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Testing a Candidate European Option Price with the Black–Scholes PDE

Article Quant Q&A · Author: Srini
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Summary

The document asks whether a proposed function of the underlying price, involving the price multiplied by the logarithm of its cube, can represent the value of a European option in the Black–Scholes framework. The accepted response says to substitute the candidate into the Black–Scholes partial differential equation and check whether it satisfies the equation. Under the stated simplifying assumption of a zero interest rate, a solution process is a martingale, linking its current value to the conditional expected terminal payoff.

A second response emphasizes that an option-pricing problem must also specify the terminal payoff at expiry: the proposed function would need to agree with that payoff at maturity. Satisfying a differential equation alone is not enough to identify the intended contract without the relevant boundary or terminal conditions. The replies do not carry out the derivative calculations or verify that the proposed function actually qualifies, so they offer a checking method rather than a completed proof.

Key ideas

  • Substitute a candidate value function and its derivatives into the Black–Scholes PDE to test whether it solves the pricing equation.
  • A valid European option valuation must match the contract’s payoff at expiry.
  • With zero interest rate, a suitable Black–Scholes solution has a martingale interpretation under the model assumptions.
  • The document gives a verification approach but does not perform the calculation for the proposed function.

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Full text
# How do I prove that a certain price is price of European option in Black-Scholes framework


# How do I prove that a certain price is price of European option in Black-Scholes framework












I want to show whether the following price at t is of a european option in Black-Scholes Framework. $$S_tlog_e (S_t^3) $$ Is it just trying to substitute the function (and partial derivates) in the Black-Scholes PDE?

## Answer by piterbarg (score 7, accepted)

https://quant.stackexchange.com/a/59194

Yes it is actually just substituting it into the Black Scholes PDE. If the PDE is satisfied, $V(t,S(t)),t\ge 0$ is a martingale and hence $V(t,S(t)) = E_t (V(T,S(T))$ so that $V(t,S(t))$ is the expected value, at time t, of an option that pays $V(T,S(T))$ at time $T$. Here I assumed $r=0$ for simplicity

## Answer by d_797 (score 1)

https://quant.stackexchange.com/a/59190

If it was a solution to a European option PDE pricing problem, at time $T$ you have $V(T,S) = S \log{S^3}$.

Based on this terminal function, you can then solve the BS PDE using a semi-analytical approach to find $V(t,S)$. My guess is it will not be the candidate you posted (haven't checked).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.