Consistent Variance Curve Models |
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Authors: | Hans Buehler |
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Institution: | (1) Quantitative Products Analytics, Global Markets Equity, Deutsche Bank AG London, 1 Great Winchester Street, London, EC2N 2EQ, UK;(2) Institut für Mathematik, MA 7-4, TU Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany |
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Abstract: | 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 briefly discuss 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. |
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Keywords: | Variance swaps Options on variance Market models Arbitrage-free term structure dynamics Heath– Jarrow– Morton theory Consistent parametrizations |
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