Variance swaps on a foreign asset

There is very little information on variance swaps on a foreign asset. There can be two kinds of contracts:

• one that pays the foreign variance in a domestic currency, this is a quanto contract as the exchange rate is implicitly fixed.
• one that pays the foreign variance, multiplied by the fx rate at maturity. This is a flexo contract, and is just about buying a variance swap from a foreign bank. The price of such a contract today is very simple, just the standard variance swap price multiplied by the fx rate today (change of measure).

For quanto contracts, it's not so obvious a priori. If we consider a stochastic volatility model for the asset, the replication formula will not be applicable directly as the stochastic volatility will appear in the quanto drift correction. Furthermore, vanilla quanto option prices can not be computed simply as under Black-Scholes, a knowledge of the underlying model is necessary.

Interestingly, under the Schobel-Zhu model, it is simple to fit an analytic formula for the quanto variance swap. The standard variance swap price is:

The quanto variance swap can be priced with the same formula using a slightly different theta:

We can use it to assess the accuracy of a naive quanto option replication where we use the ATM quanto forward instead of the forward in the variance swap replication formula.

Interestingly, the  quanto forward approximation turns out to be very accurate and the correction is important. The price without correction is the price with zero correlation, and we see it can be +/-5% off in this case.

The local vol price seems a bit off, I am not sure exactly why. It could be due the discretization, the theoretical variance should be divided by (N-1) but here we divide by N where N is the number of observations. That would still lead to a skewed price but better centered around correlation 0.

It's also a bit surprising that local vol is worse than the simpler ATM quanto forward approximation: it seems that it's extracting the wrong information to do a more precise quanto correction, likely related to the shift of stochastic volatility under the domestic measure.

Variance swaps on a foreign asset

There is very little information on variance swaps on a foreign asset. There can be two kinds of contracts:

• one that pays the foreign variance in a domestic currency, this is a quanto contract as the exchange rate is implicitly fixed.
• one that pays the foreign variance, multiplied by the fx rate at maturity. This is a flexo contract, and is just about buying a variance swap from a foreign bank. The price of such a contract today is very simple, just the standard variance swap price multiplied by the fx rate today (change of measure).

For quanto contracts, it's not so obvious a priori. If we consider a stochastic volatility model for the asset, the replication formula will not be applicable directly as the stochastic volatility will appear in the quanto drift correction. Furthermore, vanilla quanto option prices can not be computed simply as under Black-Scholes, a knowledge of the underlying model is necessary.

Interestingly, under the Schobel-Zhu model, it is simple to fit an analytic formula for the quanto variance swap. The standard variance swap price is:

The quanto variance swap can be priced with the same formula using a slightly different theta:

We can use it to assess the accuracy of a naive quanto option replication where we use the ATM quanto forward instead of the forward in the variance swap replication formula.

Interestingly, the  quanto forward approximation turns out to be very accurate and the correction is important. The price without correction is the price with zero correlation, and we see it can be +/-5% off in this case.

The local vol price seems a bit off, I am not sure exactly why. It could be due the discretization, the theoretical variance should be divided by (N-1) but here we divide by N where N is the number of observations. That would still lead to a skewed price but better centered around correlation 0.

It's also a bit surprising that local vol is worse than the simpler ATM quanto forward approximation: it seems that it's extracting the wrong information to do a more precise quanto correction, likely related to the shift of stochastic volatility under the domestic measure.

Jumps impact: Variance swap vs volatility swap

Beside the problem with the discreteness of the replication, variance swaps are sensitive to jumps. This is an often mentioned reason for the collapse of the single name variance swap market in 2008 as jumps are more likely on single name equities.

Those graphs are the result of Monte-Carlo simulations with various jump sizes using the Bates model, and using Local Volatility implied from the Bates vanilla prices. The local volatility price will be the same price as per static replication for the variance swap, and we can see it they converge when there is no jump.

The presence of jumps lead to a theoretically higher variance swap price, again, which we miss completely with the static replication. As jumps go higher, the difference is more pronounced.

Volatility swaps are a bit better behaved in this regard. Interestingly, local volatility overestimate the value in this case (which for variance swaps it underestimates the value). I also noticed that the relatively recent formula from Carr-Lee will underestimate jumps even more so than local volatility: it is more precise in the absence of jumps, very close to Heston, but less precise than local volatility when jumps increase in size.

I have added a small section around this in my paper on SSRN.

Jumps impact: Variance swap vs volatility swap

Beside the problem with the discreteness of the replication, variance swaps are sensitive to jumps. This is an often mentioned reason for the collapse of the single name variance swap market in 2008 as jumps are more likely on single name equities.

