The Black-Scholes equation is derived from the assumption that an asset's price evolves according to a stochastic process, specifically an Itô process. Here, we'll derive the partial differential equation (PDE) for a derivative security whose value depends on the underlying asset's price $S$.

Let’s assume that the price of the asset evolves according to the following Itô process:

$dS_t = \alpha(S_t, t) S_t \, dt + \beta(S_t, t) S_t \, dW_t$

Where:

• $\alpha(S_t, t)$ is the drift coefficient,
• $\beta(S_t, t)$ is the volatility coefficient,
• $W_t$ is a Wiener process.

Let $V(S_t, t)$ represent the price of the derivative (such as a European call option) that depends on the underlying asset's price $S$ and time $t$. Using the same steps as the Black-Scholes derivation, we can apply Itô’s Lemma to the function $V(S_t, t)$:

$dV = \frac{\partial V}{\partial t} \, dt + \frac{\partial V}{\partial S} \, dS + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} (dS)^2$

Substitute the Itô process for $dS$:

$dS = \alpha(S_t, t) S_t \, dt + \beta(S_t, t) S_t \, dW_t$ $(dS)^2 = \beta(S_t, t)^2 S_t^2 \, dt$

Thus, we have:

$dV = \frac{\partial V}{\partial t} \, dt + \frac{\partial V}{\partial S} \left( \alpha(S_t, t) S_t \, dt + \beta(S_t, t) S_t \, dW_t \right) + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \beta(S_t, t)^2 S_t^2 \, dt$

Now, let's rewrite this equation:

$dV = \left( \frac{\partial V}{\partial t} + \alpha(S_t, t) S_t \frac{\partial V}{\partial S} + \frac{1}{2} \beta(S_t, t)^2 S_t^2 \frac{\partial^2 V}{\partial S^2} \right) dt + \beta(S_t, t) S_t \frac{\partial V}{\partial S} \, dW_t$

In a risk-neutral world, we eliminate the stochastic term (the $dW_t$ term). The return on the derivative should equal the risk-free rate $r$. Therefore, the value of the derivative must satisfy:

$dV = r V \, dt$

Equating the deterministic terms from both expressions for $dV$:

$r V = \frac{\partial V}{\partial t} + \alpha(S_t, t) S_t \frac{\partial V}{\partial S} + \frac{1}{2} \beta(S_t, t)^2 S_t^2 \frac{\partial^2 V}{\partial S^2}$

Rearranging this equation gives us the PDE satisfied by the derivative security $V(S, t)$:

$\frac{\partial V}{\partial t} + \alpha(S_t, t) S \frac{\partial V}{\partial S} + \frac{1}{2} \beta(S_t, t)^2 S^2 \frac{\partial^2 V}{\partial S^2} = r V$

Thus, the analog of the Black-Scholes equation, where the asset price evolves according to an Itô process with drift $\alpha(S_t, t)$ and volatility $\beta(S_t, t)$, is:

$\boxed{\frac{\partial V}{\partial t} + \alpha(S, t) S \frac{\partial V}{\partial S} + \frac{1}{2} \beta(S, t)^2 S^2 \frac{\partial^2 V}{\partial S^2} = r V}$

Key Coefficients:

• $\alpha(S_t, t)$: the drift term,
• $\beta(S_t, t)^2$: the volatility term squared,
• $r$: the risk-free rate.

Do you have any questions or need further details on this?

Related Questions:

1. What is the derivation of the original Black-Scholes equation?
2. How do the assumptions of risk neutrality affect this derivation?
3. What happens if the volatility $\beta$ is a constant instead of a function of $S$ and $t$?
4. How does Itô’s Lemma contribute to the derivation of the Black-Scholes equation?
5. What is the significance of the risk-free rate $r$ in this context?

Tip:

In financial models, the assumption of constant volatility is often relaxed to account for more complex market behavior, such as in stochastic volatility models.