Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment

<Para ID="Par1">We price a contingent claim liability (claim for short) using a utility indifference argument. We consider an agent with exponential utility, who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of a proportional transaction cost ε>0 in two cases: with and without a claim. Using the heuristic computations of Whalley and Wilmott (Math. Finance 7:307–324, <CitationRef CitationID="CR42">1997</CitationRef>), under suitable technical conditions, we provide a rigorous derivation of the asymptotic expansion of the value function in powers of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon^{\frac{1}{3}}$</EquationSource> </InlineEquation> in both cases with and without a claim. Additionally, using the utility indifference method, we derive the price of the claim at the leading order of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon^{\frac{2}{3}}$</EquationSource> </InlineEquation>. In both cases, we also obtain a “nearly optimal” strategy, whose expected utility asymptotically matches the leading terms of the value function. We also present an example of how this methodology can be used to price more exotic barrier-type contingent claims. Copyright Springer-Verlag Berlin Heidelberg 2014