Geometric Brownian Motion and the Itô Convexity Adjustment

# squaring stochastic calculus and other solutions


# squaring stochastic calculus and other solutions












It is well-known that the solution to the stochastic SDE

$$ dS = S_0(\mu dt + \sigma dWt) $$

is

$$ S_t=S_0 e^{(\mu-\frac{\sigma^2}{2})t+W_t} $$

Were $\sigma=0$, this is simply the formula for continuous compounding. It stands to reason that the term $-\frac{\sigma^2}{2}t$ exists because the other way to write the solution is $$ S_t=S_0 e^{(\mu-\frac{\sigma^2}{2})t}e^{W_t} $$

or, even better,

$$ S_t=S_0 e^{\mu t}e^{-\frac{\sigma^2}{2}t}e^{W_t} $$

i.e., what re are really doing is (1) scaling by the determinstic amount; (2) scaling by the variable amount (the last term); and then (3) eliminating the bias introduced by the prior step from the last term by multiplying by the second term.

If so, it seems unfair to claim that 'returns are normally distributed' in the standard explanation. That would imply that in price-space, the price (an RV) would be multiplied by a normally distributed increment. But, in reality, what is really being said is that log P is growing by a constant term plus an (additive) normally distributed Brownina motion, i.e.,,

$$\frac{dS}{S}=\mu t + N(\frac{\mu}{t}, \frac{\sigma^2}{t})$$

in other words, in price space returns are not normally distributed, but themselves long-normally distributed (which would have the desireable property that product lof L-N RVs are themselves L-N, so it is an attractor).

But. other than convenience in stochastic calculus, what is the justification for using log-normality? Is there an empirical argument that returns are log-normally distributed?

## Answer by David Addison (score 2)

https://quant.stackexchange.com/a/39446

Ito's lemma results in a convexity adjustment since when you take a the exponential of a linear function with Gaussian noise that function becomes convex. GBM is therefore simply a solution to a differential equation in which the logarithmic returns have a normal distribution. Intuitively, the resulting "convexity" adjustment can be seen as consequence of Jensen's inequality.

I.e., write the instantaneous change of $X$ as being normally distributed:

$dX =X \,\mu \,d\tau + X\,\sigma dB_\tau$,

$\frac{dX}{X} =\mu \,d\tau + \sigma dB_\tau$,

$\int\frac{dX}{X} d\tau =\mu \,\tau + \sigma B_\tau$.

$\frac{dX}{X}$ looks related to $1 \over X $ (i.e., the derivative of $\log[X]$), so if the normal rules of calculus applied we would simply write the solution as:

$X_t = X_0 e^{ \mu \, \tau + \sigma B_\tau}$

However, the normal rules of calculus DO NOT apply since $X$ is a random variable, which intuitively means that the higher order derivatives of $f(X,\tau)$ actually do something. This problem thereofore requires Ito's lemma (or other stochastic differentiation methods) in order to describe what those higher level terms do. In the case of GBM, the term $-\sigma^2/2$ is a consequence of the second order terms of a Taylor series approximation of $X_{\tau}$.

How exactly you want to conceptualize this result is a matter of personal preference. While there are a few errors in your derivation of the distribution, the gist is that — as you allude — you can think of the solution as a lognormal distribution with a correction term or you could think of the distribution itself as being "off-normal”.

I.e.:

$\mathbb{E}[X_t] = X_0 e ^{(\mu-\frac{\sigma^2}{2})t+\sigma W_t} \equiv X_0 e ^{\mu t+ Y_t} \equiv X_0 e ^{Z_t} \equiv \mathcal{P}_t $

where $W_t \sim \mathcal{N}[0,\sqrt{\tau}]$,

$Y_t \sim \mathcal{N}[-\frac{\sigma^2}{2}\tau,\sigma \sqrt{\tau}]$,

$Z_t \sim \mathcal{N}[(\mu-\frac{\sigma^2}{2})\tau,\sigma \sqrt{\tau}]$, and

$\mathcal{P}_t \sim \mathcal{logN}[(\mu-\frac{\sigma^2}{2})\tau + \log[X_0],\sigma \sqrt{\tau}] $

Analytically, these methods are equivalent.