Monday, March 16, 2015

Volatility Swap vs Variance Swap Replication - Truncation

I have looked at jump effects on volatility vs. variance swaps. There is a similar behavior on tail events, that is, on truncating the replication.

One main problem with discrete replication of variance swaps is the implicit domain truncation, mainly because the variance swap equivalent log payoff is far from being linear in the wings.

The equivalent payoff with Carr-Lee for a volatility swap is much more linear in the wings (not so far of a straddle). So we could expect the replication to be less sensitive to the wings truncation.

I have done a simple test on flat 40% volatility:

As expected, the vol swap is much less sensitive, and interestingly, very much like for the jumps, it moves in the opposite direction: the truncated price is higher than the non truncated price.

Wednesday, March 11, 2015

Arbitrage free SABR with negative rates - alternative to shifted SABR

Antonov et al. present an interesting view on SABR with negative rates: instead of relying on a shifted SABR to allow negative rates up to a somewhat arbitrary shift, they modify slightly the SABR model to allow negative rates directly:
$$ dF_t = |F_t|^\beta v_t dW_F $$
with \( v_t \) being the standard lognormal volatility process of SABR.

Furthermore they derive a clever semi-analytical approximation for this model, based on low correlation, quite close to the Monte-Carlo prices in their tests. It's however not clear if it is arbitrage-free.

It turns out that it is easy to tweak Hagan SABR PDE approach to this "absolute SABR" model: one just needs to push the boundary F_min far away, and to use the absolute value in C(F).

It then reproduces the same behavior as in Antonov et al. paper:

"Absolute SABR" arbitrage free PDE

Antonov et al. graph
 I obtain a higher spike, it would look much more like Antonov graph had I used a lower resolution to compute the density: the spike would be smoothed out.

Interestingly, the arbitrage free PDE will also work for high beta (larger than 0.5):
beta = 0.75
It turns out to be then nearly the same as the absorbing SABR, even if prices can cross a little the 0. This is how the bpvols look like with beta = 0.75:
red = absolute SABR, blue = absorbing SABR with beta=0.75
They overlap when the strike is positive.

If we go back to Antonov et al. first example, the bpvols look a bit funny (very symmetric) with beta=0.1:

For beta=0.25 we also reproduce Antonov bpvol graph, but with a lower slope for the left wing:
bpvols with beta = 0.25
It's interesting to see that in this case, the positive strikes bp vols are closer to the normal Hagan analytic approximation (which is not arbitrage free) than to the absorbing PDE solution.

For longer maturities, the results start to be a bit different from Antonov, as Hagan PDE relies on a order 2 approximation only:

absolute SABR PDE with 10y maturity
The right wing is quite similar, except when it goes towards 0, it's not as flat, the left wing is much lower.

Another important aspect is to reproduce Hagan's knee, the atm vols should produce a knee like curve, as different studies show (see for example this recent Hull & White study or this other recent analysis by DeGuillaume). Using the same parameters as Hagan (beta=0, rho=0) leads to a nearly flat bpvol: no knee for the absolute SABR, curiously there is a bump at zero, possibly due to numerical difficulty with the spike in the density:
The problem is still there with beta = 0.1:

Overall, the idea of extending SABR to the full real line with the absolute value looks particularly simple, but it's not clear that it makes real financial sense.

Tuesday, February 24, 2015

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.