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Chapter 11: Negative Volatility and the ... > 3 The Value of a European Call Optio... - Pg. 240

240 DERIVATIVES 2 Negative Volatility ­ A Direct Approach Instead of starting, as Haug (2002) does, with the classical Merton, Black and Scholes formula, and plugging in a negative volatility directly in this formula, I suggest we take a look at the dynamic equation for the risky asset. It is dS(t) = µ dt + dB(t), S(t) (11.1) where S(t) is the spot price of the risky asset at time t, µ is the drift parameter and is the volatility parameter. B(t) is the Brownian motion stochastic process at time t under the given probability measure P , and B(t) is thus normally distributed with mean zero and variance t (under P ). Informally one can think of the increment dB(t) as normally distributed with mean zero and variance dt. The term dB(t) is thus normally distributed with mean zero and variance 2 dt. A representation for the price process S is as follows: S(t) = S(0)e (µ- 2 1 2 )t+ B(t) . (11.2) The only stochastic component in this relation is the term B(t), which has a normal distribution