## Math489/889 Fall 2010 Stochastic Processes and Advanced Mathematical Finance Problems to Practice

No Due Date: Practice Only
1. Find the mode (the value of the independent variable with the highest probability) of the lognormal probability density function. (Use parameters $\mu$ and $\sigma$.)
2. A call option on a stock is said to be in the money if the value of the stock is higher than the strike price, so that selling or exercising the call option would result in a profit (ignoring transaction costs.) Consider a stock (or more properly an index fund) whose value $S\left(t\right)$ is described by the stochastic differential equation $dS=r\cdot S\phantom{\rule{0em}{0ex}}dt+\sigma \cdot S\phantom{\rule{0em}{0ex}}dW$ corresponding to a non-zero compounded growth and pure market fluctuations proportional to the stock value and has a current market price of ${z}_{0}$? What is the probability that a call option is in the money based on a strike price $K=1.25{z}_{0}$ at expiration $T$ time units later? Evaluate for $K=1.25{z}_{0}$, $T=1∕2$, $r=0.04$, and $\sigma =0.10$ and also for $K={z}_{0}$ with the same parameters.
1. What is the price of a European call option on a non-dividend-paying stock when the stock price is \$52, the strike price is \$50, the risk-free interest rate is 12% per annum (compounded continuously), the volatility is 30% per annum, and the time to maturity is 3 months?
2. What is the price of a European call option on a non-dividend paying stock when the stock price is \$30, the exercise price is \$29, the risk-free interest rate is 5%, the volatility is 25% per annum, and the time to maturity is 4 months?
3. Show by substitution that two exact solutions of the Black-Scholes equations are
1. $V\left(S,t\right)=AS$, $A$ some constant.
2. $V\left(S,t\right)=Aexp\left(rt\right)$

Explain in financial terms what each of these solutions represents. That is, describe a simple “claim”, “derivative” or “option” for which this solution to the Black Scholes equation gives the value of the claim at any time.

4. A pharmaceutical company has a stock that is currently \$25. Early tomorrow morning the Food and Drug Administration will announce that it has either approved or disapproved for consumer use the company’s cure for the common cold. This announcement will either immediately increase the stock price by \$10 or decrease the price by \$10. Discuss the merits of using the Black-Scholes formula to value options on the stock.
5. Use the put-call parity relationship to derive the relationship between
1. The Delta of European call and the Delta of European put.
2. The Gamma of European call and the Gamma of European put.