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hal-00324636, version 1

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

D. Madan 1, Bernard Roynette () 2, Marc Yor 3

10.1142/S0219024909005580 12 (2009) 1075-1090

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.

  • 1:  Robert H. Smith School of Business
  • University of Maryland
  • 2:  Institut Elie Cartan Nancy (IECN)
  • CNRS : UMR7502 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
  • 3:  Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
  • CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot
  • Domain : Mathematics/Probability
    Mathematics/Statistics
    Statistics/Statistics Theory
  • Keywords : First and last passage times – pseudo-inverse – local time-space calculus – Black-Scholes set up
  • Internal note : Prépublication IECN 2008/41
 
  • hal-00324636, version 1
  • http://hal.archives-ouvertes.fr/hal-00324636
  • oai:hal.archives-ouvertes.fr:hal-00324636
  • From: Estelle Carciofi <>
  • Submitted on: Thursday, 25 September 2008 15:45:41
  • Updated on: Thursday, 28 April 2011 16:51:03