# 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?

## 1 Answer

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.

• Thanks, but for barrier option I would have also some constants on the boundary isn't it? And again this is a piece-wise smooth function on the boundary of a compact domain. Could you give an example for a barrier option? – quallenjäger Apr 27 '17 at 15:37
• I am surprised this was accepted as the answer, as it doesn't really address the actual question regarding well-posedness of the problem. Or did I miss something? – LocalVolatility Apr 27 '17 at 15:38
• @LocalVolatility Yes, I thought first dirchlet condition is sufficient for well-posedness – quallenjäger Apr 27 '17 at 15:54
• Sorry I didn't see the question about the well-posedness of the problem. In pratical finance applications as long as you've done the appropriate change of variable and the volatility is bounded you don't really care about the boundary conditions as long as $a$ and $b$ are away enough from $x_0$, because the probability of reaching $a$ or $b$ is very small. This is why a linear condition, which is simple to implement and is not payoff dependent, is appropriate. – Antoine Conze Apr 28 '17 at 9:46
• Only for barrier options where the boundaries are set to the barriers and therefore may be reached with high probability do you need to be careful, and the condition should be a dirichlet set to the payoff on barrier. – Antoine Conze Apr 28 '17 at 9:52