Download Option Pricing and Market Efficiency ROBERT JARROW

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Option Pricing
and Market Efficiency
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this false belief by demonstrating the intimate
relationship between these two concepts.
Understanding the relation between
market eff iciency and option pricing is
important because it generates new insights
that are relevant to applying these concepts in
practice. For market efficiency, these insights
provide new methods for testing markets for
efficiency with respect to an information
set, avoiding the curse of a joint hypothesis.
For option pricing and hedging, these new
insights give a better understanding of the
current methodology’s limitations and provide a prescription for instances when the
standard option-pricing methodology doesn’t
work, thereby facilitating its proper usage.
The insights of this article are based on
new results recently proven in mathematical
finance. (See Jarrow and Larsson [2012] and
Jarrow [2012].) In this article, we review
the definition of an informationally efficient
market and the arbitrage-free option-pricing
methodology, focusing on risk-neutral valuation. We conclude with a discussion of new
insights about option pricing usage and their
implications.
C
T
he notions of market informational
efficiency and the arbitrage-free
option pricing methodology are
two of the most successful insights
developed by finance academics and applied in
practice over the past forty years. An informationally efficient market is one where prices
ref lect information instantaneously, implying
that positive alpha trading strategies based on
that information are nonexistent. Modern
theory prices options under the assumptions
of arbitrage-free and complete markets,
implying that hedging directional and volatility risk are possible for many asset classes,
including equities, fixed-income securities,
foreign currencies, and commodities. The
importance of these concepts is evidenced by
the fact that they are standard textbook material taught worldwide in beginning finance
courses.
Most financial academics and practitioners believe that these topics are only
weakly related, if at all. Indeed, the standard
textbook presentation of market efficiency
doesn’t discuss option pricing theory. And
conversely, the standard textbook presentation of option pricing doesn’t mention market
efficiency. Surprisingly, this common belief
is false! Although not well understood, current option pricing theory is closely related
to the notion of an informationally efficient
market. The purpose of this article is to dispel
IS
is the Ronald E. and
Susan P. Lynch Professor
of Investment Management at Cornell University
in Ithaca, NY.
[email protected]
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ROBERT JARROW
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ROBERT JARROW
OPTION P RICING AND M ARKET EFFICIENCY
MARKET EFFICIENCY
Intuitively, a market is efficient with
respect to an information set F if the prices
of the assets trading in that market instanta-
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neously ref lect this information. That is, it is not possible to find positive alpha trading strategies based on
this information, because informed traders trade on this
information the moment it becomes available, causing
prices to move immediately to their correct values.
Three different information sets F are usually
identified. Current and past price and volume data
comprise the weakest information set, called weak-form
efficiency. All publicly available information from balance sheets, the financial press, and fundamental analysis comprise semi-strong-form efficiency. Finally, all
information, including private/inside information, comprises strong-form efficiency. Analysts have tested these
different forms of efficiency for more than forty years
(see Fama [1991, 1998]). The accumulated evidence
supports weak-form efficiency. Evidence is mixed for
semi-strong-form efficiency and rejects strong-form
efficiency. Active portfolio management-using portfolio
theory to identify positive alphas-would be of no use if
semi-strong-form efficiency worked well.
Tests of the informationally efficient market are
commonly subject to the curse of a joint hypothesis. To
test efficiency, analysts usually hypothesize a particular
equilibrium model (e.g., a particular three-factor assetpricing model), then begin the search for a positive alpha.
If they discover a positive alpha, they cannot know if its
existence is due to the hypothesized equilibrium model’s
invalidity or the market’s inefficiency. Either could cause
positive alphas. This is why the evidence is mixed for the
semi-strong-form case. True believers in efficiency can
always discredit the hypothesized equilibrium model.
Those who don’t believe in efficiency stop at the first
rejection.
Recently, scholars formalized the definition of
an informationally efficient market, using the tools of
mathematical finance (see Jarrow and Larsson [2012]).
They define a market as efficient with respect to an information set F if there exists an equilibrium economy
(satisfying the standard conditions of supply equal to
demand1) that supports the market’s price process, where
the price process ref lects the information set F.2 That
is, the observed price process could be generated by an
equilibrium economy consisting of rational investors
trading stocks to maximize their preferences, knowing
the information set F.
This definition does not require that we identify
the equilibrium economy generating the price process,
only that one such economy exists. This definition is
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easily seen to be consistent with Fama’s [1970] original
definition, and the manner in which market efficiency
has been tested over the subsequent years (see Fama
[1991, 1998]) using the joint-model hypothesis. Indeed,
as mentioned before, the standard approach to testing
efficiency is to assume a particular equilibrium model,
then show that the model is consistent or inconsistent
with historical observations of the price process and different information sets. If the model is consistent with
the data, we accept informational efficiency. If inconsistent, we don’t necessarily reject informational efficiency,
because of the joint hypothesis. To reject efficiency, we
must show that there is no equilibrium model consistent
with the data.
A relevant theorem was first proved by Jarrow and
Larsson [2012].
