Moment Methods for Exotic Volatility Derivatives

Volatility derivatives are designed to facilitate the trading of volatility, thus allowing one to directly take a range of tailored views. A basic contract is the variance swap which upon expiry pays the difference between a standard historical estimate of daily return variance and a fixed rate determined at inception, see (Dupire 1992), (Carr and Madan 1998) and (Derman et al. 1999). A variant on this is the corridor variance swap which differs from the standard variance swap only in that the underlyings price must be inside a specified corridor in order for its squared return to be included in the floating part of the variance swap payout (Carr and Lewis 2004). A further generalization is the conditional variance swap which pays the realized variance of an asset again within some corridor, whereby the average is taken only over the period when the spot is in the range. The advantage of conditional corridor variance swaps is that they allow one to take a view on volatility that is contingent upon the price level, gaining exposure to volatility only where required. Although conditional variance swaps appear to be traded, see (J.P. Morgan Securities 2006), this is the first article in the open literature proposing a pricing methodology.

The numerical method we present extends without difficulties to other exotic volatility contracts such as variance swaptions, see also (Carr and Lee 2007). We also take the liberty of inventing exotic volatility derivatives such as variance knockout options. These do not seem to be much traded although perhaps they should be. A variance knockout can be regarded as a variation on the theme of barrier knockout options whereby the knock-out condition is not triggered by the underlying Date: February 13, 2013.

crossing a certain level. Instead, a variance knockout vanishes in case realized variance prior to maturity exceeds a certain pre-assigned threshold. The benefit of a variance knockout over a barrier knockout is that its hedge ratios are smoother.

In this paper, we use operator methods for pricing. Our presentation is selfcontained for the specific purpose at hand but see the review article (Albanese 2006) for a more extended discussion of operator methods. One point we should stress is that the mathematical and numerical methods we present would work as efficiently no matter what underlying model for the stock price process is chosen. We select one for the purpose of discussing a concrete case and generating sample graphs, but we are confident the reader can do better if she intends to refine it. Our mathematical and numerical methods are model agnostic as they do not rely on closed form solvability and their performance is not linked to the model definition. Any model for the stock price dynamics would work just as well, the only limitation being that market models cannot be accommodated within the formalism we propose.

We believe that it is important to embed econometric estimates in the volatility process for the underlying stock price and thus make it to reproduce realistically the features of the historical process. Operator methods reviewed in (Albanese 2006) are useful in this respect as they allow one to construct models semi-parametrically or even non-parametrically while resting assured that numerical efficiency is not affected by model choice. Our working example is a 3-factor equity model which encompasses stochastic outlook dynamics, stochastic volatility, local volatility and jumps of both finite and infinite activity. This process can formally be expressed as follows:

(1) dS t = µ t (S t )dt + σ t (S t )dW t + jumps.

Here both the drift and the volatility terms can be freely specified, thus making available additional degrees of freedom and allowing for a time-homogeneous calibration.

Armed with the calibrated model we proceed to a moment method which allows us to obtain conditional moments of integrals of stochastic processes. Having the first few moments as a starting point, the distribution is extrapolated by matching moments with elementary probability distributions functions known in analytically closed form. On this basis we then price volatility derivatives and find their hedge ratios.

The article is organized as follows: The next section describes the underlying model, and the following one the moment method. Sections (4), ( 5) and (6) describe the variance swaps, corridor variance swaps and conditional variance swaps respectively, and their pricing within this framework. In section ( 7) we look at some more variance related contracts, particularly the gamma swap and variance knock-out options. A final section concludes.

Our working example, as a base model, is specified semi-parametrically and amounts to a 3-factor equity model with stochastic volatility and stochastic outlook dynamics. We find that postulating a slowly varying stochastic outlook dynamics alongside a faster mean-reverting volatility dynamics is essential to mimic the histor

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