Abstract
The American options generalize the European options in the sense that they can be exercised at any moment prior to maturity. They are part of the more general category of American-type derivatives that we shall define in Subsection 3.1 as a sequence X = (X n ) of random variables that are adapted to a given filtration (F n ), typically generated by the prices of the underlyings. The value of X n is the premium/payoff paid to the holder of the derivative if he/she exercises the option at time t n .
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- 1.
The Optional Sampling Theorem states that if M is a martingale (a submartingale) and v is a bounded stopping time then E [M v ] = M 0 (E [M v ] ≥ M 0 respectively). For the proof see for example Theorem A.129 in [17].
- 2.
For the proof see, for instance, Lemma A.125 in [17].
- 3.
In the sense that X n depends on the underlying through the values of a Markov process which (see Remark 3.15) may be the value S n of the underlying itself at time n if S is Markovian.
- 4.
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© 2012 Springer-Verlag Italia
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Pascucci, A., Runggaldier, W.J. (2012). American options. In: Financial Mathematics. Unitext(). Springer, Milano. https://doi.org/10.1007/978-88-470-2538-7_3
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DOI: https://doi.org/10.1007/978-88-470-2538-7_3
Publisher Name: Springer, Milano
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