## Option Prices as Probabilities: A New Look at Generalized Black-Scholes FormulaeDiscovered in the seventies, Black-Scholes formula continues to play a central role in Mathematical Finance. We recall this formula. Let (B ,t? 0; F ,t? 0, P) - t t note a standard Brownian motion with B = 0, (F ,t? 0) being its natural ?ltra- 0 t t tion. Let E := exp B? ,t? 0 denote the exponential martingale associated t t 2 to (B ,t? 0). This martingale, also called geometric Brownian motion, is a model t to describe the evolution of prices of a risky asset. Let, for every K? 0: + ? (t) :=E (K?E ) (0.1) K t and + C (t) :=E (E?K) (0.2) K t denote respectively the price of a European put, resp. of a European call, associated with this martingale. Let N be the cumulative distribution function of a reduced Gaussian variable: x 2 y 1 ? 2 ? N (x) := e dy. (0.3) 2? ?? The celebrated Black-Scholes formula gives an explicit expression of? (t) and K C (t) in terms ofN : K ? ? log(K) t log(K) t ? (t)= KN ? + ?N ? ? (0.4) K t 2 t 2 and ? ? |

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### Contents

Reading the BlackScholes Formula in Terms of First and Last Passage Times | 1 |

Generalized BlackScholes Formulae for Martingales in Terms of Last Passage Times | 21 |

Representation of some particular Azéma supermartingales | 65 |

An Interesting Family of BlackScholes Perpetuities | 89 |

Study of Last Passage Times up to a Finite Horizon | 115 |

Put Option as Joint Distribution Function in Strike and Maturity | 143 |

Existence and Properties of PseudoInverses for Bessel and Related Processes | 161 |

Existence of PseudoInverses for Diffusions | 203 |

Complements | 239 |

Bessel Functions and Bessel Processes | 251 |

259 | |

Further Readings | 264 |

268 | |

### Other editions - View all

Option Prices As Probabilities Christophe Profeta,Bernard Roynette,Marc Yor No preview available - 2010 |

### Common terms and phrases

admits an increasing applying assume BESQ Black-Scholes formula Brownian bridge Brownian motion started Cameron-Martin formula change of variable Chapter compute continuous local martingale decreasing deduce defined denote Dirac measure distribution function ends the proof equivalent European put Existence of Pseudo-Inverses filtration function f geometric Brownian motion Hence implies increasing function increasing pseudo-inverse inf{t integrable Itô's formula Laplace transform last passage Lebesgue measure Lemma Let M1 Lévy process log(K log(S Markov property martingale Mathematical Finance monotone convergence theorem motion with drift notation Note obtain Option Prices parameter PFH-function Prices as Probabilities process of dimension process with index Profeta proof of Theorem Proposition prove Point relation Remark resp satisfies Section semimartingale speed measure Springer Finance squared Bessel process standard Brownian motion submartingale sup{t supermartingale taking values Theorem 2.1 uniformly integrable