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doi: 10.3934/fmf.2021008
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Implied price processes anchored in statistical realizations

Dilip B. Madan 1, and  King Wang 2,

1. 

Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA
2. 

Morgan Stanley, 1585 Broadway, 5th floor, New York, NY 10036, USA

Received  August 2021 Revised  December 2021 Early access April 2022

It is observed that statistical and risk neutral densities of compound Poisson processes are unconstrained relative to each other. Continuous processes are too constrained and generally not consistent with market data. Pure jump limit laws deliver operational models simultaneously consistent with both data sets with the additional imposition of no measure change on the arbitrarily small moves. The measure change density must have a finite Hellinger distance from unity linking the two worlds. Models are constructed using the bilateral gamma and the CGMY models for the risk neutral specification. They are linked to the physical process by measure change models. The resulting models simultaneously calibrate statistical tail probabilities and option prices. The resulting models have up to eight or ten parameters permitting the study of risk reward relations at a finer level. Rewards measured by power variations of the up and down moves are observed to value negatively(positively) the even(odd) variations of their own side with the converse holding for the opposite side.

Keywords: Equivalent Lévy processes, V-shaped measure change, valuation equilibria.
Mathematics Subject Classification: Primary: 60G18, 60G51, 91G20.
Citation: Dilip B. Madan, King Wang. Implied price processes anchored in statistical realizations. Frontiers of Mathematical Finance, doi: 10.3934/fmf.2021008
References:
[1]

K. Back, Martingale pricing, Annual Reviews of Financial Economics, 2 (2010), 235-250. 

[2]

G. Bakshi, D. B. Madan and G. Panayotov, Return of claims on the upside and the viability of u-shaped pricing kernels, Journal of Financial Economics, 97 (2010), 130-154. 

[3]

O. E. Barndorff-Nielsen, Processes of normal inverse Gaussian type, Finance and Stochastics, 2 (1998), 41-68. 

[4]

O. E. Barndorff-Nielsen and A. Shiryaev, Change of Time and Change of Measure, Advanced Series on Statistical Science & Applied Probability, 13. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. doi: 10.1142/7928.

[5]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.

[6]

D. Breeden and R. Litzenberger, Prices of state-contingent claims implicit in options prices, Journal of Business, 51 (1978), 621-651. 

[7]

P. Carr, H. Geman, D. B. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332. 

[8]

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.

[9]

F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Mathematische Annalen, 300 (1994), 463-520.  doi: 10.1007/BF01450498.

[10]

E. Eberlein, Application of generalized hyperbolic L évy motions to finance, Lévy Processes: Theory and Applications, (2001).

[11]

E. Eberlein and U. Keller, Hyperbolic distributions in finance, Bernoulli, 1 (1995), 281-299. 

[12]

E. Eberlein and K. Prause, The generalized hyperbolic model: Financial derivatives and risk measures, Mathematical finance–Bachelier Congress, Springer Finance, Springer, Berlin, (2002), 245–267.

[13]

J. Harrison and D. M. Kreps, Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, 20 ((1979), 381-408.  doi: 10.1016/0022-0531(79)90043-7.

[14]

J. M. Harrison and S. R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and Their Applications, 11 (1981), 215-260. 

[15]

J. C. Jackwerth, Recovering risk aversion from option prices and realized returns, Review of Financial Studies, 13 (2000), 433-467. 

[16]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Second edition, Grundlehren der Mathematischen Wissenschaften, 288. Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05265-5.

[17]

E. Jondeau, S.-H. Poon and M. Rockinger, Financial Modeling Under Non-Gaussian Distributions, Springer Finance, Springer-Verlag London, Ltd., London, 2007.

[18]

A. Y. Khintchine, Limit laws of sums of independent random variables, ONTI, Moscow, (Russian), (1938).

[19]

P. Lévy, Théorie de l'Addition des Variables Aléatoires, Gauthier-Villars, Paris, 1937.

[20]

U. Küchler and S. Tappe, Bilateral gamma distributions and processes in financial mathematics, Stochastic Processes and Their Applications, 118 (2008), 261-283.  doi: 10.1016/j.spa.2007.04.006.

[21]

D. Madan, P. Carr and E. Chang, The variance gamma process and option pricing, Review of Finance, 2 (1998), 79-105. 

[22]

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.

