I'm trying to understand the analogy between the Black-Scholes equation (1) and the heat partial differential equation (2). I understand that (1) can be written in the form of (2) mathematically, but what does this tell us about the nature of the variables involved in (1)?
1$\begingroup$ Mainly it means (it is possible because) the BSE equation is of Parabolic Type (as opposed to Elliptical or Hyperbolic type). This classification of 2d order PDEs into three types is considered theoretically important (for ex in setting boundary conditions or in choosing numerical methods). $\endgroup$– nbbo2Jan 26 at 9:36
2$\begingroup$ The BS equation can easily be transformed into the heat equation which makes it easier to solve numerically. See this. $\endgroup$– Kurt G.Jan 27 at 14:12
1$\begingroup$ Another important point is that this "heat" equation has as time variable not $t$ but $\tau=T-t\,.$ In other words: the BS and its heat sister are solved backwards with final conditions (not forwards with initial conditions as physicists would solve the real heat equation). In fact, the PDE book by Olver explains nicely that the BS equation with initial condition is practically not solvable. $\endgroup$– Kurt G.Jan 27 at 14:13
For heat equation, it describes how heat diffuses (usually measured by temperature) through the length of the material and over time.
For Black Scholes, it describes how the value of the option diffuses over time and, instead of through some material, as the underlying "travels" across its range of possible values.