{"product_id":"option-prices-as-probabilities-a-new-look-at-generalized-black-scholes-formulae-von-cristophe-profeta-bernard-roynette-marc-yor-christophe-profeta","title":"Option Prices as Probabilities","description":"Discovered in the seventies, Black-Scholes formula continues to play a central role in Mathematical Finance. We recall this formula. Let (B ,t? 0; F ,t? 0, P) - t t note a standard Brownian motion with B = 0, (F ,t? 0) being its natural ?ltra- 0 t t tion. Let E := exp B? ,t? 0 denote the exponential martingale associated t t 2 to (B ,t? 0). This martingale, also called geometric Brownian motion, is a model t to describe the evolution of prices of a risky asset. Let, for every K? 0: + ? (t) :=E (K?E ) (0.1) K t and + C (t) :=E (E?K) (0.2) K t denote respectively the price of a European put, resp. of a European call, associated with this martingale. Let N be the cumulative distribution function of a reduced Gaussian variable: x 2 y 1 ? 2 ? N (x) := e dy. (0.3) 2? ?? The celebrated Black-Scholes formula gives an explicit expression of? (t) and K C (t) in terms ofN : K ? ? log(K) t log(K) t ? (t)= KN ? + ?N ? ? (0.4) K t 2 t 2 and ? ?\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783642103940\"\u003e\u003ch3\u003eA New Look at Generalized Black-Scholes Formulae\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9783642103940","offer_id":39443211518045,"sku":"9783642103940","price":53.49,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/241754b7-742d-412b-8c8f-1aa327fef17a.jpg?v=1782448901","url":"https:\/\/shop.autorenwelt.de\/products\/option-prices-as-probabilities-a-new-look-at-generalized-black-scholes-formulae-von-cristophe-profeta-bernard-roynette-marc-yor-christophe-profeta","provider":"Autorenwelt Shop","version":"1.0","type":"link"}