[23]

D. B. Madan and E. Seneta, The variance gamma (VG) model for share market returns, Journal of Business, 63 (1990), 511-524. 

[24]

D. B. Madan and K. Wang, Asymmetries in financial returns, International Journal of Financial Engineering, 4 (2017), 1750045, 37 pp. doi: 10.1142/S2424786317500451.

[25]

D. B. Madan and K. Wang, Exposure valuations and their capital requirements, SSRN, (2021), 3956745.

[26]

R. C. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.

[27]

J. Pitman and M. Yor, Infinitely divisible laws associated with hyperbolic functions, Canadian Journal of Mathematics, 55 (2003), 292-330.  doi: 10.4153/CJM-2003-014-x.

[28]

P. Protter and K. Shimbo, No arbitrage and general semimartingales, Institute of Mathematical Statistics Collections, Markov Processesand Related Topics: A Festschrift for Thomas G. Kurtz, 4 (2008), 267-283. 

[29] K. Sato, Lévy processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. 
[30]

W. Schoutens and J. L. Teugels, Lévy processes, polynomials and martingales, Communications in Statistics: Stochastic Models, 14 (1998), 335-349. 

show all references

References:
[1]

K. Back, Martingale pricing, Annual Reviews of Financial Economics, 2 (2010), 235-250. 

[2]

G. Bakshi, D. B. Madan and G. Panayotov, Return of claims on the upside and the viability of u-shaped pricing kernels, Journal of Financial Economics, 97 (2010), 130-154. 

[3]

O. E. Barndorff-Nielsen, Processes of normal inverse Gaussian type, Finance and Stochastics, 2 (1998), 41-68. 

[4]

O. E. Barndorff-Nielsen and A. Shiryaev, Change of Time and Change of Measure, Advanced Series on Statistical Science & Applied Probability, 13. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. doi: 10.1142/7928.

[5]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.

[6]

D. Breeden and R. Litzenberger, Prices of state-contingent claims implicit in options prices, Journal of Business, 51 (1978), 621-651. 

[7]

P. Carr, H. Geman, D. B. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332. 

[8]

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.

[9]

F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Mathematische Annalen, 300 (1994), 463-520.  doi: 10.1007/BF01450498.

[10]

E. Eberlein, Application of generalized hyperbolic L évy motions to finance, Lévy Processes: Theory and Applications, (2001).

[11]

E. Eberlein and U. Keller, Hyperbolic distributions in finance, Bernoulli, 1 (1995), 281-299. 

[12]

E. Eberlein and K. Prause, The generalized hyperbolic model: Financial derivatives and risk measures, Mathematical finance–Bachelier Congress, Springer Finance, Springer, Berlin, (2002), 245–267.

[13]

J. Harrison and D. M. Kreps, Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, 20 ((1979), 381-408.  doi: 10.1016/0022-0531(79)90043-7.

[14]

J. M. Harrison and S. R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and Their Applications, 11 (1981), 215-260. 

[15]

J. C. Jackwerth, Recovering risk aversion from option prices and realized returns, Review of Financial Studies, 13 (2000), 433-467. 

[16]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Second edition, Grundlehren der Mathematischen Wissenschaften, 288. Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05265-5.

[17]

E. Jondeau, S.-H. Poon and M. Rockinger, Financial Modeling Under Non-Gaussian Distributions, Springer Finance, Springer-Verlag London, Ltd., London, 2007.

[18]

A. Y. Khintchine, Limit laws of sums of independent random variables, ONTI, Moscow, (Russian), (1938).

[19]

P. Lévy, Théorie de l'Addition des Variables Aléatoires, Gauthier-Villars, Paris, 1937.

[20]

U. Küchler and S. Tappe, Bilateral gamma distributions and processes in financial mathematics, Stochastic Processes and Their Applications, 118 (2008), 261-283.  doi: 10.1016/j.spa.2007.04.006.

[21]

D. Madan, P. Carr and E. Chang, The variance gamma process and option pricing, Review of Finance, 2 (1998), 79-105. 

[22]

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.

[23]

D. B. Madan and E. Seneta, The variance gamma (VG) model for share market returns, Journal of Business, 63 (1990), 511-524. 

