Deep Equal Risk Pricing of Financial Derivatives with Multiple Hedging Instruments

Abstract

This paper studies the equal risk pricing (ERP) framework for the valuation of European financial derivatives. This option pricing approach is consistent with global trading strategies by setting the premium as the value such that the residual hedging risk of the long and short positions in the option are equal under optimal hedging. The ERP setup of Marzban et al.(2020) is considered where residual hedging risk is quantified with convex risk measures.The main objective of this paper is to assess through extensive numerical experiments the impact of including options as hedging instruments within the ERP framework. The reinforcement learning procedure developed in Carbonneau and Godin (2020), which relies on the deep hedging algorithm of Buehler et al. (2019b), is applied to numerically solve the global hedging problems by representing trading policies with neural networks. Among other findings, numerical results indicate that in the presence of jump risk, hedging long-term puts with shorter-term options entails a significant decrease of both equal risk prices and market incompleteness as compared to trading only the stock. Monte Carlo experiments demonstrate the potential of ERP as a fair valuation approach providing prices consistent with observable market prices. Analyses exhibit the ability of ERP to span a large interval of prices through the choice of convex risk measures which is close to encompass the variance-optimal premium.

Deep equal risk pricing of financial derivatives with non-translation invariant risk measures

“…Carbonneau and Godin (2021b) provide a tractable methodology based on deep reinforcement learning to implement the ERP framework with convex risk measures under very general conditions. Carbonneau and Godin (2021a) examine the impact of introducing options as hedging instruments within the ERP framework under convex risk measures.…”

“…Marzban et al (2020) propose to use dynamic programming which they apply on a robust optimization setting. Conversely, Carbonneau and Godin (2021b) and Carbonneau and Godin (2021a) use the deep reinforcement learning approach of Buehler et al (2019) coined as deep hedging. Other papers have relied on the deep hedging methodology for the hedging of financial derivatives: Cao et al (2020), Carbonneau (2021), Horvath et al (2021) and Lütkebohmert et al (2021).…”

“…Due to its favorable properties, several studies use the CVaR within the ERP framework: see Carbonneau and Godin (2021b) and Carbonneau and Godin (2021a). It was observed in the foremost that when only the underlying asset is used to hedge put options and conventional risk-neutral measures are used to determine the initial capital for hedging, the tail risk is much more pronounced for the short position than for the long one, especially for out-of-the-money puts.…”

“…This approach should therefore not be pursued in practice. A second possible solution to the inflated ERP prices issue which is explored in Carbonneau and Godin (2021a) consists in incorporating other hedging instruments (e.g. short-term options) within dynamic hedging schemes.…”

“…The main contribution of this manuscript is twofold. The first is to propose a modification of the deep reinforcement learning approach illustrated in Carbonneau and Godin (2021b) and Carbonneau and Godin (2021a) to handle non-translation invariant risk measures within ERP naturally and without excessive additional computational burden. This modification essentially consists in feeding varying initial hedging portfolio values with simulated risky asset paths to the deep hedging algorithm from Buehler et al (2019), and then coupling the trained neural network output with a bisection search to seek the initial hedging portfolio value equating risks for both the long and short positions.…”