{'fraction_non_alphanumeric': 0.01594746716697936, 'fraction_numerical': 0.0, 'mean_word_length': 5.0625, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Option Pricing in Jump Diffusions: A New Integral-Based Framework for European and American Options

This paper deals with the valuation of European and American put options in jump diffusion models. A new integral transform framework for solving the partial integro-differential equation (PIDE) inherent in pricing problems is proposed. In the case of European options the solution is a single integral expression independent of the distribution of the jump size. We also derive analytical expressions for the Greeks. In the case of American options, we shall characterize the general solution of the inhomogeneous PIDE. Due to the discontinuous nature of the paths of the underlying asset, the structure of the solution significantly differs from the pure diffusion setup. A formula for the limiting behavior of the early exercise boundary at expiry is also presented. As an explicit example we choose Merton's log-normal model for the distribution of the jump size and derive pricing formulae for European and American put options on a dividend paying underlying.