[24]

D. B. Madan and K. Wang, Asymmetries in financial returns, International Journal of Financial Engineering, 4 (2017), 1750045, 37 pp. doi: 10.1142/S2424786317500451.

[25]

D. B. Madan and K. Wang, Exposure valuations and their capital requirements, SSRN, (2021), 3956745.

[26]

R. C. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.

[27]

J. Pitman and M. Yor, Infinitely divisible laws associated with hyperbolic functions, Canadian Journal of Mathematics, 55 (2003), 292-330.  doi: 10.4153/CJM-2003-014-x.

[28]

P. Protter and K. Shimbo, No arbitrage and general semimartingales, Institute of Mathematical Statistics Collections, Markov Processesand Related Topics: A Festschrift for Thomas G. Kurtz, 4 (2008), 267-283. 

[29] K. Sato, Lévy processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. 
[30]

W. Schoutens and J. L. Teugels, Lévy processes, polynomials and martingales, Communications in Statistics: Stochastic Models, 14 (1998), 335-349. 

Figure 1.  Fit of Bilateral Gamma based Sato Process across maturities between a month and a quarter on SPY options as at January 2, 2010
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Figure 2.  Fit of bilateral gamma based model to the observed tail probabilities as estimated from bootstrapped data for a thousand immdeiately prior returns over a gamma distributed number of days with a mean of five days and a volatility of 3 days. The estimated number of days is 8.24
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Figure 3.  Fit of bilateral CGMY model to SPY options on April 4, 2017. The estimated number of days is 2.90
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Figure 4.  Fit of bilateral CGMY based model to tail probabilities estimated from bootstrapped return data for a gamma distributed random number of days with a mean of five days and a volatility of three days. The estimated number of days is 2.90
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Table 1.   
Distance to Unity
BG Based CGMY Based
Percentile Up Moves Down Moves Up Moves Down Moves
1 0 0.9432 0 0
5 0 1.0081 0 0
10 0 1.0840 0 0
25 0.0006 1.2174 0.9459 2.0824
50 0.0248 1.4629 1.4314 2.8091
75 0.1287 1.8979 2.2129 3.7624
90 0.2744 2.4683 2.9719 5.5502
95 0.3880 2.9356 3.8536 6.7162
99 0.8490 3.8934 40.3766 41.2440
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Distance to Unity
BG Based CGMY Based
Percentile Up Moves Down Moves Up Moves Down Moves
1 0 0.9432 0 0
5 0 1.0081 0 0
10 0 1.0840 0 0
25 0.0006 1.2174 0.9459 2.0824
50 0.0248 1.4629 1.4314 2.8091
75 0.1287 1.8979 2.2129 3.7624
