6 The Efficient Market Hypothesis (EMH), A Guide To Modeling Prices
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One thing to keep in mind here is the difference between derivatives pricing
and looking for the best process to describe the underlying stock price process.
All that matters for derivatives pricing is that there exist an equivalent martingale
measure under which the discounted underlying price process, while not
necessarily already a martingale, becomes a martingale.
Derivatives pricing does not require that the discounted underlying stock price
process actually be a martingale under the physical probability measure. The
physical probability measure is the measure that deﬁnes actual prices, not risk neutral
prices. The question that naturally arises is, which discounted underlying price
processes admit at least one EMM? What do these processes look like?
One seemingly easy way to answer this question is simply to hypothesize
that original, actual discounted underlying price processes are already
martingales under the physical probability measure! This is a very nice trick, because
in this case we would already have an EMM. (Provided we can describe its
probability structure, which may still be quite a challenge, we could implement
it.) If the above was literally true, then the no-arbitrage condition would also
automatically hold.
If discounted prices were already martingales under the physical probability
measure, then an EMM already exists and therefore there can be no-arbitrage,
due to FTAP1.
We could use that EMM to price contingent claims as well. The only
question that would remain is uniqueness of the EMM or, what is equivalent,
market completeness.
What this all amounts to in the context of the BOPM, Nу1 is that p′=p,
which may be a fairly remote possibility. The BOPM also assumes that stock
prices are well described by a Binomial process.
Turning to the empirical side, the early empirical literature focused on
something very close to this hypothesis, which we will call the martingale
hypothesis for the physical probability measure. It wasn’t exactly this hypothesis
because actual prices were not discounted. However, for the short intervals
of time over which price changes were measured, this would make little
difference since the discount factor is small.
Commodity prices and stock prices have been intensively studied for a very
long time. Many of the early empirical studies found little or no signiﬁcant
correlation between past changes in stock prices and subsequent price changes. This
ﬁnding was also conﬁrmed for futures prices. This suggested that empirical
stock price processes meet the condition of ‘independent increments’.
The ﬁrst model of prices consistent with the empirical ﬁndings was the
random walk model or what we will call, in continuous time, the arithmetic
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Brownian motion model (ABM). We will be discussing this model in detail in
Chapter 16.
Later, it was found that independence was too strong a condition, because
the results of the empirical tests seemed only to establish the weaker uncorrelated
increments property (see the Appendix, section 15.7, for this distinction).
This is when martingales entered the picture, because Samuelson (1965)
produced his theoretical paper explaining the empirical ﬁndings that many
researchers had observed up to that time.
As established in the Appendix, section 15.7, if the underlying martingale
process has ﬁnite ﬁrst and second moments, then martingales will have
uncorrelated increments. Therefore, martingales with ﬁnite ﬁrst and second
moments are consistent with the empirical evidence.
After this initial research, a vast amount of further research formulated and
tested the efficient market hypothesis (EMH). There are several variations of this
model, namely weak, semi-strong, and strong form efﬁciency.
The EMH relates the expectations of future prices to information sets.
Let IN denote the information available at time N for N=0,1,2,…, and note
that IN is an evolving information set in the sense that, I0⊆I1⊆I2⊆I3⊆… .
This just means that as time evolves, the currently available information
set—which includes all previous information—gets bigger, or at least it doesn’t
get smaller. Previously available information is also currently available.
Now we can state the EMH in its three forms in terms of martingales. We
are standing at time t=N and trying to predict the random stock price one
period ahead, at time t=N+1. We have available the information set at time
t=N, IN. What can we say?
According to the EMH, the best predictor of future stock prices is the current
price SN . The EMH is just the martingale hypothesis, where the conditioning
variable is information itself. The EMH says that the sequence of prices is a
martingale with respect to IN. That is,
E(SN+1)|IN)=SN for all N=0,1,2,3,…
The currently available information set, IN , can be in at least one of three
forms:
1. Weak-Form EMH, IN={all historical stock prices S0, S1, S2,…SN}.
2. Semi-Strong Form EMH, IN={all historical stock prices and other publicly
available information, such as accounting statements as of time t=N}.
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3. Strong-Form EMH, IN={all publicly available information and also all
private information, for example, information held by insiders}.
There are a number of interesting features of the EMH. First, it ties price
formation in with information and information ﬂows. By analyzing the nature
of information ﬂows, one can say quite a bit about the structure of equilibrium
prices. For example, are price paths continuous, or do they have predictable
or unpredictable jumps?
