# American Institute of Mathematical Sciences

doi: 10.3934/fmf.2021007
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## Quadratic variation, models, applications and lessons

 1 Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA 2 Derivative Product Strats, Morgan Stanley, 1585 Broadway, 5th floor, New York, NY 10036, USA

This paper is the private opinion of the authors and does not necessarily reflect the policy and views of Morgan Stanley. We also thank the reviewers for their constructive comments. Any remaining errors are our responsibility

Received  December 2020 Revised  August 2021 Early access November 2021

Time changes of Brownian motion impose restrictive jump structures in the motion of asset prices. Quadratic variations also depart from time changes. Quadratic variation options are observed to have a nonlinear exposure to risk neutral skewness. Joint Laplace Fourier transforms for quadratic variation and the stock are developed. They are used to study the multiple of the cap strike over the variance swap quote attaining a given percentage price reduction for the capped variance swap. Market prices for out-of-the-money options on variance are observed to be above those delivered by the calibrated models. Bootstrapped data and simulated paths spaces are used to study the multiple of the dynamic hedge return desired by a quadratic variation contract. It is observed that the optimized hedge multiple in the bootstrapped data is near unity. Furthermore, more generally, it is exposures to negative cubic variations in path spaces that raise variance swap prices, lower hedge multiples, increase residual risk charges and gaps to the log contract hedge. A case can be made for both, the physical process being closer to a continuous time change of Brownian motion, while simultaneously risk neutrally this may not be the case. It is recognized that in the context of discrete time there are no issues related to equivalence of probabilities.

Citation: Dilip B. Madan, King Wang. Quadratic variation, models, applications and lessons. Frontiers of Mathematical Finance, doi: 10.3934/fmf.2021007
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##### References:
Graph of the quadratic and cubic risk neutral variations as inferred from the CBOE VIX and SKEW indices
Volatility of Variance
 quartile bcmy cgmy bcmyssd cgmyssd 1 336 320 358 353 2 356 341 380 374 3 370 353 401 387
 quartile bcmy cgmy bcmyssd cgmyssd 1 336 320 358 353 2 356 341 380 374 3 370 353 401 387
Strike Multiple
 quartile bcmy cgmy bcmyssd cgmyssd 1 3.44 2.96 3.94 3.73 2 3.91 3.5 4.55 4.26 3 4.27 3.79 5.27 4.74
 quartile bcmy cgmy bcmyssd cgmyssd 1 3.44 2.96 3.94 3.73 2 3.91 3.5 4.55 4.26 3 4.27 3.79 5.27 4.74
 Number Variable 1 $vswap_{t}$ 2 $lcs_{t}$ 3 $gap_{t}$ 4 $rc_{t}$ 5 $vswapm_{t}$ 6 $lcsm_{t}$ 7 $gapm_{t}$ 8 $rcm_{t}$ 9 $h_{t}$ 10 $\rho _{t}$ 11 $qv_{t}$ 12 $rcs_{t}$ 13 $vm_{t}$
 Number Variable 1 $vswap_{t}$ 2 $lcs_{t}$ 3 $gap_{t}$ 4 $rc_{t}$ 5 $vswapm_{t}$ 6 $lcsm_{t}$ 7 $gapm_{t}$ 8 $rcm_{t}$ 9 $h_{t}$ 10 $\rho _{t}$ 11 $qv_{t}$ 12 $rcs_{t}$ 13 $vm_{t}$
Unit Multiple
Optimized Multiple
Quadratic on Cubic Variation Regression Result
 Constant Cubic variation Coefficient Coefficient $0.0131$ $-0.8715$ t-statistic $(63.88)$ $(-296.06)$ RSQ $97.28\%$
 Constant Cubic variation Coefficient Coefficient $0.0131$ $-0.8715$ t-statistic $(63.88)$ $(-296.06)$ RSQ $97.28\%$
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