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

Abstract : For a large class of $\mathbb{R}_{+}$ valued, continuous local martingales $(M_{t}\; t \ge 0)$, with $M_{0} =1$ and $M_{\infty} = 0$, the put quantity : $\Pi_{M} (K,t) = E \big((K-M_{t})^{+} \big)$ turns out to be the distribution function in both variables $K$ and $t$, for $K \le 1$ and $t \ge 0$, of a probability $\gamma_{M}$ on $[0,1] \times [0, \infty[$. In this paper, the first in a series of three, we discuss in detail the case where $\dis M_{t} = \mathcal{E}_{t} := \exp \big(B_{t} - \frac{t}{2}\big)$, for $(B_{t}, \; t \ge 0)$ a standard Brownian motion.
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International Journal of Theoretical and Applied Finance, World Scientific Publishing, 2009, 12, pp.1075-1090. <10.1142/S0219024909005580>
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Dernière modification le : mercredi 15 mars 2017 - 12:13:21
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### Citation

D. Madan, Bernard Roynette, Marc Yor. Put option prices as joint distribution functions in strike and maturity : the Black-Scholes case. International Journal of Theoretical and Applied Finance, World Scientific Publishing, 2009, 12, pp.1075-1090. <10.1142/S0219024909005580>. <hal-00324636>

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