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- W2499387673 abstract "6.1 IntroductionUnlike European options, American options can be exercised anytime before maturity. Note that since the American option gives more rights to its owner than the European option, its price should be larger.As before, the price of the underlying asset satisfiesdSτ=Sτ(r(τ)dτ+σ(Sτ,τ)dWτ),where the volatility may depend on time and price and where Wτ is a standard Brownian motion. There exists a probability ℙ* (the risk neutral probability) under which the discounted price of the asset is a martingale. It can be proven that under the risk neutral probability, the price of the American option of payoff P∘ and maturity T isP(St,t)=supτ∈Tt,TE*(e−∫tτr(s)dsP∘(Sτ)|Ft),(6.1)where Tt,T denotes the set of stopping times in [t, T] (see [78]).It is possible to prove that P (S, t) is also the solution to the variational inequality, which is the weak form of the following set of inequalities:∂P∂t+σ2(S,t)S22∂2P∂S2+rS∂P∂S−rP≤0inℝ+×[0,T),P≥P∘inℝ+×[0,T),(∂P∂t+σ2(S,t)S22∂2P∂S2+rS∂P∂S−rP)(P−P∘)=0inℝ+×[0,T),(6.2)with dataP|t=T=P∘.(6.3)The theory for pricing an American option is not easy: the proof of the above-mentioned result can be found in Bensoussan and Lions [13] and Jaillet, Lamberton, and Lapeyre[78]." @default.
- W2499387673 created "2016-08-23" @default.
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- W2499387673 date "2005-01-01" @default.
- W2499387673 modified "2023-09-27" @default.
- W2499387673 title "6. American Options" @default.
- W2499387673 doi "https://doi.org/10.1137/1.9780898717495.ch6" @default.
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