90 0.2744 2.4683 2.9719 5.5502
95 0.3880 2.9356 3.8536 6.7162
99 0.8490 3.8934 40.3766 41.2440
Table 2.   
Distance to Unity Regressions BG
Upside Downside
Coeff t-stat Coeff t-stat
$ Constant $ $ -0.1251 $ $ -16,84 $ $ -0.3131 $ $ -8.53 $
$ b $ $ 119.7625 $ $ 111.89 $ $ 21.3256 $ $ 18,59 $
$ c $ $ 0.0206 $ $ 7.49 $ $ 1.2206 $ $ 51.61 $
$ a $ $ 0.0035 $ $ 0.58 $ $ 0.0072 $ $ 0.22 $
$ g $ $ -0.0018 $ $ -0.40 $ $ 0.0419 $ $ 1.27 $
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Distance to Unity Regressions BG
Upside Downside
Coeff t-stat Coeff t-stat
$ Constant $ $ -0.1251 $ $ -16,84 $ $ -0.3131 $ $ -8.53 $
$ b $ $ 119.7625 $ $ 111.89 $ $ 21.3256 $ $ 18,59 $
$ c $ $ 0.0206 $ $ 7.49 $ $ 1.2206 $ $ 51.61 $
$ a $ $ 0.0035 $ $ 0.58 $ $ 0.0072 $ $ 0.22 $
$ g $ $ -0.0018 $ $ -0.40 $ $ 0.0419 $ $ 1.27 $
Table 3.   
Distance to Unity Regressions CGMY
Upside Downside
Coeff t-stat Coeff t-stat
$ Constant $ $ -2.5279 $ $ -2.58 $ $ -5.8442 $ $ -4.39 $
$ C $ $ 152.77 $ $ 206.41 $ $ 128.29 $ $ 133.18 $
$ M/G $ $ 0 $ $ -0.11 $ $ 0 $ $ 5.89 $
$ Y $ $ 1.3876 $ $ 1.47 $ $ 6.4309 $ $ 4.77 $
$ a $ $ 0.5334 $ $ 0.98 $ $ 0.7 $ $ 1.32 $
$ g $ $ 0.3370 $ $ 0.99 $ $ -0.0985 $ $ -0.20 $
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Distance to Unity Regressions CGMY
Upside Downside
Coeff t-stat Coeff t-stat
$ Constant $ $ -2.5279 $ $ -2.58 $ $ -5.8442 $ $ -4.39 $
$ C $ $ 152.77 $ $ 206.41 $ $ 128.29 $ $ 133.18 $
$ M/G $ $ 0 $ $ -0.11 $ $ 0 $ $ 5.89 $
$ Y $ $ 1.3876 $ $ 1.47 $ $ 6.4309 $ $ 4.77 $
$ a $ $ 0.5334 $ $ 0.98 $ $ 0.7 $ $ 1.32 $
$ g $ $ 0.3370 $ $ 0.99 $ $ -0.0985 $ $ -0.20 $
Table 4.   
Risk Reward Relations BG
Dependent Variable
Explanatory Variable mp mn mp-mn mp-mn
Constant $ -0.8953 $ $ -4.8001 $ $ 3.5352 $ $ 25.6063 $
$ -4.69 $ $ -13.62 $ $ 8.06 $ $ 13.81 $
$ s_{p}/s $ $ 2.6610 $ $ -0.0049 $ $ 2.8800 $ $ -0.7228 $
$ 120.02 $ $ 0.12 $ $ 56.38 $ $ -12.72 $
$ c_{p}/c $ $ -3.6327 $ $ -0.0206 $ $ -3.9597 $ $ 3.7913 $
$ -90.60 $ $ 0.28 $ $ -42.87 $ $ 33.29 $
$ q_{p}/q $ $ 1.8544 $ $ 0.0187 $ $ 2.0101 $ $ 2.9772 $
$ 73.54 $ $ 0.40 $ $ 34.60 $ $ 46.13 $
$ s_{n} $ $ -0.0403 $ $ 4.9494 $ $ -5.0418 $
$ -2.23 $ $ 147.94 $ $ -121.01 $
$ c_{n} $ $ 0.1079 $ $ -5.5852 $ $ 5.8038 $
$ 2.76 $ $ -77.38 $ $ 64.57 $
$ q_{n} $ $ -0.0639 $ $ 2.0908 $ $ -2.2128 $
$ -3.05 $ $ 53.98 $ $ -45.88 $
RSQ $ 0.9827 $ $ 0.9975 $ $ 0.9960 $ $ 0.9327 $
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Risk Reward Relations BG
Dependent Variable
Explanatory Variable mp mn mp-mn mp-mn
Constant $ -0.8953 $ $ -4.8001 $ $ 3.5352 $ $ 25.6063 $
$ -4.69 $ $ -13.62 $ $ 8.06 $ $ 13.81 $
