Given that a stock price follows the relationship dS = σSdW(t), what is the differential of the natural logarithm of the stock price?
Choose one of the following propositions:
1. d(ln(S)) = 0.5 σ^2 dt + σ dW(t)
2. d(ln(S)) = - 0.5 σ^2 dt + σ dW(t)
3. d(ln(S)) = σ^2 dt + σ dW(t)
This relationship comes from Ito's Lemma, which is a fundamental result in stochastic calculus. Ito's Lemma provides a way to find the differential of a function of a stochastic process.
To understand this, let's break it down:
dS = σS dW(t) [EQUATION 1]
where σ is the volatility of the stock, and W(t) is a standard Brownian motion (or Wiener process).
We want to find:
By Ito's Lemma, for a twice differentiable function f(S), we have:
df(S) = f'(S) dS + 0.5 f''(S) (dS)^2
Let's use f(S) = ln(S). The derivatives are:
f'(S) = 1/S
f''(S) = -1/S^2
Substituting these derivatives and the given dS into Ito's formula:
df(S) = (1/S) σS dW(t) - 0.5 (1/S^2) (σS dW(t))^2 [EQUATION 2]
Now, here's the key part:
The differential of the Brownian motion squared, (dW(t))^2, is equal to dt. That's a fundamental property of Brownian motion (quadratic variation).
(dS)^2 = (σS dW(t))^2 = σ^2 S^2 dt
So our [EQUATION 2] becomes:
d(ln(S)) = σ dW(t) - 0.5 σ^2 dt