Abstract
An American option is a contract giving its holder the right to buy (call option) or sell (put option) one unit of an underlying security of value S for a prearranged amount. This right can be exercised at any time prior to the expiration date T. In contrast, a European option can be exercised only at the expiry. Define the amount paid to the holder of an American option at the moment of exercise, the payoff, as Ψ (S, t) ≥ 0; a standard contract is a put option where Ψ = max(K − S, 0) and K is the strike price. The discounted exercise value of the option is Z(t) = Ψ (t) / B(t), where B(t) is the value at time t of $1 invested in a riskless money market account at t = 0. American option valuation can be characterised as an optimal stopping problem. The time 0 value of an American option is given by
where the supremum is taken over all the possible stopping times τ less than the expiration date T, and the expectation is taken over the risk-neutral probability density. This is the primal problem.
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References
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Lamper, D., Howison, S. (2004). Monte Carlo Valuation of American Options. In: Buikis, A., Čiegis, R., Fitt, A.D. (eds) Progress in Industrial Mathematics at ECMI 2002. The European Consortium for Mathematics in Industry, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09510-2_43
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DOI: https://doi.org/10.1007/978-3-662-09510-2_43
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