PRICING OF AMERICAN PATH‐DEPENDENT CONTINGENT CLAIMS

PRICING OF AMERICAN PATH‐DEPENDENT CONTINGENT CLAIMS We consider the problem of pricing path‐dependent contingent claims. Classically, this problem can be cast into the Black‐Scholes valuation framework through inclusion of the path‐dependent variables into the state space. This leads to solving a degenerate advection‐diffusion partial differential equation (PDE). We first estabilish necessary and sufficient conditions under which degenerate diffusions can be reduced to lower‐dimensional nondegenerate diffusions. We apply these results to path‐dependent options. Then, we describe a new numerical technique, called forward shooting grid (FSG) method, that efficiently copes with degenerate diffusion PDEs. Finally, we show that the FSG method is unconditionally stable and convergent. the FSG method is the first capable of dealing with the early exercise condition of American options. Several numerical examples are presented and discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Finance Wiley

PRICING OF AMERICAN PATH‐DEPENDENT CONTINGENT CLAIMS

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Publisher
Wiley
Copyright
Copyright © 1996 Wiley Subscription Services, Inc., A Wiley Company
ISSN
0960-1627
eISSN
1467-9965
D.O.I.
10.1111/j.1467-9965.1996.tb00111.x
Publisher site
See Article on Publisher Site

Abstract

We consider the problem of pricing path‐dependent contingent claims. Classically, this problem can be cast into the Black‐Scholes valuation framework through inclusion of the path‐dependent variables into the state space. This leads to solving a degenerate advection‐diffusion partial differential equation (PDE). We first estabilish necessary and sufficient conditions under which degenerate diffusions can be reduced to lower‐dimensional nondegenerate diffusions. We apply these results to path‐dependent options. Then, we describe a new numerical technique, called forward shooting grid (FSG) method, that efficiently copes with degenerate diffusion PDEs. Finally, we show that the FSG method is unconditionally stable and convergent. the FSG method is the first capable of dealing with the early exercise condition of American options. Several numerical examples are presented and discussed.

Journal

Mathematical FinanceWiley

Published: Jan 1, 1996

References

  • Path Dependent Options: The Case of Lookback Options
    Conze, Conze; Viswanathan, Viswanathan
  • Bessel Processes, Asian Options and Perpetuities
    Geman, Geman; Yor, Yor

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