Good evening,
I’m currently working on the following problem for my PDE class and I would like an opinion on it,
Let’s consider the Black-Scholes model with (time-varying) volatility, $sigma = sigma(t)$, and (time varying) risk free return rate,$r=r(t)$.
$$ V_t + frac{sigma^2(t)}{2}S^2 V_{SS} + r(t)V_S-r(t)V = 0 space, space S>0,space 0<t<T $$
And the following final condition: $$V(S,T) = phi(S)space , space S>0$$ where $phi$ represents the option’s payoff.
I started by considering the following variable change, $$ S = e^x$$ $$ t = T – theta $$ This allowed me to consider the following functions:
$$ U(x,theta) = V(e^x,T-theta) space,space hatsigma(theta) = sigma(T-theta) space,space hat r(theta) = r(T-theta) $$ This also turned my final condition into an initial condition, $U(x,0) = phi(e^x) $, and I derived the following transformation.
$$ U_{theta} = frac{hatsigma^2(theta)}{2}U_{xx} + Big(hat r(theta) – frac{hatsigma^2(theta)}{2}Big)U_x – hat r(theta)U space,space x in mathbb{R} space,space 0 < theta < T $$
Then, I introduced a new time variable, $$ tau(theta) = frac{1}{2} int_{0}^{theta} hatsigma^2(xi)dxi$$ I managed to prove that this function is a bijection from an interval $(0,T)$ to an interval $(0,Upsilon)$. Therefore, $tau$ is invertible and we can have $theta = theta(tau)$
With this new time variable, I defined the following functions, $$ R(tau) = hat r(theta(tau)) space,space Sigma(tau) = hatsigma(theta(tau))$$ Which then allowed me to to define $$ k(tau) = 2 frac{R(tau)}{Sigma^2(tau)} $$
Given $u(x,tau) = U(x,theta(tau))$, I derived the following new equation, $$u_{tau} = u_{xx} + (k(t)-1)u_x -k(t)u space,space x in mathbb{R} space,space 0 < tau < Upsilon $$
I then defined the following “updating factor”, $$d(tau) = e^{{int_{0}^{tau}k(xi)dxi}} $$ and a new function $$ v(x,tau) = d(tau)u(x,tau) $$ This new function allowed me to derive the following transformation, $$v_tau = v_{xx} + (k(t)-1)v_x space,space x in mathbb{R} space,space 0 < tau < Upsilon $$
I then solved the following PDE problem,
$$ psi_tau = (k(t)-1)psi_x space,space x in mathbb{R} space,space 0<tau<Upsilon $$
$$ psi(x,0) = x $$
This problem has the following solution, $$psi(x,tau) = x + int_{0}^{tau} k(xi)-1 dxi $$
With this $psi$ solution, with $psi = y$, I made a new transform with the following function, $$v(x,tau) = w(psi(x,tau),tau) $$ This transformation allowed me to achieve the heat equation, $$w_tau = w_{yy} $$ With the initial condition, $$w(y,0) = phi(e^y)$$
Having all of these transforms and functions, my main goal is to solve the first problem, given all this information above.
$$ V_t + frac{sigma^2(t)}{2}S^2 V_{SS} + r(t)V_S-r(t)V = 0 space, space S>0,space 0<t<T $$ $$V(S,T) = phi(S)space , space S>0$$
In order not to have this post being twice as long as it is, I won’t explicit any reasoning behind these proofs, but I’m glad to provide any reasoning if needed.
My question here is the following: should I start by solving the heat equation, by applying a Fourier transform, and reverse each transform I’ve done so far one by one? Or is there a simpler way to solve this? I’ve also looked into the Carr-Madan decomposition for this matter, but I haven’t learned that yet.
I was looking forward into having some kind of clue in order to have a starting point, because I’m really lost in all of this “mess”. I really appreciate if you have read this far, and I apologize for the long post.
Thank you!
