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.

A method to hedge variable annuities in the presence of basis risk is developed. A regime-switching model is considered for the dynamics of market assets. The approach is based on a local optimization of risk and is therefore very tractable and flexible. The local optimization criterion is itself optimized to minimize capital requirements associated with the variable annuity policy, the latter being quantified by the Conditional Value-at-Risk (CVaR) risk metric. In comparison to benchmarks, our method is successful in simultaneously reducing capital requirements and increasing profitability. Indeed the proposed local hedging scheme benefits from a higher exposure to equity risk and from time diversification of risk to earn excess return and facilitate the accumulation of capital. A robust version of the hedging strategies addressing model risk and parameter uncertainty is also provided.

This paper introduces a flexible risk decomposition method for life insurance contracts embedding several risk factors. Hedging can be naturally embedded in the framework. Although the method is applied to variable annuities in this work, it is also applicable in general to other insurance or financial contracts. The approach relies on applying an allocation principle to components of a Shapley decomposition of the gain and loss. The implementation of the allocation method requires the use of a stochastic on stochastic algorithm involving nested simulations. Numerical examples studying the relative impact of equity, interest rate and mortality risk for guaranteed minimal maturity benefit (GMMB) policies conclude our analysis.

The current paper provides a general approach to construct distortion operators that can price financial and insurance risks. Our approach generalizes the (Wang 2000) transform and recovers multiple distortions proposed in the literature as particular cases. This approach enables designing distortions that are consistent with various pricing principles used in finance and insurance such as no-arbitrage models, equilibrium models and actuarial premium calculation principles. Such distortions allow for the incorporation of risk-aversion, distribution features (e.g. skewness and kurtosis) and other considerations that are relevant to price contingent claims. The pricing performance of multiple distortions obtained through our approach is assessed on CAT bonds data. The current paper is the first to provide evidence that jump-diffusion models are appropriate for CAT bonds pricing, and that natural disaster aversion impacts empirical prices. A simpler distortion based on a distribution mixture is finally proposed for CAT bonds pricing to facilitate the implementation.

Variable annuity (VA) policies are typically issued on mutual funds invested in both fixed income and equity asset classes. However, due to the lack of specialized models to represent the dynamics of fixed income fund returns, the literature has primarily focused on studying long-term investment guarantees on single-asset equity funds. This article develops a mixed bond and equity fund model in which the fund return is linked to movements of the yield curve. Theoretical motivation for our proposed specification is provided through an analogy with a portfolio of rolling horizon bonds. Moreover, basis risk between the portfolio return and its risk drivers is naturally incorporated into our framework. Numerical results show that the fit of our model to Canadian VA data is adequate. Finally, the valuation of VAs is illustrated and it is found that the prevailing interest rate environment can have a substantial impact on guarantee costs.