Math 489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 4

Steve Dunbar

Due Sept 27, 2010
  1. Consider a stock whose price today is $50. Suppose that over the next year, the stock price can either go up by 6%, or down by 3%, so the stock price at the end of the year is either $53 or $48.50. The continuously compounded interest rate on a $1 bond is 4%. If there also exists a call option on the stock with an exercise price of $50, then what is the price of the call option? Also, what is the replicating portfolio?
  2. A stock price is currently $50. it is known that at the end of 6 months, it will either be $60 or $42. The risk-free rate of interest with continuous compounding on a $1 bond is 10% per year. Calculate the value of a 6-month European call option on the stock with strike price $48 and find the replicating portfolio.
  3. Consider a three-time-stage example. The first time interval is a month, then the second time interval is two months, finally, the third time interval is a month again. A stock starts at 50. In the first interval, the stock can go up by 10% or down by 3%, in the second interval the stock can go up by 5% or down by 5%, finally in the third time interval, the stock can go up by 6% or down by 3%. The continuously compounded interest rate on a $1 bond is 2% in the fist period, 3% in the second period, and 4% in the third period. Find the price of a call option with exercise price 50, with exercise date at the end of the 4 months. Also, find the replicating portfolio at each node.
  4. A long strangle option pays max(K1 S, 0,S K2) if it expires when the underlying stock value is S. The parameters K1 and K2 are the lower strike price and the upper strike price, and K1 < K2. A stock currently has price $100 and goes up or down by 20% in each time period. What is the value of such a long strangle option with lower strike 90 and upper strike 110 at expiration 2 time units in the future? Assume a simple interest rate of 10% in each time period.
  5. Your friend, the financial analyst comes to you, the mathematical economist, with a proposal: “The single period binomial pricing is all right as far as it goes, but it certainly is simplistic. Why not modify it slightly to make it a little more realistic? Specifically, assume the stock can take three values at time T, say it goes up by a factor U with probability pU, it goes down by a factor D with probability pD, where D < 1 < U and the stock stays somewhere in between, changing by a factor M with probability pM where D < M < U and pD + pM + pU = 1.” The market contains only this stock, a bond with a continuously compounded risk-free rate r and an option on the stock with payoff function f(ST ). Make a mathematical model based on your friend’s suggestion and provide a critique of the model based on the classical applied mathematics criteria of existence of solutions to the model and uniqueness of solutions to the model.