Introduction

The binomial model is a versatile tool for option pricing, but its adaptability extends beyond equities. In this chapter, we’ll explore how the binomial model can be tailored to price options on a range of assets, including stocks with dividends, stock indices, currencies, and futures. By understanding these modifications, you’ll expand your ability to evaluate a broader spectrum of financial instruments and make informed investment decisions.

The basic binomial model assumes a single underlying asset, usually a stock, with two possible price movements: up and down. To price options on different assets, we need to adjust this framework to accommodate their unique characteristics.

Options on Stocks with Dividends

For stocks that pay dividends, we modify the model by incorporating dividend payments into the calculation of stock prices at each node. The adjusted formula for calculating the stock price at each node is:

$$S_u=S_0 \times e^{(r-D) \Delta t+\sigma \sqrt{\Delta t}}$$

Where:

• $S_u$ is the stock price at the up node
• $S_0$ is the initial stock price
• $r$ is the risk-free interest rate
• $D$ is the dividend yield
• $\Delta t$ is the time step
• $\sigma$ is the volatility

Options on Stock Indices

Pricing options on stock indices, such as the S\&P 500, involves a similar modification. The key adjustment is to replace the single stock price with an index value and account for dividends if applicable. The formula for calculating the index-based option price becomes:

$$C_u=e^{-r \Delta t} \left[p C_u^u+(1-p) C_u^d\right]$$

Where:

• $C_u$ is the option price at the up node
• $p$ is the probability of an up movement
• $C_u^u$ is the option price in the up state
• $C_u^d$ is the option price in the down state

Options on Currencies

For currency options, we need to account for exchange rates. The modified binomial model includes exchange rate movements in the calculations. The formula for pricing currency options becomes:

$$S_u=S_0 \times e^{\left(r_f-r_d\right) \Delta t+\sigma \sqrt{\Delta t}}$$

Where:

• $S_u$ is the exchange rate at the up node
• $S_0$ is the initial exchange rate
• $r_f$ is the foreign risk-free interest rate
• $r_d$ is the domestic risk-free interest rate
• $\Delta t$ is the time step
• $\sigma$ is the volatility

Options on Futures:

Pricing options on futures contracts involves adjusting the model to account for the characteristics of the underlying futures instrument. The modified model considers the cost of carry and convenience yield associated with futures. The formula for pricing futures options is:

$$F_u=F_0 \times e^{(r-y) \Delta t+\sigma \sqrt{\Delta t}}$$

Where:

• $F_u$ is the futures price at the up node
• $F_0$ is the initial futures price
• $r$ is the risk-free interest rate
• $y$ is the convenience yield (or cost of carry)
• $\Delta t$ is the time step
• $\sigma$ is the volatility