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Pricing options on realized variance. (English) Zbl 1096.91022
The authors mention that one may view the modelling of quadratic variations of appropriate martingales as fundamental to the study of the risk neutral law for the price of a financial asset. After a detailed historical review they conclude that a realistic modelling of quadratic variations requires either the relaxation of independent increments or the relaxation of continuity of returns (or both). The paper focuses on pricing derivatives on quadratic variation by relaxing the continuity of returns, while retaining independent increments. The analysis is restricted to Lévy and Sato processes of returns. The following properties are studied: infinite activity, variation, complete monotonicity, self decomposability, and membership in the hierarchy of higher orders of decomposability for forward returns and realized variations. It is considered how one may reversely engineer a price process with a pre-specified skewness so that it is consistent with given quadratic variation properties. The Laplace transforms of quadratic variation are employed to price options on realized variance and volatility. The price of a contract paying the square root of the stochastic clock and the contract paying the volatility, as measured by the square root of realized quadratic variation, are compared.
MSC:
91B28Finance etc. (MSC2000)
60G18Self-similar processes
60G51Processes with independent increments; Lévy processes
60G52Stable processes