chapter3 - Chapter 3: Discrete-Time Securities Market 1...

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Unformatted text preview: Chapter 3: Discrete-Time Securities Market 1 Chapter 3: Discrete-Time Securities Market There are many uncertainties in the securities market. For some securities such as callable bond, there is time uncertainty on when call provision is exercised. In many other situations, future cash flows are not certain as well. For example, a call option written on a common stock has uncertain future cash flow. In the previous chapter, we explored an example of security market consisting of two basic assets: one is risk-less asset, another one is risky and its price movement follows a binomial tree. In this simple example we have shown that, by no-arbitrage principle, how to price other security. This chapters purpose is to extend previous discussion to a general security market consisting of many basic assets. To make the notations simple while not losing the fundamental finance ideas, we first focus on the single period securities market. Then we extend the analysis to multi-period security market. We shall define a concept no- arbitrage and then to put forward a fundamental theorem about whether a market is free of arbitrage or not. In a no-arbitrage securities market, valuation of other complicated security depends on it can be replicated (dynamically) by those basic assets. This leads to another important concept: complete . If the security can be replicated by basic assets, it can be valued by a risk-neutral probability measure , which is the last important concept of this chapter. Section 1. Single-Period Securities Market 1. Securities Market There are two times: one is time zero which is the current time; another one is time one which denotes a future time . There is only one time period from time zero and time one. Market participants observe the same information at time zero. However, they dont know the information in the future. There are N securities in this market , namely N S S S ,..., , 2 1 . These securities are also called basic assets. How to model the uncertainties of the prices of S at time one? We make use of a simple probability space ) }, ,..., , { ( 2 1 P M = . There are M possible scenarios for each basic asset. P is a probability, but it is agent-dependent. It other words, every investor of this market agrees that there are M scenarios but they might have different view (probability) of price movement of those basic assets. Here is one example of the price movement of security. Chapter 3: Discrete-Time Securities Market 2 Time Zero Time One 30 20 10 The possible outcome of the first security is written as )] ( , ), ( ), ( [ 1 2 1 1 1 M S S S & at time one, corresponding to state i . The possible outcomes of all basic security are written as a matrix: The current price of these basic assets, denoted by )] ( , ), ( ), ( [ ) ( 2 1 N S S S S & = , is known at time zero for all investors in this market....
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This note was uploaded on 11/02/2011 for the course ACTSC 446 taught by Professor Adam during the Winter '09 term at Waterloo.

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chapter3 - Chapter 3: Discrete-Time Securities Market 1...

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