Appendix: The Black–Scholes Formula

There are a number of assumptions underlying the formula:

1. There are no transaction costs and no taxes.
2. The risk-free rate is constant for the life of the option.
3. The market operates continuously (day and night).
4. The stock price moves continuously, with no sudden jumps.
5. The stock pays no cash dividends.
6. The option can be exercised only at the expiration date.
7. The underlying stock can be sold short without penalty.
8. The distribution of returns on the underlying security (the common stock) is lognormal.

Based on these assumptions, it is possible to derive their

C = P × N(d₁)-EX×e^-rt×N(d₂),

where

C = the value of the call option

N(d) = the cumulative normal probability density function

EX = the exercise price of the option

σ² = the variance per period of continuously compounded rate of return on stock

t = the time to the maturity date

r = the continuously compounded risk-free rate of interest

P = the price of the stock now