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.
Best Answer
Olver writes on p. 129 that the time reversed heat equation having a negative coefficient in front of the $\partial^2_xu$ term is ill-posed. In fact, he explains very nicely in this earlier chapter that such an equation is numerically unstable: very small changes in the intitial data can produce arbitrarily large changes in the solution at an arbitrarily small time. The BS PDE (having a negative coefficient in front of the $\partial^2_xu$ term) is analogous to the ill-posed time reveresed heat equation.
For more than 20 years I was aware of the fact that BS is solved backwards in time, i.e., you start with final data and solve for the initial option price. And yes, final data are smoothed out quickly. Olver's chapter 4.1 is indeed marvelous. Will buy that book.