Simulating Geometric Brownian Motion in Log-Price Space

# Why is it more accurate to simulate ln(S) rather than S?


# Why is it more accurate to simulate ln(S) rather than S?












Let's take a process $S$ that satisfies: \begin{equation} dS = \mu S dt + \sigma S dz \end{equation} with $dz$ a Wiener process, $\sigma$ the volatility of $S$, $\mu$ the expected return of $S$.

From Ito's lemma, we have that the process verified by $ln(S)$ is: \begin{equation} d(ln (S)) = (\mu - \sigma^2/2)dt + \sigma dz \end{equation}

Why is it more accurate to use the second equation to simulate a path for S rather than the first one?

## Answer by Kiann (score 1)

https://quant.stackexchange.com/a/44292

The specification of $ln(S)$ is based on the explicit assumption security prices and interest rates cannot go below zero.

And for the behaviour of securities, it has been well-established via empirical research that the security absolute price grows at an exponential rate rather than absolute rate.... i.e. after $T$ years, security price tends to be $S(0)e^{r_fT}$, instead of $S(0)\cdot(1 + r_tT)$.

Since the financial crisis, this assumption for funding interest rates have proven to be false, and there are multiple models where the $S$ is modeled instead of $ln(S)$.

## Answer by LvM_ (score 0)

https://quant.stackexchange.com/a/73740

In the context of Monte Carlo simulation:

For lognormal random walk, we can find an exact time stepping algorithm.

Given: $d ln S = (r-\frac{1}{2} \sigma^2)dt + \sigma dX$ (I use r instead of $\mu$ to price under the risk-neutral world)

we integrate this, to get:

$S(T) = S(t) exp((r-\frac{1}{2} \sigma^2)(T-t) + \sigma \int_t^TdX )$

$\Leftrightarrow S(t+\delta t) = S(t) exp((r-\frac{1}{2} \sigma^2)\delta t + \sigma \sqrt(\delta t) \phi )$ (over a time step $\delta t$)

Therefore, if you are trying to price an option whose payoff depends on the final asset value and is path independent, you should use the lognormal random walk and use a single time step. (it saves a lot of computation time!)

Whereas if you try to price an option which is path dependent (such as Asian Option), you can you use either SDE. (just make sure to use small time step!)

[ref: Paul Willmott on Quantitative Finance. Vol.3 Part 6 Chapter 80]