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Extra info for Das Abenteuerliche Herz. Essays III, Bd 9
Follows the standard Black-Scholes formula. Note also that follows geometric Brownian motion. e. ppB (ω,t) follows a geometric Brownian motion with no drift. The reason is that the factor eµT in the payoff of the stock at time T just multiplies the value of the stock at all time periods by a constant factor. If we modiﬁed our utility function to incorporate timediscounting, identical drift factors would appear in pA and pB , so the ratio would be unchanged. We could obtain a nonzero drift in an inﬁnite horizon model in which the agent discounts, and dividends are paid at all times and follow a geometric Brownian motion with drift.
Our proof makes extensive use of nonstandard analysis, in particular Anderson’s construction of Brownian Motion and the Itˆo Integral () and Lindstr¨om’s extension of that construction to stochastic integrals with respect to L2 martingales [35, 36, 37, 38]. It is beyond the scope of this paper to develop these methods; Anderson  and Hurd and Loeb  are suitable references. We construct our probability space, ﬁltration and Brownian Motion following Anderson’s construction . Speciﬁcally, we construct a hyperﬁnite economy as in Raimondo , with the following changes to incorporate the derivatives D1 , .
All but one portion of the proof works if Aj (ω, T ) is an arbitrary continuous function of βj (ω, T ) satisfying an exponential growth condition, and that one part works for a large class of functions of βj , but we have not identiﬁed the exact class. Of course, changing the payoff will alter the pricing formula. In Anderson and Raimondo , derivative securities paid out at time T , with the payout (measured in units of the consumption good) equalling 34 R. M. Anderson and R. C. Raimondo Hm (pB (ω, Tm ), pA1 (ω, Tm ), .
Das Abenteuerliche Herz. Essays III, Bd 9