Pricing European Options with Binomial Trees

From One Step to Many

The beauty of the binomial model is its scalability. A multi-period model is nothing more than a series of one-period models stitched together. At each node in our price tree, the same fundamental pricing relationship holds: the option's value is the discounted expected value of its two possible future states.

The following video provides an excellent high-level introduction to this idea, showing how adding more steps makes the model more realistic without introducing new mathematical complexity.

The process creates a "tree" of possible asset prices over time, which recombines at certain nodes. For example, a price movement of 'up' then 'down' results in the same final price as 'down' then 'up'.

The Backward Induction Algorithm

To price a European option on a multi-period tree, we use a clear, algorithmic process known as backward induction. It consists of four main steps:

1. Construct the Stock Price Tree: Start with the current stock price, $S_0$. For each time step, $\Delta t$, calculate the two possible subsequent prices by multiplying by the up-factor, $u$, and the down-factor, $d$. Repeat this for all $N$ time steps until you reach the option's expiration date, $T$.
2. Calculate Terminal Option Payoffs: At the final time step ($t=T$), for each possible final stock price ($S_T$), calculate the option's intrinsic value.
   • For a European call option: $C_T=\max (S_T-K,0)$
   • For a European put option: $P_T=\max (K-S_T,0)$
3. Work Backwards Through the Tree: Move to the second-to-last time step ($t=T-\Delta t$). At each node, calculate the option's value using the one-period risk-neutral valuation formula. The value at any node $C_t$ is the discounted expected value of the two possible option values at the next step, $C_{t+\Delta t,u}$ and $C_{t+\Delta t,d}$. $$C_t=e^{-r\Delta t}\left[\pi C_{t+\Delta t,u}+(1-\pi )C_{t+\Delta t,d}\right]$$ Here, $\pi$ is the risk-neutral probability: $$\pi =\frac{e^{r\Delta t}-d}{u-d}$$
4. Repeat Until Time 0: Continue this process, stepping back one period at a time, until you arrive back at $t=0$. The value you calculate at this initial node, $C_0$, is the arbitrage-free price of the European option today.

A Two-Period Example

Let's solidify this with a practical example. We'll follow a worked example for a two-period European call option. This resource lays out the entire process numerically.

As you can see, the core of the process is the repeated application of the same one-period pricing formula you've already learned.

Refining the Model's Parameters

You might be asking where the parameters $u$, $d$, and $\Delta t$ come from in a real-world application. For a model with $N$ steps over a total time to maturity of $T$ (in years), the length of each step is simply $\Delta t=T/N$.

The up and down factors, $u$ and $d$, are derived from the underlying asset's volatility, $\sigma$. A common formulation, proposed by Cox, Ross, and Rubinstein, ensures that as the number of steps increases, the model converges to the famous Black-Scholes-Merton model. The formulas are:

• Up-factor: $u=e^{\sigma \sqrt{\Delta t}}$
• Down-factor: $d=e^{-\sigma \sqrt{\Delta t}}$ (which is simply $1/u$)

The following short video explains this connection between volatility and the $u$ and $d$ factors.

With these definitions for $u$, $d$, and $\Delta t$, we can construct a binomial tree for any European option given its time to maturity, the underlying's volatility, and the risk-free rate.

Let's look at one more example, this time a 3-period model, to see all the pieces in action.

A 3-Period European Put Option

This next resource provides a complete example for a 3-period European put option.

Let's use the diagram from the Oxford PDF to test your understanding. The diagram shows that at $t=2$, when the stock price is 99, the option value is 5.76. Can you verify this?

• Step 1: Identify the two possible stock prices at $t=3$.
  • Up-move: $99\times 1.1=108.9$
  • Down-move: $99\times 0.9=89.1$
• Step 2: Calculate the option payoffs at $t=3$ for the strike price $K=110$.
  • Payoff (up): $\max (110-108.9,0)=1.1$
  • Payoff (down): $\max (110-89.1,0)=20.9$
• Step 3: Calculate the discounted expected value using $\pi =0.75$ and a discrete interest rate of $r=5\%$. $$\text{Value}=\frac{1}{1+0.05}\left[0.75\times 1.1+(1-0.75)\times 20.9\right]$$ $$\text{Value}=\frac{1}{1.05}\left[0.825+5.225\right]=\frac{6.05}{1.05}\approx 5.76$$

The calculation matches the value in the tree, confirming the backward induction process.

Conclusion

In this lesson, we have extended the one-period binomial model to a more flexible and realistic multi-period framework. You've learned the systematic algorithm of backward induction, which is the cornerstone of pricing path-dependent derivatives in discrete time.

Key Takeaways:

• A multi-period binomial model is an iteration of the one-period model, applying the same risk-neutral valuation principle at each step.
• The price of a European option is found using backward induction: calculate terminal payoffs at expiration and then work backward through the tree, node by node.
• The value at any node is the discounted expected future value, calculated using the risk-neutral probability $\pi$.
• The model parameters ($u,d$) can be calibrated to the underlying asset's volatility ($\sigma$), making the model's assumptions consistent with continuous-time finance.

We have focused exclusively on European options, which can only be exercised at expiration. In our next lesson, we will tackle American options. This will add one layer of complexity to our algorithm: at each node, we will not only calculate the discounted expected value but also compare it to the value of immediate exercise. This "early exercise" decision is what distinguishes American from European option pricing.