Power Transforms of Geometric Brownian Motion

Why Transform a Geometric Brownian Motion?

A power of a geometric Brownian motion is again an exponential process, but its drift changes with the power. This makes powers useful for computing moments and for finding martingales other than the original process. The reciprocal is the special case $p=-1$: its drift does not simply equal the negative of the original drift, because Ito's formula contributes a second-derivative correction.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ carrying a Brownian motion $B_t$ relative to the filtration. Fix a deterministic $S_0>0$ and constants $\mu,\sigma\in\mathbb{R}$. The geometric Brownian motion considered here is the positive process $S_t=S_0\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)$, which satisfies $dS_t=\mu S_t\,dt+\sigma S_t\,dB_t$. Here $\mu$ is the drift coefficient and $\sigma$ is the diffusion coefficient. All martingale statements below are under $\mathbb{P}$.

For every finite $T$, the Ito integrals used below have progressively measurable integrands in $\mathcal{L}^2$, the class of processes $H$ satisfying $\mathbb{E}\int_0^T H_s^2\,ds<\infty$. In particular, the process $S_t^p$ is adapted and continuous for each real $p$; its square has finite expectation on every finite time interval, as the moment calculation below verifies.

Dynamics of a Real Power

Result: For every real $p$, the power transform $X_t=S_t^p$ satisfies $$ dX_t=\left(p\mu+\tfrac12p(p-1)\sigma^2\right)X_t\,dt+p\sigma X_t\,dB_t. $$ The coefficient of $dt$ is its drift; the coefficient of $dB_t$ is its diffusion coefficient. The formula includes $p=0$, for which $X_t=1$.

Proof. Since $S_t>0$, the function $f(x)=x^p$ is twice continuously differentiable on the range of $S_t$, including for negative real $p$. Its derivatives are $f'(x)=px^{p-1}$ and $f''(x)=p(p-1)x^{p-2}$. Ito's formula, with its second-derivative correction displayed, gives $$ d f(S_t)=f'(S_t)\,dS_t+\tfrac12 f''(S_t)\,d[S]_t = pS_t^{p-1}(\mu S_t\,dt+\sigma S_t\,dB_t) +\tfrac12p(p-1)S_t^{p-2}\sigma^2S_t^2\,dt. $$ Here $[S]_t=\int_0^t\sigma^2 S_s^2\,ds$, since the finite-variation part has zero quadratic variation and $dB_t\,dB_t=dt$, $dB_t\,dt=0$, and $dt\,dt=0$. Collecting the $dt$ and $dB_t$ terms yields the stated equation. $\square$

Worked Example: A Fractional Power Martingale

Let $S_0=2$, $\mu=\tfrac1{10}$, and $\sigma=\tfrac12$. Find a nonconstant power transform with zero drift, and check directly that it is a martingale.

For $p=\tfrac15$, the drift multiplier is $\tfrac15\cdot\tfrac1{10}+\tfrac12\cdot\tfrac15\cdot(-\tfrac45)\cdot\tfrac14=\tfrac1{50}-\tfrac1{50}=0$. The diffusion multiplier is $p\sigma=\tfrac1{10}$. Thus $d(S_t^{1/5})=\tfrac1{10}S_t^{1/5}\,dB_t$.

To check the result independently, the explicit process is $S_t^{1/5}=2^{1/5}\exp(-t/200+B_t/10)$: indeed, $\tfrac15(\mu-\tfrac12\sigma^2)=\tfrac15(\tfrac1{10}-\tfrac18)=-\tfrac1{200}$. For $0\le s\le t$, independence of $B_t-B_s$ from $\mathcal{F}_s$ and its normal distribution give $\mathbb{E}[\exp((B_t-B_s)/10)]=\exp((t-s)/200)$. The deterministic exponential factor in the ratio $S_t^{1/5}/S_s^{1/5}$ is $\exp(-(t-s)/200)$, so $\mathbb{E}[S_t^{1/5}\mid\mathcal{F}_s]=S_s^{1/5}$. The process is integrable because its expectation is $2^{1/5}<\infty$. It is therefore a true martingale, not merely a local martingale.

