[Paper Review] Bermudan options by simulation

L. C. G. Rogers|arXiv (Cornell University)|Aug 25, 2015

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper presents a generic, simulation-based method for pricing high-dimensional Bermudan options using Markovian approximation and duality to generate tight upper and lower price bounds. By simulating the underlying process and binning states to create stopping and hedging rules, it achieves accurate bounds in seconds regardless of dimension, with results typically within 5–10% of each other and fully insensitive to the number of underlying assets.

ABSTRACT

The aim of this study is to devise numerical methods for dealing with very high-dimensional Bermudan-style derivatives. For such problems, we quickly see that we can at best hope for price bounds, and we can only use a simulation approach. We use the approach of Barraquand & Martineau which proposes that the reward process should be treated as if it were Markovian, and then uses this to generate a stopping rule and hence a lower bound on the price. Using the dual approach introduced by Rogers, and Haugh & Kogan, this approximate Markov process leads us to hedging strategies, and upper bounds on the price. The methodology is generic, and is illustrated on eight examples of varying levels of difficulty. Run times are largely insensitive to dimension.

Motivation & Objective

• To develop a general-purpose numerical method for pricing Bermudan-style derivatives in high dimensions where traditional dynamic programming fails due to the curse of dimensionality.
• To provide computationally efficient upper and lower bounds on the option price without requiring explicit solution of the value function.
• To enable practical implementation of optimal stopping and hedging strategies using only simulation of the underlying Markov process.
• To ensure the method remains effective and scalable across diverse financial derivatives, regardless of the number of underlying assets.
• To offer a framework that is robust to model uncertainty and estimation error, with bounds that reflect realistic market spreads.

Proposed method

• The method approximates the reward process as Markovian by discretizing its state space into bins, enabling simulation-based computation of transition probabilities.
• It uses the Barraquand & Martineau approach to derive a stopping rule based on the binned Markov process, yielding a lower bound on the option price.
• The dual method of Rogers and Haugh & Kogan is applied to generate a corresponding upper bound via a hedging strategy derived from the same binned process.
• Both bounds are evaluated through Monte Carlo simulation, using the same code for all problems with only the process and payoff function changed.
• The approach is dimension-agnostic: computational cost remains low even for hundreds of underlying assets, as it avoids storing or solving high-dimensional value functions.
• The method relies solely on the ability to simulate one step of the underlying Markov process, making it broadly applicable to complex derivatives.

Experimental results

Research questions

• RQ1 Can we price high-dimensional Bermudan options effectively when traditional dynamic programming becomes infeasible due to the curse of dimensionality?
• RQ2 To what extent can simulation-based methods generate tight, reliable bounds on derivative prices without solving the full value function?
• RQ3 How can we construct practical stopping and hedging rules that are both simple and effective in high-dimensional settings?
• RQ4 What is the impact of model approximation and discretization on the accuracy of price bounds in realistic financial contexts?
• RQ5 Can a generic, reusable algorithm be designed that works across diverse derivatives with minimal problem-specific tuning?

Key findings

• The method produces upper and lower price bounds that are typically within 5–10% of each other, a range consistent with estimation and model uncertainty in practice.
• Computational time remains largely insensitive to dimension, with typical runtimes of just a few tens of seconds, even for problems with hundreds of underlying assets.
• The approach yields explicit, implementable stopping and hedging rules that are derived directly from the simulation and binning process.
• The method is fully generic: the same code can price any Bermudan option by changing only the underlying process and payoff function.
• The bounds are robust to model uncertainty; the largest source of error is the Markovian approximation of the reward process, not simulation or binning.
• For problems like the fixed-strike Bermudan Asian option, the method's performance degrades when the state space is inadequately captured, highlighting the need for richer state representations.