Second, we know that square-integrable martingales have increments that
are uncorrelated (see section 15.7). This means that past information IN is useless
in predicting future prices or even subsequent price changes.
That’s why martingales predict using the current value, SN . That’s also
why martingales can do no better in predicting than the current market price.
There are no trends in prices, martingales are driftless. Any kind of trend or
correlation could be used to forecast prices and could lead to arbitrage
opportunities. Thus the martingale formulation is one that is consistent with
no-arbitrage.
All forms of the EMH conclude that it is effectively impossible to predict
future price changes from past information. The different forms of the EMH
specify what information one has access to in attempting to predict.
It is pretty clear that if a market is informationally efficient in any reasonable
sense, then there exists a positive, linear pricing mechanism (this would be
implied by efﬁciency). Then, the no-arbitrage condition must hold by the
FTAP1. That is, there can be no-arbitrage opportunities in an informationally
efﬁcient market.
Another interesting aspect of the EMH is that it says that one need not
look far and wide for at least one EMM. If actual prices form a martingale,
and discounting has minimal effect on the actual prices, then the most obvious
example of an EMM is the actual price process! The actual price process is
equivalent to itself. Of course, this can only happen if there is no-arbitrage.
What could stand in the way of the actual price process being an EMM?
The risk-free rate r >0 is one candidate. Discounting prices by the numeraire
eliminates that part of drift caused by the risk-free rate and thereby makes the
discounted price process (but not the actual price process) a martingale.
If there is an empirically measurable risk premium in stock prices, then actual
prices would form a sub-martingale, not a martingale. A risk premium in stock
prices would not create arbitrage, but rather is a reward for taking on nondiversiﬁable risk(s). If these are the only two factors contributing to positive
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drift, which is not consistent with martingale behavior, then by de-trending
the price series, one should be left with a martingale.
Even if actual (de-trended) prices do not form a martingale, the EMH
suggests a very interesting hypothesis. That hypothesis is that the actual price
process cannot be very far from being a martingale, as long as no-arbitrage
holds. Because, if there is no-arbitrage, we know that there is at least one
EMM for the discounted price process.
So the question is, if no-arbitrage holds, then do actual price processes (while
not necessarily martingales themselves) have a martingale component?
This takes us into the research literature. Instead of going that route, we
will examine some actual continuous time price processes that are consistent
with the EMH, and for which we can price options and therefore, for which
there is at least one EMM for the discounted price process (FTAP1). We will
also look for martingale components in the actual price processes. Chapter 16
begins this program.
15.7 APPENDIX: ESSENTIAL MARTINGALE PROPERTIES
Here we collect a few of the many properties of martingales that are used in
proving results that make martingales useful in applied ﬁnance. We restrict
attention to discrete-time martingales and sometimes even choose N=2. No
attempt at mathematical rigor is claimed. The intuition behind these results
is the primary concern.
We start with a discrete-time stochastic process (Xn()n=0,1,2,3,… with ﬁnite
ﬁrst and second moments E(Xn )<∞ and E(Xn2 )<∞ for all n=0,1,2,3,… and the
martingale property,
E(Xn+1()|Xn)=Xn for all n=0,1,2,3,…
(MP1)
1. Tower Property (TP)
E(X2|X0)=E{E(X2|X1)|X0}
(TP)
This is a general property of conditional expectations and doesn’t require
a martingale. The Tower Property says that, whenever we condition a
martingale process by an event earlier than the immediately preceding one,
like X0, we can break this down into a two-step process. First, condition by
the immediately preceding event. Then condition that conditional expectation
by the earlier one.
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One implication of TP is that the martingale property MP also holds for
all earlier events. Let’s see. Suppose (Xn())n=0,1,2 is a martingale.
E(X2|X0)=E{E(X2|X1)|X0}
=E{X1|X0}
=X0.
Here we used the martingale property twice. First, E(X2|X1)=X1. Second,
E(X1|X0)=X0. The TP allows us to rephrase the martingale property as,
E(Xn+1()|Xm)=Xm for all mр n=0,1,2,3,…
(MP2)
Sometimes you will see the (MP1) property stated as (MP2). By the tower
property, (MP2) follows from (MP1).
2. Double Expectations (DE)
This is another general property of conditional and unconditional expectations,
with important applications to martingales,
E{E(X2|X0)}=E(X2)
(DE)
This says that we ﬁrst calculate the conditional expectation of a random variable,
say X2, with respect to another random variable, say X0 . If we integrate out
the conditioning variable, X0, then we obtain the unconditional expectation
E(X2).
