In a previous post, I have explored the arbitrage free wiggles in the volatility surface that P. Jaeckel found in his paper. I showed that interpolating in log prices instead of prices was enough to remove the wiggles, but then, it appears that the interpolation is not guaranteed to be arbitrage free, even though it often is. On another example from P. Jaeckel paper, that I reproduced inaccurately but well enough, it is not.
Using a Harmonic spline instead of a regular cubic spline seems to be enough to make the interpolation arbitrage free in practice (although not in theory as log-prices are not convex), but it is only C1.
One can use a tension spline instead. There is an algorithm from Pruess that updates the tension parameters so as to make the interpolation monotone and convex. It can be tweaked to ensure that the interpolation follows:
$$f’^2 + f’’ \geq 0$$
This condition makes the interpolation on the log prices arbitrage free. As a first draft experiment, I just ensured numerically that the condition was valid, the resulting algorithm was fast enough and worked reasonably well.
Here is how the normalized density looks like with a simple harmonic spline on the call prices, a cubic spline in the logarithm of call prices, and an optimized tension spline in the logarithm of call prices.
logarithm of the curvature
The blank spot for the Log curve (spline) corresponds to negative curvature, that is, arbitrage. Notice how the tension spline is fully continuous, with no real spikes.
The implied volatility curve is very close to a more classic cubic spline interpolation on the implied variance.Of course, the same kind of algorithm can be applied directly to the variance with a slightly different numerical test to ensure it is arbitrage-free, it might however be easier to find a more robust and more clever algorithm in the log prices.
I always wondered if Bachelier was really worse than Black-Scholes in practice. As an experiment I fit various implied volatility surfaces with Bachelier and Black-Scholes and look at the average error in implied volatility by slice.
In theory Bachelier is appealing because slightly simpler: log returns are a bit more challenging to think about than returns. And it also takes indirectly into account the fact that OTM calls are less likely than OTM puts because of default risk, if you assume absorbing probability at strike 0.
Bachelier
Black-Scholes
The error is in general smaller for Black-Scholes for short expiries, and higher for long expiries when compared to Bachelier. Interestingly, in theory, the difference of the models is more pronounced for longer expiries. One would have imagined that for very short expiries Bachelier would be equivalent to Black, it is not the case in term of fitting the smile.
Displaced diffusion (mixing both Bachelier and Black linearly) allows to gain x2 accuracy.
Displaced Diffusion
What about SABR? Let's look at lognormal SABR, usually used for equities.
Lognormal SABR (beta=1)
The fit is much better for long expiries, but not so great for a few outliers for long maturities, it can be actually worse than the simple displaced diffusion. A normal SABR might fix that.
Normal SABR (beta=0) using Hagan lognormal formula
If one relies on the standard lognormal formula, the beta=0 SABR behaves very badly.
Normal SABR using Hagan normal formula
This is fixed using the normal (bpvol) Hagan formula directly. The fit is then better overall for long maturities as expected from Black-Scholes vs Bachelier behavior.
If one look in log scale, the conclusion is not so obvious: beta=1 produces a better fit for a majority even for long expiries, but worse for a few (30% in my case) outliers.
What's clear however, is that one should never use the Black implied volatility Hagan formula with beta=0. This leaves a question on displaced SABR. Is displaced SABR is better suited than SABR with varying beta?
Peter Jaeckel, in a recent paper (pdf), shows that something that sounds like a reasonable arbitrage free interpolation can produce wiggles in the implied volatility slice.
The interpolation in question is using some convexity preserving spline on call and put option prices directly and in strike, assuming those input prices are arbitrage free. This is very similar to Kahale interpolation (pdf).
It seemed too crazy for me so I had to try out his example. And using a harmonic spline, it does produce arbitrage free wiggles.
Wiggles in the implied volatility
If we look at the probability density (the curvature), the Harmonic spline maintains a positive density, nearly piecewise flat in log scale, while Hyman, because it preserves only monotonicity, has some negative density which are cut out from the graph.
Probability Density
In reality, if we look at the interpolation of prices in log scale, one can see that splines won't behave as expected at first on small numbers: they will give a much higher weight to the high values, producing something like a piecewise linear interpolation.
In reality, what one really wants for such data is to just interpolate the log prices with a spline, not the prices. This is the curve named "Log" in the graphs, where a simple cubic spline is used on the log prices, and fed to exp after interpolation.
Now it sounds like a reasonable arbitrage free interpolation would be to interpolate the discrete density log linearly, in a similar spirit as Hagan-West yield curve interpolation (pdf).
In general, if you interpolate very small numbers with a spline, you probably are doing something wrong.
In an earlier post, I have been quickly exploring adjoint differentiation in the context of analytical Black-Scholes. Today, I tried to mix it in a simple Black-Scholes Monte-Carlo as described in L. Capriotti paper, and measured the performance to compute delta compared to a numerical single sided finite difference delta.I was a bit surprised that even on a single underlying, without any real optimization, adjoint delta was faster by a factor of nearly 40%. I suspect this is mostly due to exp evaluations being /2.On a basket of 4 assets, the adjoint method was 3.25x faster.It’s quick to have such results on basic payoffs: it took me a few hours and worked on the first run, even though my Monte-Carlo is slightly different from the Capriotti paper. It is much more challenging to have it working across a wide variety of payoffs, and to automatize some of it.
Several papers show that the limit for large strikes of Heston is SVI.
