# Black-Scholes equation to Heat equation .(Boundary conditions)

I have been given a problem to code the heat equation which is transformed from B-S equation (European call option) .

Now the boundary conditions are for European call option: $$C(S,T)=\max(S-K,0)$$ $$C(0,t)=0$$ $$C(S,t) \sim S \space as \> S\to \infty$$ Transforming it to heat equation :$$\frac{\partial u}{\partial \tau}=\frac{\partial^2u}{\partial x^2}$$ with the initial condition: $$u(x,0)=max(e^{\frac{k+1}{2}x}-e^{\frac{k-1}{2}x},0)$$ what are the boundary conditions for this heat equation ?

• Apply the same transformation that you used to get the initial condition from the payoff function to get the transformed boundary conditions from the original ones. Mar 27 '19 at 18:02
• The first boundary condition is $0$. I am confused about the second one. Mar 28 '19 at 15:29
• @LocalVolatility , do you have any link to any such paper related to my question? I am stuck. Mar 29 '19 at 10:11
• Please add the exact change of variables you used to get from $C(S, t)$ to $u(x, \tau)$ and what your desired boundary conditions on $C(S, t)$ are, then I'll answer the question. Mar 29 '19 at 11:45
• quant.stackexchange.com/questions/84/… here the derivation was given. they used the same parameters but the boundary conditions of the heat equation is not given. I get weird boundary conditions. Mar 29 '19 at 18:40