Stochastic Processes and

Advanced Mathematical Finance

Homework 4

Due Sept 27, 2010

- 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?
- 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.
- 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.
- A long strangle option pays $max\left({K}_{1}-S,0,S-{K}_{2}\right)$ if it expires when the underlying stock value is $S$. The parameters ${K}_{1}$ and ${K}_{2}$ are the lower strike price and the upper strike price, and ${K}_{1}<{K}_{2}$. 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.
- 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 ${p}_{U}$, it goes down by a factor $D$ with probability ${p}_{D}$, where $D<1<U$ and the stock stays somewhere in between, changing by a factor $M$ with probability ${p}_{M}$ where $D<M<U$ and ${p}_{D}+{p}_{M}+{p}_{U}=1$.” 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 $f\left({S}_{T}\right)$. 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.