Math 489/889 Final Name:________________________________
Thursday, December 17, 2009

1. (15 points) A stock has a constant volatility of 18% and the risk-free interest rate (compounded continuously) is 6%. What is the value of an option to buy the stock for $25 in two years time, given the current stock price is $20?

2. 1. (10 points) Using the Black-Scholes formula, write the formula for the value of a strap, a derivative composed of a put option with strike price $K$ and two call options on the same underlying security with the same strike price and maturity date.
   2. (10 points) Draw a graph of the final or terminal value of this composite derivative.
   3. (5 points) What mathematical property of the Black-Scholes equation allows you to write the formula for the value of a strap, as above?

3. (20 points) Find a numerical approximation at $t=0.2,0.4,0.6,0.8,1.0$ to the solution of the Stochastic Differential Equation: $$dX=\left(1-X\right)dt+\sqrt{X}dW,X\left(0\right)=0.0$$

   (Remark: With some general parameters, this stochastic differential equation is a model of a “mean-reverting square-root process that models asset prices”.) Use $dt=0.2$, the table of net totals of randomly generated coin flips below, and recall that as in the notes $dW\approx \sqrt{dt}\left(S\left(Ndt\right)∕\sqrt{Ndt}\right)$.

   $j$	$X_j$	$\left(1-X_j\right)$	$\sqrt{X_j}$	$\left(1-X_j\right)dt+\sqrt{X_j}dW$	$X_{j+1}$
   0
   1
   2
   3
   4
   5

   $n$	0	10	20	30	40	50	60	70	80	90	100
   $S_n$	0	-4	-6	-10	-8	-8	-6	-8	-6	-10	-8

4. (20 points) Use the put-call parity relationship to derive the relationship between:
   1. the Vega of a European call option and the Vega of a European put option. (Recall that the Vega is the rate of change of an option value with respect to changes in the volatility of the underlying security.) Show your work.
   2. the Theta of a European call option and the Theta of a European put option. (Recall that the Theta is the rate of change of an option value with respect to time.) Show your work.

5. (30 points) A Brownian motion with drift has parameter $r=-0.1$ and standard deviation parameter $\sigma =2$. It starts at $x_0=2.82$.
   1. What are the probability distributions of the position at $t=4,8,12$?
   2. What are the probabilities that the stochastic process is negative at $t=4,8,12$?

6. 1. (5 points) A stock price is currently $20. Tomorrow, important news is expected that will either immediately increase the price by $5 or decrease the price by $5. Discuss the merits of using the Black-Scholes formula to value options on the stock.
   2. (5 points) Why is “plain” Brownian Motion not an adequate model of stock market prices?