Theorem (Market Efficiency). The market is efficient with respect to an information set F if and only if the
economy satisfies no arbitrage 3 and no dominance using the
information set F.
No arbitrage is the standard notion that there does
not exist a trading strategy that requires zero initial
investment, never loses money, but with strictly positive
probability makes money under some circumstances.
No dominance means that there do not exist two different traded assets with the same initial cost, with one
traded asset always earning at least as large returns as the
other, and in some circumstances strictly larger returns.
Although related notions, no dominance is weaker. If
there are short-sale restrictions in an economy, no arbitrage may not apply, but no dominance does.
The theorem states that, if using the information
set F follows arbitrage or dominance in the economy,
then the market is not informationally efficient, that is,
there can be no equilibrium economy with the observed
price process. This makes sense, because if there is arbitrage or dominance, then either investors are not holding
their optimal portfolios (because trading more generates
more wealth), or supply does not equal demand (because
no one wants to hold the dominated assets).
This theorem is important for testing market efficiency. It lets us side-step the joint-model hypothesis
to reject market efficiency. If we can find an arbitrage
opportunity or a dominance trading strategy based on
the information set F, then the market is inefficient.
In essence, such a demonstration precludes all possible
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equilibrium models. Yet this theorem still does not provide a method for accepting market efficiency. To accept
market efficiency using this theorem, we must exhaust
all possible arbitrage and dominance-trading strategies,
which is an impossible task. However, using some additional insights from mathematical finance, it is possible
to accept market efficiency using a conditional joint
model hypothesis. This argument follows a review of
the arbitrage-free option-pricing methodology.
THE FUNDAMENTAL THEOREMS
OF ASSET PRICING
Arbitrage-free option pricing theory is based on
three fundamental theorems. The first fundamental
theorem relates no arbitrage to the existence of riskneutral probabilities that make the asset’s price, normalized by the value of a money market account, nearly a
martingale.
Consider a market with trading over the time
horizon [0,T ] in a stock with market price St and a
money market account with value
t
Bt = e ∫0
rsds
d
where rs is the default-free spot interest rate at time s.
Let both price processes ref lect the information set F.
Let the probabilities generating the randomness in these
price processes be denoted P. These are called the statistical probabilities.
Theorem (First Fundamental Theorem). There is
no arbitrage in the sense of no free lunch with vanishing risk
(NFLVR) if and only if there exist probabilities Q, which
agree with P on zero-probability events, such that BStt is a local
martingale.
The probabilities Q are called risk-neutral probabilities. Two concepts in this theorem need further
explanation. NFLVR is a technical extension of the standard definition of no arbitrage. It also excludes limiting
arbitrage opportunities, that is, trading strategies that are
almost arbitrage opportunities, in that they may have a
very small probability of loss. (See Jarrow and Protter
[2008] for a complete discussion.)
Recall that a martingale is a stochastic process in
which the current value equals the expected future value.
Martingales ref lect fair games, because deviations from
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expectations only occur by chance and are unpredictable.
A local martingale is a generalization of a martingale. It
is a stochastic process that is a martingale if it stops early
at a set of random times (before time T).4 The idea is
that, although it is not a true martingale, it is a martingale
if stopped early. We call a local martingale that is not a
martingale a strict local martingale.
Although technical in construction, there is an
economic interpretation for the difference between a
stock price that is a local martingale and a true martingale. A stock price that is a local martingale can have a
price bubble. A stock price that is a martingale cannot.
To understand this economic interpretation, we must
introduce additional notation.
Let ST be the stock’s liquidation value at time T,
and (for simplicity) suppose that the stock pays no dividends over its life. The stock’s fundamental value is defined
as the amount we would pay to buy the stock, hold it
until the liquidation date, and never resell it.
Given no arbitrage, the first fundamental theorem
tells us that risk-neutral probabilities Q exist. There could
be many of these probability measures. Choose the one the
market uses to price derivatives, which can be obtained by
calibration. The fundamental value is defined as
T
⎛
⎞
− rsds
d ⎟
⎜
⎜
⎟
t
⎟
t ⎜⎜ T
⎟
⎜
⎟
⎝
⎠
fvt = E S e
∫
(1)
where Et (⋅) is the time t expectation using the riskneutral probabilities Q.
This expression represents the expected discounted
cash f low from holding the stock until the liquidation
date T where discounting is done using the default-free
spot interest rate. (The interest earned on the money
market account.) The risk of the liquidation value is
incorporated into the present value computation through
using the risk-neutral probabilities, not the statistical
probabilities.
Finally, an asset price bubble is defined as the difference between the stock’s market price and its fundamental value, that is,
β t = St
f t ≥0
fv
(2)
We can now connect the difference between local
martingales and martingales as they relate to asset-price
bubbles. It follows easily from this last expression that:
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Corollary (Local Martingales and Price Bubbles).
Given no arbitrage in the sense of NFLVR, there is an assetprice bubble, βt > 0, if and only if the risk-neutral probabilities
Q only make BStt a local martingale and not a martingale.
