This Demonstration shows two approaches to using binomial option pricing for a European call option on an underlying asset with nonconstant volatility. The holder of the option has the right but not the obligation to purchase an asset at some fixed maturity date

in the future for a price agreed upon today (known as the strike price

).

The binomial trees employ standard pricing techniques similar to those first set out in [1]. These trees split the time until maturity into

time steps of size

. At each time step, the value of the underlying asset goes up or down, which scales the value by

and

. This is carried out for all nodes (points in the tree) until the option expires. This process effectively simulates possible values of the underlying asset at time

.

At maturity, the option can be exercised or not. The payoff of the option

is calculated for each possible value of the underlying asset at time

,

, where

is the value of the underlying asset at time

. The value of the option is then calculated using backward induction. That is, for each node in the tree, the present value of the possible outcomes in the next step is taken. Let

be the option value at node

, where

represents the total number of down moves that have occurred and

is the total number of moves. So,

is the value of the option one period before maturity if the underlying asset has only increased in value since the option contract was entered into. This is repeated to

to get the price of the option at the inception of the contract (

).

is the risk-neutral probability of an up move;

.

Typically

is constant throughout the tree. The issue that arises when volatility of the underlying asset is not constant (as shown in the left-hand tree) is that the up and down moves also become nonconstant. This means the tree that represents the process of the underlying asset's value will not recombine. This results in

possible outcomes at time

as opposed to

if the tree recombined. Although

converges to the true price as

,

typically needs to be large to obtain good approximations. This makes valuation using a non-recombining tree impractical.

One solution to the issue created by nonconstant volatility in binomial option pricing is shown in the right-hand tree. Instead of having constant time steps

and nonconstant up and down movements (

), we configure nonconstant time steps so that the up and down movements become constant. This is a modified version of a model presented by Guthrie in [2], who builds it to represent a real options approach to investment decisions, where an investor receives signals of a project's true value prior to investment. These signals decrease in how much they reveal about a project's true value over time.