Put Option as Joint Distribution Function in Strike and Maturity

Christophe Profeta (), Bernard Roynette () and Marc Yor
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Christophe Profeta: Université Nancy I
Bernard Roynette: Université Nancy I
Marc Yor: Université Paris VI

Chapter Chapter 6 in Option Prices as Probabilities, 2010, pp 143-159 from Springer

Abstract: Abstract For a large class of ℝ+-valued, continuous local martingales (M t ,t≥0), with M 0=1 and M ∞=0, the put quantity: $\Pi_{M}(K,t)=\mathbb{E}\left[(K-M_{t})^{+}\right]$ turns out to be the distribution function in both variables K and t, for K≤1 and t≥0, of a probability γ M on [0,1]×[0,+∞[. We discuss in detail, in this Chapter, the case where $(M_{t}=\mathcal{E}_{t}:=\exp(B_{t}-\frac{t}{2}),t\geq0)$ , for $(B_{t},\;t\ge 0)$ a standard Brownian motion, and give an extension to the more general case of the semimartingale $\mathcal{E}^{\sigma,-\nu }_{t}:=\exp \big(\sigma B_{t}-\nu t\big)$ , (σ≠0,ν>0).

Date: 2010
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DOI: 10.1007/978-3-642-10395-7_6

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