Utility based asset pricing under high risk aversion

Abstract

This work aims to present three methods of pricing an asset.<br />A price of a derivative is the amount of money a buyer agrees to pay to the seller at time 0 in order to receive the derivative at maturity time T. If the market is complete, this price is uniquely given by the initial wealth of the portfolio in stocks and bonds that recreates the terminal payoff (replication). But in reality transaction costs or non-traded assets cause that the financial market is not complete. Then different prices consistent with the No Arbitrage Condition exist as each corresponds to a different martingale measure. The superreplication price is defined as the supremum of these martingale measures and therefore unrealistically high, but all risks and uncertainty is removed. Hence we want to find another way of pricing in an incomplete market but remain risk averse.<br />For these purposes we introduce the utility indifference price after explaining the concept of utility maximization and giving a short definition on risk aversion. This price considers the risk aversion and can also depend on the agent s initial wealth. Unlike the superreplication price the utility indifference price is not linear in the number of units of the claim, but converges to the superreplication price if the risk aversion tends to infinity. This statement is also proved. The utility indifference price can be considered as an interpolation between the totally risk averse superreplication price and the marginal utility price, which we introduce as the third price. By means of two examples all these properties will be verified.