There is an interesting course on Machine Learning on Coursera, it does not require much knowledge and yet manages to teach quite a lot.
I was struck by the fact that most techniques and ideas apply also to problems in quantitative finance.
Linear regression: used for example in the Longstaff-Schwartz approach to price Bermudan options with Monte-Carlo. Interestingly the teacher insists on feature normalization, something we can forget easily, especially with the polynomial features.
Gradient descent: one of the most basic minimizer and we use minimizers all the time for model calibration.
Regularization: in finance, this is sometimes used to smooth out the volatility surface, or can be useful to add stability in calibration. The lessons are very practical, they explain well how to find the right value of the regularization parameter.
Neural networks: calibrating a model is very much like training a neural network. The backpropagation is the same thing as the adjoint differentiation. It’s very interesting to see that it is a key feature for Neural networks, otherwise training would be much too slow and Neural networks would not be practical. Once the network is trained, it is evaluated relatively quickly forward. It’s basically the same thing as calibration and then pricing.
Support vector machines: A gaussian kernel is often used to represent the frontier. We find the same idea in the particle Monte-Carlo method.
Principal component analysis: can be applied to the covariance matrix square root in Monte-Carlo simulations, or to “compress” large baskets, as well as for portfolio risk.
It’s also interesting to hear the teacher repeating that people should not try possible improvements at random (often because they have only one idea) but analyze before what makes the most sense. And that can imply digging in the details, looking at what’s going on 100 samples.
While it sounds like a straightforward remark, I have found that people (including myself) tend to do the same mistakes in finance. We might use some quadrature, find out it does not perform that well in some cases, replace it with another one that behaves a bit better, without investigating the real issue: why does the first quadrature break? is the new quadrature really fixing the issue?
Quasi-Random numbers (like Sobol) are a relatively popular way in finance to improve the Monte-Carlo convergence compared to more classic Pseudo-Random numbers (like Mersenne-Twister). Behind the scenes one has to be a bit more careful about the dimension of the problem as the Quasi-Random numbers depends on the dimension (defined by how many random variables are independent from each other).
For a long time, Sobol was limited to 40 dimensions using the so called Bratley-Fox direction numbers (his paper actually gives the numbers for 50 dimensions). Later Lemieux gave direction numbers for up to 360 dimensions. Then, P. Jäckel proposed some extension with a random initialization of the direction vectors in his book from 2006. And finally Joe & Kuo published direction numbers for up to 21200 dimensions.
But there are very few studies about how good are real world simulations with so many quasi-random dimensions. A recent paper "Fast Ninomiya-Victoir Calibration of the Double-Mean-Reverting Model" by Bayer, Gatheral & Karlsmark tests this for once, and the results are not so pretty:
With their model, the convergence with Sobol numbers becomes worse when the number of time-steps increases, that is when the number of dimension increases. There seems to be even a threshold around 100 time steps (=300 dimensions for Euler) beyond which a much higher number of paths (2^13) is necessary to restore a proper convergence. And they use the latest and greatest Joe-Kuo direction numbers.
Still the total number of paths is not that high compared to what I am usually using (2^13 = 8192). It's an interesting aspect of their paper: the calibration with a low number of paths.
Recently, some instabilities were noticed in the Carr-Lee seasoned volatility swap price in some situations.
The Carr-Lee seasoned volatility swap price involve the computation of a double integral. The inner integral is really the problematic one as the integrand can be highly oscillating. I first found a somewhat stable behavior using a specific adaptive Gauss-Lobatto implementation (the one from Espelid) and a change of variable. But it was not very satisfying to see that the outer integral was stable only with another specific adaptive Gauss-Lobatto (the one from Gander & Gauschi, present in Quantlib). I tried various choices of adaptive (coteda, modsim, adaptsim,...) or brute force trapezoidal integration, but either they were order of magnitudes slower or unstable in some cases. Just using the same Gauss-Lobatto implementation for both would fail...
I then noticed you could write the integral as a Fourier transform as well, allowing the use of FFT. Unfortunately, while this worked, it turned out to require a very large number of points for a reasonable accuracy. This, plus the tricky part of defining the proper step size, makes the method not so practical.
