Chapter 14: Black-Scholes Option Pricing Formula

Question 1

6 Assumptions underlying the Black-Scholes

Answer 1

1. The price of the underlying share follows a geometric Brownian motion
2. There are no risk-free arbitrage opportunities.
3. The risk-free rate of interest is constant, the same for all maturities and the same for borrowing or lending.
4. Unlimited short selling (i.e. negative holdings) is allowed.
5. There are not taxes or transaction costs
6. The underlying assets can be traded continuously and in infinitesimally small numbers of units.

Question 2

5 Ways in which the Black-scholes assumptions are not realistic

Answer 2

1. Share prices can jump. (the geometric Brownian motion has continuous sample paths, no jumps)
2. The risk-free rate of interest does vary and in an unpredictable way.
3. Unlimited short selling may not be allowed, except perhaps at penal rates of interest. These problems can be mitigated by holding mixtures of derivatives which reduce the need for short selling.
4. Share can normally only be dealt with in integer multiples of one unit, not continuously, and dealings attract transaction costs: invalidating assumptions 4, 5 and 6.
5. Distributions of share returns tend to have fatter tails than suggested by the log-normal model, invalidating assumption 1.

Question 3

Explain why the prices actually quoted in the market may differ from the theoretical Black-Scholes price

Answer 3

• Buy/Sell spread - the market maker makes a margin
• Odd results can arise from different views on implied volatility
• Market participants may make other adjustments to allow for the extent to which the BS assumptions and reality diverge.