Chapter 14 - The Black-Scholes Option Pricing Formula

Question 1

Assumptions of Black-Scholes (6)

Answer 1

1. The underlying share price follows geometric Brownian motion.
2. The market is arbitrage-free.
3. The risk-free rate r is constant, the same for all maturities and the same for all borrowing and lending.
4. Assets may be held in any amount. Unlimited short selling (negative holdings) is allowed.
5. There are no taxes or transaction costs.
6. Assets may be bought and sold at any time t>0. Assets can be traded continuously and in infitesimally small number of units

An implication of these is that the market in the underlying share is complete: that is, all derivative securities have payoffs which can be replicated.

Question 2

Validity of assumptions

Answer 2

1. Share prices can jump. This invalidates assumption 1 since geometric Brownian motion has continuous sample paths.
2. The risk-free rate of interest does vary and in an upredictable way.
3. Unlimited short selling may not be allowed, except perhaps at penal rates of interest.
4. Shares can normally onyl be dealt in integer multiples of one unit, not continuously, and dealings attract transaction costs.
5. Distributions of share returns tend to have fatter tails than suggested by the log-normal model.

Question 3

Conditions for fair derivative pricing

Answer 3

For any derivative to be fairly priced, it must:

• satisfy the boundary conditions, ie have the correct payoff at expiry and
• satisfy the Black Scholes PDE