Pricing path-dependent options under the Hawkes jump diffusion process

Xingchun Wang

doi:10.3934/jimo.2022024

Article Contents

2023, Volume 19, Issue 3: 1911-1930. Doi: 10.3934/jimo.2022024

This issue Previous Article Imitative innovation or independent innovation strategic choice of emerging economies in non-cooperative innovation competition Next Article Saddle-point type optimality criteria, duality and a new approach for solving nonsmooth fractional continuous-time programming problems

Pricing path-dependent options under the Hawkes jump diffusion process

Xingchun Wang

School of International Trade and Economics, University of International Business and Economics, Beijing 100029, China

Received: July 31, 2021

Revised: December 31, 2021

Early access: February 2022

Published: March 2023

This study was supported by the National Natural Science Foundation of China (No. 11701084)

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Abstract

In this paper, we investigate the pricing of a path-dependent option with default risk under the Hawkes jump diffusion process. For each asset, its dynamics are driven by a Hawkes jump diffusion process, and their diffusive components, Hawkes jumps as well as jump amplitudes are all correlated. In the proposed pricing framework, we obtain the prices of fader options with/without default risk in closed form. Finally, we present numerical examples to illustrate the prices of fader options with default risk.

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Mathematics Subject Classification: Primary: 60G51, 91B70; Secondary: 65K10.

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Figure 1. Fader option prices against strike prices. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively

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Figure 2. Fader option prices against initial levels of the intensity. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively

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Figure 3. Fader option prices against the values of $ \theta $. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively

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Figure 4. Fader option prices against the values of $ \delta $. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively

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Figure 5. Fader option prices against initial values of issuer's assets. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively

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Figure 6. Fader option prices against the levels of issuer's debt. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively

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