Chapter 13: Binomial Trees

The One-Step Binomial Model

A simple yet powerful approach: assume the stock price can move either up or down over a single time step of length $\Delta t$.

• Stock price $S_0$ goes up to $S_0\cdot u$ with probability $p$
• Stock price $S_0$ goes down to $S_0\cdot d$ with probability $1-p$

Example: $S_0=20$, $u=1.1$, $d=0.9$, $T=3$ months, $r=12\%$, $K=21$.

Up state: $S_u=22$, call payoff = $\max (22-21,0)=1$ Down state: $S_d=18$, call payoff = $\max (18-21,0)=0$

No-Arbitrage Argument

Create a riskless portfolio: long $\Delta$ shares of stock and short 1 call:

$\Delta \cdot S_u-f_u=\Delta \cdot S_d-f_d$

Solving:

$\Delta =\frac{f_u-f_d}{S_{0}u-S_{0}d}$

The portfolio earns the risk-free rate, so:

$S_0\Delta -f=(S_{0}u\Delta -f_u)\cdot e^{-rT}$

Solving for $f$, the call price:

$f=S_0\Delta -(S_{0}u\Delta -f_u)e^{-rT}=0.633$

Risk-Neutral Valuation

The same result emerges if we define risk-neutral probabilities:

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

Then the option value is the expected payoff discounted at the risk-free rate:

$f=e^{-r\Delta t}\cdot [p\cdot f_u+(1-p)\cdot f_d]$

Key insight: In a risk-neutral world, all investors are indifferent to risk. The expected return on all assets is the risk-free rate. The option price is the same in the real world and the risk-neutral world.

For our example: $p=(e^{0.12\times 0.25}-0.9)/(1.1-0.9)=0.6523$

$f=e^{-0.12\times 0.25}\times [0.6523\times 1+0.3477\times 0]=0.633$

Two-Step Binomial Trees

Extend to two steps. At each node:

1. Compute the option values at the final nodes (maturity)
2. Work backwards using risk-neutral valuation

European options: One pass backward through the tree American options: At each node, compare the value of immediate exercise with the computed value (take the maximum)

Matching Volatility

The parameters $u$ and $d$ are chosen to match the stock's volatility $\sigma$:

$u=e^{\sigma \sqrt{\Delta t}},d=e^{-\sigma \sqrt{\Delta t}}=\frac{1}{u}$

The risk-neutral probability:

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

Delta ($\Delta$)

Delta is the ratio of the change in option price to the change in stock price:

$\Delta =\frac{f_u-f_d}{S_{0}u-S_{0}d}$

For the one-step example: $\Delta =(1-0)/(22-18)=0.25$

Delta is the number of shares to hold to create a riskless hedge. It changes at each node in the tree.

Increasing the Number of Steps

As the number of time steps $n$ increases:

• The binomial pricing converges to the Black–Scholes–Merton price
• A tree with 30 steps usually gives adequate accuracy
• DerivaGem can handle up to 500 steps

Cox-Ross-Rubinstein Formulas

The binomial tree is consistent with Black–Scholes–Merton as $\Delta t\to 0$ when:

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

For a European call with $n$ steps, the price is:

$c=e^{-rn\Delta t}{\sum }_{j=0}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)p^j(1-p{)}^{n-j}\max (S_0{u}^j{d}^{n-j}-K,0)$

Options on Other Assets

The binomial tree adapts to other underlyings by adjusting the risk-neutral probability $p$:

Appendix: From Binomial to Black–Scholes–Merton

As $n\to \infty$ and $\Delta t\to 0$, the binomial model converges to the lognormal stock price distribution assumed by Black–Scholes–Merton. The binomial sum converges to the Black–Scholes–Merton pricing formula.