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Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2010


Problem Statement

Consider a two-time-stage example. Each time stage is a year. A stock starts at 50. In each year, the stock can go up by 10% or down by 3%. The interest rate on a $1 bond is constant at 6% each year. Find the price of a call option with exercise price 50, with exercise date at the end of the second year. Also, find the replicating portfolio at each node.

Solution

The solution is shown on the tree diagrammed below.

We can also use the risk-neutral probabilities to price the binomial tree recursively as in the diagram below:

 exp(0.06)- 0.97 p = ----------------= 0.706435 1.1 - 0.97

and

1 - p = .293565

Pricing a European call

The portfolio amounts are also indicated at each node.


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