Interestingly, I stumbled onto a surface where the Hagan SABR fit was perfect as well as the SVI fit, while the Heston fit was not.
Originally, I knew that, on this data, the SVI fit was perfect. Until today, I just never tried to fit a lognormal SABR on the same data. I did a small test with random values of the SABR parameters alpha, rho, nu, and found out that in deed, the SVI fit is always perfect on SABR.
Is this just because the Taylor expansion of SVI will match the Taylor expansion of SABR up to the fourth order? It seems that the wings are also not too far off in general so there could be more to it.
Gatheral actually devised an SVI formula that uses SABR like variables in a talk here.
Of course, the reverse is not true. SVI has more parameters and provides in general a better fit than SABR.
With a small time expansion, it is easy to derive a reasonable initial guess, without resorting to some global minimizer.
Like Forde did for Heston, one can find the 5 Schobel-Zhu parameters through 5 points at coordinates (0,0), (x0,t1), (-x0,t1), (x0,t2), (-x0,t2), where x0 is a chosen the log-moneyness, for example, 0.1 and t1, t2 relatively short expiries (for example, 0.1, 0.25).
We can truncate the small time expansion so that the polynomial in (x,t) is fully captured by those 5 points. In practice, I have noticed that using a more refined expansion with more terms resulted not only in more complex formulas to lookup the original stochastic volatility parameters, but also in an increased error, because of the redundancy of parameters in the polynomial expansion. My previous Schobel-Zhu expansion becomes just:
In practice, I have found that the procedure works rather well.
On some more extreme surfaces, where theta=0, the error in kappa and theta is higher. Interestingly, I received a few real world surfaces like this, where theta=0, which I found a bit puzzling. I wondered if it was because those surfaces were preprocessed with SABR, that has no mean reversion, but I could not fit those exactly with SABR.
Small time expansions for Heston can be useful during the calibration of the implied volatility surface, in order to find an initial guess for a local minimizer (for example, Levenberg-Marquardt). Even if they are not so accurate, they capture the dynamic of the model parameters, and that is often enough.
I noticed, however, that on some surfaces, the Lorig expansion was quickly very inaccurate (LPP1 for order-1, LPP2 for order-2, LPP3 for order-3). Those surfaces seem to be the ones were the Feller condition is largely violated. In practice, in my set of volatility surfaces for 10 different equities/indices, the best fit is always produced by Heston parameters where the Feller condition is violated.
T=0.5, Feller condition largely violated
T=0.5, Feller condition slightly violated
Out of curiosity, I calibrated my surfaces feeding the order-1 approximation to the differential evolution, in order to find my initial guess, and it worked for all surfaces.
The order-3 formula, even though it is more precise at the money, was actually more problematic for calibration: it failed to find a good enough initial guess in some cases, maybe because the reference data should be truncated, to possibly keep the few shortest expiries, and close to ATM strikes.
The paper implied vol for any local stochastic vol model from Lorig et al. presents a very generic and simple formula to compute implied volatility expansions up to order-2 (there is actually an order-3 formula available in their Mathematica CDF file).
I tried it on the Schobel-Zhu stochastic volatility model. This model is an interesting alternative to Heston. I found that, in practice, the implied volatility surface fit was as good, while the simulation under the QE scheme is quite faster (and simpler) than Heston. Here is the result of applying their technique on Schobel-Zhu:
And this is how it behaves on some realistic input:
T=0.1
T=0.5
In practice, while not extremely good, it seems to be enough for Calibration to find an initial guess via differential evolution.
Last week, my work laptop died after spilling out some water on its keyboard inadvertently. Fortunately, its SSD was intact.
As the laptop SSD connector (SATA) and power follow the desktop computers standards, and as I use Linux, I just plugged the SSD to my home desktop and booted off the SSD. I had to change slightly the X configuration but otherwise everything worked. Linux is great for that
The same way, I just plugged the SSD to the desktop at work and it worked. Instead of carrying a laptop, I carry now my small and ultra-light SSD.
The experience is so good, that it seems to me it should be an option to everybody. Before, hard drives were very sensitive to travel, but now with SSDs, I don’t really see why we should carry a screen and a keyboard unless we use them (for example in a travel). Also I find that I actually notice the speed difference between my desktops and former laptop, which I never really paid too much attention to before.
At first I did not make use of the Brownian Bridge technique in Heston QMC, because the variance process is not simulated like a Brownian Motion under the Quadratic Exponential algorithm from Andersen.
It is, however, perfectly possible to use the Brownian Bridge on the asset process. Does it make a difference? In my small test, it does not seem to make a difference. An additional question would be, is it better to take first N for the asset and next N for the variance or vice versa or intertwined? Intertwined would seem the most natural (this is what I used without Brownian Bridge, but for simplicity I did Brownian bridge on first N).
By contrast, Schobel-Zhu QE scheme can make full use of the Brownian Bridge technique, in the asset process as well as in the variance process. Here is a summary of the volatility process under the QE scheme from van Haastrecht:
Another nice property of Schobel-Zhu is that the QE simulation is as fast as Euler, and therefore 2.5x faster than the Heston QE.
I calibrated the model to the same surface, and the QMC price of a ATM call option seems to have a similar accuracy as Heston QMC. But we can see that the Brownian Bridge does increase accuracy in this case. I was surprised that accuracy was not much better than Heston, but maybe it is because I did yet not implement the Martingale correction, while I did so in the Heston case.
