analytic solution of Black-Scholes PDE

Here we present an analytical solution for the Black-Scholes partial differential equation,

rf=ft+rxfx+12σ2x22fx2,f=f(t,x), (1)

over the domain 0<x<, 0tT, with terminal condition f(T,x)=ψ(x), by reducing this parabolic PDE to the heat equation of physics.

We begin by making the substitution:


which is motivated by the fact that it is the portfolio value discounted by the interest rate r (see the derivation of the Black-Scholes formula) that is a martingaleMathworldPlanetmath. Using the product ruleMathworldPlanetmath on f=ertu, we derive the PDE that the function u must satisfy:


or simply,

0=ut+rxux+12σ2x22ux2. (2)

Next, we make the substitutions:


These changes of variables can be motivated by observing that:

  • The underlying process described by the variable x is a geometric Brownian motion (as explained in the derivation of the Black-Scholes formula itself), so that logx describes a Brownian motionMathworldPlanetmath, possibly with a drift. Then logx should satisfy some sort of diffusion equation (well-known in physics).

  • The evolution of the system is backwards from the terminal state of the system. Indeed, the boundary conditionMathworldPlanetmath is given as a terminal state, and the coefficient of u/t is positive in equation (2). (Compare with the standard heat equation, 0=-u/t+u/x, which describes a temperature evolving forwards in time.) So to get to the heat equation, we have to use a substitution to reverse time.





substituting in equation (2), we find:

0=-us+(r-12σ2)uy+12σ22uy2. (3)

The first partial derivativeMathworldPlanetmath with respect to y does not cancel (unless r=12σ2) because we have not take into account the drift of the Brownian motion. To cancel the drift (which is linear in time), we make the substitutions:


Under the new coordinate system (z,τ), we have the relations amongst vector fields:


leading to the following of equation (3):



uτ=12σ22uz2,u=u(τ,z), (4)

which is one form of the diffusion equation. The domain is on -<z< and 0τT; the initial condition is to be:


The original function f can be recovered by


The fundamental solution of the PDE (4) is known to be:


(derived using the Fourier transformMathworldPlanetmath); and the solution u with initial condition u0 is given by the convolution:


In terms of the original function f:


(τ=T-t) which agrees with the result derived using probabilistic methods (

Title analytic solution of Black-Scholes PDE
Canonical name AnalyticSolutionOfBlackScholesPDE
Date of creation 2013-03-22 16:31:34
Last modified on 2013-03-22 16:31:34
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 6
Author stevecheng (10074)
Entry type Derivation
Classification msc 60H10
Classification msc 91B28
Related topic ExampleOfSolvingTheHeatEquation
Related topic BlackScholesPDE
Related topic BlackScholesFormula