BFS 2002

Poster Presentation

There has recently been much interest in products that provide exposure to the realised volatilities or variances of asset returns (or covariances between asset returns), while avoiding direct exposure to the underlying assets themselves. These products are attractive to investors who either wish to hedge volatility risk or who wish to take a view on future realised volatilities. We consider the pricing of this family of products and especially volatility and variance swaps. We take a stochastic volatility model as our starting point and under risk-neutral valuation we provide closed form formulae for volatility-average and variance swaps, and show how other related products can be priced. A general pricing equation is introduced for derivatives that depend on four state variables: the asset value S, the time t, the volatility , and a running average, denoted by I, which represents our knowledge to date of the average that will determine the payoff. We consider an asymptotic analysis under which we derive approximate solutions to this equation, valid when volatility is fast mean-reverting over the typical lifetime of options and other contracts. We illustrate the procedure with a mean-reverting lognormal model while others can also be considered. The analysis is simpler for strictly volatility products, in particular, having a solution for the value of the volatility swap to first order, we are able to compare with the explicit results obtained before.
      http://www.mth.kcl.ac.uk/~avraam AND http://www.maths.ox.ac.uk/mfg