What is the final explicit solution for S obtained in this clip?

Conditions

• $\mu$ and $\sigma$ are constant.
• The ansatz $S=f(W,t)$ is valid.
• The ODE for $\phi(t)$ is solved.

Reasoning, step by step

1. Solve the ODE for $\phi(t)$ to get $\phi(t) = c_1 e^{\left(\mu - \frac{1}{2}\sigma^2\right)t}$.
2. Substitute $\phi(t)$ back into $f(W,t) = \phi(t)e^{\sigma W}$.
3. Combine the exponents to get $S = c_1 e^{\sigma W + \left(\mu - \frac{1}{2}\sigma^2\right)t}$.
4. Identify $c_1 = S_0$ using the initial condition at $t=0$.

Example

The bottom line on the board writes '$S = f(W,t) = c_1 \exp (\sigma W + (\mu - 1/2 \sigma^2)t)$'.

Common misconceptions

• Forgetting the $-\frac{1}{2}\sigma^2$ term in the exponent, which arises from Itô's lemma.
• Assuming $c_1$ is always 1, rather than being determined by the initial value $S_0$.

Watch the explanation

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