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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!

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  • $\begingroup$ I also need to find solutions that are of the form of phi(t)*theta(x) and figure out what terminal value problems I can solve with these solutions? Can you assist? I am so lost. $\endgroup$ – galeshapley Feb 12 at 19:51
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    $\begingroup$ If the solution is of the form theta(x) that means it is not a function of t. So the derivative with respect to t namely $f_t(t,x)$ can be set to zero; the equation becomes a ODE. Similarly if the solution is of the form phi(t), you can set all the derivatives with respect to x to zero, resulting in an extremely simple equation, easy to solve. $\endgroup$ – noob2 Feb 12 at 19:57
  • $\begingroup$ So, I will end up with $\theta(x)=-\frac{1}{2} \sigma^{2}(x) f_{x x}(x)-x$ and with r(t)f(t)=phi(t). Is this right? $\endgroup$ – galeshapley Feb 12 at 20:11
  • $\begingroup$ And for the part that I asked in the comments, should I just be multiplying these solutions together? I am not sure what terminal vlaue problem I can solve with these. $\endgroup$ – galeshapley Feb 12 at 20:17
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    $\begingroup$ $f_{t}(t, x)=-\frac{1}{2} \sigma^{2} x^{2} f_{x x}(t, x)-r x f_{x}(t, x)+r f(t, x)$ I think this is actually the formula i want to work with $\endgroup$ – galeshapley Feb 12 at 20:28
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$f(t,x)=\theta(x)$ 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 $\phi=h$

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  • $\begingroup$ Thank you! this is helpful. So when it asks what can be solved if f(T,x)=theta(x)*phi(t) what does this mean? Does this mean we can solve stuff with any of the solutions above? $\endgroup$ – galeshapley Feb 15 at 23:56

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