The Ito's Lemma in Simple Terms

Given that a stock price follows the relationship \( dS = \sigma S dW(t) \), we want to find the differential of the natural logarithm of the stock price. Choose one of the following propositions:

1. \( d(\ln(S)) = 0.5 \sigma^2 dt + \sigma dW(t) \)
2. \( d(\ln(S)) = - 0.5 \sigma^2 dt + \sigma dW(t) \)
3. \( d(\ln(S)) = \sigma^2 dt + \sigma dW(t) \)

Derivation Using Ito's Lemma:

This relationship comes from Ito's Lemma, a fundamental result in stochastic calculus. Ito's Lemma provides a way to find the differential of a function of a stochastic process.

Given:

\[ dS = \sigma S dW(t) \quad \text{[EQUATION 1]} \]

where \( \sigma \) is the volatility of the stock, and \( W(t) \) is a standard Brownian motion (or Wiener process). We aim to find \( d(\ln(S)) \).

By Ito's Lemma, for a twice differentiable function \( f(S) \), we have:

\[ df(S) = f'(S) dS + 0.5 f''(S) (dS)^2 \]

Using \( f(S) = \ln(S) \), the derivatives are:

\[ f'(S) = \frac{1}{S}, \quad f''(S) = -\frac{1}{S^2} \]

Substituting these derivatives and the given \( dS \) into Ito's formula:

\[ df(S) = \frac{1}{S} \sigma S dW(t) - 0.5 \frac{1}{S^2} (\sigma S dW(t))^2 \quad \text{[EQUATION 2]} \]

Now, here's the key part:

• The differential of the Brownian motion squared, \( (dW(t))^2 \), is equal to \( dt \). This is a fundamental property of Brownian motion (quadratic variation).

Thus:

\[ (dS)^2 = (\sigma S dW(t))^2 = \sigma^2 S^2 dt \]

Substituting this into [EQUATION 2], we get:

\[ d(\ln(S)) = \sigma dW(t) - 0.5 \sigma^2 dt \]

Conclusion:

The correct proposition is:

\[ d(\ln(S)) = - 0.5 \sigma^2 dt + \sigma dW(t) \]