# American Institute of Mathematical Sciences

June  2022, 1(2): 189-217. doi: 10.3934/fmf.2021007

## 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 Published  June 2022 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, 2022, 1 (2) : 189-217. doi: 10.3934/fmf.2021007
##### References:
 [1] B. C. Boniece, G. Didier and F. Sabzikar, On fractional Lévy processes: Tempering, sample path properties and stochastic integration, Journal of Statistical Physics, 178 (2020), 954-985.  doi: 10.1007/s10955-019-02475-1. [2] S. Boyarchenko and S. Levendorski, Option pricing for truncated Lévy processes, International Journal of Theoretical and Applied Finance, 3 (2000), 549-552. [3] M. Broadie and A. Jain, The effects of jumps and discrete sampling on volatility and variance swaps, International Journal of Theoretical and Applied Finance, 11 (2008), 761-797.  doi: 10.1142/S0219024908005032. [4] P. Carr, H. Geman, D. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332. [5] P. Carr, H. Geman, D. Madan and M. Yor, Pricing options on realized variance, Finance and Stochastics, 9 (2005), 453-475.  doi: 10.1007/s00780-005-0155-x. [6] P. Carr, H. Geman, D. B. Madan and M. Yor, Self-decomposability and option pricing, Mathematical Finance, 17 (2007), 31-57.  doi: 10.1111/j.1467-9965.2007.00293.x. [7] P. Carr and R. Lee, Robust Replication of Volatility Derivatives, Working Paper, Courant Institute of Mathematical Sciences, New York University, 2009. [8] P. Carr and R. Lee, Hedging variance options on continuous semimartingales, Finance and Stochastics, 14 (2010), 179-207.  doi: 10.1007/s00780-009-0110-3. [9] P. Carr, T. Lee and M. Lorig, Robust replication of volatility and hybrid derivatives on jump diffusions, Mathematical Finance, 31 (2021), 1394-1422.  doi: 10.1111/mafi.12327. [10] P. Carr, R. Lee and L. Wu, Variance swaps on time-changed Lévy processes, Finance and Stochastics, 16 (2012), 335-355.  doi: 10.1007/s00780-011-0157-9. [11] P. Carr and D. B. Madan, Towards a theory of volatility trading, Option Pricing, Interest Rates and Risk Management, Handb. Math. Finance, Cambridge Univ. Press, Cambridge, 2001, 458-476. doi: 10.1017/CBO9780511569708.013. [12] P. Carr and and L. Wu, A tale of two indices, Journal of Derivatives, 13 (2006), 13-29. [13] P. Carr and L. Wu, Variance Risk Premiums, Review of Financial Studies, 22 (2009), 1311-1341. [14] M. Davis, J. Obloj and V. Raval, Arbitrage Bounds for Weighted Variance Swap Prices, Mathematical Finance, 24 (2013), 821-854. [15] K. Demeterfi, E. Derman, M. Kamal and J. Zou, A guide to volatility and variance swaps, Journal of Derivatives, 4 (1999), 9-32. [16] E. Eberlein and D. B. Madan, The Distribution of Returns at Longer Horizons, Recent Advances in Financial Engineering; Proceedings of the KIER-TMU workshop, Eds. M. Kijima, C. Hara, Y. Muromachi, H. Nakaoka and K. Nishide, World Scientific, Singapore, 2010. [17] J. Gatheral, The Volatility Surface: A Practitioner's Guide, Wiley, New York, 2006. [18] H. Geman, D. Madan and M. Yor, Time changes for Lévy processes, Mathematical Finance, 11 (2001), 79-96.  doi: 10.1111/1467-9965.00108. [19] H. Geman, D. Madan and M. Yor, Stochastic volatility, jumps and hidden time changes, Finance and Stochastics, 6 (2002), 63-90.  doi: 10.1007/s780-002-8401-3. [20] A. Y. Khintchine, Limit Laws of Sums of Independent Random Variables, ONTI, Moscow, (Russian), 1938. [21] P. Lévy, Théorie de l'Addition des Variables Aléatoires, Gauthier-Villars, Paris, 1937. [22] D. B. Madan and W. Schoutens, Self similarity in long horizon asset returns, Mathematical Finance, 30 (2020), 1368-1391. [23] D. B. Madan and W. Schoutens, Arbitrage free approximations to candidate volatility surface quotations, Journal of Risk and Financial Management, 12 (2019), 69.  