Mathematics of Financial Markets
Mathematical finance is a child of the 20th century. It was born on 29 March 1900 with the presentation of Louis Bachelier’s doctoral dissertation Théorie de la speculation [1]. Now, one hundred years later, it is the basis of a huge industry, at the cent
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1. Background Mathematical finance is a child of the 20th century. It was born on 29 March 1900 with the presentation of Louis Bachelier's doctoral dissertation Theorie de la speculation [1]. Now, one hundred years later, it is the basis of a huge industry, at the centre of modern global economic development, and the source of a great deal of interesting mathematics. Further, the theory and applications have proceeded in parallel in an unusually closely-linked way. This article aims to give the flavour of the mathematics, to describe how the confluence of mathematical ideas, economic theory and computer technology proved so effective, and to indicate how the theory relates to the practice of 'financial engineering'. Bachelier's extraordinary thesis was years, and in some respects decades, ahead of its time. For example it introduces Brownian motion as a model for stock prices five years before Einstein's classic paper [l 0] on that subject. Brownian motion is a continuous-path stochastic process (B(t), t ::': 0) such that (a) B(O) = 0, (b) the increments (B(t4)- B(t3)), (B(t2)- B(t1)) are independent for t1 _::=: t2 _::=: t3 _::=: t4 and (c) the increment (B(t2) - B(ti)) is normally distributed with mean zero and variance t2 - t1. It is simultaneously a Markov process and a martingale, though neither of those concepts had been named or clearly formulated in 1900. A martingale M(t) is the mathematical representation of a player's fortune in a fair game. The defining property is that, fort > s, E[M(t)j.j?f] = M(s), where E[ .j.¥,] represents the conditional expectation given.~, the 'information up to times': the expected fortune at some later time t is equal to the current fortune M (s). Bachelier arrived at this by economic reasoning. Arguing that stock markets have symmetry in that every trade involves a buyer and a seller, and that there cannot be any consistent bias in favour of one or the other, he formulated his famous dictum 'L'esperance mathematique du speculateur est nulle'.
This is tantamount to the martingale property. Assuming that the price process B(t) is Markovian, Bachelier introduced the transition density p(x, t; y, s) defining the probabilities of moving from state y to state x: P [B(t)
E
[x, x
+ dx]IB(s)
= y] = p(x, t; y, s)dx.
* Work supported by the Austrian Science Foundation (FWF) under grant WittgensteinPrize Z36-MAT.
B. Engquist et al. (eds.), Mathematics Unlimited — 2001 and Beyond © Springer-Verlag Berlin Heidelberg 2001
362
M . DAVJS
He assumed that this would be temporally and spatially homogeneous, i.e. that p(x, t; y, s) = q(x- y, t - s) for some function q. By arguing that, starting at B(O) = 0, the transition probability should satisfy P[B(t + s) E dx] = J P[B(t +s) E dxiB(s) = y]P[B(s) = y] dy he obtained what is now known as the Chapman-Kolmogorov equation q(x , s
+ t) =
i:
q(x- y, t)q(y, s)dy.
(1.1)
He then showed that ( 1.1) is satisfied by the Brownian transition function
x2)
q(x, t) = -1- exp ( - .Jfiit 2t
(1 .2)
(not worrying about uniqueness), and went on
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