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Atlanta Fed Working Papers


Derivatives on Volatility: Some Simple Solutions Based on Observables

Steven L. Heston and Saikat Nandi
Working Paper 2000-20
November 2000

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Proposals to introduce derivatives whose payouts are explicitly linked to the volatility of an underlying asset have been around for some time. In response to these proposals, a few papers have tried to develop valuation formulae for volatility derivatives—derivatives that essentially help investors hedge the unpredictable volatility risk. This paper contributes to this nascent literature by developing closed-form/analytical formulae for prices of options and futures on volatility as well as volatility swaps. The primary contribution of this paper is that, unlike all other models, our model is empirically viable and can be easily implemented.

More specifically, our model distinguishes itself from other proposed solutions/models in the following respects: (1) Although volatility is stochastic, it is an exact function of the observed path of asset prices. This is crucial in practice because nonobservability of volatility makes it very difficult (in fact, impossible) to arrive at prices and hedge ratios of volatility derivatives in an internally consistent fashion, as it is akin to not knowing the stock price when trying to price an equity derivative. (2) The model does not require an unobserved volatility risk premium, nor is it predicated on the strong assumption of the existence of a continuum of options of all strikes and maturities as in some papers. (3) We show how it is possible to replicate (delta hedge) volatility derivatives by trading only in the underlying asset (on whose volatility the derivative exists) and a risk-free asset. This bypasses the problem of having to trade numerously many options on the underlying asset, a hedging strategy proposed in some other models.

JEL classification: G12, G13

Key words: volatility, options, hedge


The views expressed here are the authors’ and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors’ responsibility.

Please address questions regarding content to Steven L. Heston, Goldman Sachs & Company, Asset Management Division, 32 Old Slip, New York, New York 10005, 212-357-1989, 212-357-6563 (fax), steven.heston@gs.com, or Saikat Nandi, Research Department, Federal Reserve Bank of Atlanta, 104 Marietta Street, N.W., Atlanta, Georgia 30303, 404-498-7094, 404-498-8810 (fax), saikat.u.nandi@atl.frb.org.

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