The most common model for the price of a non-dividend paying stock comes from a model of the return of a stock, which is assumed to follow an Ito process, as follows (see, for example, Hull (1993), Chapter 9 and Wilmott (2001)):
| \frac{\Delta S}{S} = Normal \big(\mu \Delta t,\sigma\sqrt {\Delta t} \big) |
Equation 1
where DS is the change in the stock price at some small time interval Dt, m is the proportional return of the stock (usually expressed as an annual proportion, like 10%) and s is the stock price volatility (approximately equal to the standard deviation of the yearly return of the stock). The model has a lot of intuitive appeal because it reflects that the change in price of the stock is proportional to the stock price at that moment.
Integrating Equation 1 over time, we get:
| log \bigg(\frac {S_t}{S_0}\bigg) = Normal \Bigg(\bigg(\mu-\frac{\sigma^2}{2}\bigg)t,\sigma \sqrt t \Bigg) |
Equation 2
Alternatively:
| log(S_t) = Normal \Bigg( log ~ S_0 + \bigg( \mu - \frac {\sigma^2}{2} \bigg)t,\sigma \sqrt t \Bigg) |
Equation 3
The relationship between consecutive periods in a time series is:
Log(S_t) = Normal \bigg(log (S_t-1) + \bigg( \mu - \frac {\sigma^2}{2} \bigg), \sigma \bigg) | or | S_t = S_{t-1} Exp \bigg[ Normal \bigg(\mu - \frac{\sigma^2}{2} , \sigma \bigg)\bigg] |
ASIDE: If you attempted to guess the above formula from Central Limit Theorem principles you will be surprised to see the inclusion of a s2/2 component in Equation 2 for the mean return. It is there because the return in each short period is a function of the stock price at that moment, and therefore cannot be simply added up in log space. The proof of the formula is outside the scope of this guide: it is a result of applying Ito's Lemma (see, for example Hull (1993, page 208), a mathematical result crucial in Black and Scholes work on valuing of options.
From the definition of a Lognormal distribution, if log(St) is normally distributed, then St must be lognormally distributed. Thus, under this model, the price of a stock will be lognormal.
If the stock had no variance, Equation 3 would look like this:
log(S_t) = log ~ S_0 + \mu t or S_t = S_0 e ^{\mu t}
If m is defined as the yearly continuously compounded return, then the daily return md is determined by solving:
e ^{\mu} = e ^{\mu_t 365} i.e., simply md = m/365
Lognormal Random Walk model offers an example.
An extension to this model is to consider mean reversion, which means that the stock price tends to move back towards the expected return as a function of how far away it is at any moment from that expected return. The section Stock price with mean reversion extends the above model to include mean reversion.
The links to the Lognormal Random Walk software specific models are provided here:
The Lognormal Random Walk model is also the underpinning behind the Black Scholes formula, and can therefore be used to value call options.
