Options, Futures, and Other Derivatives, 10th Edition

14.6 ITÔ’S LEMMA

The price of a stock option is a function of the underlying stock’s price and time. More generally, we can say that the price of any derivative is a function of the stochastic variables underlying the derivative and time. A serious student of derivatives must, therefore, acquire some understanding of the behavior of functions of stochastic variables. An important result in this area was discovered by the mathematician K. Itô in 1951,6 and is known as Itô’s lemma.

6 See K. Itô, “On Stochastic Differential Equations,” Memoirs of the American Mathematical Society, 4 (1951): 1–51.

Suppose that the value of a variable x follows the Itô process

$$dx=a\left(x,t\right)dt+b\left(x,t\right)dz$$(14.11)

where dz is a Wiener process and a and b are functions ...