One immediate application of DE is to show that the means of a martingale
process are constant. Start with a martingale process (Xn()), n=0,1,2.
E{E(X2|X1)}=E(X2)
by (DE)
E(X2|X1)=X1
by (MP1)
Therefore,
E{E(X2|X1)}=E(X1).
So we conclude that E(X2)=E(X1).
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This justiﬁes the statement that a martingale process neither increases nor
decreases on average. The same statement is that the means of a martingale
process are constant.
3. Uncorrelated Martingale Increments (UCMI)
This is probably the most important property of martingales from the point
of view of ﬁnance. Recall that we assume that our martingales have ﬁnite
means and variances. Then the result is that the changes of a square-integrable
martingale process, while not necessarily independent, are uncorrelated.
Consider the two martingale increments (changes) X2–X1 and X1–X0. First
note that E(X2–X1)=E(X2)–E(X1)=0 because of the constant means property.
For the same reason, E(X1–X0)=E(X1)–E(X0)=0.
The deﬁnition of the covariance between two random variables X and Y
is Cov(X,Y)=E{(X–E(X))(Y–E(Y))}. In our case X=X2–X1 and Y=X1–X0 so,
for our case, Cov(X,Y)=E{XY} because E(X)=E(Y)=0.
Then,
Cov(X2–X1,X1–X0)=E{(X2–X1)(X1–X0)}
=E{X2–X1)X1}–E{(X2–X1)X0}.
We will consider each term separately. For the ﬁrst term, E{(X2–X1)X1},
E{(X2–X1)X1} =E{E{(X2–X1)X1|X1}} by (DE),
=E{E{(X2X1–X12)|X1}} by expanding all terms,
=E{E(X2X1|X1)–E(X12|X1)} by linearity of
conditional expectation,
2
=E{X1E(X2|X1)–X1 } by factoring out the constant
X1 in E(X2|X1), and noting that
E(X12|X1)=X12,
=E{X1X1–X12} by (MP1)
=E{X12–X12}
=0.
Next we consider the second term E{(X2–X1)X0},
E{(X2–X1)X0} =E{E{(X2–X1)X0|X0}} again by (DE),
=E{E{(X2X0–X1X0)|X0}} by expanding all terms,
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=E{E(X2X0|X0)–E(X1X0|X0}} by linearity of
conditional expectations,
=E{X0E(X2|X0)–X0 E(X1|X0)} by factoring out the
constant X0 in E(X1|X0),
=E{X0X0–X0X0} by (MP2)
=E{X02–X02}
=0.
Combining both terms, we have proved that Cov(X2–X1,X1–X0)=0, so that
the martingale increments X2–X1 and X1–X0 are uncorrelated.
4. Degrees of ‘Independence’: Independent, Uncorrelated, and
Orthogonal Random Variables
Understanding the EMH requires an intimate knowledge of the types of
correlation between random variables that can occur. So we collect the three
basic notions here.
A. Two random variables X() and Y() are said to be independent if
E[X()*Y()]=E[X()]*E[Y()].
B. Two random variables X() and Y() are said to be orthogonal if
E[X()*Y()]=0.
C. Two random variables X() and Y() are said to be uncorrelated if
COV(X(),Y())=E[(X()–E(X())(Y()–E(Y())]=0.
This means that, after subtracting their means, X′()=X()–E(X()) and
Y ′()=Y()–E(Y()) are orthogonal.
The strongest form of unrelatedness is independence. Then comes
orthogonality. Finally, there is uncorrelatedness. Thus, martingales enjoy a weak
form of unrelatedness between their increments. Weaker than independence
of increments, which some processes exhibit.
n
n
n
EQUIVALENT MARTINGALE MEASURES
n
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
n
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KEY CONCEPTS
Primitive Arrow–Debreu Securities and Option Pricing.
Exercise 1, Pricing B(0,1).
Exercise 2, Pricing ADu() and ADd().
Pricing a European Call Option.
Pricing any Contingent Claim.
Equivalent Martingale Measures (EMMs).
Introduction and Examples.
Deﬁnition of a Discrete-Time Martingale.
Martingales and Stock Prices.
The Equivalent Martingale Representation of Stock Prices.
The Equivalent Martingale Representation of Option Prices.
Discounted Option Prices.
Summary of the EMM Approach.
