In addition to Assumptions 1, 2, 3, 4, 5 and 6 from Section 5 required to set up the differential equation in Section 8, we will now further simplify our model by assuming that the parameters involved (interest rates, dividend yields, volatility) are constant (Assumptions 9, 11 and thus 7 from Section 5) despite the fact that these assumptions are quite unrealistic. These were the assumptions for which Fischer Black and Myron Scholes first found an analytic expression for the price of a plain vanilla option, the famous Black-Scholes option pricing formula. For this reason, we often speak of the Black-Scholes world when working with these assumptions. In the Black-Scholes world, solutions of the Black-Scholes differential equation (i.e. option prices) for some payoff profiles (for example for plain vanilla calls and puts) can be given in closed form. We will now present two elegant methods to derive such closed form solutions.
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- Integral Forms and Analytic Solutions in the Black-Scholes World
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