Using Trees to Price Mortgages and Other Interest Rate Derivatives

The pricing of interest rate derivatives is a topic of significant interest to banks and hedge funds, particularly in light of the recent 2023 banking crisis. Rudimentary models for pricing interest rate derivatives, such as the binomial no-arbitrage model, often fall short in capturing the complex dynamics of the modern financial markets. In this paper, I propose an extension to the simple binomial no-arbitrage model by introducing a model that accounts for both volatility jumps and yield curve steepness jumps. This extended model, which I refer to as the two-factor tree model, offers a more comprehensive and realistic representation of the dynamic interest rate environment, thus providing a valuable tool for market participants seeking to hedge against interest rate and yield curve risk. We are interested in discrete-time models because they allow us to price instruments for which closed-form solutions in continuous time do not exist. By implementing this two-factor tree, we are able to price a wide range of interest rate derivatives, including mortgages, options, swaps, CDS, and futures contracts, with high precision. The code accompanying this paper contains a memory efficient and fast implementation of this tree, designed with the needs of practitioners in mind. The implementation leverages the power of Python and the NumPy and numba libraries for optimal performance. It is capable of pricing mortgages, bonds, European and American interest rate options on our two-factor tree in sub second times, making it suitable for real-time applications.