Andreasen Huge extrapolation

There are not many arbitrage free extrapolation schemes. Benaim et al. extrapolation is one of the few that claims it. However, despite the paper's title, it is not truely arbitrage free. The density might be positive, but the forward is not preserved by the implied density. It can also lead to wings that don't obey Lee's moments condition.

On a Wilmott forum, P. Caspers proposed the following counter-example based on extrapolating SABR: \( \alpha=15\%, \beta=80\%, \nu=50\%, \rho=-48\%, f=3\%, T=20.0 \). He cut this smile at 2.5% and 6% and used the BDK extrapolation scheme with mu=nu=1.

A truly arbitrage free extrapolation can be obtained through Andreasen Huge volatility interpolation, making sure the grid is wide enough to allow extrapolation. Their method is basically a one step finite difference implicit Euler scheme applied to a local volatility parameterization that has as many parameters than prices. The method is presented with piecewise constant local volatility, but actually used with piecewise linear local volatility in their example.

Smile

Density with piecewise linear local volatility

There is still a tiny oscillation that makes the density negative, but one understands why typical extrapolations fail on the example: the change in density must be very steep.
Note that moving the left extrapolation point even closer to the forward might fix BDK negative density, but we are already very close, and we can really wonder if going closer is really a good idea since we would effectively use a somewhat arbitrary extrapolation in most of the interpolation zone.

It turns out that we can also use a cubic spline local volatility with linear extrapolation, and the density would look then:

Density with cubic spline local volatility

Interestingly, the right side of the density is much better captured.
The wiggle persists, although it is smaller. This is likely due to the fact that I am using a cubic spline on top of the finite difference prices (in order to have a C2 density). Using a better C2 convexity preserving interpolation would likely remove this artefact.

Those figures also show why relying just on extrapolation to fix SABR is not necessarily a good idea: even a real arbitrage free extrapolation will make a not so meaningful density. The proper solution is to really use Hagan's arbitrage free SABR PDE, which would be as nearly fast in this case.

Andreasen Huge extrapolation

There are not many arbitrage free extrapolation schemes. Benaim et al. extrapolation is one of the few that claims it. However, despite the paper's title, it is not truely arbitrage free. The density might be positive, but the forward is not preserved by the implied density. It can also lead to wings that don't obey Lee's moments condition.

On a Wilmott forum, P. Caspers proposed the following counter-example based on extrapolating SABR: \( \alpha=15\%, \beta=80\%, \nu=50\%, \rho=-48\%, f=3\%, T=20.0 \). He cut this smile at 2.5% and 6% and used the BDK extrapolation scheme with mu=nu=1.

A truly arbitrage free extrapolation can be obtained through Andreasen Huge volatility interpolation, making sure the grid is wide enough to allow extrapolation. Their method is basically a one step finite difference implicit Euler scheme applied to a local volatility parameterization that has as many parameters than prices. The method is presented with piecewise constant local volatility, but actually used with piecewise linear local volatility in their example.

Smile

Density with piecewise linear local volatility

There is still a tiny oscillation that makes the density negative, but one understands why typical extrapolations fail on the example: the change in density must be very steep.
Note that moving the left extrapolation point even closer to the forward might fix BDK negative density, but we are already very close, and we can really wonder if going closer is really a good idea since we would effectively use a somewhat arbitrary extrapolation in most of the interpolation zone.

It turns out that we can also use a cubic spline local volatility with linear extrapolation, and the density would look then:

Density with cubic spline local volatility

Interestingly, the right side of the density is much better captured.
The wiggle persists, although it is smaller. This is likely due to the fact that I am using a cubic spline on top of the finite difference prices (in order to have a C2 density). Using a better C2 convexity preserving interpolation would likely remove this artefact.

Those figures also show why relying just on extrapolation to fix SABR is not necessarily a good idea: even a real arbitrage free extrapolation will make a not so meaningful density. The proper solution is to really use Hagan's arbitrage free SABR PDE, which would be as nearly fast in this case.

Unintuitive behavior of the Black-Scholes formula - negative volatilities in displaced diffusion extrapolation

I am looking at various extrapolation schemes of the implied volatilities. An interesting one I stumbled upon is due to Kahale. Even if his paper is on interpolation, there is actually a small paragraph on using the same kind of function for extrapolation. His idea is to simply lookup the standard deviation \( \Sigma \) and the forward \(f\) corresponding to a given market volatility and slope:
$$ c_{f,\Sigma} = f N(d_1) - k N(d_2)$$
with
$$ d_1 = \frac{\log(f/k)+\Sigma^2/2}{\Sigma} $$
We have simply:
$$ c'(k) = - N(d_2)$$

He also proves that we can always find those two parameters for any \( k_0 > 0,  c_0 > 0,  -1 < c_0' < 0 \)

Then I had the silly idea of trying to match with a put  instead of a call for the left wing (as those are out-of-the-money, and therefore easier to invert numerically). It turns out that it works in most cases in practice and produces relatively nice looking extrapolations, but it does not always work. This is because contrary to the call, the put value is bounded with \(f\).
$$ p_{f,\Sigma} = k N(-d_2) - f N(-d_1)$$
Inverting \(p_0'\) is going to lead to a specific \(d_2\), and you are not guaranteed that you can push \(f\) high and have \(p_{f,\Sigma}\) large enough to match \(p_0\). As example we can just take \(p_0 \geq k N(-d_2)\) which will only be matched if \(f \leq 0\).

This is slightly unintuitive as put-call parity would suggest some kind of equivalence. The problem here is that we would need to consider the function of \(k\) instead of \(f\) for it to work, so we can't really work with a put directly.

Here are the two different extrapolations on Kahale own example:

Extrapolation of the left wing with calls (blue doted line)

Extrapolation of the left wing with puts (blue doted line)

Displaced diffusion extrapolation is sometimes advocated. It is not the same as Kahale extrapolation: In Kahale, only the forward variable is varying in the Black-Scholes formula, and there is no real underlying stochastic process. In a displaced diffusion setting, we would adjust both strike and forward, keeping put-call parity at the formula level. But unfortunately, it suffers from the same kind of problem: it can not always be solved for slope and price. When it can however, it will give a more consistent extrapolation.

I find it interesting that some smiles can not be extrapolated by displaced diffusion in a C1 manner except if one allows negative volatilities in the formula (in which case we are not anymore in a pure displaced diffusion setting).

Extrapolation of the left wing using negative displaced diffusion volatilities (blue dotted line)