Derivatives

CHAPTER 16Black–Scholes Model

Aims

• To demonstrate how option prices change with changes in the price of the underlying asset, its volatility, interest rates and time to maturity.
• To establish upper and lower bounds for the price of European calls and puts.
• To show how the Black–Scholes formula is used to price European calls and puts.
• To show how options and the underlying asset (e.g. stock-Z) can be combined into a portfolio which does not change in value when there is a small change in the price of the underlying asset – this is delta hedging.
• To explain implied volatility and its use in options trading.

16.1 DETERMINANTS OF OPTION PRICES

It can be shown that the option premium varies second-by-second as the stock price, the risk-free interest rate and the volatility of the stock change, over time. Let us develop some intuitive arguments which give some insight into the determination of European option prices. We consider each factor in turn, holding all the other factors constant. This intuitive approach will help us understand the mathematical formulas for option prices which we present later. In each case we assume the investor has a long options position (i.e. has purchased a call or a put) and we only consider European stock options (where the stock pays no dividends). The results are summarised in Table 16.1.

TABLE 16.1 Factors affecting the option premia

Long European call option	Long European put option
Time to expiration, T	+	+
Current stock price, ...