2 Ways for an Accurate Barrier with Finite Difference

I had explored the issue of pricing a barrier using finite difference discretization of the Black-Scholes PDE a few years ago. Briefly, for explicit schemes, one just need to place the barrier on the grid and not worry about much else, but for implicit schemes, either the barrier should be placed on the grid and the grid  truncated at the barrier, or a fictitious point should be introduced to force the correct price at the barrier level (0, typically).

The fictitious point approach is interesting for the case of varying rebates, or when the barrier moves around. I first saw this idea in the book "Paul Wilmott on Quantitative Finance".

Recently, I noticed that Hagan made use of the ficitious point approach in its "Arbitrage free SABR" paper, specifically he places the barrier in the middle of 2 grid points. There is very little difference between truncating the grid and the fictitious point for a constant barrier.

In this specific case there is a difference because there are 2 additional ODE solved on the same grid, at the boundaries. I was especially curious if one could place the barrier exactly at 0 with the fictitious point, because then one would potentially need to evaluate coefficients for negative values. It turns out you can, as values at the fictitious point are actually not used: the mirror point inside is used because of the mirror boundary conditions.

So the only difference is the evaluation of the first derivative at the barrier (used only for the ODE): the fictitious point uses the value at barrier+h/2 where h is the space between two points at the same timestep, while the truncated barrier uses a value at barrier+h (which can be seen as standard forward/backward first order finite difference discretization at the boundaries). For this specific case, the fictitious point will be a little bit more precise for the ODE.

2 Ways for an Accurate Barrier with Finite Difference

I had explored the issue of pricing a barrier using finite difference discretization of the Black-Scholes PDE a few years ago. Briefly, for explicit schemes, one just need to place the barrier on the grid and not worry about much else, but for implicit schemes, either the barrier should be placed on the grid and the grid  truncated at the barrier, or a fictitious point should be introduced to force the correct price at the barrier level (0, typically).

The fictitious point approach is interesting for the case of varying rebates, or when the barrier moves around. I first saw this idea in the book "Paul Wilmott on Quantitative Finance".

Recently, I noticed that Hagan made use of the ficitious point approach in its "Arbitrage free SABR" paper, specifically he places the barrier in the middle of 2 grid points. There is very little difference between truncating the grid and the fictitious point for a constant barrier.

In this specific case there is a difference because there are 2 additional ODE solved on the same grid, at the boundaries. I was especially curious if one could place the barrier exactly at 0 with the fictitious point, because then one would potentially need to evaluate coefficients for negative values. It turns out you can, as values at the fictitious point are actually not used: the mirror point inside is used because of the mirror boundary conditions.

So the only difference is the evaluation of the first derivative at the barrier (used only for the ODE): the fictitious point uses the value at barrier+h/2 where h is the space between two points at the same timestep, while the truncated barrier uses a value at barrier+h (which can be seen as standard forward/backward first order finite difference discretization at the boundaries). For this specific case, the fictitious point will be a little bit more precise for the ODE.

SABR with the new Hagan PDE Approach

At a presentation of the Thalesians, Hagan has presented a new PDE based approach to compute arbitrage free prices under SABR. This is similar in spirit as Andreasen-Huge, but the PDE is directly on the density, not on the prices, and there is no one-step procedure: it's just like a regular PDE with proper boundary conditions.

I was wondering how it compared to Andreasen Huge results.

My first implementation was quite slow. I postulated it was likely the Math.pow function calls. It turns out they could be reduced a great deal. As a result, it's now quite fast. But it would still be much slower than Andreasen Huge. Typically, one might use 40 time steps, while Andreasen Huge is 1, so it could be around a 40 to 1 ratio. In practice it's likely to be less than 10x slower, but still.

While looking at the implied volatilities I found something intriguing with Andreasen Huge: the implied volatilities from the refined solution using the corrected forward volatility look further away from the Hagan implied volatilitilies than without adjustment, and it's quite pronounced at the money.

Interestingly, the authors don't plot that graph in their paper. They plot a similar graph of their own closed form analytic formula, that is in reality used to compute the forward volatility. I suppose that because they calibrate and price through their method, they don't really care so much that the ATM prices don't match Hagan original formula.

We can see something else on that graph: Hagan PDE boundary is not as nice as Andreasen Huge boundary for high strikes (they use a Hagan like approx at the boundaries, this is why it crosses the Hagan implied volatilities there).

If we use a simple option gamma = 0 boundary in Andreasen Huge, this results in a very similar shape as the Hagan PDE. This is because the option price is effectively 0 at the boundary.
Hagan chose a specifically taylored Crank-Nicolson scheme. I was wondering how it fared when I reduced the number of time-steps.

The answer is: not good. This is the typical Crank-Nicolson issue. It could be interesting to adapt the method to use Lawson-Morris-Goubet or TR-BDF2, or a simple Euler Richardson extrapolation. This would allow to use less time steps, as in practice, the accuracy is not so bad with 10 time steps only.

What I like about the Hagan PDE approach is that the implied vols and the probability density converge well to the standard Hagan formula, when there is no negative density problem, for example for shorter maturities. This is better than Andreasen Huge, where there seems to be always 1 vol point difference. However their method is quite slow compared to the original simple analytic formula.

Update March 2014 - I have now a paper around this "Finite Difference Techniques for Arbitrage Free SABR"