IMFC5 - Chapter 5 Arbitrage-Free Pricing in One-Period...

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Unformatted text preview: Chapter 5 Arbitrage-Free Pricing in One-Period Finite Models The simplest financial models involving random evolution of prices are those in which there are only two trading times and the prices of the basic securities are modelled as random variables on a finite probability space. Although such models are too simplistic to accurately model real-world financial markets, they will allow us to introduce some important ideas (such as risk-neutral pricing) in the simplest possible setting. Moreover, these very basic models can be used as building blocks in more sophisticated models. 5.1 One-Period Binomial Model In the one-period binomial model there are two times, t = 0 and t = 1, a bank, and a single stock that pays no dividends. We can borrow or invest any amount of money at the bank between t = 0 and t = 1 at the one-period interest rate r 0. Capital in the bank account evolves according to the rule B 1 = B (1 + r ) , where B is the initial capital; B > 0 corresponds to an investment, whereas B < indicates a loan. The initial price per share of the stock is S . It is assumed that we can buy or sell short as many shares of the stock as we please at time 0 at the price S per share. The price S 1 of the stock at time 1 is not known at time 0. We assume that there are two possible values of S 1 . Consequently it is appropriate to model S 1 as a random variable on a probability space ( , P ) in which contains two elements. We think of these two elementary events as head and tail resulting from the toss of a coin. An appropriate sample space is therefore = { H,T } . We do not assume that the coin is fair, i.e. we do not assume that P ( H ) = P ( T ). We do, however, assume that P ( H ) > , P ( T ) > . 131 We also assume that S 1 ( H ) negationslash = S 1 ( T ); otherwise the stock is just another bank account (and the no-arbitrage principle would tell us that the interest rate for the stock must also be r .) We refer to P as the reference probability measure , or the actual probability measure , in order to distinguish it from another probability measure P , called a pricing measure (or risk-neutral measure ), that will be introduced later to compute arbitrage-free prices. This model is extremely simple. It is certainly a stretch to believe that the price of the stock at the terminal time can take only two possible values, and that these values are known at the initial time. However, it is very useful to study (and completely understand) the one-period binomial model. Indeed, many of the important features of this model have analogues in more sophisticated models. In particular, one-period binomial models can be strung together to form multi-period binomial models, which are somewhat more realistic. A continuous-time model can be obtained taking a suitable limit of multi-period binomial models. The key to analyzing multi-period models, is the complete analysis of one-period models....
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This note was uploaded on 11/08/2011 for the course 21 270 taught by Professor Schaffer during the Spring '10 term at Carnegie Mellon.

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IMFC5 - Chapter 5 Arbitrage-Free Pricing in One-Period...

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