Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

Andrey Itkin; Dmitry Muravey

doi:10.3934/fmf.2021002

Article Contents

2022, Volume 1, Issue 1: 53-79. Doi: 10.3934/fmf.2021002 open access

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Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

Andrey Itkin^1, , and
Dmitry Muravey^2,

1.
Tandon School of Engineering, New York University, 1 Metro Tech Center, 10th floor, Brooklyn NY USA
2.
Moscow State University, Moscow, Russia

^* Corresponding author: Andrey Itkin
^* Corresponding author: Andrey Itkin

Received: December 31, 2020

Revised: March 31, 2021

Published: March 2022

Dmitry Muravey acknowledges support by the Russian Science Foundation under the Grant number 20-68-47030

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Abstract

We continue a series of papers devoted to construction of semi-analytic solutions for barrier options. These options are written on underlying following some simple one-factor diffusion model, but all the parameters of the model as well as the barriers are time-dependent. We managed to show that these solutions are systematically more efficient for pricing and calibration than, e.g., the corresponding finite-difference solvers. In this paper we extend this technique to pricing double barrier options and present two approaches to solving it: the General Integral transform method and the Heat Potential method. Our results confirm that for double barrier options these semi-analytic techniques are also more efficient than the traditional numerical methods used to solve this type of problems.

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Mathematics Subject Classification: Primary: 35Q79, 35Q62, 60G15, 60H30.

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Figure 1. Contours of integration of Eq.(18) in the complex plane $ p \in \mathbb{C} $ with poles at $ p_1^\pm, p_2^\pm, \ldots $.

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Figure 2. Prices of Call double barrier options obtained in the test by using three methods.}

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Figure 3. Absolute difference in prices of Call double barrier options obtained in the test by using three methods.

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Table 1. Parameters of the test

$ S_0 $	$ \sigma_0 $	$ \sigma_K $	$ r $	$ a_L $	$ b_L $	$ a_H $	$ b_H $,
50	0.3	0.7	0.01	20	5	71	-2

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Table 2. Comparison of double barriers Call option prices obtained by various methods

T	0.038	0.083	0.333	0.500	1.000	2.000	5.000	0.038	0.083	0.333	0.500	1.000	2.000	5.000
K	MC							GIT
45	5.0174	5.0379	5.1500	5.2212	5.4473	5.8825	7.1820	5.0173	5.0375	5.1498	5.2244	5.4478	5.8912	7.1946
50	0.1732	0.2583	0.5216	0.6370	0.8941	1.2815	2.4866	0.1735	0.2587	0.5243	0.6407	0.8984	1.2932	2.5056
55	0.0000	0.0000	0.0000	0.0000	0.0012	0.0075	0.0766	0.0000	0.0000	0.0000	0.0000	0.0012	0.0083	0.0852
K	FD							HP
45	5.0173	5.0374	5.1483	5.2218	5.4416	5.8804	7.1465	5.0173	5.0375	5.1498	5.2244	5.4478	5.8912	7.1946
50	0.1757	0.2567	0.5224	0.6391	0.8972	1.2918	2.4999	0.1735	0.2587	0.5243	0.6407	0.8984	1.2932	2.5056
55	0.0000	0.0000	0.0000	0.0001	0.0017	0.0103	0.1221	0.0000	0.0000	0.0000	0.0000	0.0012	0.0083	0.0852

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