Equal risk pricing of derivatives with deep hedging

Abstract

This article provides a universal and tractable methodology based on deep reinforcement learning to implement the equal risk pricing framework for financial derivatives pricing under very general conditions. The equal risk pricing framework entails solving for a derivative price which equates the optimally hedged residual risk exposure associated, respectively, with the long and short positions in the contingent claim. The solution to the hedging optimization problem considered, which is inspired from the [Marzban, S., Delage, E. and Li, J.Y., Equal risk pricing and hedging of financial derivatives with convex risk measures. arXiv preprint arXiv:2002.02876, 2020.] framework relying on convex risk measures, is obtained through the use of the deep hedging algorithm of [Buehler, H., Gonon, L., Teichmann, J. and Wood, B., Deep hedging. Q. Finance, 2019, 19, 1271–1291]. Consequently, the current paper's approach allows for the pricing and the hedging of a very large number of contingent claims (e.g. vanilla options, exotic options, options with multiple underlying assets) with multiple liquid hedging instruments under a wide variety of market dynamics (e.g. regime-switching, stochastic volatility, jumps). A novel ε-completeness measure allowing for the quantification of the residual hedging risk associated with a derivative is also proposed. The latter measure generalizes the one presented in [Bertsimas, D., Kogan, L. and Lo, A.W., Hedging derivative securities and incomplete markets: an ε-arbitrage approach. Oper. Res., 2001, 49, 372–397.] based on the quadratic penalty. Monte Carlo simulations are performed under a large variety of market dynamics to demonstrate the practicability of our approach, to perform benchmarking with respect to traditional methods and to conduct sensitivity analyses. Numerical results show, among others, that equal risk prices of out-of-the-money options are significantly higher than risk-neutral prices stemming from conventional changes of measure across all dynamics considered. This finding is shown to be shared by different option categories which include vanilla and exotic options.