Binomial Trees

Boosting Binomial Exotic Option Pricing with Tensor Networks ArXiv ID: 2505.17033 “View on arXiv” Authors: Maarten van Damme, Rishi Sreedhar, Martin Ganahl Abstract Pricing of exotic financial derivatives, such as Asian and multi-asset American basket options, poses significant challenges for standard numerical methods such as binomial trees or Monte Carlo methods. While the former often scales exponentially with the parameters of interest, the latter often requires expensive simulations to obtain sufficient statistical convergence. This work combines the binomial pricing method for options with tensor network techniques, specifically Matrix Product States (MPS), to overcome these challenges. Our proposed methods scale linearly with the parameters of interest and significantly reduce the computational complexity of pricing exotics compared to conventional methods. For Asian options, we present two methods: a tensor train cross approximation-based method for pricing, and a variational pricing method using MPS, which provides a stringent lower bound on option prices. For multi-asset American basket options, we combine the decoupled trees technique with the tensor train cross approximation to efficiently handle baskets of up to $m = 8$ correlated assets. All approaches scale linearly in the number of discretization steps $N$ for Asian options, and the number of assets $m$ for multi-asset options. Our numerical experiments underscore the high potential of tensor network methods as highly efficient simulation and optimization tools for financial engineering. ...

Gaussian Recombining Split Tree ArXiv ID: 2405.16333 “View on arXiv” Authors: Unknown Abstract Binomial trees are widely used in the financial sector for valuing securities with early exercise characteristics, such as American stock options. However, while effective in many scenarios, pricing options with CRR binomial trees are limited. Major limitations are volatility estimation, constant volatility assumption, subjectivity in parameter choices, and impracticality of instantaneous delta hedging. This paper presents a novel tree: Gaussian Recombining Split Tree (GRST), which is recombining and does not need log-normality or normality market assumption. GRST generates a discrete probability mass function of market data distribution, which approximates a Gaussian distribution with known parameters at any chosen time interval. GRST Mixture builds upon the GRST concept while being flexible to fit a large class of market distributions and when given a 1-D time series data and moments of distributions at each time interval, fits a Gaussian mixture with the same mixture component probabilities applied at each time interval. Gaussian Recombining Split Tre Mixture comprises several GRST tied using Gaussian mixture component probabilities at the first node. Our extensive empirical analysis shows that the option prices from the GRST align closely with the market. ...