Old Problems, Classical Methods, New Solutions

A. Lipton

Chapter 16 in Options — 45 Years since the Publication of the Black–Scholes–Merton Model:The Gershon Fintech Center Conference, 2023, pp 331-375 from World Scientific Publishing Co. Pte. Ltd.

Abstract: We use a powerful extension of the classical method of heat potentials, recently developed by the present author and his collaborators, to solve several significant problems of financial mathematics. We consider the following problems in detail: (a) calibrating the default boundary in the structural default framework to a constant default intensity, (b) calculating default probability for a representative bank in the mean-field framework, (c) finding the hitting time probability density of an Ornstein–Uhlenbeck process. Several other problems, including pricing American put options and finding optimal mean-reverting trading strategies, are mentioned in passing. Besides, two nonfinancial applications — the supercooled Stefan problem and the integrate-and-fire neuroscience problem — are briefly discussed as well.

Keywords: Options; Call; Put; Stock; Equity; Bond; Debt; Dividend; Investment; Diversification; Volatility; Black–Scholes; Merton Model; Stochastic; Swap; Commodity; Index; Contingent Claims; Exotic Option (search for similar items in EconPapers)
JEL-codes: C C02 G1 G11 G12 G15 (search for similar items in EconPapers)
Date: 2023
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