Stock and Money Market Models

Suppose that m risky assets are traded. These will be referred to as stocks. Their prices at time n 0,1, 2, are denoted by Si(n), , Sm(n). In addition, investors have at their disposal a risk-free asset, that is, an investment in the money market. Unless stated otherwise, we take the initial level of the risk-free investment to be one unit of the home currency, A(0) 1. However, in some numerical examples and exercises we shall often take A(0) 100 for convenience. Because the money market...

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Figure 3.3 Two-step binomial tree of stock prices In both figures S 0 1 for simplicity. Figure 3.4 Three-step binomial tree of stock prices Exercise 3.13 Find d and u if 5 1 can take two values, 87 or 76, and the top possible value of 5 2 is 92. Suppose that the risk-free rate under continuous compounding is 14 , the time step t is one month, 5 0 22 dollars and d 0.01. Find the bounds on the middle value of 5 2 consistent with Condition 3.2. Suppose that 32, 28 and x are the possible values of...

CEs sxy

Figure 7.8 Time value CE S - S - X of a European call option For a European option we have to wait until the exercise time T to realise the payoff. The risk that the stock price will rise above X in the meantime may be considerable, which reduces the value of the option. Figure 7.9 Time value PE S - X - S of a European put option The time value of an American call option is the same as that of a European call if there are no dividends and Figure 7.8 applies. For an American put a typical graph...

Proof

Compare the following two formulae for S m S m S n 1 K n 1 1 K n 2 1 K m . Both of them follow from Definition 3.1. In each of the following three scenarios find the one-step returns, assuming that K 1 K 2 Given that K 1 10 or -10 , and K 0, 2 21 , 10 or -1 , find a possible structure of scenarios such that K 2 takes at most two different values. The lack of additivity is often an inconvenience. This can be rectified by introducing the logarithmic return on a risky security, motivated by...

Forward Contracts

A forward contract is an agreement to buy or sell an asset on a fixed date in the future, called the delivery time, for a price specified in advance, called the forward price. The party to the contract who agrees to sell the asset is said to be taking a short forward position. The other party, obliged to buy the asset at delivery, is said to have a long forward position. The principal reason for entering into a forward contract is to become independent of the unknown future price of a risky...

Binomial Tree Model

We shall discuss an extremely important model of stock prices. On the one hand, the model is easily tractable mathematically because it involves a small number of parameters and assumes an identical simple structure at each node of the tree of stock prices. On the other hand, it captures surprisingly many features of real-world markets. The model is defined by the following conditions. The one-step returns K n on stock are identically distributed independent random variables such that at each...

Basic Notions and Assumptions

Suppose that two assets are traded one risk-free and one risky security. The former can be thought of as a bank deposit or a bond issued by a government, a financial institution, or a company. The risky security will typically be some stock. It may also be a foreign currency, gold, a commodity or virtually any asset whose future price is unknown today. Throughout the introduction we restrict the time scale to two instants only today, t 0, and some future time, say one year from now, t 1. More...

Preface

True to its title, this book itself is an excellent financial investment. For the price of one volume it teaches two Nobel Prize winning theories, with plenty more included for good measure. How many undergraduate mathematics textbooks can boast such a claim Building on mathematical models of bond and stock prices, these two theories lead in different directions Black-Scholes arbitrage pricing of options and other derivative securities on the one hand, and Markowitz portfolio optimisation and...