Deep Learning-Based BSDE Solver for Libor Market Model with Applications to Bermudan Swaption Pricing and Hedging

The Libor market model, also known as the BGM Model, is a term structure model of interest rates. It is widely used for pricing interest rate derivatives, especially Bermudan swaptions, and other exotic Libor callable derivatives. For numerical implementation the pricing of derivatives with Libor market models is mainly carried out with Monte Carlo simulation. The PDE grid approach is not particularly feasible due to the “Curse of Dimensionality”. The standard Monte Carlo method for American swaption pricing more or less uses regression to estimate the expected value as a linear combination of basis functions as demonstrated in the classical paper by Longstaff and Schwartz [Longstaff01]. However, [Longstaff01] only provides the lower bound for American option price. Another complexity arises from applying Monte Carlo simulation is the computation of the sensitivities of the option, the so-called “Greeks”, which are fundamental for a trader's hedging activity. Recently, an alternative numerical method based on deep learning and backward stochastic differential equations (BSDEs) appeared in quite a few researches [Weinan17, Jiequn17]. For European style options the feedforward deep neural networks (DNN) show not only feasibility but also efficiency to obtain both prices and numerical Greeks. The standard LMM implementation requires dimension five or higher in factor space even after PCA, which cannot be solved by traditional PDE solvers, such as finite differences or finite elements methods. In this paper, a new backward DNN solver is proposed for Bermudan swaptions. Our approach is representing financial pricing problems in the form of high dimensional stochastic optimal control problems, FBSDEs, or equivalent PDEs. We demonstrate that using backward DNN the high-dimension Bermudan swaption pricing and hedging can be solved effectively and efficiently. A comparison between Monte Carlo simulation and the new method for pricing vanilla interest rate options manifests the superior performance of the new method. We then use this method to calculate prices and Greeks of Bermudan swaptions as a prelude for other Libor callable derivatives