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- Replication and Hedging /
- Variance Swaps Greeks

# Variance Swaps Greeks

In this section we consider the Greeks for variance swaps, giving information about the sensitivity of variance swaps to various market variables. We work directly by differentiating the mark-to-market value of the variance swap contract.

The value of a variance swap, per unit vega-notional, at time t is given by:

where K_{0} is the initial (fixed) variance swap strike and σ_{Expected, t} is the expected realised volatility, at time t, between trade inception and maturity. This expected volatility is then calculated in the same way as the mark-to-market P&L as a combination accrued realised variance to date and future implied variance.

Note also that intra-day there is a term representing the square of the move which will act to give the variance swap delta on an intra-day basis:

where σ_{0,t-1} is the realised variance accrued to day t-1, and K_{t,T} is the strike of a variance swap on day t expiring at T. S_{t-1} is the value of the underlying at the close on day t-1, and S_{t} is the value of the underlying at the valuation time on day t.

The Greeks of the variance swap can then be calculated by differentiating P_{t}.

**Gamma**

Gamma comes only from the exposure to realised volatility on each day:

Since the dollar gamma is achieved by scaling the gamma by the spot squared, this gives a constant dollar gamma as expected.

**Theta**

Calculating the theta of the variance swap gives:

In particular, if the variance strike doesn’t change, theta remains constant. Note, that since T is measured in days this value represents a daily theta.

Annualising gives a value for theta of:

which can then be shown to satisfy the formula taking σ^{2} as the implied variance from the variance swap strike.

__Vega__

We can calculate exposure to volatility in terms of (i) sensitivity to changes in ATM volatility or (ii) variance strike or (iii) implied variance (strike squared).

**(i) **To compute the vega in terms of sensitivity to ATM volatility, we must make some assumptions about how the variance swap strike relates to ATM volatility.

If we assume the Derman approximation: then

This approximation also allows the calculation of sensitivity to the skew or the skew squared:

--> increasing skew will increase the value of the variance swap, and do so by more, if there is more time remaining until expiry.

**(ii) / (iii)** Computing sensitivities to the strike (or strike squared) is more straightforward and needs no such assumptions about the skew surface:

These all tell us that the exposure to implied variance (or volatility) decreases with time as the accrued realised volatility is locked in to the P&L.

**Delta**

Firstly, assuming that the variance strike K has no sensitivity to the underlying, the variance swap can be seen to take on delta only intra-day:

This represents the replication of the log contract which will have to be done at the end of the day to capture that day’s realised variance.

If the variance strike itself has a dependency on the underlying (implied variance is directional) then the variance swap acquires other sources of delta, in addition to the intra-day delta.

For example, using Derman’s approximation where σ_{ATM} is the Black-Scholes ATM IV (or ATM forward volatility if rates or dividends are non-zero).

Calculating delta (only on the close to avoid the intra-day delta discussed above) gives:

We have which implies if skew is linear.

Therefore delta =

Thus with a ‘normally shaped’ negatively sloping skew, the delta of the variance swap will be negative, at least if the skew curve is sticky with strike.

This fits with the intuition that IV will tend to go down as the underlying rallies – which is exactly what a (negative) linear skew represents.

Other sensitivities to consider include:

**Jumps**

The replication argument assumes a continuous diffusion process without jumps. In reality jumps can happen, and the volatility associated with them is not guaranteed to be captured by the replication process. However, the error associated with this is of order jump cubed and is therefore small enough in practice to be ignored. Although, in the case of a single-stock where an M&A event is anticipated, the probability of a large jump increases and this may be reflected in an increase in variance swap prices.

**Stochastic Interest Rates**

Our replication argument is constructed using forwards. In reality the variance swap pays out on the volatility of the spot underlying. If rates (and dividends) are deterministic, this should make no difference to the variance swap strike. If rates are uncertain, this will act to change the relationship between spot and forward volatility, altering the variance swap strike. Except for very long-dated variance the effect on variance swap prices should be small, and in practice can offset the effect of jumps.

**Convexity**

As explained, it is really convexity and not skew which acts to increase the variance swap strike above ATM volatility.