Optimal Portfolio with Options in a Framework of Mean-CVaR

Abstract

TheModern Portfolio Theory has its benchmark which is theMarkowitz’s Mean-Variance portfolio optimisation. However the Markowitz’s Mean-Variance optimisation of portfolios with options (even simple products such as call or put) is a particularly difficult task, as the asymmetric returns of options and financial products with embedded options require the use of asymmetric risk measures. Furthermore, returns of options are a positive function of the standard deviation of the return of the underlying assets; the variance becomes not an appropriate risk measure for portfolio with options. Recently, Value at Risk (VaR) and Conditional Value at Risk (CVaR) have gained acceptance in world financial markets as appropriate risk measures in risk management. Furthermore, CVaR has advantage to be a coherent risk measure. We analyse the problem of computing the optimal portfolios in a framework of Mean-CVaR. Simulation methods are considered the appropriate choices, especially when portfolios with options or instruments with embedded options are analyzed. We present the optimal frontier Mean-CVaR for an application of a portfolio composed of four correlated assets and two calls and two puts. Comparison is made with the Mean-Variance and Mean-Variance approach for this application portfolio.