Mathematical Finance by Mark H. A. Davis

Author:Mark H. A. Davis
Language: eng
Format: epub
ISBN: 9780191092039
Publisher: OUP Oxford
Published: 2018-11-26T16:00:00+00:00

The mathematics behind all this is sophisticated. The value of the option in the holding region satisfies the same partial differential equation as in the European BS case, but we have to determine where the boundary between the holding and exercise regions lies. This is determined by the requirement that the expected discounted value of the option payoff should be maximized when the option is exercised at the first boundary hitting time. All this was worked out in 1965 by MIT mathematician Henry McKean as an appendix to a paper by Samuelson. McKean was a key figure in the development of stochastic calculus and indeed had collaborated directly with Itô. However, 1965 was pre-BS and it was only in the 1980s that the complete arbitrage theory of American options was finalized, by A. Bensoussan and I. Karatzas.

For American options there is no pricing formula similar to the BS formula for the European case. Solutions have to be obtained computationally. There is a very simple solution when the price process is represented by a binomial tree, and this is one of the reasons why the introduction of the binomial tree was such a breakthrough. In the tree, one starts at the end, where exercise values are known, and works backwards. At any node N in the tree, the option value is the discounted average under risk-neutral probability q of the values at the two nodes to which the price may move from N. Call that the ‘continuation value’ at N. The ‘immediate exercise value’ at N is [K − S]+ for the put option, where S is the underlying price at N. Now replace the continuation value at N, and all other nodes at the same time step, by the immediate exercise value if the latter is bigger, and carry on working backwards. This simple modification to the algorithm produces the American value at the initial time. The nodes where immediate exercise is better define the exercise region. As we pointed out earlier, by constructing binomial trees with more and more smaller and smaller steps we can approximate geometric Brownian motion and hence the BS American price. In practice, the binomial tree is replaced by a trinomial tree, using just the same algorithm, because of the extra flexibility in calibration. Figure 16 was produced this way.

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