Equity volatility: The technicalities of a variance swap
An investment research paper (Conditional Variance Swaps, Product Note, JPMorgan Securities, April 3 2006) by analysts at JPMorgan explains how variance swaps emerged.
The researchers note that the Black-Scholes model was developed in the 1970s to price simple call and put options, and a key point of the model was that market makers could delta hedge – cancel out price movements in the underlying asset – their options positions to remove exposure to market direction.
The residual risk remaining in the portfolio was in the level of implied volatility exposure used to price the options. Naively, note the analysts, the exposure at expiry is assumed to be the difference between what was originally priced as the implied volatility and the subsequent realized volatility of the underlying over the life of the delta-hedged option.
Although this is indeed a big component of final profit and loss, delta-hedged options also show a large amount of path dependency, which means the price is not independent of the underlying. Despite this, it was noticed that the path dependency could be reduced by holding a portfolio of options. Moreover, by holding the portfolio weighted by the inverse of the square of the strikes, the path dependency could be largely eliminated, giving the portfolio a profit-and-loss profile based on the difference between (the squares of) implied volatility and the subsequent realized volatility.