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Some content that appears in printmay not be available in electronic books. ISBN pbk. Finance—Mathematical models. Brownian motion process. W54 Models are, for the most part, caricatures of reality, but if they are good, like good caricatures,they portray, though perhaps in a disturbed manner, some features of the real world. ContentsPreface xiii1 Brownian Motion 1 1. Contents ix T 4. PrefaceThis is a text which presents the basics of stochastic calculus in an ele-mentary fashion with plenty of practice.
Elementary ordinary calculushas been successfully taught along these lines for decades. Useful side readings are given with eachtopic. Annexes provide background and elaborate some more technicalaspects. The technical prerequisites are elementary probability theoryand basic ordinary calculus. OUTLINEThe sequence of chapters in this text is best explained by working back-wards from the ultimate use of Brownian motion calculus, namely thevaluation of an option.
Because of the absence of downside risk, options are widely usedfor risk management and investment. The key task is to determine whatit should cost to buy an option prior to the payoff date. What makesthe payoff uncertain is the value of the so-called underlying asset of theoption on the date of payoff. In the standard case the underlying assetis a share of stock.
So the be-haviour of the stock price needs to be modelled. In the standard case itis assumed to be driven by a random process called Brownian motion,denoted B. The basic model for the stock price is as follows. This is the Tsubject of Chapter 1. Then the value of S T needs to be obtainedfrom the above equation.
That requires stochastic calculus rules whichare set out in Chapter 4, and methods for solving stochastic differentialequations which are described in Chapter 5. Once all that is in place, amethod for the valuation of an option needs to be devised. Two methodsare presented.
One is based on the concept of a martingale which isintroduced in Chapter 2. Chapter 7 elaborates on the methodology forthe change of probability that is used in one of the option valuationmethods.
The references have been selected with great care, to suit a variety ofbackgrounds, desire and capacity for rigour, and interest in application. Preface xvThey should serve as a companion. In view ofthe perceived audience, several well-known texts that are mathematicallymore demanding have not been included. In the interest of readability,this text uses the Blackwood Bold font for probability operations; aprobability distribution function is denoted as P, an expectation as E, avariance as Var, a standard deviation as Stdev, a covariance as Cov, anda correlation as Corr.
Chapter 1 introducesthe properties of Brownian motion as a random process, that is, the truetechnical features of Brownian motion which gave rise to the theoryof stochastic integration and stochastic calculus. Annex A presents anumber of useful computations with Brownian motion which require nomore than its probability distribution, and can be analysed by standardelementary probability techniques. Later that centuryit was postulated that the irregular motion is caused by a very largenumber of collisions between the pollen and the molecules of the liq-uid which are microscopically small relative to the pollen.
The hitsare assumed to occur very frequently in any small interval of time, in-dependently of each other; the effect of a particular hit is thought tobe small compared to the total effect. Around Louis Bachelier,a doctoral student in mathematics at the Sorbonne, was studying thebehaviour of stock prices on the Bourse in Paris and observed highlyirregular increments.
In the s Norbert Wiener, a mathematicalphysicist at MIT, developed the fully rigorous probabilistic frameworkfor this model. This kind of increment is now called a Brownian motionor a Wiener process. The position of the process is commonly denoted 1 This is meant in the mathematical sense, in that it can be positive or negative. Brownian motion is widely used to model randomness ineconomics and in the physical sciences.
With time onthe horizontal axis, and B t on the vertical axis, at each time t, B t isthe position, in one dimension, of a physical particle. It is a random vari-able. The probabilitydistribution depends only on the time spacing; it is the same for all timeintervals that have the same length. On the left is an element from the domain,on the right the corresponding function value in the range. Anormally distributed random variable is also known as a Gaussian ran-dom variable, after the German mathematician Gauss.
However, the lim-ited liability of shareholders rules this out. Butas time progresses the standard deviation increases, the density spreadsout, and that probability is no longer negligible. Half a century later,when research in stock price modelling began to take momentum, itwas judged that it is not the level of the stock price that matters to in-vestors, but the rate of return on a given investment in stocks.
The random term has an expected valueof 1. Thus the expected value of the stock price at time t, given S 0 , Now consider a particlewhich moves along in time as follows.
The reasonfor this choice will be made clear shortly. It is assumed that successiveincrements are independent of one another. This process is known as asymmetric because of the equal probabilities random walk.
Connecting these positions by straight lines gives a contin-uous path. The position at any time between the discrete time points isobtained by linear interpolation between the two adjacent discrete timepositions. The complete picture of all possible discrete time positions isgiven by the nodes in a so-called binomial tree, illustrated in Figure 1. At time-point n, the node which is at the end of a path thathas j up-movements is labelled n, j , which is very convenient fordoing tree arithmetic.
The number of paths ending at node n, j is given by aPascal triangle. This has the same shape as the binomial tree.
Brownian Motion Calculus / Edition 1
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Brownian Motion Calculus
Brownian Motion Calculus by Ubbo F. Wiersema (Trade Paper)