Suppose an asset's price evolves according to the Itô process where are positive constants. Find the analog of the Black-Scholes equation. That is, find a PDE (partial differential equation) satisfied by the value of a derivative security whose value depends on the underlying asset . Use the same assumptions made to derive Black-Scholes, except taking the expression above for the evolution of . Enter the coefficient functions in the appropriate boxes below. Express your answers in terms of alpha for , beta for , gamma for , for the risk-free rate, and , as needed

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.

Suppose an asset's price evolves according to the Itô process where are positive constants. Find the analog of the Black-Scholes equation. That is, find a PDE (partial differential equation) satisfied by the value of a derivative security whose value depends on the underlying asset. Use the same assumptions made to derive Black-Scholes, except taking the expression above for the evolution of.

Explore the derivation of the partial differential equation (PDE) for a derivative security under the Itô process. Learn how the Black-Scholes equation is modified to incorporate stochastic calculus, the Itô process, and key financial concepts. Suitable for advanced financial mathematics students.

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