Pricing Rules of Thumb

It is necessary to have prices available for the entire strip of (OTM) options to calculate the true theoretical price of a VS

--> But reasonable approximations for VS prices can be made under certain assumptions about the skew.

Flat skew

Flat skew --> all strikes trade at identical IVs --> VS level = constant IV level.

Useless as flat skew --> zero vol of vol --> vol cannot change --> P&L of a vatiance swap could only be zero.

Linear skew

If skew is assumed to be linear, at least for strikes relatively close to the money, then Derman’s approximation can be used.

Derman's approximation:

- presupposes a linear put skew

- assumes call skew is flat

- calculates the VS strike as a function of 3 variables:

ATM (forward) volatility
slope of the skew
maturity of the swap

Derman s approximation

In practice, this approximation tends to work best for short-dated index variance (up to 1y).

As maturity increases --> OTM strikes have greater effect on the VS price --> contribution of skew more important.

--> inability of Derman's approximation to account for skew convexity can make it less accurate.

For single-stocks --> convexity can be more significant, even at shorter dates --> approximation can be less successful.

In general, the approximation tends to underestimate the VS price.

Example of Derman's approximation:

Index trading at 100, maturity = 0.5y, 6m forward price = 102.5.

For K = 90 --> IV = 26%

For K = 100 --> IV = 22%

For K = 102.5 --> IV = 21%

Slope of skew = (26% - 22%)/10%

K_var = 21% * (1.24)^0.5= 23.38 --> 2.4 vegas above ATM forward vol

Log-linear skew

In reality, volatility skew is not linear across all option strikes – and more accurate VS approximations can be used.

Log linear skew

Using the previous example, we can calculate the log-linear approximation:

ß = -4% / ln(0.9) = 0.38 --> Kvar = 23.55% --> slightly higher than the value using the linear Derman's approximation .

Gatheral’s formula

Gatheral expresses the VS strike as an integral (summation) of the IVs accross the entire range of strikes.

This formula characterises the skew curve in terms of the BS d₂ parameter, which measures the ‘moneyness’ of the associated OTM option.

This leads to potentially powerful methods of variance swap approximation: by fitting a quadratic, or higher order polynomial to the skew surface parameterised in terms of d₂, it is then possible to directly calculate a theoretical variance swap price from this parameterisation.

Example:

If skew curve is quadratic in variable z = d₂ : σ²(z) = σ² + αz+ βz² then theoretical VS strike is K_var= σ₀²T + βT

- in ‘d2-space’ the linear component of the skew, α, has no effect on the VS price

- base level of volatility σ₀ affects the swap strike

_-the convexity β affects the swap strike.