Deep equal risk pricing of illiquid derivatives with multiple hedging instruments

Alexandre Carbonneau and Frédéric Godin

Abstract

This paper leverages the equal risk pricing (ERP) framework for the valuation of illiquid financial derivatives. Such a method sets the derivative price as the premium, which leads to equal residual optimal hedging risk for agents hedging the long and short positions on the derivative. The inclusion of multiple hedging instruments such as liquid vanilla options in the hedging procedure ensures that information contained in the price of these liquid assets is transferred to the price of the illiquid asset. This property leads to genuine market-consistent pricing, which allows the use of observables (liquid instrument prices) to determine nonobservable prices (ie, that of illiquid derivatives) to be maximized. The ERP problem is solved numerically through deep reinforcement learning. Several numerical experiments are provided to study the properties of equal risk prices of derivatives when multiple options are used as hedging instruments. Notably, the equal risk prices produced with option hedges are typically lower than those obtained through hedges relying exclusively on the underlying asset, which is caused by a reduction in the level of market incompleteness when options become available for trading.