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Old Problems, Classical Methods, New Solutions

In: Options — 45 years since the Publication of the Black–Scholes–Merton Model The Gershon Fintech Center Conference

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  • A. Lipton

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.

Suggested Citation

  • A. Lipton, 2023. "Old Problems, Classical Methods, New Solutions," World Scientific Book Chapters, in: David Gershon & Alexander Lipton & Mathieu Rosenbaum & Zvi Wiener (ed.), Options — 45 years since the Publication of the Black–Scholes–Merton Model The Gershon Fintech Center Conference, chapter 16, pages 331-375, World Scientific Publishing Co. Pte. Ltd..
    • Handle: RePEc:wsi:wschap:9789811259142_0016
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    Keywords

    Options; Call; Put; Stock; Equity; Bond; Debt; Dividend; Investment; Diversification; Volatility; Black–Scholes; Merton Model; Stochastic; Swap; Commodity; Index; Contingent Claims; Exotic Option;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G15 - Financial Economics - - General Financial Markets - - - International Financial Markets
    • G1 - Financial Economics - - General Financial Markets
    • C - Mathematical and Quantitative Methods
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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