\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Quadratic variation, models, applications and lessons

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

Abstract Full Text(HTML) Figure(1) / Table(13) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: 60G18, 60G51, 91G20.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Graph of the quadratic and cubic risk neutral variations as inferred from the CBOE VIX and SKEW indices

    Table 1.  Quadratic Variation Composition

     | Show Table
    DownLoad: CSV

    Table 2.  Regression Coefficients of quadratic variation

     | Show Table
    DownLoad: CSV

    Table 3.   

     | Show Table
    DownLoad: CSV

    Table 4.  Volatility of Variance

    quartilebcmycgmybcmyssdcgmyssd
    1336320358353
    2356341380374
    3370353401387
     | Show Table
    DownLoad: CSV

    Table 5.  Strike Multiple

    quartilebcmycgmybcmyssdcgmyssd
    13.442.963.943.73
    23.913.54.554.26
    34.273.795.274.74
     | Show Table
    DownLoad: CSV

    Table 7.   

    NumberVariable
    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$vm_{t}$
     | Show Table
    DownLoad: CSV

    Table 8.  Unit Multiple

     | Show Table
    DownLoad: CSV

    Table 9.  Optimized Multiple

     | Show Table
    DownLoad: CSV

    Table 10.   

     | Show Table
    DownLoad: CSV

    Table 11.   

     | Show Table
    DownLoad: CSV

    Table 12.   

     | Show Table
    DownLoad: CSV

    Table 13.   

     | Show Table
    DownLoad: CSV

    Table 14.  Quadratic on Cubic Variation Regression Result

    ConstantCubic variation Coefficient
    Coefficient$0.0131$$-0.8715$
    t-statistic$(63.88)$$(-296.06)$
    RSQ$97.28\%$
     | Show Table
    DownLoad: CSV
  • [1] B. C. BonieceG. 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. CarrH. GemanD. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332. 
    [5] P. CarrH. GemanD. 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. CarrH. GemanD. 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. CarrT. 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. CarrR. 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. DavisJ. Obloj and V. Raval, Arbitrage Bounds for Weighted Variance Swap Prices, Mathematical Finance, 24 (2013), 821-854. 
    [15] K. DemeterfiE. DermanM. 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. GemanD. Madan and M. Yor, Time changes for Lévy processes, Mathematical Finance, 11 (2001), 79-96.  doi: 10.1111/1467-9965.00108.
    [19] H. GemanD. 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. SatoLévy 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. 
  • 加载中
Open Access Under a Creative Commons license

Figures(1)

Tables(13)

SHARE

Article Metrics

HTML views(398) PDF downloads(200) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return