Variance and Volatility Derivatives in the Heston Model
Rather than gaining exposure to the market's volatility through vanilla call and put options, investors can take views on the future realized volatility directly by trading derivatives on variance and volatility. 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. At expiry the receiver of the "floating leg" pays (or owes) the difference between the realized variance (or volatility) and the agreed upon strike. At inception the strike is generally chosen such that the fair value of the swap is zero. This strike is referred to as fair variance (or fair volatility).
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.
The issuer of a variance swap is exposed to unlimited risk, if the volatility of the underlying becomes very large. Variance swaps are thus often capped, especially if the swap is written on a single stock rather than an index.
More exotic volatility derivatives are recent financial inventions that prove attractive to investors with their easily understood payoffs and increased flexibility in volatility trading and risk management. Representive products include conditional and corridor variance swaps, options on realized variance and covariance and correlation swaps. They can be used as investment tools for investors with specific views on future volatility, and to enable them to deal with risk exposures directly without taking positions in the underlying and/or delta-hedging.
Conditional and corridor variance swaps are contingent on the market level of the underlying asset: variance is only accumulated when the asset price stays within a certain range. They can be useful for investors to protect themselves against market movements. Buyers pay less than a "vanilla" variance swap to get protection, and sellers reduce their sensitivity to large moves in the asset price and volatility at the same time.
Options on realized variance or volatility can be used by investors for trading volatility while protecting themselves from large spikes. For European style options on realized variance, the payoff functions will take the general form as those of options on stocks and will depend on realized variances and prescribed strikes.
In the Heston model, arguably the most popular model of stochastic volatility, closed-form solutions exist for the prices of various contingent claims on realized variance and volatility. For example, analytical formulae can be derived for the fair variance of vanilla, capped, conditional and corridor variance swaps, the fair volatility of a volatility swap, and the fair value of European options on realized variance. In the Heston model, both the price and the variance are assumed to be stochastic. The price process resembles a geometric Brownian motion and the variance process is a mean reverting square-root process first introduced in short term interest models.
FINCAD Analytics value variance and volatility swaps in the Heston model. To find out more information about FINCAD products and services, contact a FINCAD Representative