Binomial Tree Option Pricing › The One-Step Binomial Model

The central idea behind option pricing in this framework is replication. Rather than trying to guess the probability of the stock going up or down, we construct a portfolio of basic assets whose payoff exactly matches the derivative's payoff in every possible future state. This replicating portfolio consists of two components: the underlying stock and a risk-free bond. If we can perfectly replicate the derivative using these tradeable instruments, then the derivative must have the same price as the portfolio, otherwise arbitrage opportunities would exist.

Suppose we hold $\Delta$ shares of stock and invest $B$ dollars in a risk-free bond earning rate $r$ over the period. The parameter $\Delta$ represents the number of shares we hold, which can be fractional, and $B$ represents our cash position in the risk-free bond, which can be positive (lending) or negative (borrowing). At expiration, after one time period $T$, this portfolio is worth:

• Up state: $\Delta \cdot S_u + B \cdot e^{rT} = \Delta \cdot u S_0 + B \cdot e^{rT}$
• Down state: $\Delta \cdot S_d + B \cdot e^{rT} = \Delta \cdot d S_0 + B \cdot e^{rT}$

For this portfolio to replicate the derivative, we need:

$$\begin{aligned} \Delta \cdot u S_0 + B \cdot e^{rT} &= f_u \\ \Delta \cdot d S_0 + B \cdot e^{rT} &= f_d \end{aligned}$$

where:

• $\Delta$: number of shares of stock in the portfolio
• $S_0$: current stock price
• $u, d$: up and down movement factors
• $B$: amount invested in the risk-free bond
• $r$: risk-free interest rate
• $T$: time to expiration
• $f_u, f_d$: option payoffs in the up and down states

This is a system of two linear equations in two unknowns ($\Delta$ and $B$). We can solve this system using straightforward algebraic manipulation. The strategy is to first eliminate one variable to find the other, then substitute back to complete the solution.

Subtracting the second equation from the first eliminates $B$, since the bond value $B \cdot e^{rT}$ appears identically in both equations:

$$\begin{aligned} (\Delta \cdot u S_0 + B \cdot e^{rT}) - (\Delta \cdot d S_0 + B \cdot e^{rT}) &= f_u - f_d && \text{(subtract down state from up state)} \\ \Delta \cdot S_0 (u - d) &= f_u - f_d && \text{(simplify to isolate } \Delta \text{)} \end{aligned}$$

where:

• $\Delta$: number of shares in the portfolio
• $S_0$: current stock price
• $u, d$: up and down movement factors
• $B$: bond investment
• $r$: risk-free rate
• $T$: time to expiration
• $f_u, f_d$: option payoffs

This gives us the hedge ratio, which represents the number of shares needed to replicate the derivative:

$$\Delta = \frac{f_u - f_d}{S_0(u - d)} = \frac{f_u - f_d}{S_u - S_d}$$

where:

• $\Delta$: the hedge ratio (number of shares to hold)
• $f_u - f_d$: the range of option payoffs
• $S_u, S_d$: stock prices in up and down states
• $S_u - S_d$: the range of stock prices
• $S_0$: current stock price
• $u, d$: up and down movement factors

This formula has a beautiful interpretation: the hedge ratio equals the change in option value divided by the change in stock price. This is exactly the discrete-time analog of the delta we encountered in continuous-time option pricing. It tells us how many shares to hold to hedge the derivative, and it measures the sensitivity of the option price to movements in the underlying stock. For a call option, $\Delta$ is positive because the option gains value when the stock rises. For a put option, $\Delta$ is negative because the option loses value when the stock rises.

Once we know $\Delta$, we can solve for $B$ by substituting back into either of our original equations. Using the up-state equation:

$$\begin{aligned} B \cdot e^{rT} &= f_u - \Delta \cdot u S_0 && \text{(from up state equation)} \\ B &= e^{-rT}(f_u - \Delta \cdot u S_0) && \text{(solve for } B \text{)} \end{aligned}$$

where:

• $B$: present value of the bond investment
• $e^{-rT}$: discount factor
• $f_u - \Delta \cdot u S_0$: portfolio cash needed to match the payoff in the up state
• $f_u, f_d$: option payoffs
• $\Delta$: hedge ratio
• $u, d$: up and down factors
• $S_0$: current stock price
• $r$: risk-free interest rate
• $T$: time to expiration

The bond position $B$ is typically negative for call options, meaning we borrow money to fund the stock purchase. For put options, $B$ is typically positive, meaning we lend money while shorting stock.

