By Yves Achdou
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Let u(S, t) be such that for all positive numbers €, • ue-€l°z2(S+2) £ C°([0, T]- L 2 (R+)) n L2(0, T; V), . e. Proof. ), and for two positive parameters z and ^, the function wnere It is easy to see that V e C°([0, T7]; R), and for all t, \HO > 0. We have also ^r(O) = 0. 3 Mathematical Analysis 37 We choose two values Si < 1 and £2 > 1 such that, for S e (0, Si) U (£2, +00), 4rmax R+x|0 ,ri |cr2 + Saff - r < ±| log(S)|. We define Si = (0, Si) U (S2, +00) and
Find P e C°([0, T}\ L 2 (R+)) n L2(0, T; V), such that |f e L2(0, 7; V), satisfying In order to apply the abstract theory of Lions and Magenes , we need the following estimate. 10 (Garding's inequality). 1, there exists a nonnegative Proof. 21) and calling R = max,e[0,r] r(f), where X = (R + a + Caa)2/(CT2). This achieves the proof. 11. 24) /i«5 a unique solution, and we have the estimate, for all t, 0 < t < T, Proof. The proof is given in . 25) and integrating in time between 0 and t. 11 applies for any European option with a payoff function in L 2 (R + ), in particular to vanilla puts.
Since the American option gives its owner more rights than the corresponding European option, its price should be higher. Consider, for example, an American vanilla put: if Pt were less than K — St, then one could buy a put and a share of the underlying asset and immediately exercise the option, making a risk-free immediate benefit of K — St — Pt > 0; this is ruled out by the no-arbitrage assumption, so we see that Pt > K — St. More generally, the value at time t of an American option with payoff Q° is always larger than Q°(St).