doi: 10.3390/jrfm12020069. [24] D. B. Madan, W. Schoutens and K. Wang, Measuring and monitoring the efficiency of markets, International Journal of Theoretical and Applied Finance, 20 (2017), 1750051, 32 pp. doi: 10.1142/S0219024917500510. [25] D. B. Madan and K. Wang, Asymmetries in financial returns, International Journal of Financial Engineering, 4 (2017), 1750045, 37 pp. doi: 10.1142/S2424786317500451. [26] D. B. Madan and K. Wang, Signed infinitely divisible, signed probability models in finance, Available at SSRN, (2020a), paper no. 3489946. [27] D. B. Madan and K. Wang, Pricing and hedging option on assets with options on related assets, Available at SSRN, (2020b), paper no. 3641658. [28] D. B. Madan and K. Wang, The structure of financial returns, Finance Research Letters, 40 (2021), 101665.  doi: 10.1016/j.frl.2020.101665. [29] D. B. Madan and K. Wang, Option implied VIX, skew and kurtosis term structure indices, Available at SSRN, (2020d), paper no. 3654563. [30] D. B. Madan and K. Wang, Pricing product options and using them to complete markets for functions of two underlying asset prices, Journal of Risk and Financial Management, 14 (2021), 355.  doi: 10.3390/jrfm14080355. [31] D. B. Madan and K. Wang, Option surface econometrics with applications, SSRN, (2021b), paper no. 3768817. [32] D. B. Madan and M. Yor, Representing the CGMY and meixner Lévy processes as time-changed Brownian motions, Journal of Computational Finance, 12 (2008), 27-47.  doi: 10.21314/JCF.2008.181. [33] A. Neuberger, The log contract, Journal of Portfolio Management, 20 (1994), 74-80. [34] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. [35] W. Zheng and Y. K. Kwok, Fourier transform algorithms for pricing and hedging discretely sampled exotic variance products and volatility derivatives under additive processes, Journal of Computational Finance, 18 (2014), 3-30.

show all references

##### References:
 [1] B. C. Boniece, G. Didier and F. Sabzikar, On fractional Lévy processes: Tempering, sample path properties and stochastic integration, Journal of Statistical Physics, 178 (2020), 954-985.  doi: 10.1007/s10955-019-02475-1. [2] S. Boyarchenko and S. Levendorski, Option pricing for truncated Lévy processes, International Journal of Theoretical and Applied Finance, 3 (2000), 549-552. [3] M. Broadie and A. Jain, The effects of jumps and discrete sampling on volatility and variance swaps, International Journal of Theoretical and Applied Finance, 11 (2008), 761-797.  doi: 10.1142/S0219024908005032. [4] P. Carr, H. Geman, D. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332. [5] P. Carr, H. Geman, D. Madan and M. Yor, Pricing options on realized variance, Finance and Stochastics, 9 (2005), 453-475.  doi: 10.1007/s00780-005-0155-x. [6] P. Carr, H. Geman, D. B. Madan and M. Yor, Self-decomposability and option pricing, Mathematical Finance, 17 (2007), 31-57.  doi: 10.1111/j.1467-9965.2007.00293.x. [7] P. Carr and R. Lee, Robust Replication of Volatility Derivatives, Working Paper, Courant Institute of Mathematical Sciences, New York University, 2009. [8] P. Carr and R. Lee, Hedging variance options on continuous semimartingales, Finance and Stochastics, 14 (2010), 179-207.  doi: 10.1007/s00780-009-0110-3. [9] P. Carr, T. Lee and M. Lorig, Robust replication of volatility and hybrid derivatives on jump diffusions, Mathematical Finance, 31 (2021), 1394-1422.  doi: 10.1111/mafi.12327. [10] P. Carr, R. Lee and L. Wu, Variance swaps on time-changed Lévy processes, Finance and Stochastics, 16 (2012), 335-355.  