I had heard before of the Filon quadrature, which I thought was more of a curiosity. The main idea is to integrate exactly x^n * cos(k*x). One then relies on a piecewise parabolic approximation of the function f to integrate f(x) * cos(k*x). Interestingly, a very similar idea has been used in the Sali quadrature method for option pricing, except one integrates exactly x^n * exp(-k*x^2).
It turned out to be remarkable on that problem, combined with a simple adaptive Simpson like method to find the right discretization. Then as if by magic, any outer integration quadrature worked.
Without knowing that it was a well known general concept, I first noticed the use of the Lamperti transform in the Andersen-Piterbarg “Interest rate modeling” book p.292 “finite difference solutions for general phi”.
Today, I found a paper referencing this paper explicitly when presenting the transformation of a stochastic process to a unit diffusion in “Density estimates for solutions to one dimensional Backward SDE’s”. In addition it also references one exercise of the Karatzas-Schreve book “Brownian motion and Stochastic calculus”, which presents again the same idea, without calling it Lamperti transform.
I stumbled upon an unexpected problem: the one touch barrier formula can break down under negative rates. While negative rates can sound fancy, they are actually quite real on some markets. Combined with relatively low volatilities, this makes the standard Black-Scholes one touch barrier formula blow up because somewhere the square root of a negative number is taken.
At first, I had the idea to just floor the number to 0. But then I needed to see if this rough approximation would be acceptable or not. So I relied on a TR-BDF2 discretization of the Black-Scholes PDE, where negative rates are not a problem.
Later, I was convinced that we ought to be able to find a closed form formula for the case of negative rates. I went back to the derivation of the formula, the book from Kwok is quite good on that. The closed form formula just stems from being the solution of an integral of the first passage time density (which is a simpler way to compute the one touch price than the PDE approach). It turns out that, then, the closed form solution to this integral with negative rates is just the same formula with complex numbers (there are actually some simplifications then).
It is a bit uncommon to use the cumulative normal distribution on complex numbers, but the error function on complex numbers is more popular: it's actually even on the wikipedia page of the error function. And it can be computed very quickly with machine precision thanks to the Faddeeva library.
With this simple closed form formula, there is no need anymore for an approximation. I wrote a small paper around this here.
Later a collegue made the remark that it could be interesting to have the bivariate complex normal distribution for the case of partial start one touch options or partial barrier option rebates (not sure if those are common). Unfortunately I could not find any code or paper for this. And after asking Prof. Genz (who found a very elegant and fast algorithm for the bivariate normal distribution), it looks like an open problem.
In this case, it turns out that the Vogt initial guess method (guess via asymptotes and minimum variance) is actually very good as long as one has a good way to lookup the asymptotes (the data is not always convex, while SVI is) and as long as rho is not close to -1, that is for long maturity affine like smiles, where SVI is actually more difficult to calibrate properly due to the over-parameterisation in those cases.
Still after looking at all of this, one has a sense that, even though it works on a wide variety of surfaces, it could break down because of the complexity: are asymptotes ok, is rho close to -1? how close? is ATM better or maximum curvature better? how do we impose boundaries on a and sigma with Levenberg-Marquardt? (truncation should not be too close to the transform, but how far?)
This is where the Quasi-Explicit method from Zeliade is very interesting: it is simpler, not necessarily to code, but the method itself. There are things to take care of (solving at each boundary), but those are mathematically well defined. The only drawback is performance, as it can be around 40 times slower. But then it’s still not that slow.
The variance under SVI becomes linear when the log-moneyness is very large in absolute terms. The lognormal SABR formula with beta=0 or beta=1 has a very different behavior. Of course, the theoretical SABR model has actually a different asymptotic behavior.
As an illustration, we calibrate SABR (with two different values of beta) and SVI against the same implied volatility slice and look at the wings behavior.
While the Lee moments formula implies that the variance should be at most linear, something that the SABR formula does not respect. It is in practice not the problem with SABR as the actual Lee boundary: V(x) < 2|x|/T (where V is the square of the implied volatility and x the log-moneyness) is attained for extremely low strikes only with SABR, except maybe for very long maturities.
A related behavior is the fact that the lognormal SABR formula can actually match steeper curvatures at the money than SVI for given asymptotes.
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, I 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.
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.
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.
