Martingales

Charles S. Tapiero

Martingales origins

• Its origin lies in the history of games of chance …. Girolamo Cardano proposed an elementary theory of gambling in 1565, Liber de Ludo Aleae The Book of Games of Chance). The notion of “fair game” is 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. To the extent to which you depart from that equality, it is in your opponent’s favour, you are a fool, and if in your own, you are unjust

• This is the essence of the Martingale

• Tomorrow’s price is expected to be today’s and thus it is also its best forecast

• Non-overlapping price changes are uncorrelated at all lead and lags, which further implies the ineffectiveness of all linear forecasting rules

• The Martingale was long 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.

• If this is the case, how can a Martingale account for the tradeoff between risk and return

What does a Martingale Assume?

Note 1

• Martingales are in fact a very powerful tool. Through a numeraire accounting we can obtain relative prices that are a Martingale

• Including historic probabilities

• (Girsanov Theorem)

The martingale property and No arbitrage

• The martingale property is one of the fundamental mathematical properties which underlies many important results in finance. For example, the « Fundamental 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 information is reflected in the market prices (Fama 1970).

• This means that at any one time, the current price fully represents all the information. Of course ever since Fama, we are aware of pricing anomalies

Implication

• This property is also translated in mathematical terms by stating that the prices are defined by a martingale. In this sense, efficient markets are equated to the existence of a martingale. The proof that a process is a martingale is thus extremely useful for it justifies the use of assumptions that are so fundamental to financial theory. As a result, an important number of results in finance depend on the underlying stochastic process being a martingale.

• Under a particular probability measure and not the historical probability measure.

E p t T t p t( ) ( ) ( )

Risk neutral pricing: A convenience made possible by Martingales

• It is very convenient in pricing securities to act as if all expected returns equal the risk-free rate, which is the same as if all investors are risk neutral. This is called the principle of risk-neutral pricing.

• To price risk-neutrality, one must change the probability measure to what is called naturally a “risk-neutral probability.

• Such a risk neutral probability exists if there are no arbitrage opportunities in the market which is a mild assumption.

Martingales are closely associated to

Complete Financial Markets! • Rational expectations• Law of the single price• No long term memory• Novikov condition for Martingales (Dybvig

and Huang have shown that this is equivalent to a solvability constraint, which is thus quite intuitive)

• ….. Martingales… a consequence of no arbitrage… Any violation of this condition perturbs the basic assumptions of theoretical finance

Convenience with Martingales:

Risk neutral pricing

• It is very convenient in pricing securities to act as if all expected returns equal the risk-free rate, which is the same as if all investors are risk neutral. This is called the principle of risk-neutral pricing.

• To price risk-neutrality, one must change the probability measure to what is called naturally a “risk-neutral probability.

• Such a risk neutral probability exists if there are no arbitrage opportunities in the market which is a mild assumption.

Note

• Martingales are in fact a very powerful tool. Through a numeraire accounting we can obtain relative prices that are a Martingale

• Including historic probabilities

• (Girsanov Theorem)

Definition of Martingales

Let ( ), and ( ),X t t T Y t t T be two stochastic processes.

( ),X t t T is said to be a Martingale with respect to ( ),X t t T

if for all t, X(t) is measurable with respect to the sigma-algebra ( ),Y t s t

generated by the filtration ( ),Y s s t and if in addition we have:

1. ( )E X t

2. ( 1) ( ), ( )E X t Y s s t X t

Proposition

If ( ),X t t T is a Martingale, we have:

1. ( ) (0)EX t EX

2. ( ) ( ), ( )E X t u Y s s t X t where u is any integer

Proposition

Let ( ),X t t T is a Martingale, with respect to the filtration

( )F t then

1. ( ) (0)EX t EX

2. ( ) ( ) ( )E X t u F t X t

ExampleDefine the random walk:

( 1) ( ) ( 1)

1( 1)

1

V t V t U t

U t

and ( ) ( ), ( ); ( )F t X s Y s s t X t

We calculate:

( 1) ( ) ( ) ( ) ( 1) ( )E V t F t E V t F t E U t F t

Since V(t) is an adapted process with respect to the filtration F(t),

( 1) ( ) ( ) ( 1) ( )E V t F t V t E U t F t

Thus, it may be a Martingale if:

( 1) ( ) 0E U t F t

Otherwise, it is not

Proposition

Let ( ),X t t T be a Martingale with respect

to the filtration ( )F t and let be a convex function;

then the process ( ( ))X t is a sub-Martingale with respect to F(t).

