9

Binomial Option Pricing Tutorial and Spreadsheets

This tutorial introduces binomial option pricing, and offers an Excel spreadsheet to help you better understand the principles.  Additionally, a spreadsheet that prices Vanilla and Exotic options with a binomial tree is provided.

Binomial option pricing is based on a no-arbitrage assumption, and is a mathematically simple but surprisingly powerful method to price options. Rather than relying on the solution to stochastic differential equations (which is often complex to implement), binomial option pricing is relatively simple to implement in Excel and is easily understood.

No-arbitrage means that markets are efficient, and investments earn the risk-free rate of return.

Binomial trees are often used to price American put options, for which (unlike European put options) there is no close-form analytical solution.

Price Tree for Underlying Asset

Consider a stock (with an initial price of S0) undergoing a random walk. Over a time step Δt, the stock has a probability p of rising by a factor u, and a probability 1-p of  falling in price by a factor d. This is illustrated by the following diagram.

One Step Binomial Model

Cox, Ross and Rubenstein Model

Cox, Ross and Rubenstein (CRR) suggested a method for calculating p, u and d.  Other methods exist (such as the Jarrow-Rudd or Tian models), but the CRR approach is the most popular.

Over a small period of time, the binomial model acts similarly to an asset that exists in a risk neutral world. This results in the following equation, which implies that the effective return of the binomial model (on the right-hand side) is equal to the risk-free rate

Additionally, the variance of a risk-neutral asset and an asset in a risk neutral world match. This gives the following equation.

The CRR model suggests the following relationship between the upside and downside factors.

Rearranging these equations gives the following equations for p, u and d.

The values of p, u and d given by the CRR model means that the underlying initial asset price is symmetric for a multi-step binomial model.

Two-Step Binomial Model

This is a two-step binomial lattice.

Two-Step Binomial Model

At each stage, the stock price moves up by a factor u or down by a factor d.  Note that at the second step, there are two possible prices, u d S0 and d u S0. If these are equal, the lattice is said to be recombining. If they are not equal, the lattice is said to be non-recombining.

The CRR model ensures a recombining lattice; the assumption that u = 1/d means that u d S0 = d u S0 = S0, and that the lattice is symmetrical.

Multi-Step Binomial Model

The multi-step binomial model is a simple extension of the principles given in the two-step binomial model. We simply step forward in time, increasing or decreasing the stock price by a factor u or d each time.

Multi-Step Binomial Model

Each point in the lattice is called a node, and defines an asset price at each point in time. In reality, many more stages are usually calculated than the three illustrated above, often thousands.

Payoffs for Option Pricing

We will consider the following payoff functions.

Put: VN = max(X – SN, 0)

Call: VN = max(SN – X, 0)

VN is the option price at the expiry node N, X is the strike or exercise price, SN is the stock price at the expiry node N.

We now need to discount the payoffs back to today. This involves stepping back through the lattice, calculating the option price at every point.

This is done with an equation that varies with the type of option under consideration. For example, European and American options are priced with the equations below.

European Put: Vn = e-rΔt(p Vu + ( 1 – p ) Vd )

European Call: Vn = e-rΔt(p Vu + ( 1 – p ) Vd

American Put: Vn = max(X – Sn, e-rΔt ( p Vu + ( 1 – p ) Vd ))

American Call: Vn = max(Sn – X, e –rΔt ( p Vu + ( 1 – p ) Vd ))

N is any node before expiry.

Binomial Option Pricing in Excel

This Excel spreadsheet implements a binomial pricing lattice to calculate the price of an option. Simply enter some parameters as indicated below.

Excel will then generate the binomial lattice for you. The spreadsheet is annotated to improve your understanding.

Note that the stock price is calculated forward in time.  However, the option price is calculated backwards from the expiry time to today (this is known as backwards induction).

The spreadsheet also compares the Put and Call price given by the binomial option pricing lattice with that given by the analytic solution of the Black-Scholes equation; for many time steps in the lattice, the two prices converge.

Excel Spreadsheet for Binomial Option Pricing

Pricing Vanilla and Exotic Options with Binomial Tree in Excel

This Excel spreadsheet prices several types of options (European, American, Shout, Chooser, Compound) with a binomial tree. The spreadsheet also calculate the Greeks (Delta, Gamma and Theta). The number of time steps is easily varied – convergence is rapid.

The algorithms are written in password-protected VBA . If you’d like to see and edit the VBA, purchase the unprotected spreadsheet at http://investexcel.net/buy-spreadsheets/.

Excel Spreadsheet to Price Vanilla, Shout, Compound and Chooser Options

9 Responses to "Binomial Option Pricing Tutorial and Spreadsheets"

1. Lisa says:

Hi I was wondering whether you have any spreadsheets that calculate the price of an option using the binomial option pricing model (CRR) (including dividend yield)..and then a comparison against the black scholes price (for the same variables) could be shown on a graph (showing the convergence)

• Samir says:

I’ve hacked together this worksheet. It compares prices of European options given by analytical equations and a binomial tree. You can change the number of binomial steps to compare the convergence against the analytical solution

European Option – Analytical vs CRR

2. josphat says:

hello,

thanks alot for that explanation.

Do you know how to get the implied volatility of american options through binomial tree? can you point me to a paper illustrating this please.

• Samir says:

In this spreadsheet I’ve backed out the implied volatility of an American (or European) option from a binomial tree using a simple Goal Seek: Implied Volatility from Binomial Tree

When I get time I’ll write a spreadsheet that uses Newton-Raphson or a Bisection method on a binomial tree

Samir

4. Chang Fee says:

This stuff is a bit over my head. I’d like to find a way to tell what the delta of any given stock option is. For instance, if you were looking at Puts on Amazon:

http://finance.yahoo.com/q/op?s=AMZN+Options

How would you find the delta of the \$230 May Puts?

Is there anything else that would be wise to look at?

Thanks so much, from an Options Newbie!

CF

5. oot says:

This is great and helpful. Thank you for your contribution to the community.

6. JO says:

Hi Samir, am writing a paper over the Binomial method for my school. I would like to have your permission to copy the two step Binomial graphic onto my paper. It will be referenced following the APA citation guide.

Thanks in anticipation to your favorable response.