Boundary Conditions for Finite Difference Black–Scholes Pricing
Summary
The document discusses how to choose boundary conditions when numerically solving the Black–Scholes partial differential equation on a finite underlying-price grid. The appropriate conditions depend on the payoff and on how the finite grid corresponds to the economically relevant domain. For a call, the response suggests setting the value to zero at the low-price boundary and approximating the high-price value by the underlying price. It also describes a zero second-derivative condition as a practical alternative for a range of payoffs.
Barrier claims need special treatment: if the grid boundary is the barrier, the value there should match the payoff specified at the barrier. For a uniform grid, the answer recommends transforming the underlying to log-price so the coefficient of the second-derivative term is constant, which can improve numerical convergence. These are practical numerical guidelines, not a general proof of well-posedness; accuracy still depends on payoff details, boundary placement, and the discretization scheme.
Key ideas
- Finite-grid boundary conditions should reflect the option payoff and the modeled price domain.
- A call can use a zero value at the low-price boundary and an underlying-price approximation at the high boundary.
- A zero-curvature boundary condition can work practically for several payoff types.
- For a barrier option, impose the specified barrier payoff at the barrier boundary.
- Using log-price coordinates makes the second-derivative coefficient constant for Black–Scholes on a uniform grid.
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# Finite Difference Method for Black-Scholes-Formula
# Finite Difference Method for Black-Scholes-Formula
Using finite difference method for the Black-Scholes-Partial Differential Equation one need to impose some boundary conditions on the edge of the grid, i.e for a Grid on $D=[a,b]\times R^+$ one need to impose the boundary condition on $V(a,t)$ and $V(b,t)$, where $V(x,t)$ is described by Black-Scholes-Partial Differential Equation with $x$ and $t$ representing respectively the underlying price and the time-to-maturity of the derivative $V$.
Question: By imposing those boundary condition, does it has impact on the well-posedness of the problem? From non-stochastic partial differential equation we know that adding boundary condition could make the initial value problem ill-posed. Is there any "rule" of choosing boundary conditions ensuring the problem to be well posed?
## Answer by Antoine Conze (score 3, accepted)
https://quant.stackexchange.com/a/33927
It depends on the type of payoff you want to price. If it is a call option, you know that $V(0,t) = 0$ and $V(x,t) \approx x$ when $x \rightarrow +\infty$ so you can use a dirichlet condition $V(a,t) = 0$ and $V(b,t) = b$. Alternatively you can use a linear condition $\frac{\partial^2V}{\partial x^2} = 0$ which in practice works fine for a variety of payoffs. The only case where you have to be carefull is when you price barrier options, for instance an up and out option, in which case $b$ will be set to the barrier and you have to use the dirichlet condition $V(b,t) = $ payoff on barrier.
Note that to improve numerical convergence of the scheme it is better to have constant coefficients if front of the $\frac{\partial^2V}{\partial x^2}$ term in the PDE when you use a uniform grid, so in the case of the Black & Scholes PDE you should work in $y = \log(x)$ space.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.