volatile portfolio is described by thevariance of the synthesized portfolio. Transactioncosts as well as the volatile portfolio risk depend on the timelag between two consecutive transactions. Minimizing their sumyields the optimal length of the hedge interval. In this model,prices of vanilla options can be computed from a solution to afully nonlinear parabolic equation in which a diffusioncoefficient representing volatility nonlinearly depends on thesolution itself giving rise to explaining the volatility smileanalytically. We derive a robust numerical scheme for solving thegoverning equation and perform extensive numerical testing of themodel and compare the results to real option market data. Impliedrisk and volatility are introduced and computed for large optiondatasets. We discuss how they can be used in qualitative andquantitative analysis of option market data

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