Option Prices as Probabilities: A New Look at Generalized by Christophe Profeta,Bernard Roynette,Marc Yor

By Christophe Profeta,Bernard Roynette,Marc Yor

came upon within the seventies, Black-Scholes formulation keeps to play a primary position in Mathematical Finance. We keep in mind this formulation. permit (B ,t? zero; F ,t? zero, P) - t t notice a typical Brownian movement with B = zero, (F ,t? zero) being its traditional ?ltra- zero t t tion. permit E := exp B? ,t? zero denote the exponential martingale linked t t 2 to (B ,t? 0). This martingale, often known as geometric Brownian movement, is a version t to explain the evolution of costs of a dicy asset. permit, for each ok? zero: + ? (t) :=E (K?E ) (0.1) ok t and + C (t) :=E (E?K) (0.2) ok t denote respectively the cost of a eu placed, resp. of a ecu name, linked to this martingale. permit N be the cumulative distribution functionality of a discounted Gaussian variable: x 2 y 1 ? 2 ? N (x) := e dy. (0.3) 2? ?? the distinguished Black-Scholes formulation offers an particular expression of? (t) and okay C (t) in phrases ofN : ok ? ? log(K) t log(K) t ? (t)= KN ? + ?N ? ? (0.4) okay t 2 t 2 and ? ?

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