Friday, August 01, 2014

SVI and long maturities issues

On long maturities equity options, the smile is usually very much like a skew:  very little curvature. This usually means that the SVI rho will be very close to -1, in a similar fashion as what can happen for the the correlation parameter of a real stochastic volatility model (Heston, SABR).

In terms of initial guess, we looked at the more usual use cases and showed that matching a parabola at the minimum variance point often leads to a decent initial guess if one has an ok estimate of the wings. We will see here that we can do also something a bit better than just a flat slice at-the-money in the case where rho is close to -1.

In general when the asymptotes lead to rho < -1, it means that we can't compute b from the asymptotes as there is in reality only one usable asymptote, the other one having a slope of 0 (rho=-1). The right way is to just recompute b by matching the ATM slope (which can be estimated by fitting a parabola at the money). Then we can try to match the ATM curvature, there are two possibilities to simplify the problem: s >> m or m >> s.

Interestingly, there is some kind of discontinuity at m = 0:
  • when m = 0, the at-the-money slope is just b*rho. 
  • when m != 0 and m >> s, the at-the-money slope is b*(rho-1). 
In general it is therefore a bad idea to use m=0 in the initial guess. It appears then that assuming m >> s is better. However, in practice, with this choice, the curvature at the money is matched for a tiny m, even though actually the curvature explodes (sigma=5e-4) at m (so very close to the money). This produces this kind of graph:
This apparently simple issue is actually a core problem with SVI. Looking back at our slopes but this time in the moneyness coordinate, the slope at m is b*rho while the slope at the money is b*(rho-1) if m != 0. If s is small, as the curvature at m is just b/s this means that our there will always be this funny shape if s is small. It seems then that the best we can do is hide it: let m>max(moneyness) and compute the sigma to match the ATM curvature. This leads to the following:
This is all good so far. Unfortunately running a minimizer on it will lead to a solution with a small s. And the bigger picture looks like this (QE is Zeliade Quasi-Explicit, Levenberg-Marquardt would give the same result):
Of course a simple fix is to not let s to be too small, but how do we defined what is too small? I have found that a simple rule is too always ensure that s is increasing with the maturity supposing that we have to fit a surface. This rule has also a very nice side effect that spurious arbitrages will tend to disappear as well. On the figure above, I can bet that there is a big arbitrage at k=m for the QE result.

Thursday, July 31, 2014

More SVI Initial Guesses

In the previous post, I showed one could extract the SVI parameters from a best fit parabola at-the-money. It seemed to work reasonably well, but I found some real market data where it can be much less satisfying.

Sometimes (actually not so rarely) the ATM slope and curvatures can't be matched given rho and b found through the asymptotes. As a result if I force to just match the curvature and set m=0 (when the slope can't be matched), the simple ATM parabolic guess looks shifted. It can be much worse than this specific example.

It is then a bit clearer why Vogt looked to match the lowest variance instead of ATM. We can actually also fit a parabola at the lowest variance (MV suffix in the graph) instead of ATM. It seems to fit generally better.

I also tried to estimate the asymptotic slopes more precisely (using the slope of the 5-points parabola at each end), but it seems to not always be an improvement.

However this does not work when rho is close to -1 or 1 as there is then no minimum. Often, matching ATM also does not work when rho is -1 or 1. This specific case, but quite common as well for longer expiries in equities need more thoughts, usually a constant slice is ok, but this is clearly where Zeliade's quasi explicit method shines.

So far it still all looks good, but then looking at medium maturities (1 year), sometimes all initial guesses don't look comforting (although Levenberg-Marquardt minimization still works on those - but one can easily imagine that it can break as well, for example by tweaking slightly the rho/b and look at what happens then).

There is lots of data on this 1 year example. One can clearly see the problem when the slope can not be fitted ATM (SimpleParabolicATM-guess), and even if by chance when it can (TripleParabolicATM-guess), it's not so great.
Similarly fitting the lowest variance leads only to a good fit of the right wing and a bad fit everywhere else.

Still, as if by miracle, everything converges to the best fit on this example (again one can find cases where some guesses don't converge to the best fit). I have added some weights +-20% around the money, to ensure that we capture the ATM behavior accurately (otherwise the best fit is funny).

It is interesting to see that if one minimizes the min square sum of variances (what I do in Vogt-LM, it's in theory slightly faster as there is no sqrt function cost) it results in an ugly looking steeper curvature, while if we just minimize the min square sum of volatilities (what I do in SimpleParabolicMV_LM), it looks better.

Tuesday, July 29, 2014

Another SVI Initial Guess

The SVI formula is:
$$w(k) = a + b ( \rho (k-m) + \sqrt{(k-m)^2+ \sigma^2}$$
where k is the log-moneyness, w(k) the implied variance at a given moneyness and a,b,rho,m,sigma the 5 SVI parameters.

A. Vogt described a particularly simple way to find an initial guess to fit SVI to an implied volatility slice a while ago. The idea to compute rho and sigma from the left and right asymptotic slopes. a,m are recovered from the crossing point of the asymptotes and sigma using the minimum variance.

Later, Zeliade has shown a very nice reduction of the problem to 2 variables, while the remaining 3 can be deduced explicitly. The practical side is that constraints are automatically included, the less practical side is the choice of minimizer for the two variables (Nelder-Mead) and of initial guess (a few random points).

Instead, a simple alternative is the following: given b and rho from the asymptotic slopes, one could also just fit a parabola at-the-money, in a similar spirit as the explicit SABR calibration, and recover explicitly a, m and sigma.

To illustrate I take the data from Zeliade, where the input is already some SVI fit to market data.
3M expiry - Zeliade data

4Y expiry, Zeliade data
We clearly see that ATM the fit is better for the parabolic initial guess than for Vogt, but as one goes further away from ATM, Vogt guess seems better.

Compared to SABR, the parabola itself fits decently only very close to ATM. If one computes the higher order Taylor expansion of SVI around k=0, powers of (k/sigma) appear, while sigma is often relatively small especially for short expiries: the fourth derivative will quickly make a difference.

On implied volatilities stemming from a SABR fit of the SP500, here is how the various methods behave:
1M expiry on SABR data
4Y expiry on SABR data
As expected, because SABR (and thus the input implied vol) is much closer to a parabola, the parabolic initial guess is much better than Vogt. The initial guess of Vogt is particularly bad on long expiries, although it will still converge quite quickly to the true minimum with Levenberg-Marquardt.

In practice, I have found the method of Zeliade to be very robust, even if a bit slower than Vogt, while Vogt can sometimes (rarely) be too sensitive to the estimate of the asymptotes.

The parabolic guess method could also be applied to always fit exactly ATM vol, slope and curvature, and calibrate rho, b to gives the best overall fit. It might be an idea for the next blog post.