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On the valuation of a fair price in case of an non perfectly liquid market

Le : 06/06/2007 11h00
Par : L.A. Bordag (Halmstad University, Sweden)
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Résumé : Abstract The famous Black-Scholes model is based on the idea of an ideal market, it means, arbitrage-free, frictionless and liquid market. On the ideal market all participants are price-taker, nobody can change an asset price by own trading strategy. Forty years ago, as this theory was developed, the price-taker assumption was true for the most participants on the market. Nowadays we have well established instruments to valuate derivative products. It is one of the pre-conditions for the growing up of large hedge fonds. Other-sides the trading strategies of hedge fonds, assurances and large banks influence the asset prices. Such actors on the market, called program trader, can not be treated as price-taker. They have remarkable frictions and liquidity problems on the market. To take into account these developments a lot of new advanced models are introduced now. All these models try to incorporate feed-back effects and frictions in linear BlackScholes model. From mathematical point of view we leave the nice linear world of Black-Scholes parabolic equation and obtain instead different types nonlinear equations. The studied model, which was recently developed by Frey&Co, was suggested to design a perfect hedging strategy for a large trader. In this case the implementation of a hedging strategy affects the price of the underlying security, we have a typical feed-back effect. This feedback-effect leads to a strong nonlinear version of the Black-Scholes partial differential equation. Using the Lie group theory we reduce the partial differential equation in special cases to ordinary differential equations. The found Lie group of the model equation gives rise to invariant solutions. Families of exact invariant solutions for special values of parameters are described. These solutions were used to test numerical schemes for solving a nonlinear Black-Scholes equation.