### CVaR-minimising hedging by a smoothing method

#### Abstract

Minimisation of a conditional value-at-risk (CVaR) is a non-smooth stochastic minimisation problem with the non-smoothness caused by a plus function in the integrand of the objective function. We study the performance of several smoothing approximations of the plus function in the CVaR minimisation, using an example of a one period CVaR-minimising hedging. The smooth plus function that outperforms others is identified. The convergence of the solution of the smoothed CVaR minimisation problem with increase in the sample size and with decrease in the smoothing parameter is illustrated. The performance of the one period CVaR-minimising hedging and delta-gamma hedging are compared in terms of several commonly used performance criteria. Numerical simulations show that the magnitude and the probability of large losses are smaller in the CVaR-minimising hedge, compared to the delta-gamma hedge. This often occurs at the expense of a deteriorated expected return and increased variance of the hedge. We identify the situations when the CVaR-minimising hedge outperforms the delta-gamma hedge according to all performance criteria.

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#### Keywords

hedging, CVaR-minimisation, smoothing method

DOI: http://dx.doi.org/10.21914/anziamj.v52i0.3942

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