Implied price processes anchored in statistical realizations

Dilip B. Madan; King Wang

doi:10.3934/fmf.2021008

Article Contents

2022, Volume 1, Issue 3: 321-342. Doi: 10.3934/fmf.2021008 open access

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$ G $

-expectation approach to stochastic ordering

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

Published: September 2022

Abstract / Introduction Full Text(HTML) Figure(4) / Table(5) Related Papers Cited by

Abstract

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:

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

Citation:

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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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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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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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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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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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