Transforming Black–Scholes Option Boundaries into Heat Equation Boundaries

Summary

The document asks how to translate boundary conditions for a European call option into boundary conditions for the heat equation obtained by transforming the Black–Scholes partial differential equation. It states the call’s terminal payoff, the zero-price boundary at an underlying price of zero, and the asymptotic behavior as the underlying price becomes large. It also gives a transformed heat equation and its initial condition.

No transformed boundary conditions or derivation are supplied; the text poses the problem rather than resolving it. The useful concept is that a PDE transformation must carry the original payoff and spatial boundaries into the new variables, while preserving the relevant limiting behavior. Applying those conditions requires the precise change of variables connecting price and time to the heat equation’s space and time coordinates, which the document does not fully specify.

Key ideas

• The European call payoff supplies the terminal condition for the Black–Scholes equation.
• The original problem also specifies behavior at zero underlying price and at very large prices.
• The document presents a transformed heat equation and initial condition but leaves the new boundary conditions unanswered.
• Boundary conditions must be translated using the exact coordinate and dependent-variable transformation.

Tags

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


# 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 ?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.