The Reciprocal and Its Correction Term

Setting $p=-1$ in the power formula gives the reciprocal dynamics. Equivalently, applying Ito's formula directly to $f(x)=1/x$ uses $f'(x)=-x^{-2}$ and $f''(x)=2x^{-3}$. The correction term is essential:

d\left(\frac1{S_t}\right) =\left(-\mu+\sigma^2\right)\frac1{S_t}\,dt -\sigma\frac1{S_t}\,dB_t.

The drift is $-\mu+\sigma^2$, not just $-\mu$. The extra $\sigma^2$ comes from $\tfrac12 f''(S_t)\sigma^2S_t^2\,dt$. In particular, when $\sigma\ne0$, the reciprocal has zero drift precisely when $\mu=\sigma^2$.

Worked Example: A Reciprocal That Is a Martingale

Take $S_0=4$, $\mu=\tfrac14$, and $\sigma=\tfrac12$. Compute the dynamics of $Y_t=1/S_t$ and verify its martingale property.

The reciprocal drift is $-\tfrac14+(\tfrac12)^2=-\tfrac14+\tfrac14=0$, and its diffusion coefficient is $-\tfrac12Y_t$. Hence $dY_t=-\tfrac12Y_t\,dB_t$. The original process is $S_t=4\exp(t/8+B_t/2)$, so $Y_t=\tfrac14\exp(-t/8-B_t/2)$. For $s\le t$, the conditional expectation of $\exp(-(B_t-B_s)/2)$ is $\exp((t-s)/8)$, which cancels the factor $\exp(-(t-s)/8)$. Therefore $\mathbb{E}[Y_t\mid\mathcal{F}_s]=Y_s$. Also $\mathbb{E}[Y_t]=\tfrac14$, so $Y_t$ is integrable and is a true martingale.

Check: the zero-drift condition can also be read from the general power coefficient at $p=-1$: $-\mu+\tfrac12(-1)(-2)\sigma^2=-\mu+\sigma^2=0$. This agrees with the direct reciprocal calculation.

Exactly When Is a Power a Martingale?

Theorem: Let $p\in\mathbb{R}$, and define $\kappa_p=p\mu+\tfrac12p(p-1)\sigma^2$. Then $S_t^p$ is a local martingale if and only if $\kappa_p=0$. It is a true martingale if and only if $\kappa_p=0$ as well. If $\sigma\ne0$, the zero-drift powers are $p=0$ and $p=1-\dfrac{2\mu}{\sigma^2}$; these may coincide. If $\sigma=0$, the condition is $p\mu=0$.

Proof. The explicit power is $$ S_t^p=S_0^p\exp\left(\left(p\mu-\tfrac12p\sigma^2\right)t+p\sigma B_t\right). $$ For $0\le s\le t$, the Brownian increment $B_t-B_s$ is independent of $\mathcal{F}_s$ and is normal with mean $0$ and variance $t-s$. The normal moment-generating function therefore gives $$ \mathbb{E}[S_t^p\mid\mathcal{F}_s] =S_s^p\exp\left(\left(p\mu-\tfrac12p\sigma^2+\tfrac12p^2\sigma^2\right)(t-s)\right) =S_s^p e^{\kappa_p(t-s)}. $$ This conditional expectation exists: the same formula with $s=0$ gives a finite expectation for every real $p$ and every finite $t$. Consequently $S^p$ is a martingale exactly when $\kappa_p=0$.

For the local martingale assertion, the Ito dynamics already show that when $\kappa_p=0$, $S^p$ is a stochastic integral plus its initial value, and the conditional-expectation calculation proves it is a true martingale. Conversely, if $S^p$ is a local martingale, subtracting its stochastic-integral part in the Ito equation shows that $A_t=\int_0^t\kappa_p S_s^p\,ds$ is a continuous finite-variation local martingale. Such a process is constant: its quadratic variation is zero, and a continuous local martingale with zero quadratic variation is constant (by Ito's formula for its square after localization). If $\kappa_p\ne0$, then $A_t$ is not constant, since $S_s^p>0$ and continuity gives $\int_0^t S_s^p\,ds>0$ for every $t>0$. Hence a local martingale must have $\kappa_p=0$. Finally, when $\sigma\ne0$, factoring gives $\kappa_p=\tfrac12\sigma^2p(p-1+2\mu/\sigma^2)$; when $\sigma=0$, it gives $\kappa_p=p\mu$. $\square$

