Lecture 9 - Black-Scholes-Merton Model

Question 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?

Answer 1

The Black-Scholes-Merton option pricing model assumes that the probability distribution of the stock price in one year (or at any other future time) is lognormal.
It assumes that the continuously compounded rate of return on the stock during the year is normally distributed.

Question 2

Explain the principle of risk-neutral valuation

Answer 2

The price of an option or other derivative when expressed in terms of the price of the underlying stock is independent of risk preferences. Options, therefore, have the same value in a risk-neutral world as they do in the real world. We may therefore assume that the world is risk neutral for the purpose of valuing options. This simplifies the analysis.
In a risk-neutral world, all securities have an expected return equal to the risk-free rate.
Also, in a risk-neutral world, the appropriate discount rate to use for expected future cash flows is the risk-free interest rate.

Question 3

What is implied volatility? How can it be calculated?

Answer 3

The implied volatility is the volatility that makes the Black-Scholes-Merton price of an option equal to its market price.
The implied volatility is calculated using an iterative procedure.
A simple approach is the following. Suppose we have two volatilities one too high (i.e., giving an option price greater than its market price) and the other too low (i.e., giving an option price lower than the market price). By testing the volatility that is halfway between the two, we get new too-high volatility or new too-low volatility. We can use this procedure repeatedly to bisect the range and converge on the correct implied volatility. Other more sophisticated approaches, (e.g., involving the Newton-Raphson procedure) are used in practice.

Question 4

Assumptions of the Black-Scholes-Merton model

Answer 4

• Expected return and volatility are constant
• Short selling without restrictions is possible
• No transaction costs or taxes. All securities are perfectly divisible.
• No dividends
• No arbitrage opportunities
• Security trading is continuous
• The risk-free rate is constant and the same for all maturities
• Constant volatility

Question 5

How can you estimate volatility from historical data?

Answer 5

To estimate the volatility σ of a stock price empirically, the stock price is observed at fixed intervals of time (e.g., every day, every week, or every month). For each time period, the natural logarithm of the ratio of the stock price at the end of the time period to the stock price at the beginning of the time period is calculated. The volatility is estimated as the standard deviation of these numbers divided by the square root of the length of the time period in years. Usually, days when the exchanges are closed are ignored in measuring time for the purposes of volatility calculations.