Aspects of Mathematical Finance by Marc Yor

By Marc Yor

This selection of essays relies on lectures given on the "Académie des Sciences" in Paris via the world over popular specialists in mathematical finance. the gathering develops, in easy but rigorous phrases, a few difficult themes equivalent to chance measures, the thought of arbitrage, dynamic types regarding basic stochastic procedures like Brownian movement and Lévy procedures. The publication additionally encompasses a description of the trainings of French monetary analysts.

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39, no. 1, pp. 73–96 Stochastic Clock and Financial Markets H. Geman Abstract Brownian motion played a central role throughout the twentieth century in probability theory. The same statement is even truer in finance, with the introduction in 1900 by the French mathematician Louis Bachelier of an arithmetic Brownian motion (or a version of it) to represent stock price dynamics. This process was ‘pragmatically’ transformed by Samuelson in ([48, 49]; see also [50]) into a geometric Brownian motion ensuring the positivity of stock prices.

Merton [44] considered the addition of a jump component to the classical geometric Brownian motion model for the pricing of options on stocks. Bates [7], Bakshi, Cao and Chen [4] propose models that contain a diffusion component in addition to a low or finite activity jump part. The diffusion component accounts for high activity in price fluctuations while the jump component is used to account for rare and extreme movements. By contrast, GMY account for the small, high activity and rare large moves of the price process in a unified and connected manner: all motion occurs via jumps.

3 We do not write about the increasing research interest in statistical financial modelling as it is a subject in itself. 28 P. Barrieu, N. El Karoui More precisely, when the market conditions are (t0 , x0 ) and for the sake of simplicity the short rate is assumed to be null, then 1 CT (T, K) = σ 2 (T, K) K 2CKK (T, K), C(t0 , x0 ) = (x0 − K)+ . 2 (5) where C(T, K) is the price of the call option with parameters (T, K). This formula is especially attractive as it gives a simple relation between ‘quoted Call option prices’.

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