Valuing perpetual American put options under the Heston Model
Research Seminar by Dr. Paul Johnson
Date: 31 March 2025 11:40-12:40
Location: Corvinus University of Budapest, building E, Institute of Finance – E.279.1.
The valuation of American options under stochastic volatility has attracted considerable interest due to the complexity of calibrating the market price of options with different strike prices and maturities. In this paper, we consider the pricing of perpetual American put options under the Heston model and derive novel (and accurate) asymptotic approximations using perturbation techniques for the option price, including the optimal exercise boundary. We demonstrate the difficulty and inefficiency of obtaining accurate valuations for the full (elliptic) partial differential equation problem with finite-difference methods. This leads us to simplify the problem by assuming small volatility, which usefully reduces the problem to be of parabolic type in one of the dimensions, thereby reducing the computational task considerably, and yet replicates the solution of the full problem well. This, in turn, leads to a further asymptotic and even simpler approach found by developing a quite straightforward series solution of the parabolised system, based on small displacements of the variance from its long-run mean (a critical region in parameter space). This approach, also, when compared with the full benchmark solution, yields remarkably useful results but at virtually no computational cost.
Joint work with Yuwei Qi and Prof. Peter Duck.
Dr. Paul Johnson, Lecturer in Financial Mathematics, The University of Manchester, is part of the Mathematical Modelling in Finance and Economics Group in which they model financial systems with uncertain price and uncertain physical flow, leading to non-linear PDEs which must be solved numerically. He has investigated such systems in the world of finance, banking, renewable energy, mining, and Revenue Management systems.
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