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On page 300 of Hull's Options, Futures and Other Derivatives

It is tempting to suggest that a stock price follows a generalized Wiener process; that is, that it has a constant expected drift rate and a onstant variance rate. However, this model fails to capture a key aspect of stock prices. This is that the expected percentage return required by investors from a stock is independent of the stock’s price. If investors require a 14% per annum expected return when the stock price is $\\\$$ 10, then, ceteris paribus, they will also require a 14% per annum expected return when it is $50.

Clearly, the assumption of constant expected drift rate is inappropriate and needs to be replaced by the assumption that the expected return (i.e., expected drift divided by the stock price) is constant. If $S$ is the stock price at time $t$, then the expected drift rate in $S$ should be assumed to be $\mu S$ for some constant parameter m. This means that in a short interval of time, $\Delta t$, the expected increase in S is $\mu S \Delta t$. The parameter m is the expected rate of return on the stock.

An Ito process can be written as $dx = a(x,t)dt + b(x,t)dz$, $dz$ is a basic Wiener process that has a drift rate of zero and a variance of $1.0$. I thought $a(x,t)$ is the drift of the Ito process $dx$, but after reading this, I know it is also the expected drift. Do we not need any calculation to get the expected drift of $dx$? If we need some calculation, how? Why does the expected return equal the expected drift divided by the stock price?

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  • $\begingroup$ Under mild conditions (namely $\mathbb{E}\left[ \int_0^t a(x,t)^2 dt \right] < \infty$), the process $\int_0^s a(x,t)dW$ is a martingale with mean 0. See this answer: quant.stackexchange.com/a/15799/55385. $\endgroup$
    – Achrbot
    Nov 30, 2023 at 13:49

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