Those graphs are the result of Monte-Carlo simulations with various jump sizes using the Bates model, and using Local Volatility implied from the Bates vanilla prices. The local volatility price will be the same price as per static replication for the variance swap, and we can see it they converge when there is no jump.

The presence of jumps lead to a theoretically higher variance swap price, again, which we miss completely with the static replication. As jumps go higher, the difference is more pronounced.

Volatility swaps are a bit better behaved in this regard. Interestingly, local volatility overestimate the value in this case (which for variance swaps it underestimates the value). I also noticed that the relatively recent formula from Carr-Lee will underestimate jumps even more so than local volatility: it is more precise in the absence of jumps, very close to Heston, but less precise than local volatility when jumps increase in size.

I have added a small section around this in my paper on SSRN.

Variance Swap Replication : Discrete or Continuous?

People regularly believe that Variance swaps need to be priced by discrete replication, because the market trades only a discrete set of options.

In reality, a discrete replication will misrepresent the tail, and can be quite arbitrary. It looks like the discrete replication as described in Derman Goldman Sachs paper is in everybody's mind, probably because it's easy to grasp. Strangely, it looks like most forget the section "Practical problems with replication" on p27 of his paper, where you can understand that discrete replication is not all that practical.

Reflecting on all of this, I noticed it was possible to create more accurate discrete replications easily, and that those can have vastly different hedging weights. It is a much better idea to just replicate the log payoff continuously with a decent model for interpolation and extrapolation and imply the hedge from the greeks.

I wrote a small paper around this here.

Variance Swap Replication : Discrete or Continuous?

People regularly believe that Variance swaps need to be priced by discrete replication, because the market trades only a discrete set of options.

In reality, a discrete replication will misrepresent the tail, and can be quite arbitrary. It looks like the discrete replication as described in Derman Goldman Sachs paper is in everybody's mind, probably because it's easy to grasp. Strangely, it looks like most forget the section "Practical problems with replication" on p27 of his paper, where you can understand that discrete replication is not all that practical.

Reflecting on all of this, I noticed it was possible to create more accurate discrete replications easily, and that those can have vastly different hedging weights. It is a much better idea to just replicate the log payoff continuously with a decent model for interpolation and extrapolation and imply the hedge from the greeks.

I wrote a small paper around this here.

GTK 3.0 / Gnome 3.0 annoyance

It's quite incredible that Gnome 3.0 was almost an identical mess as KDE 4.0 had been a year or two earlier. Both are much better now, more stable, but both also still have their issues, and don't feel like a real improvement over Gnome 2.0 or KDE 3.5.

Now the main file manager for Gnome 3.0, Nautilus has buttons with nearly identical icons that mean vastly different things, one is a menu, the other is a list view. Also it does not integrate with other desktops well from a look and feel perpective, here is a screenshot under XFCE (KDE would not look better).The push for window buttons inside the toolbar makes for a funny looking window. In Gnome Shell, it's not much better, plus there are some windows with a dark theme and some with a standard theme all mixed together.

On the left is Caja: an updated GTK 2.0 version of Nautilus. I find it more functional, I don't really understand the push to remove most options from the screen in Gnome 3.0. The only positive thing I can see on for the new Nautilus, is the grey color for the left side, which looks more readable and polished.

Interestingly, Nautilus within Ubuntu Unity feels better, it has a real menu and standard looking window. I suppose they customized it quite a bit.

When it comes to HiDPI support, Gnome shell is often touted has having one of the best. Well maybe for laptop screens, but certainly not for larger screens, where it just double everything and everything just looks too big. XFCE is actually decent on HiDPI screens.

GTK 3.0 / Gnome 3.0 annoyance

It's quite incredible that Gnome 3.0 was almost an identical mess as KDE 4.0 had been a year or two earlier. Both are much better now, more stable, but both also still have their issues, and don't feel like a real improvement over Gnome 2.0 or KDE 3.5.

Now the main file manager for Gnome 3.0, Nautilus has buttons with nearly identical icons that mean vastly different things, one is a menu, the other is a list view. Also it does not integrate with other desktops well from a look and feel perpective, here is a screenshot under XFCE (KDE would not look better).The push for window buttons inside the toolbar makes for a funny looking window. In Gnome Shell, it's not much better, plus there are some windows with a dark theme and some with a standard theme all mixed together.

On the left is Caja: an updated GTK 2.0 version of Nautilus. I find it more functional, I don't really understand the push to remove most options from the screen in Gnome 3.0. The only positive thing I can see on for the new Nautilus, is the grey color for the left side, which looks more readable and polished.

Interestingly, Nautilus within Ubuntu Unity feels better, it has a real menu and standard looking window. I suppose they customized it quite a bit.

When it comes to HiDPI support, Gnome shell is often touted has having one of the best. Well maybe for laptop screens, but certainly not for larger screens, where it just double everything and everything just looks too big. XFCE is actually decent on HiDPI screens.