Since the replicating portfolio has the same payoff as the derivative in both states, by the no-arbitrage principle from Part III Chapter 4, the derivative must have the same price as the portfolio:

$$f_0 = \Delta \cdot S_0 + B$$

where:

• $f_0$: current option price
• $\Delta$: number of shares held in the replicating portfolio
• $S_0$: current stock price
• $B$: value of the risk-free bond investment

If the derivative traded at any other price, an arbitrage opportunity would exist. If the derivative were cheaper than the replicating portfolio, we could buy the derivative, sell the portfolio, and pocket the difference while having perfectly offsetting positions. Conversely, if the derivative were more expensive, we could sell the derivative, buy the portfolio, and again lock in a risk-free profit. Market forces ensure that such opportunities are quickly eliminated, establishing the pricing relationship above.

Now we can derive an explicit formula for the option price by substituting our expressions for $\Delta$ and $B$ into the portfolio value equation $f_0 = \Delta S_0 + B$:

$$\begin{aligned} f_0 &= \Delta S_0 + e^{-rT}(f_u - \Delta u S_0) && \text{(substitute } B \text{)} \\ &= e^{-rT}f_u + \Delta S_0 (1 - u e^{-rT}) && \text{(group terms)} \\ &= e^{-rT}f_u + \frac{f_u - f_d}{u - d} (1 - u e^{-rT}) && \text{(substitute } \Delta \text{)} \\ &= e^{-rT}\left[f_u + \frac{(f_u - f_d)(e^{rT} - u)}{u - d}\right] && \text{(factor out } e^{-rT} \text{)} \\ &= e^{-rT}\left[\frac{e^{rT} - d}{u - d} f_u + \frac{u - e^{rT}}{u - d} f_d\right] && \text{(simplify)} \end{aligned}$$

where:

• $f_0$: current option price
• $\Delta$: number of shares held in the replicating portfolio
• $S_0$: current stock price
• $\frac{e^{rT} - d}{u - d}$: weight assigned to the up payoff
• $\frac{u - e^{rT}}{u - d}$: weight assigned to the down payoff
• $r$: risk-free interest rate
• $T$: time to expiration
• $u, d$: up and down factors
• $f_u, f_d$: option payoffs

This final expression is remarkable: the option price is a discounted weighted average of the two possible payoffs. The weights depend only on the up and down factors and the risk-free rate, not on any probability of the stock going up or down. This observation leads us to one of the most important concepts in derivative pricing.

The formula above has a beautiful interpretation. Define the risk-neutral probability $p$:

$$p = \frac{e^{rT} - d}{u - d}$$

where:

• $p$: risk-neutral probability of an up move
• $u, d$: up and down factors
• $r$: risk-free rate
• $T$: time step

Then the pricing formula becomes:

$$f_0 = e^{-rT}[p \cdot f_u + (1-p) \cdot f_d]$$

where:

• $f_0$: current option price
• $p$: risk-neutral probability
• $1-p$: probability of a down move
• $e^{-rT}$: risk-free discount factor
• $f_u, f_d$: option payoffs in up and down states
• $r$: risk-free interest rate
• $T$: time step duration

This elegant formula states that the derivative price equals the discounted expected payoff, where the expectation uses the probability $p$ rather than the real-world probability of an up move. The expression $p \cdot f_u + (1-p) \cdot f_d$ computes the expected payoff under the risk-neutral probability measure, and the factor $e^{-rT}$ discounts this expected value back to the present. We never needed to know or estimate the actual probability of the stock rising; the risk-neutral probability emerges naturally from the no-arbitrage requirement.

For $p$ to be a valid probability (between 0 and 1), we need:

$$d < e^{rT} < u$$

where:

• $d$: down movement factor
• $u$: up movement factor
• $r$: risk-free interest rate
• $T$: time step duration

This condition ensures no arbitrage and has an intuitive economic interpretation. The term $e^{rT}$ represents the growth factor of a risk-free investment over the period. The condition states that this risk-free growth must fall strictly between the down move $d$ and the up move $u$. If $e^{rT} \geq u$, the risk-free bond would always outperform the stock, making the stock unattractive at any positive price. Conversely, if $e^{rT} \leq d$, the stock would always outperform the risk-free bond, meaning investors would borrow at the risk-free rate and invest in the stock for guaranteed profit. Either case would create an arbitrage opportunity, which cannot persist in equilibrium.