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Decisions in Economics and Finance

Erschienen in:

21.08.2019

From volatility smiles to the volatility of volatility

verfasst von: Bernard Dumas, Elisa Luciano

Erschienen in: Decisions in Economics and Finance | Ausgabe 2/2019

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Abstract

The paper reviews models of the option surface and reduced-form models for stochastic volatility in continuous time, under the risk-neutral measure. It defines “forward volatilities,” analogous to forward interest rates in the theory of the term structure, and provides a proof that the forward volatility is a conditional expected value, under the risk-neutral measure, of the future spot volatility. The theory developed here is the analog of Heath–Jarrow–Morton bond-pricing theory. The link is established between forward volatilities and so-called “model-free” volatility measures such as the VIX.

Vorheriger Artikel Estimation of volatility in a high-frequency setting: a short review

Nächster Artikel Markovian lifts of positive semidefinite affine Volterra-type processes

If more information arrives into a market and gets incorporated into prices, the conditional volatility is reduced when the information arrives (uncertainty is reduced at that point since consumers learn). But, as this happens day in and day out, there is more movement in price: Unconditional volatility increases.

Stochastic variance can receive a price in the financial market (over and above consumption risk) only if the utility of the representative investor (imagining there exists one) is not a time-additive von Neumann–Morgenstern utility. Volatility is a “delayed risk:” A change in volatility will only have an effect on the process after the immediate period of investment.

Note that we wrote the coefficients in the V equation in the percentage form.

See also Chapter 13 in Dumas and Luciano (2017).

No similar theory exists for American-type options.

Evidently, since in practice options are traded only for a finite number of maturities and strikes, in any specific practical circumstance, the surface available is one for a discrete grid only. Interpolation methods can be used to fill in the gaps.

The mapping is not defined for \(\mathcal {K}=0\). It needs to be extended by taking the limit \(\mathcal {K}\rightarrow 0\).

This heuristic proof was given in an unpublished note by Dumas (1995) and is a generalization of the appendix in Derman and Kani (1994). See also Carr and Madan (1998) and Britten-Jones and Neuberger (2000). A general and rigorous proof for semimartingales is to be found in Carmona and Nadtochiy (2009).

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Titel: From volatility smiles to the volatility of volatility
verfasst von: Bernard Dumas
Elisa Luciano
Publikationsdatum: 21.08.2019
Verlag: Springer International Publishing
Erschienen in: Decisions in Economics and Finance / Ausgabe 2/2019
Print ISSN: 1593-8883
Elektronische ISSN: 1129-6569
DOI: https://doi.org/10.1007/s10203-019-00263-w

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