The Black-Scholes-Merton Model answers
• The original Black-Scholes and Merton papers on stock option pricing were published in which year?
• 1983
• 1984
• 1974
• 1973
Answer: D
The correct answer is 1973. By coincidence this is also the year that organized trading in call options started. Put option trading started a few years later.
• Which of the following is a definition of volatility?
• The standard deviation of the return, measured with continuous compounding, in one year
• The variance of the return, measured with continuous compounding, in one year
• The standard deviation of the stock price in one year
• The variance of the stock price in one year
Answer: A
Volatility when multiplied by the square root of t is the standard deviation of the return in a short period of time of length t. It is also the standard deviation of the continuously compounded return in one year.
• A stock price is $100. Volatility is estimated to be 20% per year. What is an estimate of the standard deviation of the change in the stock price in one week?
• $0.38
• $2.77
• $3.02
• $0.76
Answer: B
The estimate is
• What does N(x) denote?
• The area under a normal distribution from zero to x
• The area under a normal distribution up to x
• The area under a normal distribution beyond x
• The area under the normal distribution between -x and x
Answer: B
The normal distribution runs from minus infinity to plus infinity. N(x) is the area under the distribution between minus infinity and x.
• What was the original Black-Scholes-Merton model designed to value?
• A European option on a stock providing no dividends
• A European or American option on a stock providing no dividends
• A European option on any stock
• A European or American option on any stock
Answer: A
The original Black-Scholes-Merton model was designed to value a European option on a stock paying no dividends.
• A stock provides an expected return of 10% per year and has a volatility of 20% per year. What is the expected value of the continuously compounded return in one year?
• 6%
• 8%
• 10%
• 12%
Answer: B
The expected value of the continuously compounded return per year is In this case it is 0.1 – 0.22/2 = 0.08 or 8%.
• When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 6%, the volatility is 20% and the time to maturity is 3 months which of the following is the price of a European call option on the stock
• 20N(0.1)-19.7N(0.2)
• 20N(0.2)-19.7N(0.1)
• 19.7N(0.2)-20N(0.1)
• 19.7N(0.1)-20N(0.2)
Answer: B
The formula for the option price is
In this case S0 =K=20, r=0.06, =0.2, and T=0.25 so that Ke-rT=20e-0.06×0.25=19.7. Also
d1 = [ln(1)+(0.06+0.04/2)×0.25]/(0.2×0.5) =0.2 and d2=0.2 – 0.2×0.5=0.1.
B is therefore the correct answer.
• When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 6%, the volatility is 20% and the time to maturity is 3 months which of the following is the price of a European put option on the stock
• 19.7N(-0.1)-20N(-0.2)
• 20N(-0.1)-20N(-0.2)
• 19.7N(-0.2)-20N(-0.1)
• 20N(-0.2)-20N(-0.1)
Answer: A
The formula for the option price is
In this case S0 = K =20, r = 0.06, =0.2, and T=0.25 so that Ke-rT=20e-0.06×0.25=19.7. Also
d1 = [ln(1)+(0.06+0.04/2)×0.25]/(0.2×0.5) =0.2 and d2=0.2 – 0.2×0.5=0.1.
A is therefore the correct answer.
• The volatility of a stock is 18% per year. Which is closest to the volatility per month?
• 1.5%
• 3.0%
• 5.2%
• 6.3%
Answer: C
The volatility per month is the volatility per year multiplied by the square root of 1/12 . The square root of 1/12 is 0.2887 and 18% multiplied by this is 5.2%.
• When the Black-Scholes-Merton and binomial tree models are used to value an option on a non-dividend-paying stock, which of the following is true?
• The binomial tree price converges to a price slightly above the Black-Scholes-Merton price as the number of time steps is increased
• The binomial tree price converges to a price slightly below the Black-Scholes-Merton price as the number of time steps is increased
• Either A or B can be true
• The binomial tree price converges to the Black-Scholes-Merton price as the number of time steps is increased
Answer: D
The binomial tree valuation method and the Black-Scholes-Merton formula are based on the same set of assumptions. As the number of time steps is increased the answer given by the binomial tree approach converges to the answer given by the Black-Scholes-Merton formula.