Saturday, July 25, 2015

Bumping Correlations

In his book "Monte Carlo Methods in Finance", P. Jäckel explains a simple way to clean up a correlation matrix. When a given correlation matrix is not positive semi-definite, the idea is to do a singular value decomposition (SVD), replace the negative eigenvalues by 0, and renormalize the corresponding eigenvector accordingly.

One of the cited applications is "stress testing and scenario analysis for market risk" or "comparative pricing in order to ascertain the extent of correlation exposure for multi-asset derivatives", saying that "In many of these cases we end up with a matrix that is no longer positive semi-definite".

It turns out that if one bumps an invalid correlation matrix (the input), that is then cleaned up automatically, the effect can be a very different bump. Depending on how familiar you are with SVD, this could be more or less obvious from the procedure,

As a simple illustration I take the matrix representing 3 assets A, B, C with rho_ab = -0.6, rho_ac = rho_bc = -0.5.

1.00000  -0.60000  -0.50000
-0.60000   1.00000  -0.50000
-0.50000  -0.50000   1.00000

For those rho_ac and rho_bc, the correlation matrix is not positive definite unless rho_ab in in the range (-0.5, 1). One way to verify this is to use the fact that positive definiteness is equivalent to a positive determinant. The determinant will be 1 - 2*0.25 - rho_ab^2 + 2*0.25*rho_ab.

After using P. Jaeckel procedure, we end up with:

1.00000  -0.56299  -0.46745
-0.56299   1.00000  -0.46745
-0.46745  -0.46745   1.00000

If we bump now rho_bc by 1% (absolute), we end up after cleanup with:

1.00000  -0.56637  -0.47045
-0.56637   1.00000  -0.46081
-0.47045  -0.46081   1.00000

It turns out that rho_bc has changed by only 0.66% and rho_ac by -0.30%, rho_ab by -0.34%. So our initial bump (0,0,1) has been translated to a bump (-0.34, -0.30, 0.66). In other words, it does not work to compute sensitivities.

One can optimize to obtain the nearest correlation matrix in some norm. Jaeckel proposes a hypersphere decomposition based optimization, using as initial guess the SVD solution. Higham proposed a specific algorithm just for that purpose. It turns out that on this example, they will converge to the same solution (if we use the same norm). I tried out of curiosity to see if that would lead to some improvement. The first matrix becomes

1.00000  -0.56435  -0.46672
-0.56435   1.00000  -0.46672
-0.46672  -0.46672   1.00000

And the bumped one becomes

1.00000  -0.56766  -0.46984
-0.56766   1.00000  -0.46002
-0.46984  -0.46002   1.00000

We find back the same issue, rho_bc has changed by only 0.67%, rho_ac by -0.31% and rho_ab by -0.33%. We also see that the SVD correlation or the real near correlation matrix are quite close, as noticed by P. Jaeckel.

Of course, one should apply the bump directly to the cleaned up matrix, in which case it will actually work as expected, unless our bump produces another non positive definite matrix, and then we would have correlation leaking a bit everywhere. It's not entirely clear what kind of meaning the risk figures would have then.

Bumping Correlations

In his book "Monte Carlo Methods in Finance", P. Jäckel explains a simple way to clean up a correlation matrix. When a given correlation matrix is not positive semi-definite, the idea is to do a singular value decomposition (SVD), replace the negative eigenvalues by 0, and renormalize the corresponding eigenvector accordingly.

One of the cited applications is "stress testing and scenario analysis for market risk" or "comparative pricing in order to ascertain the extent of correlation exposure for multi-asset derivatives", saying that "In many of these cases we end up with a matrix that is no longer positive semi-definite".

It turns out that if one bumps an invalid correlation matrix (the input), that is then cleaned up automatically, the effect can be a very different bump. Depending on how familiar you are with SVD, this could be more or less obvious from the procedure,

As a simple illustration I take the matrix representing 3 assets A, B, C with rho_ab = -0.6, rho_ac = rho_bc = -0.5.

1.00000  -0.60000  -0.50000
-0.60000   1.00000  -0.50000
-0.50000  -0.50000   1.00000

For those rho_ac and rho_bc, the correlation matrix is not positive definite unless rho_ab in in the range (-0.5, 1). One way to verify this is to use the fact that positive definiteness is equivalent to a positive determinant. The determinant will be 1 - 2*0.25 - rho_ab^2 + 2*0.25*rho_ab.

After using P. Jaeckel procedure, we end up with:

1.00000  -0.56299  -0.46745
-0.56299   1.00000  -0.46745
-0.46745  -0.46745   1.00000

If we bump now rho_bc by 1% (absolute), we end up after cleanup with:

1.00000  -0.56637  -0.47045
-0.56637   1.00000  -0.46081
-0.47045  -0.46081   1.00000

It turns out that rho_bc has changed by only 0.66% and rho_ac by -0.30%, rho_ab by -0.34%. So our initial bump (0,0,1) has been translated to a bump (-0.34, -0.30, 0.66). In other words, it does not work to compute sensitivities.

One can optimize to obtain the nearest correlation matrix in some norm. Jaeckel proposes a hypersphere decomposition based optimization, using as initial guess the SVD solution. Higham proposed a specific algorithm just for that purpose. It turns out that on this example, they will converge to the same solution (if we use the same norm). I tried out of curiosity to see if that would lead to some improvement. The first matrix becomes

1.00000  -0.56435  -0.46672
-0.56435   1.00000  -0.46672
-0.46672  -0.46672   1.00000

And the bumped one becomes

1.00000  -0.56766  -0.46984
-0.56766   1.00000  -0.46002
-0.46984  -0.46002   1.00000

We find back the same issue, rho_bc has changed by only 0.67%, rho_ac by -0.31% and rho_ab by -0.33%. We also see that the SVD correlation or the real near correlation matrix are quite close, as noticed by P. Jaeckel.

Of course, one should apply the bump directly to the cleaned up matrix, in which case it will actually work as expected, unless our bump produces another non positive definite matrix, and then we would have correlation leaking a bit everywhere. It's not entirely clear what kind of meaning the risk figures would have then.

Monday, July 13, 2015

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.

Tuesday, July 07, 2015

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)

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)