UNCERTAINTY, GAMES OF CHANCE AND MARTINGALES

4.2 UNCERTAINTY, GAMES OF CHANCE AND MARTINGALES

Games of chance, such as betting in Monte Carlo or any casino, are popular metaphors to represent the ongoing exchanges of stock markets, where money is thrown to chance. Its historical origins can be traced to Girolamo Cardano who proposed an elementary theory of gambling in 1565 (Liber de Ludo Aleae – The Book of Games of Chance). The notion of ‘fair game’ was clearly stated: ‘The most fundamental principle of all in gambling is simply equal conditions, e.g. of opponents, of bystanders, of money, of situation, of the dice box, and of the die itself. The extent to which you depart from that equality, if it is in your opponent’s

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favour, you are a fool, and if in your own, you are unjust’. This is the essence of the Martingale (although Cardano did not use the word ‘martingale’). It was in Bachelier’s thesis in 1900 however that a mathematical model of a fair game, the martingale, was proposed. Subsequently J. Ville, P. Levy, J.L. Doob and others have constructed stochastic processes. The ‘concept of a fair game’ or martingale, in money terms, states that the expected profit at a given time given the total past capital is null with probability one. Gabor Szekely points out that a martingale is also a paradox. Explicitly,

If a share is expected to be profitable, it seems natural that the share is worth buying, and if it is not profitable, it is worth selling. It also seems natural to spend all one’s money on shares which are expected to be the most profitable ones. Though this is true, in practice other strategies are followed, because while the expected value of our money may increase (our expected capital tends to infinity), our fortune itself tends to zero with probability one. So in Stock Exchange business, we have to be careful: shares that are expected to be profitable are sometimes worth selling.

Games of dice, blackjack, roulette and many other games, when they are fair, corrected for the bias each has, are thus martingales. ‘Fundamental finance theory’ subsumes as well that under certain probability measures, asset prices turn out to have the martingale property. Intuitively, what does a martingale assume?

r Tomorrow’s price is today’s best forecast. r Non-overlapping price changes are uncorrelated at all leads and lags.

The martingale is considered to be a necessary condition for an efficient asset market, one in which the information contained in past prices is instantly, fully and perpetually reflected in the asset’s current price. A technical definition of a martingale can be summarized as the presumption that each process event (such as a new price) is independent and can be summed (i.e. it is integrable) and has the property that its conditional expectation remains the same (i.e. it is time-invariant).

That is, if Φ t ={p 0 , p 1 ,..., p t } are an asset price history at time t = 0, 1, 2, . . . expressing the relevant information we have at this time regarding the time series, also called the filtration. Then the expected next period price at time t + 1 is equal to the current price

E (p t +1 |p 0 , p 1 , p 2 ,..., p t )=p t which we also write as follows:

E (p t +1 |Φ t )=p t for any time t

If instead asset prices decrease (or increase) in expectation over time, we have a super-martingale (sub-martingale):

E (p t +1 |Φ t ) ≤ (≥) p t

Martingales may also be defined with respect to other processes. In particular, if { p t , t ≥ 0} and {y t , t ≥ 0} are two processes denoting, say, price and interest

83 rate processes, we can then say that { p t , t ≥ 0} is a martingale with respect to

UNCERTAINTY , GAMES OF CHANCE AND MARTINGALES

{y t , t ≥ 0} if:

E {| p t |} < ∞ and E ( p t +1 |y 0 , y 1 ,..., y t )=p t , ∀t Of course, by induction, it can be easily shown that a martingale implies an

invariant mean:

E (p t +1 ) = E( p t ) = · · · = E( p 0 ) For example, given a stock and a bond process, the stock process may turn out to

be a martingale with respect to the bond (a deflator) process, in which case the bond will serve as a numeraire facilitating our ability to compute the value of the stock.

Martingale techniques are routinely applied in financial mathematics and are used to prove many essential and theoretical results. For example, the first ‘funda- mental theorem of asset pricing’, states that if there are no arbitrage opportunities, then properly normalized security prices are martingales under some probability measure. Furthermore, efficient markets are defined when the relevant informa- tion is reflected in market prices. This means that at any one time, the current price fully represents all the information, i.e. the expected future price p(t + T ) conditioned by the current information and using a price process normalized to

a martingale equals the current price. ‘The second fundamental theorem of asset pricing’ states in contrast that if markets are complete, then for each numeraire used there exists one and only one pricing function (which is the martingale measure). Martingales and our ability to construct price processes that have the martingale properties are thus extremely useful to price assets in theoretical fi- nance as we shall see in Chapter 6.

Martingales provide the possibility of using a risk-neutral pricing framework for financial assets. Explicitly, when and if it can be used, it provides a mechanism for valuing assets ‘as if investors were risk neutral’. It is indeed extremely con- venient, allowing the pricing of securities by using their expected returns valued at the risk-free rate. To do so, one must of course, find the probability measure, or equivalently find a discounting mechanism that renders the asset values a mar- tingale. Equivalently, it requires that we determine the means to replicate the payoff of an uncertain stream by an equivalent ‘sure’ stream to which a risk-free discounting can be applied. Such a risk-neutral probability exists if there are no arbitrage opportunities. The martingale measures are therefore associated with a pricing of an asset which is unique only if markets are complete. This turns out to be the case when the assumptions made regarding market behaviours include:

r rational expectations, r law of the single price, r no long-term memory, r no arbitrage.

The problem in applying rational expectations to financial valuation is that it may not be always right, however. The interaction of markets can lead to

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instabilities due to very rapid and positive feedback or to expectations that are becoming trader- and market-dependent. Such situations lead to a growth of volatility, instabilities and perhaps, in some special cases, to bubbles and chaos. George Soros, the hedge fund financier has also brought attention to the concept of ‘reflexivity’ summarizing an environment where conventional traditional finance theory no longer holds and therefore theoretical finance does not apply. In these circumstances, ‘there is no hazard in uncertainty’. A trader’s ability to ‘identify

a rational behaviour’ in what may seem irrational to others can provide great opportunities for profit making. The ‘law of the single price’, claiming that two cash flows of identical char- acteristics must have, necessarily, the same price (otherwise there would be an opportunity for arbitrage) is not always satisfied as well. Information asymmetry, for example, may violate such an assumption. Any violation of these assumptions perturbs the basic assumptions of theoretical finance, leading to incomplete mar- kets. In particular, we apply this ‘law’ in constructing portfolios that can replicate risky assets. By hedging, i.e. equating these portfolios to a riskless asset, it be- comes possible to value the assets ‘as if they were riskless’. This approach will

be developed here in greater detail and for a number of situations. We shall attend to these issues at some length in subsequent chapters. At this point, we shall turn to defining terms often used in finance: random walks and stochastic processes.