FIN5DER 1 Tutorial 9 Questions Problem 15

FIN5DER 1 Tutorial 9 Questions

Problem 15.1.

What does the Black–Scholes–Merton stock option pricing model assume about the probability distribution of the stock price in one year? What does it assume about the probability distribution of the continuously compounded rate of return on the stock during the year?

Problem 15.2.

The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price change in one trading day?

Problem 15.3.

Explain the principle of risk-neutral valuation.

Problem 15.4.

Calculate the price of a three-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum.

Problem 15.5.

What difference does it make to your calculations in Problem 15.4 if a dividend of $1.50 is expected in two months?

Problem 15.6.

What is implied volatility? How can it be calculated?

Problem 15.7.

A stock price is currently $40. Assume that the expected return from the stock is 15% and its volatility is 25%. What is the probability distribution for the rate of return (with continuous compounding) earned over a two-year period?

Problem 15.13.

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, the volatility is 30% per annum, and the time to maturity is three months?

Problem 15.14.

What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per annum, and the time to maturity is six months?

Problem 19.2.

What does it mean to assert that the delta of a call option is 0.7? How can a short position in 1,000 options be made delta neutral when the delta of each option is 0.7?

Problem 19.3.

Calculate the delta of an at-the-money six-month European call option on a non-dividend-paying stock when the risk-free interest rate is 10% per annum and the stock price volatility is 25% per annum.

Problem 19.5.

What is meant by the gamma of an option position? What are the risks in the situation where the gamma of a position is large and negative and the delta is zero?

Problem 19.23.

Use the put-call parity relationship to derive, for a non-dividend-paying stock, the relationship between:

(a) The delta of a European call and the delta of a European put.

(b) The gamma of a European call and the gamma of a European put.

(c) The Vega of a European call and the Vega of a European put.