Ronnie Sircar, Thaleia Zariphopoulou
This paper is a contribution to the valuation of derivative securities in a stochastic volatility framework. The derivatives to be priced are of European type and their payoff can depend in general on both the level of the stock and the volatility at expiration.
A utility-based method is first developed which yields the price by comparing the maximal expected utilities with and without trading the derivative. The price turns out to be the solution of a certain quasilinear partial differential equation with the nonlinearity reflecting the unhedgable risk component. In general, the indifference price quasilinear PDE does not have a closed form solution. Therefore, it is desirable to obtain bounds that carry some natural insight and also numerical and asymptotic approximations to the indifference price.
Two sets of bounds on the indifference prices are deduced. The first pair is a direct consequence of the monotonic behavior of the nonlinearities with respect to the square of the correlation. The second set of bounds involve more sophisticated analysis. They are still derived from appropriate sub and super solutions to the pricing equation which are constructed by the so called sequential risk factorization method.
The problem is also analyzed by asymptotic methods in the limit of the volatility being a fast mean reverting process. The analysis relates the utility based valuation and the no arbitrage approach in such incomplete markets.