CHAPTER 47Asset Price Dynamics
Aims
- To explain a standard Wiener process and how this leads to a stochastic process for the stock price
, known as a geometric Brownian motion (GBM). - To show how Ito's lemma can be used to move from a stochastic process for the stock price
to a stochastic differential equation (SDE) for any non-linear function 
. - To explain the statistical relationship between a stochastic variable which is lognormal
and the variable itself, 
. In particular, the relationship between their expected values and variances.
A great deal of analytic work in pricing derivative securities and in constructing hedge portfolios uses continuous time stochastic processes. Any variable (such as the stock price) which changes over time in a random way is said to be stochastic. In the real world we observe discrete changes in stock prices but if the time interval of observation is small enough, this approximates to a continuous time process. The Black–Scholes option pricing formula was ...
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