Traditionally, investors gain exposure to the market's volatility through standard call and put options, derivatives that also depend on the price level of the underlying asset. By trading derivatives on variance and volatility, investors can take views on the future realized volatility directly. The simplest such instruments are variance and volatility swaps.

A **volatility swap** is a forward contract on future realized price volatility. Similarly, a** variance swap** is a forward contract on future realized price variance, variance being the square of volatility. In both cases, at inception of the trade, the strike is usually chosen such that the fair value of the swap is zero. This strike is then referred to as fair volatility or fair variance, respectively. At expiry the receiver of the floating leg pays (or owes) the difference between the realized variance (or volatility) and the agreed-upon strike, times some notional amount which is not exchanged.

Both swaps provide "pure" exposure to volatility alone, unlike vanilla options in which the volatility exposure depends on the price of the underlying asset. These swaps can thus be used to speculate on future realized volatility, to trade the spread between realized and implied volatility, or to hedge the volatility exposure of other positions.

## Technical Details

Variance swaps are theoretically simpler than volatility swaps; they can be hedged with a static position in European call and put options (with suitably chosen strikes), together with a dynamic position in the underlying asset. Volatility swaps, on the other hand can only be hedged with a dynamic portfolio of European options. But the fact that the instruments may be hedged implies that they may be valued in a model-independent manner. The price can be calculated from market-observed prices of European options of different strikes, thus the effects of the volatility smile are accounted for by construction.

Variance swaps can be replicated using a static portfolio of European vanilla options, along with an equity position. For this reason, variance swaps are more popular than volatility swaps - for which there exist only approximate static replication strategies.

The variance swap replication is accomplished using a portfolio of options with different strikes. The construction of this portfolio can be understood intuitively in the Black-Scholes model - the sensitivity of a European option to the variance of the underlying asset price depends on the asset price. This "variance vega" is largest when the underlying price is closest to the strike of the option, and is also an increasing function of the strike.

The variance vega of a portfolio of options that replicates the variance swap payoff must be independent of the underlying price. To achieve this, each option has to be weighted by the inverse of the strike squared. The variance exposure of the portfolio is largely independent of the underlying asset price, as long as the price lies within the range of option strikes. That is, as the spacing between strikes is decreased and the range of strikes in the options portfolio increases, the variance exposure becomes entirely independent of the underlying stock price. The rigorous derivation shows that such a portfolio indeed replicates the payoff of a variance swap.

## Analysis Supported

FINCAD variance and volatility swap functions can be used for the following analysis:

- calculation of fair variance for a variance swap and fair volatility for a volatility swap in a model independent manner, through replication arguments; calculation of the number of European options of each strike needed to build the replicating portfolio; calculation of risk statistics for fair variance and fair volatility respectively.
- calculation of fair value for both variance and volatility swaps, given the realized variance or volatility to date, and the fair variance or volatility for the remaining life of the swap; calculation of risk statistics for both variance and volatility swaps.

FINCAD Analytics also value variance and volatility swaps in the Heston model. To find out more information about FINCAD products and services, contact a FINCAD Representative.