Isaac Wilson

Isaac Wilson
FIN 622
Extra Credit
Multi-Period Binomial Equation

The Multi-Period Binomial process is a straight forward way to approximate the value of a derivative based on the possible future prices of its underlying asset.
As an example, an asset worth $98 could increase by 32% in one period, or decrease by an identical percentage in one period. The option can be exercised at the end of two periods. Given a risk free interest rate of 6%, what initial premium does the binomial model predict for a PUT option having a $100 exercise price?

To start, calculate the value of the asset at each possible future node.
UUa (up,up) is $98 x 1.32 x 1.32 = $170.76
UDa (up,down) is $98 x 1.32 x .68 = $87.96
DDa (down,down) is $98 x .68 x .68 = $45.32

Then, find what the value of the PUT option would be at each node. In this case, anything BELOW the $100 strike price is 'in the money'. So,
UUo = $100 - $170.76 ~ $0
UDo = $100 - $87.96 = $12.04
DDo = $100 - $45.32 = $54.68

Next, solve for Pi = [1 +.06(risk free interest rate) - .68 (down node, d)]/[1.32(up node, u) - .68] = .59375

Finally, plug Pi and the values of the derivative at the UD and DD nodes into equation 7.7 in the book which solves for the value of the option today.

Vo = (2*.59375*(1-.59375)*12.04 + ((1-.59375)^2)*54.68)/(1+.06)^2
= 13.20

So, from the possible future prices of the underlying asset, we were able to approximate the current value of the derivative itself.