Dynamics of Several Stocks

We first extend the continuous-time model of stock dynamics presented in Chapter 11 to the case of several correlated stocks. This model will then be used in our analysis of stock portfolios.

Suppose there are n assets The price p-, of the rth asset, for / = 1, 2, .3, , . ., n, is governed by a standard geometric Brownian motion equation where Zj denotes a Wiener process, but with variance parameter erf rather than 1 This is equivalent to the standard model for a single stock The new element here is that the assets are correlated through the Wiener process components . In particular,

We define the covariance matrix S as that with components cr/y, and we use the convention of = ati. We usually assume that S is nonsingular.

From Chapter 11, each asset / has a lognormal distribution, and at time

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