Monte Carlo & Inverse Cumulative Normal Distribution

In most financial Monte-Carlo simulations, there is the need of generating normally distributed random numbers. One technique is to use the inverse cumulative normal distribution function on uniform random numbers. There are several different popular numerical implementations:

• Wichura AS241 (1988)
• Moro "The full Monte" (1995)
• Acklam (2004)
• Shaw breakless formula optimized for GPUs (2011)

W. Shaw has an excellent overview of the accuracy of the various methods in his paper Refinement of the normal quantile.

But what about performance? In Monte-Carlo, we could accept a slighly lower accuracy for an increase in performance.

I tested the various methods on the Euler full truncation scheme for Heston using a small timestep (0.01). Here are the results with Sobol quasi-rng:

AS241          0.9186256922511046 0.42s
MORO           0.9186256922459066 0.38s
ACKLAM         0.9186256922549364 0.40s
ACKLAM REFINED 0.9186256922511045 2.57s
SHAW-HYBRID    0.9186256922511048 0.68s

In practice, the most accurate algorithm, AS241, is of comparable speed as the newer but less accurate algorithms of MORO and ACKLAM. Acklam refinement to go to double precision (which AS241 is) kills its performance.

What about the Ziggurat on pseudo rng only? Here are the results with Mersenne-Twister-64, and using the Doornik implementation of the Ziggurat algorithm:


AS241  0.9231388565879476  0.49s
ZIGNOR 0.9321405648313437  0.44s

There is a more optimized algorithm, VIZIGNOR, also from Doornik which should be a bit faster. As expected, the accuracy is quite lower than with Sobol, and the Ziggurat looks worse. This is easily visible if one plots the implied volatilities as a function of the spot for AS241 and for ZIGNOR.

AS241 implied volatility on Mersenne-Twister

ZIGNOR implied volatility on Mersenne-Twister

Zignor is much noisier.

Note the slight bump in the scheme EULER-FT-BK that appears because the scheme, that approximates the Broadie-Kaya integrals with a trapeze (as in Andersen QE paper), does not respect martingality that well compared to the standard full truncated Euler scheme EULER-FT, and the slightly improved EULER-FT-MID where the variance integrals are approximated by a trapeze as in Van Haastrecht paper on Schobel-Zhu:

This allows to leak less correlation than the standard full truncated Euler.

Monte Carlo & Inverse Cumulative Normal Distribution

In most financial Monte-Carlo simulations, there is the need of generating normally distributed random numbers. One technique is to use the inverse cumulative normal distribution function on uniform random numbers. There are several different popular numerical implementations:

• Wichura AS241 (1988)
• Moro "The full Monte" (1995)
• Acklam (2004)
• Shaw breakless formula optimized for GPUs (2011)

W. Shaw has an excellent overview of the accuracy of the various methods in his paper Refinement of the normal quantile.

But what about performance? In Monte-Carlo, we could accept a slighly lower accuracy for an increase in performance.

I tested the various methods on the Euler full truncation scheme for Heston using a small timestep (0.01). Here are the results with Sobol quasi-rng:

AS241          0.9186256922511046 0.42s
MORO           0.9186256922459066 0.38s
ACKLAM         0.9186256922549364 0.40s
ACKLAM REFINED 0.9186256922511045 2.57s
SHAW-HYBRID    0.9186256922511048 0.68s

In practice, the most accurate algorithm, AS241, is of comparable speed as the newer but less accurate algorithms of MORO and ACKLAM. Acklam refinement to go to double precision (which AS241 is) kills its performance.

What about the Ziggurat on pseudo rng only? Here are the results with Mersenne-Twister-64, and using the Doornik implementation of the Ziggurat algorithm:


AS241  0.9231388565879476  0.49s
ZIGNOR 0.9321405648313437  0.44s

There is a more optimized algorithm, VIZIGNOR, also from Doornik which should be a bit faster. As expected, the accuracy is quite lower than with Sobol, and the Ziggurat looks worse. This is easily visible if one plots the implied volatilities as a function of the spot for AS241 and for ZIGNOR.

AS241 implied volatility on Mersenne-Twister

ZIGNOR implied volatility on Mersenne-Twister

Zignor is much noisier.

Note the slight bump in the scheme EULER-FT-BK that appears because the scheme, that approximates the Broadie-Kaya integrals with a trapeze (as in Andersen QE paper), does not respect martingality that well compared to the standard full truncated Euler scheme EULER-FT, and the slightly improved EULER-FT-MID where the variance integrals are approximated by a trapeze as in Van Haastrecht paper on Schobel-Zhu:

This allows to leak less correlation than the standard full truncated Euler.