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).
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© 2010 Springer-Verlag Berlin Heidelberg
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Profeta, C., Roynette, B., Yor, M. (2010). Put Option as Joint Distribution Function in Strike and Maturity. In: Option Prices as Probabilities. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10395-7_6
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DOI: https://doi.org/10.1007/978-3-642-10395-7_6
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10394-0
Online ISBN: 978-3-642-10395-7
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