A lattice method for option evaluation with regime-switching asset correlation structure

Christoforidou Amalia; Christian-Oliver Ewald

doi:10.3934/jimo.2020042

Article Contents

2021, Volume 17, Issue 4: 1729-1752. Doi: 10.3934/jimo.2020042

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A lattice method for option evaluation with regime-switching asset correlation structure

Adam Smith Business School, University of Glasow, Glasgow G12 8QQ, UK

Received: January 2016

Revised: January 2017

Early access: March 2020

Published: July 2021

The first authors are supported by the Adam Smith Business School of the University of Glasow

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Abstract

This paper develops a lattice method for option evaluation in the presence of regime shifts in the correlation structure of assets, aiming at investigating whether the option prices reflect such shifts. We try to investigate whether option prices reflect switches in the correlation between the underlying asset of an option and risk-free rates.We develop and test two models.In the first model we allow all the parameters to follow a regime-switching process while in the second model, in order to isolate the regime-switching correlation effect on the option prices, we allow only the correlation to follow a regime-switching process. We use pentanomial lattices to represent the evolution of the regime-switching underlying assets. This is then applied in our empirical analysis, which focuses on crude oil. We use grid- and patternsearch based techniques to fit our models. Our findings suggest that prices of market traded options reflect the regime-switches and that a model which considers these switches produces significantly more accurate results than a single-regime model. We demonstrate that there is an asymmetry between parameter values obtained from historical data (backward looking) and those that are implied by traded options (for- ward looking) by employing the Kim filter to estimate our model.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Binomial tree

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Figure 2. Pentanomial tree

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Figure 3. Sensitivity of Call Option value to Transition Probabilities

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Figure 4. Sensitivity of Call Option Value to the Regime Parameters

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Figure 5. Sensitivity of option price to the regime resistance of the two regimes

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Figure 6. Sensitivity of Option Price to the regime switching correlation

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Table 1.

Jump	Probability	Underlying Asset Price
Up	$ \pi_{u} $	$ S_{u} $
Horizontal	$\pi_{0} $	$ S $
Down	$ \pi_{d} $	$ S_{d} $

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Table 2.

Event	Probability	Underlying Asset Price
Event	Probability	Asset 1	Asset 2
$ E_{1} $	$ \pi_{1} $	$ S_{p}u_{p} $	$ S_{b}u_{b} $
$ E_{2} $	$ \pi_{2} $	$ S_{p}u_{p} $	$ S_{b}d_{b} $
$ E_{3} $	$ \pi_{3} $	$ S_{p}d_{p} $	$ S_{b}d_{b} $
$ E_{4} $	$ \pi_{4} $	$ S_{p}d_{p} $	$ S_{b}u_{b} $
$ E_{5} $	$ \pi_{5} $	$ S_{p} $	$ S_{b} $

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Table 3.

Strike Price	Market Price	Predicted by Lattice	Predicted by BS
16	4.9	3.881616	3.9936
17	2.85	3.127282	3.0613
18	2.35	2.373772	2.2162
19	1.65	1.629733	1.5029
20	0.95	0.949999	0.9500
21	0.4	0.399999	0.5587
22	0.2	0.200003	0.3059
23	0.2	0.136097	0.1565
24	0.1	0.105053	0.0750
25	0.15	0.088113	0.0339
Sum of absolute differences		1.110163	2.573241

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Table 4.

Strike Price	Market Price	Predicted by Lattice	Predicted by BS
16	5	4.4153631	4.3208036
17	3.8	3.7998973	3.5464351
18	2.45	2.3996588	2.8585724
19	2.25	1.9270998	2.2636265
20	1.6	1.5997621	1.7622281
21	1.35	1.3087541	1.3499542
22	0.95	1.0205032	1.0186664
23	1.05	1.1713058	0.7580323
24	0.6	0.6000609	0.5569055
25	0.5	0.4960518	0.4043912
Sum of absolute differences		0.4890502	1.1145081

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Table 5.

Strike Price	Market Price	Predicted by Lattice	Predicted by BS
16	5	4.4153631	4.3208036
17	3.8	3.7998973	3.5464351
18	2.45	2.3996588	2.8585724
19	2.25	1.9270998	2.2636265
20	1.6	1.5997621	1.7622281
21	1.35	1.3087541	1.3499542
22	0.95	1.0205032	1.0186664
23	1.05	1.1713058	0.7580323
24	0.6	0.6000609	0.5569055
25	0.5	0.4960518	0.4043912
Sum of absolute differences		0.4890502	1.1145081

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Table 6.

Strike	Price	Predicted
16	4.9	1.625144372
17	2.85	1.269327353
18	2.35	0.940833993
19	1.65	0.626509398
20	0.95	0.420399871
21	0.4	0.245524212
22	0.2	0.158808617
23	0.2	0.079890983
24	0.1	0.045583587
25	0.15	0.023566284

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