Its payoff at expiration is equal to where: that is, the holder of a volatility swap receives
for every point by which the underlying's annualised realised volatility
for every point the realised volatility falls short of the strike.
[1] The underlying is usually a financial instrument with an active or liquid options market, such as foreign exchange, stock indices, or single stocks.
Unlike an investment in options, whose volatility exposure is contaminated by its price dependence, these swaps provide pure exposure to volatility alone.
This is truly the case only for forward starting volatility swaps.
Volatility swaps are more commonly quoted and traded than the very similar but simpler variance swaps, which can be replicated with a linear combination of options and a dynamic position in futures.
[1] That means, inevitably, a static replication (a buy-and-hold strategy) of a volatility swap is impossible.
chosen to minimise the expect expected squared deviation of the two sides: then, if the probability of negative realised volatilities is negligible, future volatilities could be assumed to be normal with mean
: then the hedging coefficients are: Definition of the annualized realized volatility depends on traders viewpoint on the underlying price observation, which could be either discretely or continuously in time.
sampling points of the observed underlying prices, says,
denotes an annualized factor which commonly selected to be the number of the observed price in a year i.e.
is the expiry date of the volatility swap defined by
The continuous version of the annualized realized volatility is defined by means of the square root of quadratic variation of the underlying price log-return: where
Once the number of price's observation increase to infinity, one can find that
In general, for a specified underlying asset, the main aim of pricing swaps is to find a fair strike price since there is no cost to enter the contract.
One of the most popular approaches to such fairness is exploiting the Martingale pricing method, which is the method to find the expected present value of given derivative security with respect to some risk-neutral probability measure (or Martingale measure).
And how such a measure is chosen depends on the model used to describe the price evolution.
Mathematically speaking, if we suppose that the price process
follows the Black-Scholes model under the martingale measure
is the volatility swap payoff at expiry in the discretely sampled case (which is switched to
due to the zero price of the swap, defining the value of a fair volatility strike.
For instance, we obtain the closed-form pricing formula once the probability distribution function of
is known, or compute it numerically by means of the Monte Carlo method.
Alternatively, Upon certain restrictions, one can utilize the value of the European options to approximate the solution.
[3] Regarding the argument of Carr and Lee (2009),[3] in the case of the continuous- sampling realized volatility if we assumes that the contract begins at time
is arbitrary (deterministic or a stochastic process) but independent of the price's movement i.e. there is no correlation between
the Black-Scholes formula for European call option written on
, then by the auxilarity of the call option chosen to be at-the-money i.e.
which is resulted from applying Taylor's series on the normal distribution parts of the Black-Scholes formula.