Why Finite Difference Methods Can Outperform Binomial Trees for Option Pricing

Summary

The document compares binomial trees with finite difference methods (FDM) for option valuation, especially American puts. It explains that a recombining binomial tree is a special case of an explicit FDM scheme, while more general FDM approaches can handle features that limit basic trees. Examples include local volatility, flexible boundary conditions for barrier options, and Crank–Nicolson schemes, which may converge faster as the time step is refined.

A second answer offers practical considerations: trees can be efficient for a small number of valuations, but may be less convenient when cash dividends, many options, or American exercise are involved. These are qualitative observations rather than a controlled benchmark, and the post gives no numerical comparison or detailed implementation guidance. The relative efficiency depends on the model, payoff, grid or tree design, and valuation workload.

Key ideas

• A recombining binomial tree can be viewed as a particular explicit finite difference scheme.
• Finite difference methods can represent local volatility and varied boundary conditions.
• Crank–Nicolson methods may converge faster with time-step refinement than explicit approaches.
• Finite difference methods can be useful for barrier options and repeated valuations.
• The document reports that dividends and American exercise can complicate tree-based valuation.

Tags

# Binomial Trees vs FDM


# Binomial Trees vs FDM












Binomial trees as the number of time steps is increased (or equivalently as the time step tends to 0), converge to the exact value for an option. So why do people use FDM for pricing options (for example an American Put), if Binomial Trees give already accurate results and converges quickly?

## Answer by Antoine Conze (score 4)

https://quant.stackexchange.com/a/44315

Actually recombining binomial trees are only a particular case of an explicit FDM scheme. But they have obvious limitations, the foremost being that they cannot accomodate local volatilities. Also 1/2 explicit 1/2 implicit FDM schemes (Crank-Nicolson) have faster convergence with respect to the size of the time step. And FDM schemes can accomodate all sorts of boundary conditions including Dirichlet which is necessary to accurately price barrier options.

## Answer by Amanda (score 1)

https://quant.stackexchange.com/a/60197

Our company chose to use FDM for calculating American Options. According to colleagues I talked with, binomial trees are efficient and accurate When there are a small number of option values. But it has a couple of weaknesses:

(1) Binomial tree models are generally inefficient when cash dividends should be taken into consideration;

(2) Compared with FDM, binomial trees are less efficient for multiple options valuations;

(3) Additionally, binomial trees are inefficient in valuing American options compared with European options.

Please correct me if I'm wrong.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.