Estimating Quadratic Covariation

We revisit robust replication theory of volatility derivatives and introduce a broader class which may be considered as the second generation of volatility derivatives. One of them is a swap contract on the quadratic covariation between an asset price and the model-free implied variance (MFIV) of the asset. It can be replicated in a model-free manner and its fair strike may be interpreted as a model-free measure for the covariance of the asset price and the realized variance. The fair strike is given in a remarkably simple form, which enable to compute it from the Black–Scholes implied volatility surface. We call it the model-free implied leverage (MFIL) and give several characterizations. In particular, we show its simple relation to the Black–Scholes implied volatility skew by an asymptotic method. Further to get an intuition, we demonstrate some explicit calculations under the Heston model. We report some empirical evidence from the time series of the MFIV and MFIL of the Nikkei stock average.

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Estimating the quadratic covariation of an asynchronously observed semimartingale with jumps

Annals of the Institute of Statistical Mathematics ◽

10.1007/s10463-014-0473-x ◽

2014 ◽

Vol 67 (4) ◽

pp. 707-743 ◽

Cited By ~ 7

Author(s):

Markus Bibinger ◽

Mathias Vetter

Keyword(s):

Quadratic Covariation

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The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025714500301 ◽

2014 ◽

Vol 17 (04) ◽

pp. 1450030 ◽

Cited By ~ 4

Author(s):

Litan Yan ◽

Junfeng Liu ◽

Chao Chen

Keyword(s):

Banach Space ◽

Brownian Motion ◽

Fractional Brownian Motion ◽

Local Time ◽

Borel Function ◽

Time Dependent ◽

Hurst Index ◽

Dependent Case ◽

Measurable Functions ◽

Quadratic Covariation

In this paper, we study the generalized quadratic covariation of f(BH) and BH defined by [Formula: see text] in probability, where f is a Borel function and BH is a fractional Brownian motion with Hurst index 0 < H < 1/2. We construct a Banach space [Formula: see text] of measurable functions such that the generalized quadratic covariation exists in L2(Ω) and the Bouleau–Yor identity takes the form [Formula: see text] provided [Formula: see text], where [Formula: see text] is the weighted local time of BH. These are also extended to the time-dependent case, and as an application we give the identity between the generalized quadratic covariation and the 4-covariation [g(BH), BH, BH, BH] when [Formula: see text].

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