Time-Independent Solutions to the Black-Scholes PDE

Summary

This exchange asks which Black-Scholes partial differential equation solutions can depend only on the underlying price or only on time, given a terminal payoff. The response argues that a solution independent of time must equal its terminal payoff at every time, so it is compatible with the PDE only when that payoff itself satisfies the equation as a time-constant solution. A solution independent of the underlying price must match the terminal payoff across all underlying prices; this is possible only for a constant payoff, yielding a constant solution.

A further answer points to separation of variables for the related heat equation as a broader method for exploring PDE solutions. The exchange is brief and does not derive the conditions by substituting candidate functions into the PDE or discuss boundary conditions and solution domains. Its claims should therefore be read as a basic terminal-condition observation rather than a complete classification of Black-Scholes solutions.

Key ideas

• A time-independent candidate must agree with the terminal payoff at every time.
• A price-independent candidate can match the terminal condition only when the payoff is constant.
• Separation of variables for the heat equation is suggested as a way to study related PDE solutions.
• The exchange does not provide a full derivation or address boundary conditions.

Tags

# Black Scholes Separable Solutions


# Black Scholes Separable Solutions












I want to find all the solutions of the Black Scholes PDE that are of the form $f(x,t)=\theta(x)$ or $f(x,t)=\phi(t)$.

Can someone explain and help with this? I know the PDE formula is

$f_{t}(t, x)=-\frac{1}{2} \sigma^{2}x^2 f_{x x}(t, x)-r x f_{x}(t, x)+rf(t, x)$

$f(T, x)=h(x)$

Thank you!

## Answer by Valometrics.com (score 1)

https://quant.stackexchange.com/a/51147

$f(t,x)=\theta(x)$ means that the actual price deos not depend on time so: $$f(t,x)=f(T,x)=h(x)$$ so the only solution is $h(x)$.

$f(t,x)=\phi(t)$ means that the price deos not depend on the price.it means that at time T: $$f(T,x)=\phi(t)=h(x)$$ so if h depend on x, there is no solution and if h is constant, there is one solution $\phi=h$

## Answer by d_797 (score 1)

https://quant.stackexchange.com/a/59279

You may find it useful to look at the method of "separation of variables" for the heat equation (it is often used for a PDE on a bounded domain, but could still be useful).

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