Using Ito’s Lemma to Find the Distribution of a Transformed Stock

Summary

The document applies Ito’s lemma to the cube root of a stock price modeled in a Black–Scholes market. Differentiating the transformation and substituting the stock’s drift and volatility into Ito’s formula gives a stochastic differential equation for the transformed price. Its diffusion coefficient is one third of the original volatility, while its drift includes an adjustment from the curvature of the cube root function.

Because the transformed process has proportional drift and diffusion, its solution is lognormal under the stated model assumptions. The transformed drift is therefore not simply the original drift: it depends on both the original drift and volatility. The answer calls this the risk-neutral drift, though the document’s displayed derivation has notation and sign inconsistencies, so its formula should be checked against Ito’s lemma before use. The result also relies on the Black–Scholes assumptions and does not establish lognormality for arbitrary price processes.

Key ideas

• Ito’s lemma derives the dynamics of a nonlinear transformation of a security price.
• The cube root transformation scales the original proportional volatility by one third.
• The transformation’s drift changes because Ito’s lemma adds a term involving curvature and variance.
• A geometric Brownian motion remains lognormal after this transformation under the model assumptions.
• The displayed derivation contains notation and sign inconsistencies that warrant checking.

Tags

# Proving lognormality of security in Black-Scholes market


# Proving lognormality of security in Black-Scholes market












Can someone prove that for some security $S_t$ with drift $\mu$ and volatility $\sigma^2$ in a Black-Scholes market we have that $Y_t = (S(t))^{1/3} \sim \text{Lognormal}$, w.r.t. the risk-neutral measure $\mathbb{Q}$? Also, would the drift in this case just be $\mu$?

## Answer by maarten de goede (score 4)

https://quant.stackexchange.com/a/72019

To prove this and similar transformations of securities we resort to Ito's Lemma. Let us define $f(t, S) = (S)^{1/3}$ with derivatives $\frac{\delta f(t, S)}{\delta t} = 0, \frac{\delta f(t, S)}{\delta S} = \frac{1}{3} S^{-\frac{2}{3}}, \frac{\delta^2 f(t, S)}{\delta S^2} = -\frac{2}{9}S^{-\frac{5}{3}}$.

Simply filling in these values in Ito's formula yields

\begin{align} dY(t) &= 0 \cdot dt \;+ \; \frac{1}{3}S(t)^{-\frac{2}{3}} \;+\;\frac{1}{2} \cdot \frac{2}{9} S(t)^{-\frac{5}{3}}S^2(t)\sigma^2dt, \\ &=S^{\frac{1}{3}}\left( \frac{1}{3} S^{-1}(\delta Sdt\;+\;\sigma S dW(t)) - \frac{1}{9} \sigma^2 S^{-2} S^2 dt \right), \\ &= Y(t) \left( \left( \frac{1}{3} \delta \; + \; - \frac{1}{9} \sigma^2 \right) dt + \frac{1}{3} \sigma dW(t) \right). \end{align}

Which implies $Y(t) \sim \text{Lognormal}$ Here $\left(\frac{1}{3} \delta \; - \frac{1}{9} \sigma^2 \right)$ should be recognised as the risk-neutral drift and not simply $\mu$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.