Pricing and hedging of contingent claims in incopmplete markets by modeling losses as conditional value at risk in (formula)-gain loss opportunities

Abstract

We combine the principles of risk aversion and no-arbitrage pricing and propose an alternative way for pricing and hedging contingent claims in incomplete markets. We re-consider the pricing problem under the condition that losses are modeled by the measure of CVaR in the concept of λ gain-loss opportunities. The proposed model enables investors to specify their preferences by putting restrictions on the parameter λ that stands for risk aversion. Using CVaR as a measure of risk enables us to account for extreme losses and yield a conservative result. The pricing problem is studied in discrete time, multi-period, stochastic linear optimization environment with a finite probability space. We extend our model to include the perspectives of writers and buyers of the contingent claims. We use duality to establish a pricing interval of the contingent claims excluding CVaR-λ gain-loss opportunities in the market. Duality results also provide a way for passing to appropriate martingale measures and we express the pricing interval also in terms of martingale measures. This pricing interval is shown to be tighter than the no-arbitrage bounds. We also present a numerical study of our work with respect to the risk aversion parameter λ and in various levels of confidence. We compute prices of the the writers and buyers of 48 European call and put options on the S&P500 index on September 10, 2002 using the remaining options as market traded assets. It is possible to say that our proposed model yields good bounds as most of the bounds we obtained are very close to the true bid and ask values.