$ s_{p}/s $ $ 2.6610 $ $ -0.0049 $ $ 2.8800 $ $ -0.7228 $
$ 120.02 $ $ 0.12 $ $ 56.38 $ $ -12.72 $
$ c_{p}/c $ $ -3.6327 $ $ -0.0206 $ $ -3.9597 $ $ 3.7913 $
$ -90.60 $ $ 0.28 $ $ -42.87 $ $ 33.29 $
$ q_{p}/q $ $ 1.8544 $ $ 0.0187 $ $ 2.0101 $ $ 2.9772 $
$ 73.54 $ $ 0.40 $ $ 34.60 $ $ 46.13 $
$ s_{n} $ $ -0.0403 $ $ 4.9494 $ $ -5.0418 $
$ -2.23 $ $ 147.94 $ $ -121.01 $
$ c_{n} $ $ 0.1079 $ $ -5.5852 $ $ 5.8038 $
$ 2.76 $ $ -77.38 $ $ 64.57 $
$ q_{n} $ $ -0.0639 $ $ 2.0908 $ $ -2.2128 $
$ -3.05 $ $ 53.98 $ $ -45.88 $
RSQ $ 0.9827 $ $ 0.9975 $ $ 0.9960 $ $ 0.9327 $
Table 5.   
Risk Reward Relations CGMY
Dependent Variable
Explanatory Variable mp mn mp-mn mp-mn
Constant $ 30.6410 $ $ 10.3237 $ $ 5.2518 $ $ 26.5127 $
$ 5.56 $ $ 1.35 $ $ 0.69 $ $ 4.08 $
$ s_{p}/s $ $ 49.0703 $ $ 23.2615 $ $ 15.0069 $ $ 1.7536 $
$ 108.85 $ $ 37.14 $ $ 24.08 $ $ 26.56 $
$ c_{p}/c $ $ -224.1851 $ $ -128.8385 $ $ -28.0084 $ $ 10.8180 $
$ -72.01 $ $ -29.79 $ $ -6.51 $ $ 175.89 $
$ q_{p}/q $ $ 302.6475 $ $ 181.1018 $ $ 20.0436 $ $ 3.4408 $
$ 63.64 $ $ 27.41 $ $ 3.05 $ $ 19.16 $
$ f_{p}/f $ $ -126.1078 $ $ -76.1065 $ $ -5.5934 $ $ -2.1609 $
$ -60.63 $ $ -26.34 $ $ -1.95 $ $ -16.55 $
$ s_{n} $ $ -2.0138 $ $ 22.3100 $ $ -22.9098 $
$ -7.37 $ $ 58.79 $ $ -60.68 $
$ c_{n} $ $ 8.4408 $ $ -61.4901 $ $ 61.8770 $
$ 6.23 $ $ -32.67 $ $ 33.05 $
$ q_{n} $ $ -8.4234 $ $ 61.0450 $ $ -58.6393 $
$ -4.30 $ $ 22.45 $ $ -21.67 $
$ f_{n} $ $ 2.6993 $ $ -20.8879 $ $ 19.2105 $
$ 3.19 $ $ -17.75 $ $ 16.41 $
RSQ $ .9903 $ $ .9729 $ $ .7313 $ $ .7530 $
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Risk Reward Relations CGMY
Dependent Variable
Explanatory Variable mp mn mp-mn mp-mn
Constant $ 30.6410 $ $ 10.3237 $ $ 5.2518 $ $ 26.5127 $
$ 5.56 $ $ 1.35 $ $ 0.69 $ $ 4.08 $
$ s_{p}/s $ $ 49.0703 $ $ 23.2615 $ $ 15.0069 $ $ 1.7536 $
$ 108.85 $ $ 37.14 $ $ 24.08 $ $ 26.56 $
$ c_{p}/c $ $ -224.1851 $ $ -128.8385 $ $ -28.0084 $ $ 10.8180 $
$ -72.01 $ $ -29.79 $ $ -6.51 $ $ 175.89 $
$ q_{p}/q $ $ 302.6475 $ $ 181.1018 $ $ 20.0436 $ $ 3.4408 $
$ 63.64 $ $ 27.41 $ $ 3.05 $ $ 19.16 $
$ f_{p}/f $ $ -126.1078 $ $ -76.1065 $ $ -5.5934 $ $ -2.1609 $
$ -60.63 $ $ -26.34 $ $ -1.95 $ $ -16.55 $
$ s_{n} $ $ -2.0138 $ $ 22.3100 $ $ -22.9098 $
$ -7.37 $ $ 58.79 $ $ -60.68 $
$ c_{n} $ $ 8.4408 $ $ -61.4901 $ $ 61.8770 $
$ 6.23 $ $ -32.67 $ $ 33.05 $
$ q_{n} $ $ -8.4234 $ $ 61.0450 $ $ -58.6393 $
$ -4.30 $ $ 22.45 $ $ -21.67 $
$ f_{n} $ $ 2.6993 $ $ -20.8879 $ $ 19.2105 $
$ 3.19 $ $ -17.75 $ $ 16.41 $
RSQ $ .9903 $ $ .9729 $ $ .7313 $ $ .7530 $
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