## 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.

### 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.

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.

### 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.

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.

## Wednesday, July 16, 2014

### New SABR Formulae

In a talk at the Global Derivatives conference of Amsterdam (2014), Pat Hagan presented some new SABR formulas, supposedly close to the arbitrage free PDE behavior.

I tried to code those from the slides, but somehow that did not work out well on his example, I just had something very close to the good old SABR formulas. I am not 100% sure (only 99%) that it is due to a mistake in my code. Here is what I was looking to reproduce:
 Pat Hagan Global Derivatives example

Fortunately, I then found in some thesis the idea of using Andersen & Brotherton-Ratcliffe local volatility expansion. In deed, the arbitrage free PDE from Hagan is equivalent to some Dupire local volatility forward PDE (see http://papers.ssrn.com/abstract=2402001), so Hagan just gave us the local volatility expansion to expand on  (the thesis uses Doust, which is not so different in this case).

And then it produces on this global derivatives example the following:

The AB suffix are the new SABR formula. Even though the formulas are different, that looks very much like Hagan's own illustration (with a better scale)!

I have a draft paper around this and more practical ideas to calibrate SABR:
http://papers.ssrn.com/abstract=2467231

### New SABR Formulae

In a talk at the Global Derivatives conference of Amsterdam (2014), Pat Hagan presented some new SABR formulas, supposedly close to the arbitrage free PDE behavior.

I tried to code those from the slides, but somehow that did not work out well on his example, I just had something very close to the good old SABR formulas. I am not 100% sure (only 99%) that it is due to a mistake in my code. Here is what I was looking to reproduce:
 Pat Hagan Global Derivatives example

Fortunately, I then found in some thesis the idea of using Andersen & Brotherton-Ratcliffe local volatility expansion. In deed, the arbitrage free PDE from Hagan is equivalent to some Dupire local volatility forward PDE (see http://papers.ssrn.com/abstract=2402001), so Hagan just gave us the local volatility expansion to expand on  (the thesis uses Doust, which is not so different in this case).

And then it produces on this global derivatives example the following:

The AB suffix are the new SABR formula. Even though the formulas are different, that looks very much like Hagan's own illustration (with a better scale)!

I have a draft paper around this and more practical ideas to calibrate SABR:
http://papers.ssrn.com/abstract=2467231

## Thursday, July 03, 2014

### Heston or Schobel-Zhu issues with short expiries

It's relatively well known that Heston does not fit the market for short expiries. Given that there are just 5 parameters to fit a full surface, it's almost logical that one part of the surface of it is not going to fit well the market.
I was more surprised to see how bad Heston or Schobel-Zhu were to fit a single short expiry volatility slice. As an example I looked at SP500 options with 1 week expiry. It does not really matter if one forces kappa and rho to constant values (even to 0) the behavior is the same and the error in fit does not change much.

 Schobel-Zhu fit for a slice of maturity 1 week
In this graph, the brown, green and red smiles corresponds to Schobel-Zhu fit using an explicit guess (matching skew & curvature ATM), using Levenberg-Marquardt on this guess, and using plain differential evolution.
What happens is that the smiles flattens to quickly in the strike dimension. One consequence is that the implied volatility can not be computed for extreme strikes: the smile being too low, the price becomes extremely small, under machine epsilon and the numerical method (Cos) fails. There is also a bogus angle in the right wing, because of numerical error. I paid attention to ignore too small prices in the calibration by truncating the initial data.
 Heston fit, with Lord-Kahl (exact wings)

SABR behaves much better (fixing beta=1 in this case) in comparison (As I use the same truncation as for Schobel-Zhu, the flat left wing part is ignored).
 SABR fit for a slice of maturity 1 week
For longer expiries, Heston & Schobel-Zhu, even limited to 3 parameters, actually give a better fit in general than SABR.

### Heston or Schobel-Zhu issues with short expiries

It's relatively well known that Heston does not fit the market for short expiries. Given that there are just 5 parameters to fit a full surface, it's almost logical that one part of the surface of it is not going to fit well the market.
I was more surprised to see how bad Heston or Schobel-Zhu were to fit a single short expiry volatility slice. As an example I looked at SP500 options with 1 week expiry. It does not really matter if one forces kappa and rho to constant values (even to 0) the behavior is the same and the error in fit does not change much.

 Schobel-Zhu fit for a slice of maturity 1 week
In this graph, the brown, green and red smiles corresponds to Schobel-Zhu fit using an explicit guess (matching skew & curvature ATM), using Levenberg-Marquardt on this guess, and using plain differential evolution.
What happens is that the smiles flattens to quickly in the strike dimension. One consequence is that the implied volatility can not be computed for extreme strikes: the smile being too low, the price becomes extremely small, under machine epsilon and the numerical method (Cos) fails. There is also a bogus angle in the right wing, because of numerical error. I paid attention to ignore too small prices in the calibration by truncating the initial data.
 Heston fit, with Lord-Kahl (exact wings)

SABR behaves much better (fixing beta=1 in this case) in comparison (As I use the same truncation as for Schobel-Zhu, the flat left wing part is ignored).
 SABR fit for a slice of maturity 1 week
For longer expiries, Heston & Schobel-Zhu, even limited to 3 parameters, actually give a better fit in general than SABR.