## Event Date:

## Event Date Details:

Refreshments served at 3:15 PM

## Event Location:

- South Hall 5607F

Dr. Vilmos Prokaj (UCSB)

Title: Shadow price for power utility

Abstract:Consider the Black-Scholes model with one risky asset. The investor goal is to maximize her expected discounted utility from consumption. This is the classical Merton problem, which was investigated in frictionless model as well as when the investor faces proportional transaction costs. Recently, Kallsen and Muhle-Karbe proposed a new way to look at the problem using the notion of the shadow price or shadow market. The latter is a frictionless market with a price process (this is the shadow price) evolving in the bid-ask spread of the original market. The shadow market allows more admissible strategies, therefore when the optimal policy of the shadow market is admissible in the original market it is obviously optimal in both markets. That way one can find the optimal policy of a market with transaction cost by finding a suitable frictionless model and solving the Merton problem there. Kallsen and Muhle-Karbe demonstrated their method of finding the shadow market in the case of logarithmic utility. What makes the logarithmic utility easy to handle is the fact that under quite general assumption the consumption is a fixed proportion of wealth, determined by the impatience factor of the question. This is not true for power utility and that prevented them to extend the method to this case. In a joint work with Attila Herczegh, we overcame this difficulty by showing that the marginal rate of substitution is a good candidate for the shadow price. As a byproduct we got the asymptotic expansion of the non-trading region and relative consumption rate for small transaction costs.