Verifying a Black-Scholes Call Price Solves the Pricing PDE

Summary

The document clarifies how to check that a closed-form call option price satisfies the Black-Scholes partial differential equation. The price is a deterministic function of the current underlying price and time, so the verification proceeds by calculating its ordinary partial derivatives with respect to those variables and substituting them into the PDE. The exercise is a direct analytical check of the formula.

It distinguishes this task from deriving a stochastic price process with Itô’s lemma. Itô calculus is not needed merely to verify that a given deterministic pricing function solves the PDE. The excerpt gives conceptual guidance but no derivative calculations, boundary-condition check, or full proof; those details would be required for a complete verification.

Key ideas

• A closed-form option price can be treated as a deterministic function of price and time when checking the PDE.
• Compute the function’s partial derivatives and substitute them into the Black-Scholes equation.
• Itô’s lemma is unnecessary for this direct substitution check.
• A complete proof may also require checking boundary or terminal conditions.

Tags

# Show that the equation solves the Black-Scholes PDE


# Show that the equation solves the Black-Scholes PDE












I have the solution as given

Based on this, I have to show that this solves the Black-Scholes formula

It means that I should take the partial derivatives of the solution above and then receive the differential equation of Black-Scholes.

Anyone can give me an intuition how should I do that? Should I use Ito's lemma to compute the derivatives?

## Answer by Yiannis (score 4)

https://quant.stackexchange.com/a/17860

The above equation is the price of a call option. It has nothing stochastic inside it. It only depends on the current price and the time. So no Ito is needed. You should just compute the derivatives of your solution v (like you do for any deterministic multivariable function), plug them into the PDE and verify that it's satisfied.

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