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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.

Pricing path-dependent options under the Hawkes jump diffusion process