Question

Consider the non-arbitrage theorem.

a) Formulate the non-arbitrage theorem in the case of 3 assets and 2 states of nature. [10 marks]

b) Let P be the risk neutral (also called “risk adjusted”) probability measure. How can the probability measure P be used to calculate asset prices at time t in terms of asset prices at time (t+1)? [5 marks]

Answer 1

a)The assumption of no arbitrage (NA) is compelling because it appeals to the most basic beliefs about human behavior, namely that there is someone who prefers having more wealth to having less. Since, save for some anthropologically interesting societies, a preference for wealth appears to be a ubiquitous human characteristic, it is certainly a minimalist requirement. NA is also a necessary condition for an equilibrium in the financial markets. If there is an arbitrage opportunity, then demand and supply for the assets involved would be infinite, which is inconsistent with equilibrium. The study of the implications of NA is the meat and potatoes of modern finance. The early observations of the implications of NA were more specific than the general theory we will describe. The law of one price (LOP) is the most important of the special cases of NA, and it is the basis of the parity theory of forward exchange. The LOP holds that two assets with identical payoffs must sell for the same price. We can illustrate the LOP with a traditional example drawn from the theory of international finance. If s denotes the current spot price of the Euro in terms of dollars, and f denotes the currently quoted forward price of Euros one year in the future, then the LOP implies that there is a lockstep relation between these rates and the domestic interest rates in Europe and in the United States. Consider individuals who enter into the following series of transactions. First, they loan $1 out for one year at the domestic interest rate of r, resulting in a payment to them one year from now of (1 + r). Simultaneously, they can enter into a forward contract guaranteeing that they will deliver Euros in one year. With f as the current one-year forward price of Euros, they can guarantee the delivery of (1 + r)f Euros in one year’s time. Since this is the amount they will have in Euros in one year, they can borrow against this amount in Europe: letting the Euro interest rate be re, the amount they will be able to borrow isLastly, since the current spot price of Euros is s Euros per dollar, they can convert this amount into dollars to be paid to them today. This circle of lending domestically and borrowing abroad and using the forward and spot markets to exchange the currencies will be an arbitrage if the above amount differs from the $1 with which the investor began. Hence, NA generally and the LOP in particular require that (1 + r)f = (1 + re)s, which is to say that having Euros a year from now by lending domestically and exchanging at the forward rate is equivalent to buying Euros in the current spot market and lending in the foreign bond market. Not surprisingly, as a practical matter, the above parity equation holds nearly without exception in all of the foreign currency markets. In other words, at least for the outside observer, none of this kind of arbitrage is available. This lack of arbitrage is a consequence of the great liquidity and depth of these markets, which permit any perceived arbitrage opportunity to be exploited at arbitrary scale. It is, however, not unusual to come across apparent arbitrage opportunities of mispriced securities, typically when the securities themselves are only available in limited supply.2 While the LOP is a nice illustration of the power of assuming NA, it is somewhat misleading in that it does not fully capture the implications of removing arbitrage opportunities