doi: 10.1007/s00780-011-0157-9. [11] P. Carr and D. B. Madan, Towards a theory of volatility trading, Option Pricing, Interest Rates and Risk Management, Handb. Math. Finance, Cambridge Univ. Press, Cambridge, 2001, 458-476. doi: 10.1017/CBO9780511569708.013. [12] P. Carr and and L. Wu, A tale of two indices, Journal of Derivatives, 13 (2006), 13-29. [13] P. Carr and L. Wu, Variance Risk Premiums, Review of Financial Studies, 22 (2009), 1311-1341. [14] M. Davis, J. Obloj and V. Raval, Arbitrage Bounds for Weighted Variance Swap Prices, Mathematical Finance, 24 (2013), 821-854. [15] K. Demeterfi, E. Derman, M. Kamal and J. Zou, A guide to volatility and variance swaps, Journal of Derivatives, 4 (1999), 9-32. [16] E. Eberlein and D. B. Madan, The Distribution of Returns at Longer Horizons, Recent Advances in Financial Engineering; Proceedings of the KIER-TMU workshop, Eds. M. Kijima, C. Hara, Y. Muromachi, H. Nakaoka and K. Nishide, World Scientific, Singapore, 2010. [17] J. Gatheral, The Volatility Surface: A Practitioner's Guide, Wiley, New York, 2006. [18] H. Geman, D. Madan and M. Yor, Time changes for Lévy processes, Mathematical Finance, 11 (2001), 79-96.  doi: 10.1111/1467-9965.00108. [19] H. Geman, D. Madan and M. Yor, Stochastic volatility, jumps and hidden time changes, Finance and Stochastics, 6 (2002), 63-90.  doi: 10.1007/s780-002-8401-3. [20] A. Y. Khintchine, Limit Laws of Sums of Independent Random Variables, ONTI, Moscow, (Russian), 1938. [21] P. Lévy, Théorie de l'Addition des Variables Aléatoires, Gauthier-Villars, Paris, 1937. [22] D. B. Madan and W. Schoutens, Self similarity in long horizon asset returns, Mathematical Finance, 30 (2020), 1368-1391. [23] D. B. Madan and W. Schoutens, Arbitrage free approximations to candidate volatility surface quotations, Journal of Risk and Financial Management, 12 (2019), 69.  doi: 10.3390/jrfm12020069. [24] D. B. Madan, W. Schoutens and K. Wang, Measuring and monitoring the efficiency of markets, International Journal of Theoretical and Applied Finance, 20 (2017), 1750051, 32 pp. doi: 10.1142/S0219024917500510. [25] D. B. Madan and K. Wang, Asymmetries in financial returns, International Journal of Financial Engineering, 4 (2017), 1750045, 37 pp. doi: 10.1142/S2424786317500451. [26] D. B. Madan and K. Wang, Signed infinitely divisible, signed probability models in finance, Available at SSRN, (2020a), paper no. 3489946. [27] D. B. Madan and K. Wang, Pricing and hedging option on assets with options on related assets, Available at SSRN, (2020b), paper no. 3641658. [28] D. B. Madan and K. Wang, The structure of financial returns, Finance Research Letters, 40 (2021), 101665.  doi: 10.1016/j.frl.2020.101665. [29] D. B. Madan and K. Wang, Option implied VIX, skew and kurtosis term structure indices, Available at SSRN, (2020d), paper no. 3654563. [30] D. B. Madan and K. Wang, Pricing product options and using them to complete markets for functions of two underlying asset prices, Journal of Risk and Financial Management, 14 (2021), 355.  doi: 10.3390/jrfm14080355. [31] D. B. Madan and K. Wang, Option surface econometrics with applications, SSRN, (2021b), paper no. 3768817. [32] D. B. Madan and M. Yor, Representing the CGMY and meixner Lévy processes as time-changed Brownian motions, Journal of Computational Finance, 12 (2008), 27-47.  doi: 10.21314/JCF.2008.181. [33] A. Neuberger, The log contract, Journal of Portfolio Management, 20 (1994), 74-80. [34] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. [35] W. Zheng and Y. K. Kwok, Fourier transform algorithms for pricing and hedging discretely sampled exotic variance products and volatility derivatives under additive processes, Journal of Computational Finance, 18 (2014), 3-30.
Graph of the quadratic and cubic risk neutral variations as inferred from the CBOE VIX and SKEW indices
Quadratic Variation Composition
Regression Coefficients of quadratic variation
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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