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Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing
March 2022, 1(1): 53-79. doi: 10.3934/fmf.2021002
Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit
Andrey Itkin 1,, and Dmitry Muravey 2,
Tandon School of Engineering, New York University, 1 Metro Tech Center, 10th floor, Brooklyn NY USA
Moscow State University, Moscow, Russia
* Corresponding author: Andrey Itkin
Received January 2021 Revised April 2021 Published March 2022 Early access May 2021
Fund Project: Dmitry Muravey acknowledges support by the Russian Science Foundation under the Grant number 20-68-47030
Figure(3) / Table(2)
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.
Keywords: Double barrier options, time-dependent boundaries, time-dependent rebates-at-hit, semi-analytical solution, one-factor models.
Mathematics Subject Classification: Primary: 35Q79, 35Q62, 60G15, 60H30.
Citation: Andrey Itkin, Dmitry Muravey. Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit. Frontiers of Mathematical Finance, 2022, 1 (1) : 53-79. doi: 10.3934/fmf.2021002
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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 $.
Figure Options
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Figure 2. Prices of Call double barrier options obtained in the test by using three methods.}
Figure 3. Absolute difference in prices of Call double barrier options obtained in the test by using three methods.
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
K FD HP
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Andrey Itkin Dmitry Muravey