Asymptotic option price with bounded expected loss

This paper studies the problem of option pricing in an incomplete market, where the exact replication of an option may not be possible. In an incomplete market, we suppose a situation where a hedger wants to invest as little as possible at the beginning, but he/she wants to have the expected squared loss at the end not exceeding a certain constant. We study this problem when the log of the underlying asset price process is compound Poisson, which converges to a Brownian motion with drift. In the limit, we use the mean-variance approach to find a hedging strategy which minimizes the expected squared loss for a given initial investment. Then we find the asymptotic minimum investment with the expected squared loss bounded by a given upper bound. Some numerical results are also provided.