Consistent Variance Curve Models

21 Pages Posted: 10 Apr 2005 Last revised: 17 Feb 2009

Date Written: February 15, 2006


We introduce a general approach to model a joint market of stock price and a term structure of variance swaps in an HJM-type framework. In such a model, strongly volatility-dependent contracts can be priced and risk-managed in terms of the observed stock and variance swap prices.

To this end, we introduce equity forward variance term-structure models and derive the respective HJM-type arbitrage conditions. We then discuss finite-dimensional Markovian representations of the fixed time-to-maturity forward variance swap curve and derive consistency results for both the standard case and for variance curves with values in a Hilbert space. For the latter, our representation also ensures non-negativity of the process.

We then give a few examples of such variance curve functionals and discuss briefly completeness and hedging in such models. As a further application, we show that the speed of mean-reversion in some standard stochastic volatility models should be kept constant when the model is recalibrated.

Keywords: Variance Swaps, Stochastic Volatility, HJM, Term Structure Models, Markov, Heston, Mean-Reversion, Fitted Heston

JEL Classification: D89

Suggested Citation

Buehler, Hans, Consistent Variance Curve Models (February 15, 2006). Finance Stochastics, Vol. 10, No. 2, 2006, Available at SSRN:

Hans Buehler (Contact Author)

JP Morgan ( email )

4/F, 25 Bank Street
London, E14 5JP
United Kingdom

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