Static Pricing of Exotic Derivatives Under Conditional Value-at-Risk (CVaR) in Incomplete Markets

Abstract

This paper optimizes portfolios using Conditional Value-at-Risk (CVaR) as a risk measure with tradable assets such as cash, call and put options on the S&P 500. It develops a static pricing model for the buy-and-hold strategy and evaluates dynamic strategies. The CVaR model is compared to the mean-variance model, demonstrating its ability to minimize CVaR while achieving required returns and showing sensitivity to parameters, namely, the shape of the variance-gamma distribution, volatility, confidence level, and required return. Although the mean-variance model results in lower standard deviation, the CVaR-based optimization consistently yields lower CVaR values. For portfolios with liabilities, the model incorporates hedging strategies for standard European call option and Exotic options such as quadratic, log, digital, sine, and butterfly spread options in an incomplete market. The CVaR-based approach demonstrates risk reduction and robustness in the backtesting.