The Efﬁcient Market Hypothesis (EMH), A Guide to Modeling Prices.
Essential Martingale Properties.
END OF CHAPTER EXERCISES FOR CHAPTER 15
1. (Pricing AD securities in a Binomial model)
Make the usual assumptions of the BOPM, N=1. Suppose that u′=8%,
r′=5% and that d′=–3%. The current stock price is S0=$100. Assume that
the term to expiration is one year, so everything is already annualized.
a. Calculate P0(ADu()).
b. Calculate P0(ADd()).
2. Assume the same scenario as in exercise 1 with the same parameters.
a. Price an at-the-money European call option.
b. Price an at-the-money European put option.
3. Write out the single-period, discrete version of European Put-Call Parity
and conﬁrm that the results of your calculations conform to it.
4. Consider a generalized option with payoff function MAX[ST2 –K, 0]. Use
AD securities to price it in the usual Binomial framework.
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SELECTED CONCEPT CHECK SOLUTIONS
Concept Check 3
E(W1|W0)=E(W0+X1()|W0)
=E(W0|W0)+E(X1()|W0)
=W0+E(X1()|W0)
=W0+E(X1())
=W0+0
=W0.
We used the fact that X1() is independent of W0 in the fourth equality. Our
conclusion is that the property E(W1()|W0)=W0 holds.
CHAPTER 16
OPTION PRICING IN
CONTINUOUS TIME
16.1 Arithmetic Brownian Motion (ABM)
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16.2 Shifted Arithmetic Brownian Motion
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16.3 Pricing European Options under Shifted Arithmetic
Brownian Motion with No Drift (Bachelier)
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16.3.1 Theory (FTAP1 and FTAP2)
542
16.3.2 Transition Density Functions
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16.3.3 Deriving the Bachelier Option Pricing Formula 547
16.4 Defining and Pricing a Standard Numeraire
551
16.5 Geometric Brownian Motion (GBM)
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16.5.1 GBM (Discrete Version)
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16.5.2 Geometric Brownian Motion (GBM),
Continuous Version
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16.6 Itô’s Lemma
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16.7 Black–Scholes Option Pricing
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16.7.1 Reducing GBM to an ABM with Drift
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16.7.2 Preliminaries on Generating Unknown
Risk-Neutral Transition Density Functions
from Known Ones
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16.7.3 Black–Scholes Options Pricing from Bachelier
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16.7.4 Volatility Estimation in the Black–Scholes
Model
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16.8 Non-Constant Volatility Models
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16.8.1 Empirical Features of Volatility
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16.8.2 Economic Reasons for why Volatility is not
Constant, the Leverage Effect
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16.8.3 Modeling Changing Volatility, the Deterministic
Volatility Model
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16.8.4 Modeling Changing Volatility, Stochastic
Volatility Models
16.9 Why Black–Scholes Is Still Important
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In this chapter we are going to give an introduction to continuous-time ﬁnance.
This can be a daunting and rather technical subject but we will try to cut
through all of that and stress the intuition. At the same time, we want to work
through some important examples to get the ﬂavor of the continuous-time
framework.
Many of the same ideas from the discrete-time framework we have been
discussing carry over to the continuous-time case. Risk-neutral valuation carries
over, and dynamic hedging which we discussed in the BOPM, N>1 is the
very essence of continuous-time trading and hedging.
Equivalent martingales measures (EMMs) form the foundation for the
modern approach to pricing derivatives, as we have discussed in Chapter 15.
The fundamental theorems of asset pricing, FTAP1 and FTAP2 , apply in this
continuous-time context as well.
We will begin with the prototype of all continuous time models, and that
is arithmetic Brownian motion (ABM). ABM is the most basic and important
stochastic process in continuous time and continuous space, and it has many
desirable properties including the strong Markov property, the martingale
property, independent increments, normality, and continuous sample paths.
Of course, here we want to focus on options pricing rather than the pure
mathematical theory. The idea here is to partially prepare you for courses in
mathematical ﬁnance. The details we have to leave out are usually covered
in such courses.
16.1 ARITHMETIC BROWNIAN MOTION (ABM)
ABM is a stochastic process {Wt()}tу0 deﬁned on a sample space (⍀,ℑW,℘W ).
We won’t go into all the details as to exactly what (⍀,ℑW,℘W ) represents
but you can think of the probability measure, ℘W, which is called Wiener
measure, to be deﬁned in terms of the transition density function p(T,y;t,x) for
=T–t,