Applying Zero-Curvature Boundaries in Black–Scholes PDE Schemes

Summary

The document asks how to impose a zero second-derivative condition at the right boundary when numerically solving the Black–Scholes partial differential equation with a Crank–Nicolson scheme. It contrasts adding a discrete approximation of the curvature condition to the system with substituting zero curvature into the PDE at the boundary, which leaves a first-order equation there.

The question highlights a modeling and discretization choice: the two formulations impose different boundary information, so they need not produce the same numerical solution. However, the supplied document contains only the question and no accepted answer, derivation, or numerical comparison. It therefore does not establish which treatment is appropriate, how boundary closure should be implemented, or how either choice affects accuracy. Resolving those points requires additional analysis of the PDE, the option payoff and far-field boundary behavior, and the numerical scheme.

Key ideas

• The question concerns a zero-curvature condition at the right edge of a Black–Scholes PDE grid.
• One approach discretizes the second-derivative condition and adds it to the numerical system.
• Another substitutes zero curvature into the PDE to form a boundary equation.
• The document poses the equivalence and correctness issue but provides no answer or supporting evidence.

Tags

# zero curvature boundary condition


# zero curvature boundary condition












Assume I am solving numerically Black Scholes PDE $$u_t+0.5\sigma^2s^2u_{ss}+rsu_s-ru=0$$ and I decided to have boundary condition on the right boundary as $u_{ss}=0$. One way is to write the discrete approximation for $u_{ss}$, for example, in the case of Crank-Nicolson: $$0.5\frac{u^{n+1}_{i-1}-2u^{n+1}_{i}+u^{n+1}_{i+1}}{\delta x^2}+0.5\frac{u^{n}_{i-1}-2u^{n}_{i}+u^{n}_{i+1}}{\delta x^2}=0$$ and add this to the discretization system . Another way to to put this back into equation itself and get $$u_t+rsu_s-ru=0$$ on that boundary. I might be lucky to write the closed formula on that boundary so no discretization is needed. In the second approach I basically assume the equation is solved on that boundary, along with the fact the second derivative is zero. In the first approach, I only assume the second derivative is zero. They don't look equivalent to me. So what approach is correct and why?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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