Brownian Motion and Geometric Brownian Motion in Option Pricing

# Ito's lemma and Lognormal Property


# Ito's lemma and Lognormal Property












What would be the difference between:

\begin{align} dS = udt + \sigma dz \end{align}

and

\begin{align} dS=u*S*dt + \sigma*S*dzdS \end{align} Is that the former is in absolute terms and the latter is in relative terms with the stock price?

Therefore, if I want to derive the lognormal property for ( $G =\ln S => dG = (u-\sigma^2/2)*dt + \sigma*dz$) pricing an option, can the first equation be used and how? In John Hull book is done by using the second one.

Thank you.

## Answer by StackG (score 0, accepted)

https://quant.stackexchange.com/a/57511

The answer to

\begin{align} dS = \mu dt + \sigma dW_t \end{align} is simply

\begin{align} S(t) - S(0) = \mu t + \sigma W_t \end{align}

(as discussed here in the first page, for example)

## Answer by gte (score 2)

https://quant.stackexchange.com/a/57661

Perhaps it might help if we define the difference between Brownian Motion (BM) and Geometric Brownian Motion (GBM). BM has independent, identically distributed increments while GBM has independent, identically distributed ratios between successive factors. The definition is inherited from that of arithmetic random walks, which are modelled as sums of random terms, and geometric random walks, modelled as products of random factors.

Let's look at them a bit more in detail.

The BM differential equation is:

$dS_{t} = \mu dt + \sigma dW_{t}$

where the first term, $\mu dt$, is the drift term and the second term $ \sigma dW_{t}$ is the diffusion term characterised by the Wiener process $W_{t}$.

To resolve it, we add integrals on both sides:

$\int_{t=0}^T dS_{t} =\mu \int_{t=0}^T dt + \sigma \int_{t=0}^T dW_{t}$

Here, the last term $\int_{t=0}^T dW_{t}$ is your random variable, i.e. shock.

Let us now look at the GBM. As we said earlier, the GBM is characterised by i.i.d ratios between successive factors. We define it as

$ \frac{dS_{t}}{S_{t}} = \mu dt + \sigma dW_{t}$

Here, $ \frac{dS_{t}}{S_{t}}$ is the finite-time price. To resolve, we take logs and after applying ito's lemma we obtain

$d(logS_{t}) = (\mu - \frac{1}{2} \sigma^2)dt+\sigma dW_{t}$

now we can add integrals, as we have a normal diffusion:

$\int_{t=0}^Td(logS_{t}) = (\mu - \frac{1}{2} \sigma^2)\int_{t=0}^Tdt+\sigma \int_{t=0}^TdW_{t}$

hence

$log S_{T}-log S_{0} = (\mu - \frac{1}{2} \sigma^2)T + \sigma \int_{t=0}^TdW_{t}$

Here, $(\mu - \frac{1}{2} \sigma^2)T$ is the mean of the log price after T years, and $\sigma \int_{t=0}^TdW_{t}$ is the shock, i.e. the variance after T years (normally distributed with mean 0 and variance 1).

Finally, we have the rate of return equal to

$\frac{S_{t}}{S_{0}} = exp((\mu - \frac{1}{2} \sigma^2)T + \sigma \int_{t=0}^TdW_{t})$

which is log-normally distributed.

I hope this helps!