Discrete hedging of American-type options using local risk minimization

Local risk minimization and total risk minimization discrete hedging have been extensively studied for European options [e.g., Schweizer, M., 1995. Variance-optimal hedging in discrete time. Mathematics of Operation Research 20, 1-32; Schweizer, M., 2001. A guided tour through quadratic hedging approaches. In: Jouini, E., Cvitanic, J., Musiela, M., Option pricing, interest rates and risk management, Cambridge University Press, pp. 538-574]. In practice, hedging of options with American features is more relevant. For example, equity linked variable annuities provide surrender benefits which are essentially embedded American options. In this paper we generalize both quadratic and piecewise linear local risk minimization hedging frameworks to American options. We illustrate that local risk minimization methods outperform delta hedging when the market is highly incomplete. In addition, compared to European options, distributions of the hedging costs are typically more skewed and heavy-tailed. Moreover, in contrast to quadratic local risk minimization, piecewise linear risk minimization hedging strategies can be significantly different, resulting in larger probabilities of small costs but also larger extreme cost. (C) 2007 Published by Elsevier B.V.