This is a direct outcome of Jensen’s inequality:

( ( 1)) ( ) ( ( 1)) ( ) ( ( ))E X t F t E X t F t X t

Fama, 1970

Market efficient if

( ) ( ) ( )E P t T F t P t

That is, the expected future price equals Current price

Forward Rates

( , ) ( , ) ( )

( 1, 1) ( ) ( , )

Y T t E X t t T F t

E X T t F t Y T t

And thus, rational expectations can be written by

( 1, 1) ( )

( ) ( 1) ( )

E Y T t F t

E E X t T F t F t

And therefore

( ) ( 1) ( )

( ) ( ) ( , )

E E X t T F t F t

E X t T F t Y T t

In other words, forward rates are the best estimates of prices

Martingales

x x xp

r

q

p q r p q r

t t t

t

1 0 01

0

1

0 0 0 1

,

, , ,

w. p.

w. p.

w. p.

Let

Then x t p q tt ( ); 0

Is an F(t) measurable Martingale

Proof

E x t p q E x t p q

x t p q E

x t p q p q

x t p q

t t t t t

t t

t

t

1 1 1

1

1

( )( ) / ( )( ) /

( )( ) ( )

( )( ) ( )

( )

F F

The following processes are also Martingales

x t p q tt2 0 ( );

x t p q t p q p q tt2 2 2 0 ( ) ( ) ( ) ;

s s q p tx t , / , 0

Example:

SaS w p

b

a b

bS w pa

a b

t

t

t

1

1

1

1

. .

. .

Is a Martingale

Proof

E S S S S E S S St t t t t t 1 1 0 1, ,...,

1

1 1

( 1) (1 )

t t t t

t t

b aE S S aS bS

a b a bb a a b

S Sa b

The Wiener Process is a Martingale

E W t h E W t h W t E W t( ) ( ) ( ) ( ) ( ) ( ) ( ) F t F t F t

And

E W t h W t E W t h W t( ) ( ) ( ) ( ) ( ) F t 0

The process

X t W t t( ) ( ) 2 Is a Martingale

E X t s E W t s t s

E W t s W t W t s W t W t t s

E W t s W t E W t s E W t t s

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

F(t) F(t)

F(t)

F(t) F(t) F(t)

2

2 2

2 2

2

2

Calculations

Due to independence of increments of the filtration F we can write:

E W t s W t E W t s W t s( ) ( ) ( ) ( ) 2 2F(t)

E W t s W t Z t E W t s( ) ( ) ( ) ( ) ( ) F(t) F t

And E W t s W t W t( ) ( ) ( ) F(t) 2

By conditional expectation E W t W t( ) ( )2 2F(t)

Which leads to: E X t s s W t W t t s W t t X t( ) ( ) ( ) ( ) ( ) ( ) F(t) 2 2 2 2

The process

X t W tt

( ) exp ( )

2

2Is a Martingale

Calculations

E X t s E W t st s

X t e p W t s W ts

( ) ( ) exp ( )( )

( )

( ) ( ( ) ( )) ( )

F t F t

F t

2

2

2

2 E x

Independence and conditional expectation make it possible

E X t s X t e p W t s W ts

X ts

e p W t s W t

( ) ( ) ( ) ( ( ) ( )) ( )

( ) exp ( ( ) ( ))

F t F tE X(t) x

= E x

2

2

2

2

Which leads to:

E X t s X t( ( ) ( ) F t

Girsanov’s theorem and the Price of Risk

( )( ) ( ) ( )

( )

dp tt dt t dB t

p t

Price of risk is ( ) ( ) ( ) / ( )t t r t t

Thus,

*

( )( ) ( ) ( ) ( ) ( )

( )

= ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

= ( ) ( ) ( )

dp tt t r t dt t dB t

p t

r t dt t t dt t dB t

r t dt t t dt dB t

r t dt t dB t

Defining B t*( ) is as we have defined it above, i .e. a Brownian motion relative to the probability

measure P* , which implies that the expected rate of return equals the riskless rate r t( ) when

the measure is P* . Therefore, we have risk

neutral pricing under P* which is the risk neutral probability.

Rational Expectationsand Martingales

• Rational expectations have important implications to economics and finance theory, presuming a certain behavior of markets. In simple terms, rational expectations mean that economic agents can forecast the “mean”. In other words, they select forecasts that minimize the forecast error (in other words, the mean error is null).

• Explicitly, say that stands for an information set (a time series, a stock price record, financial variables etc.)

. A forecast is an estimate based on the information set written for convenience by the function f(.) such that: with a forecast error: where y is the actual record of the series investigated.

x

txxxx ,...,, 21

)(xfy yy

Conditions for Rational Expectations

0)( E0))(()( IyEEyE

IxxexE ,0),cov()(

• One of the essential problems resulting from the application of the rational expectation model in finance is that it may be wrong.

• As a result, a "model error" can be made which requires that either another model be used or that we construct models that are robust, tolerating such structural model errors.

• The interaction of markets can lead to instabilities due to very rapid and positive feedback or to expectations that are becoming trader and markets dependent.

• Such situations lead to a growth of volatility, instabilities and perhaps, in some special cases, to bubbles and chaos.

• George Soros, the famed and wealthy hedge fund financier has also brought attention to a concept of "reflexivity" summarizing an environment where conventional traditional finance theory does no longer hold.

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