Chapter 21: Basic Numerical Procedures

Why Numerical Methods?

Closed-form solutions (like Black–Scholes–Merton) exist only for simple European options. Numerical methods are needed for:

• American options (early exercise)
• Exotic options with path-dependence
• Complex payoffs
• Options on multiple assets
• Models with stochastic volatility or jumps

The three main approaches:

1. Binomial Trees

Standard Binomial Tree

Already introduced in Chapter 13. The key is choosing parameters:

$u=e^{\sigma \sqrt{\Delta t}},d=e^{-\sigma \sqrt{\Delta t}},p=\frac{e^{r\Delta t}-d}{u-d}$

For Dividends

The tree node for the stock price before a known dividend $D$ at time $\tau$: $S=S^{\ast }+D{e}^{-r\tau }$ (where $S^{\ast }$ is the ex-dividend stock price)

Extensions

• Trinomial trees: Three branches per node (up, middle, down) — offers more flexibility
• Implied/Adaptive trees: Chosen to match observed market prices (implied volatility surface)
• Control variate technique: For American options: price American and European using the same tree, adjust the American price by the difference between the true European price and the tree-computed European price

2. Monte Carlo Simulation

Generate random paths for the underlying asset and compute the average discounted payoff.

Path Generation

Stock price at time $T$:

$S_T=S_0\exp \left[\left(\mu -\frac{{\sigma }^2}{2}\right)T+\sigma \epsilon \sqrt{T}\right]$

where $\epsilon \sim N(0,1)$ is a random draw. For path-dependent options, generate intermediate prices:

$S_{t+\Delta t}=S_t\exp \left[\left(\mu -\frac{{\sigma }^2}{2}\right)\Delta t+\sigma \epsilon \sqrt{\Delta t}\right]$

Standard Error

After $M$ trials, the standard error of the estimate:

$\text{SE}=\frac{\omega }{\sqrt{M}}$

where $\omega$ is the standard deviation of the discounted payoffs. To halve the error, quadruple the number of trials. For typical accuracy of 0.01, about 100,000 trials are needed.

Advantages of Monte Carlo

• Easily handles path dependence
• Works well for high-dimensional problems (basket options, multi-factor models)
• Simple to implement and understand

Disadvantages

• Computationally intensive
• Difficult for American/exercisable options (needs special techniques)
• Gives a price estimate with associated standard error

3. Finite Difference Methods

Solve the Black–Scholes–Merton PDE directly by discretizing time and the stock price on a grid.

The PDE

$\frac{\partial f}{\partial t}+rS\frac{\partial f}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}f}{\partial S^2}=rf$

Implicit vs. Explicit Methods

The Crank–Nicolson scheme is usually preferred because it's unconditionally stable with $O(\Delta t^2)$ accuracy.

Advantages of Finite Difference

• Efficiently handles American options (intrinsic value check at each grid point)
• Provides option prices for all stock prices and times simultaneously
• No simulation noise

Disadvantages

• Complexity increases exponentially with dimension (curse of dimensionality)
• Requires careful implementation of boundary conditions

Comparison of Methods