This argument also verifies the integrability needed for the Ito integrals in the dynamics. For any real $p$ and finite $T$, $\mathbb{E}\int_0^T(p\sigma S_s^p)^2\,ds =p^2\sigma^2\int_0^T\mathbb{E}[S_s^{2p}]\,ds<\infty$: the expectation inside the integral is $S_0^{2p}e^{\kappa_{2p}s}$, a finite continuous function on $[0,T]$. Thus the integrand is in $\mathcal{L}^2$, and its Ito integral is a square-integrable martingale. This is consistent with, and strengthens, the conditional-expectation check.

Worked Example: A Square Martingale While the Process Is Not

Let $S_0=2$, $\mu=-\tfrac18$, and $\sigma=\tfrac12$. Determine whether $S_t$ and $S_t^2$ are martingales.

For $p=1$, $\kappa_1=\mu=-\tfrac18$, so the conditional expectation formula gives $\mathbb{E}[S_t\mid\mathcal{F}_s]=S_s e^{-(t-s)/8}$. For $t>s$, this is not $S_s$, so $S$ is not a martingale. For $p=2$, $\kappa_2=2(-\tfrac18)+\tfrac12(2)(1)(\tfrac14)=-\tfrac14+\tfrac14=0$. The power equation is $d(S_t^2)=S_t^2\,dB_t$, and the theorem shows that $S^2$ is a true martingale.

Check: the explicit process is $S_t=2\exp(-t/4+B_t/2)$, so $S_t^2=4\exp(-t/2+B_t)$. Its expectation is $4\exp(-t/2)\exp(t/2)=4=S_0^2$, as required. Meanwhile $\mathbb{E}[S_t]=2e^{-t/8}$, which is not constant. The finite-time second moment is also finite, as the displayed exponential formula shows.

Parameter Cases and a Common Pitfall

When $\sigma\ne0$, the nonconstant candidate power is $p=1-2\mu/\sigma^2$, in addition to the constant transform $p=0$. If the candidate equals zero, the two roots coincide; this occurs when $\mu=\sigma^2/2$. When $\sigma=0$, the process is deterministic, $S_t=S_0e^{\mu t}$, and $S_t^p=S_0^pe^{p\mu t}$. It is constant exactly when $p\mu=0$, agreeing with the theorem. The division by $\sigma^2$ in the nondegenerate formula must not be used in this deterministic case.

A common mistake is to use the ordinary chain rule on the reciprocal and omit Ito's correction. That gives the incorrect drift $-\mu$ and would predict that $1/S_t$ is a martingale when $\mu=0$. For example, with $\mu=0$ and $\sigma=\tfrac12$, the correct drift is $-0+(\tfrac12)^2=\tfrac14$, not zero. The conditional-expectation formula for $p=-1$ confirms it: $\mathbb{E}[1/S_t\mid\mathcal{F}_s]=(1/S_s)e^{(t-s)/4}$. Thus the reciprocal is not a martingale in that case. The correction is forced by quadratic variation, $dB_t\,dB_t=dt$, rather than by the ordinary chain rule.

Finally, a zero-drift Ito equation first identifies a local martingale; it does not in general guarantee a true martingale for an arbitrary stochastic process. Here the explicit lognormal conditional expectation proves the stronger statement, for every real power and every finite time horizon. In this constant-coefficient geometric Brownian family, every power that is a local martingale is therefore a true martingale.

Check Your Understanding

Use the power drift coefficient and the conditional-expectation formula to justify your answers.

For $\sigma\ne0$, solve $\kappa_p=0$ and explain what happens if the two roots coincide.
For $p=-1$, derive the reciprocal drift and state the condition for it to be a martingale.
If $\sigma=0$, determine when $S_t^p$ is a martingale, including $p=0$.
For $S_0=1$, $\mu=\tfrac18$, and $\sigma=\tfrac12$, find the nonconstant zero-drift power, if one exists.
Explain why omitting the second-derivative term gives the wrong reciprocal dynamics.