ABSTRACT
In Chapter 7, we extend the discrete time model to the continuous time model and derive the Black-Scholes formula, which is one of the most representative formulae in finance theory developed by Black and Scholes (1973) and Merton (1973b,1977). We showed in Chapter 6 that the price of a financial derivative can be calculated by introducing a risk neutral measure in the binomial model; i.e., it is defined in the discrete and finite time interval. We show how to extend this to the continuous time interval by increasing the time intervals to infinity. Based on this extension, this chapter briefly introduces It’s lemma which holds the fundamental position in stochastic calculus. It’s lemma leads us to the Black- Scholes formula which enables to calculate the call and put options. Further, we introduce the implied volatility, an important parameter used in option pricing. We show how the implied volatility is calculated from the Black-Scholes formula and market prices of call and put options.
