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The Perpetual American Put Option for Jump-Diffusions

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Book cover Energy, Natural Resources and Environmental Economics

Part of the book series: Energy Systems ((ENERGY))

Abstract

We solve a specific optimal stopping problem with an infinite time horizon, when the state variable follows a jump-diffusion. The novelty of the paper is related to the inclusion of a jump component in this stochastic process. Under certain conditions, our solution can be interpreted as the price of an American perpetual put option. We characterize the continuation region when the underlying asset follows this type of stochastic process. Our basic solution is exact only when jump sizes cannot be negative. A methodology to deal with negative jump sizes is also demonstrated.

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Notes

  1. 1.

    i.e., if it does not explode. The Brownian motion is known not to explode.

  2. 2.

    If the exponential function inside the two different integrals can be approximated by the two first terms in its Taylor series expansion, which could be reasonable if the Lévy measure ν has short and light tails, then we have \(\frac{dY (t)} {Y (t-)} \approx {\mathit{dX}}_{t}\).

  3. 3.

    There is, in general, too high a degree of uncertainty compared to the number of underlying assets (here two) for the model to be dynamically complete. However, a pure jump model with deterministic jump sizes is known to be complete.

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Author information

Authors and Affiliations

  1. Department of Finance and Management Science, Norwegian School of Economics and Business Administration (NHH), Helleveien 30, 5045, Bergen, Norway

    Knut K. Aase

  2. Centre of Mathematics for Applications (CMA), University of Oslo, Oslo, Norway

    Knut K. Aase

Authors
  1. Knut K. Aase
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Correspondence to Knut K. Aase .

Editor information

Editors and Affiliations

  1. Business Administration, Department of Science and Management Sci, Norwegian School of Economics and, Helleveien 30, Bergen, 5045, Norway

    Endre Bjørndal

  2. , Department of Science and Management Sci, Norwegian School of Economics and Busine, Helleveien 30, Bergen, 5045, Norway

    Mette Bjørndal

  3. Dept. Industrial & Systems, Engineering, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Panos M. Pardalos

  4. Business Administration, Dept Science, Management Science, Norwegian School of Economics and, Helleveien 30, Bergen, 5045, Norway

    Mikael Rönnqvist

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Aase, K.K. (2010). The Perpetual American Put Option for Jump-Diffusions. In: Bjørndal, E., Bjørndal, M., Pardalos, P., Rönnqvist, M. (eds) Energy, Natural Resources and Environmental Economics. Energy Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12067-1_28

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  • DOI: https://doi.org/10.1007/978-3-642-12067-1_28

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-12066-4

  • Online ISBN: 978-3-642-12067-1

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