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
What does a Martingale Assume?
• 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
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
E  p( t  T )  ( t )   p( t )
• 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.
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
T } and
Let { X (t), t ÎÎ
{ X (t ), t Î T }
{Y (t ), t
T } be two stochastic processes.
is said to be a Martingale with respect to { X (t ), t Î T }
if for all t, X(t) is measurable with respect to the sigma-algebra s {Y (t ), s £ t}
generated by the filtration {Y ( s ), s £ t } and if in addition we have:
1. E { X (t ) } <¥
1) Y (s ), s t}
2. E { X (t £
X (t )
Proposition
If
{ X (t), t Î T }
is a Martingale, we have:
1. EX (t )  EX (0)
u ) Y (s ), s t }
2. E { X (t £
X (t ) where u is any integer
Proposition
Let { X (t ), t Î T } is a Martingale, with respect to the filtration
{ F (t )}
then
1. EX (t )  EX (0)
2. E { X (t u ) F (t )}
X (t )
Example
Define the random walk:
V (t 
1) V (t ) U (t 1)
ì1
U (t 1) í
î-1
s { X (s),Y (s ); s t } X (t )
and { F (t )} £
We calculate:
E éùéùéù
V (t 1) F (t ) E V (t ) F (t )
ëûëûëû
E U (t 1) F (t )
Since V(t) is an adapted process with respect to the filtration F(t),
E éùéù
V (t 1) F (t ) V (t ) E U (t 1) F (t )
ëûëû
Thus, it may be a Martingale if:
E éù
ëûU (t 1) F (t ) 0
Otherwise, it is not
Proposition
Let { X (t ), t Î 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:
E {³
( X (t 1)) F (t)}
{E ( X (t 1)) F (t)} ( X (t))
Fama, 1970
Market efficient if
E éù
ëûP(t T ) F ( t)
P(t)
That is, the expected future price equals
Current price
Forward Rates
Y (T , t )  E éë X (t , t  T ) F (t ) ùû 
E éë X (T - 1, t  1) F (t ) ùû  Y (T , t )
And thus, rational expectations can be written by
E éëY (T - 1, t  1) F (t ) ùû 
E éë E éë X (t  T ) F (t  1) ùû F (t ) ùû
And therefore
E éë E éë X (t  T ) F (t  1) ùû F (t ) ùû 
E éë X (t  T ) F (t ) ùû  Y (T , t )
In other words, forward rates are the best estimates of prices
Martingales
Let
ì x t 1  x t   t , x 0  0

ì1 w. p. p

í    0 w. p. r
í
t

 -1 w. p. q
î
î
p ³ 0, q ³ 0, r ³ 0, p  q  r  1
Then  xt - t( p - q); t ³ 0
Is an F(t) measurable Martingale
Proof
E  x t 1 - (t  1)( p - q ) / Ft   E  x t   t - (t  1)( p - q ) / Ft  
 x t - (t  1)( p - q )  E (  t ) 
 x t - (t  1)( p - q )  ( p - q ) 
 x t - t ( p - q)
The following processes are also
Martingales
{

x t2
2
xt

- t ( p - q); t ³ 0

- t ( p - q)
{
2


}
 t ( p - q) 2 - ( p  q) ; t ³ 0
}
s x t , s  q / p, t ³ 0
Example:
ì
aS t -1
St  í
bS t -1
î
Is a Martingale
1- b
w. p.
a-b
a -1
w. p.
a-b
Proof




E St 1 St ,St -1 ,..., S0  E St 1 St  St
1- b
a -1
E  St 1  St   aSt
 bSt

a -b
a -b
b(a - 1)  a (1 - b)
 St
 St
a -b
The Wiener Process is a Martingale
{
}
{
}
{
}
E W ( t  h) F ( t )  E W ( t  h ) - W ( t ) F ( t )  E W ( t ) F ( t )
And
E{W(t  h) - W(t ) F(t)}  E{W(t  h) - W(t )}  0
The process
X (t )  W (t ) - t
2
Is a Martingale
Calculations




E X (t  s) F(t)  E W (t  s) 2 F(t) - (t  s) 

E {W (t  s) - W (t )}

E {W (t  s) - W (t )}  2W (t  s)W (t ) - W (t ) 2 F(t) - (t  s) 
2
2





F(t)  2 E W (t  s) F(t) - E W (t ) 2 F(t) - (t  s)
Due to independence of increments of the
filtration F we can write:

 
E {W (t  s) - W (t )} F(t)  E {W (t  s) - W (t )}
2



E {W (t  s)W (t )} F(t)  Z (t ) E W (t  s) F( t)
And

2
 s


E {W (t  s)W (t )} F(t)  W (t ) 2
By conditional expectation


E W (t ) 2 F(t)  W (t ) 2
Which leads to:
E X (t  s) F(t)  s  2W (t ) 2 - W (t ) 2 - (t  s)  W (t ) 2 - t  X (t )


The process
ì
 2 t 

X (t )  expíW (t ) 
2 

î

Is a Martingale
Calculations
é ì
ù
 2 (t  s) 
E X (t  s) F( t)  E exp íW (t  s)  F( t)  
2

ë î
û


é
ù
ì
 2 s 
 E  X (t )exp í (W (t  s) - W (t ))  F( t) 
2 
î
ë
û
Independence and conditional
expectation make it possible
é
ù
ì
 2s 
E X (t  s) F(t )  X (t )E X(t)expí (W (t  s) - W (t ))  F(t ) 
2 
ë
û
î
  2s 
= X (t ) exp  Eexp{ (W (t  s) - W (t ))} 
 2 
Which leads to:
E X(t  s F(t)  X(t )
Girsanov’s theorem and the
Price of Risk
dp(t )
  (t )dt  s (t )dB(t )
p(t )
Price of risk is  (t )   (t ) - r (t ) / s (t )
Thus,
dp(t )
   (t )s (t )  r (t )  dt  s (t )dB(t ) 
p(t )
=r (t )dt   (t )s (t )dt  s (t )dB(t ) 
 r (t )dt  s (t )   (t )dt  dB(t )  
=r (t )dt  s (t )dB* (t )
*
B
( t ) is as we have defined it above,
Defining
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 Expectations
and 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 {x} stands for an information set
(a time series, a stock price record, financial
variables etc.)
{x}  {x1, x2 ,..., xt }
. A forecast is an estimate based on the information
set written for convenience by the function f(.) such
that: y  f (x) with a forecast error:   y - y where
y is the actual record of the series investigated.
Conditions for Rational
Expectations
E ( )  0
E(y)  E(E( y I ))  0
E(x)  cov(e, x)  0, x Î I
• 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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