I am currently reading Peter J. Olver's book Introduction to Partial Differential Equations (which is an absolutely fantastic read so far, I would recommend anyone interested in PDEs to check it out). In chapter 8, on page $299$ the Black-Scholes equation is introduced, and is written in classical evolution equation form as $$\partial_tu(t,x)=-\frac{\sigma^2x^2}{2}\partial_x^2u(t,x)-rx\partial_xu(t,x)+ru(t,x)\tag{1}$$
Where
- $u$: Current value of a financial option
- $t$: time
- $x$ price of the underlying asset
- $r$: interest rate
- $\sigma$: volatility
All quantities are $\geq 0$.
The author writes
Observe that the Black-Scholes equation is a backwards diffusion process, since, upon solving for $\partial_tu(t,x)$, the coefficient in front of the diffusion term $\partial_x^2u(t,x)$ is negative. This implies that the initial value problem is well-posed only when time runs backwards. In other words, given a prescribed value of the option at some specified time in the future, we can use the Black-Scholes equation to determine its current value. However, ill-posedness implies that we cannot predict future values from the current worth of the portfolio.
I am unsure of what he means by "cannot predict". I mean, let's say I pick some future time $t_*$ and I want to compute $u(t_*,x)$. Why can't I just couple $\boldsymbol{(1)}$ with the straightforward conditions
$$u(t,0)=0~~,~~u(0,x)=f(x)$$ And just solve the equation numerically to compute $u(t_*,x)$? Does Olver mean that a solution won't exist, or won't be unique? Or that the numerical solution won't converge? Can someone offer an intuitive explanation of why that might be the case?
Thanks.