An Elementary Introduction to Mathematical Finance, Third by Sheldon M. Ross

By Sheldon M. Ross

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However, Brownian motion appears to have two major flaws when used to model stock or commodity prices. First, since the price of a stock is a normal random variable, it can theoretically become negative. Second, the assumption that a price difference over an interval of fixed length has the same normal distribution no matter what the price at the beginning of the interval does not seem totally reasonable. For instance, many people might not think that the probability a stock presently selling at $20 would drop to $15 (a loss of 25%) in one month would be the same as the probability that when the stock is at $10 it would drop to $5 (a loss of 50%) in one month.

Covariance and Correlation 15 From this, we see that Cov(X, Y ) > 0 ⇐⇒ P{X = 1, Y = 1} > P{X = 1}P{Y = 1} P{X = 1, Y = 1} > P{Y = 1} P{X = 1} ⇐⇒ P{Y = 1 | X = 1} > P{Y = 1}. ⇐⇒ That is, the covariance of X and Y is positive if the outcome that X = 1 makes it more likely that Y = 1 (which, as is easily seen, also implies the reverse). The following properties of covariance are easily established. For random variables X and Y, and constant c: Cov(X, Y ) = Cov(Y, X ), Cov(X, X ) = Var(X ), Cov(cX, Y ) = c Cov(X, Y ), Cov(c, Y ) = 0.

1. 5 45 The Cameron-Martin Theorem For an underlying Brownian motion process with variance parameter σ 2 , let us use the notation E μ to denote that we are taking expectations under the assumption that the drift parameter is μ. Thus, for instance, E 0 would signify that the expectation is taken under the assumption that the drift parameter of the Brownian motion process is 0. The following is known as the Cameron-Martin theorem. 1 Let W be a random variable whose value is determined by the history of the Brownian motion up to time t.

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