In quantitative finance, we often wish to recover the volatility of asset prices given by a noisy It\=o semimartingale. Existing estimates, however, lose accuracy when the jumps are of infinite variation, as is suggested by empirical evidence . In this paper, we show that when the efficient prices are given by an unknown time-changed L\'evy process, the rate of time change, which plays the role of the volatility, can be estimated well under arbitrary jump activity. We further show that our estimate remains valid for the volatility in the general semimartingale model, obtaining convergence rates as good as any previously implied in the literature.
In quantitative finance, we often wish to model the behaviour of asset prices given by a noisy Ito semimartingale; unfortunately, this model is too complex to identify from price data . In this paper, we therefore consider efficient prices given by a time-changed Levy process; this model is both identifiable, and replicates salient features of price data. We give a new estimate of the rate process in this model, which governs its volatility. Our estimate obtains minimax convergence rates, and is unaffected by arbitrary jump activity. Furthermore, it remains valid for the volatility in the general semimartingale model, obtaining convergence rates as good as any previously implied in the literature.
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recover the volatility
model the behaviour
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It\=o semimartingale. Existing estimates, however, lose accuracy when the jumps are of infinite variation, as is suggested by empirical evidence
Ito semimartingale; unfortunately, this model is too complex to identify from price data
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show that when the efficient prices are given by an unknown
therefore consider efficient prices given by a
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L\'evy process, the rate of time change, which plays the role of the volatility, can be estimated well under
Levy process; this model is both identifiable, and replicates salient features of price data. We give a new estimate of the rate process in this model, which governs its volatility. Our estimate obtains minimax convergence rates, and is unaffected by
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We further show that our estimate
Furthermore, it