Put option prices as joint distribution functions in strike and maturity : the Black-Scholes case

For a large class of R+ valued, continuous local martingales (Mt t ≥ 0), with M0 = 1 and M∞ = 0, the put quantity : ΠM(K, t) = E ( (K −Mt) ) turns out to be the distribution function in both variables K and t, for K ≤ 1 and t ≥ 0, of a probability γM on [0, 1]× [0,∞[. In this paper, the first in a series of three, we discuss in detail the case where Mt = Et := exp ( Bt − t 2 ) , for (Bt, t ≥ 0) a standard Brownian motion.