The first fundamental theorem of asset pricing only
guarantees the existence of a risk-neutral probability Q.
It does not guarantee that this probability is unique.
There could be more than one. The second fundamental
theorem of asset pricing tells us that this probability
is unique if and only if the market is complete. The
market is said to be complete if an arbitrary derivative’s
random payoff at time T, say CT, can be obtained with
a self-financing trading strategy in the stock and money
market account.5 For example, CT could be the payoff
from a European call option on the stock.
Theorem (Second Fundamental Theorem).
Assuming no arbitrage in the sense of NFLVR, the market
is complete if and only if the risk-neutral probability Q is
unique.
tingale. By the corollary, if it is a martingale, the market
is efficient. If it is a strict local martingale, the market is
inefficient. Conditional upon finding a “good” stockprice process, we can accept that the market is efficient
with respect to the information set F.
To illustrate this approach, we hypothesize that the
stock price follows a constant elasticity of variance (CEV)
process, given by the stochastic differential equation:
dSt = βStαdW
Wt
(St )dt
where Wt is a standard Brownian motion and α,β are
strictly positive constants. For the parameter α:
if
α ∈(0, 1]St iis a martinga
n le
if
α ∈(1, ∞ ) St iis a strict local martingale
Two immediate corollaries of the third fundamental theorem relate market efficiency to the risk-neutral probabilities. The first uses the theorem on market
efficiency.
Of course, the α = 1 case is geometric Brownian
motion, which underlies the Black–Scholes–Merton
model.
We can test how well the CEV process fits historical stock-price data. If the CEV process is accepted
as a good stock-price model, then we can estimate the
parameter α test to see if it is greater than or less than
one. If α is less than or equal to one, the stock price is a
martingale. In this case, using the corollary, we accept
the hypothesis that the market is efficient.
Of course, this method’s hinges on finding a stochastic process that fits the historical data well. This is
not a limitation, because there are a plethora of available
stochastic processes.
The second corollary relates market efficiency
to the existence of stock-price bubbles in a complete
market.
Corollary (Market Efficiency). The market is efficient with respect to an information set F if and only if there exists
risk-neutral probabilities Q such that BStt is a martingale.
Corollary (Bubbles). Given a complete market, the
market is efficient with respect to an information set F if and
only if there are no stock-price bubbles.
This is an important corollary, because it enables
a new way to test for market efficiency, avoiding the
joint-model hypothesis. It provides a method based on
a conditional joint-model hypothesis, which lets us accept
market efficiency.
This new method first involves finding a stochastic
process that explains past stock prices well. This is the
“conditional” part of the test. We then test the stochastic
process, to see if it is a martingale or a strict local mar-
This is a useful corollary for understanding when
risk-neutral valuation does not apply.
The third fundamental theorem of asset pricing
characterizes the conditions under which the risk-neutral probabilities make the normalized stock process a
martingale, not just a local martingale. This theorem
was first proven by Jarrow and Larsson [2012].
Theorem (Third Fundamental Theorem). There
is no arbitrage in the sense of NFLVR and no dominance if
and only if there exist risk-neutral probabilities Q such that BStt
is a martingale.
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RISK-NEUTRAL VALUATION
This section reviews the standard approach used
for pricing derivatives. First, we assume both that the
market satisfies no arbitrage (in the sense of NFLVR)
and that the market is complete. Following the first and
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second fundamental theorems, this implies that there
exists unique, risk-neutral probabilities Q such that BStt
is a local martingale. Note that, at this point, the first
two fundamental theorems do not guarantee that BStt is
a martingale.
Next, the standard approach values a derivative
with a random payoff CT at time T using risk-neutral
valuation (RNV), which is
⎛
T
−
C t = Et ⎜⎜ CT e ∫t
⎜⎝
⎞
rsdds ⎟
⎟
⎟⎠
(3)
where Et (⋅) is the time t expectation using the riskneutral probabilities Q.
In RNV, the derivative’s time t price Ct is its
expected discounted payoff at time T, where the discount rate is the spot interest rate.
The uniqueness of the Q is important here. Otherwise, the derivative would not have a unique price.
This formula implies the following three facts,
which we state as lemmas.
Lemma 1. Given no arbitrage (in the sense of NFLVR)
and a complete market, RNV holds if and only if the riskneutral probabilities Q make BStt a martingale.
Lemma 2. Given no arbitrage (in the sense of NFLVR)
and a complete market, RNV holds if and only if the market
is efficient with respect to an information set F.
Lemma 3. Given no arbitrage (in the sense of NFLVR)
and a complete market, RNV holds if and only if there are no
price bubbles.
The standard approaches for pricing derivatives
(e.g., the Black–Scholes [1973], Merton [1973] model for
equity options, and the Heath–Jarrow–Morton [1992]
model for interest rate derivatives), impose conditions
on the evolution of the underlying assets such that the
market is complete and the risk-neutral probabilities
make the relevant normalized asset prices martingales.
Alternatively stated, by construction, these models
assume that the market is efficient with respect to an
information set F, or equivalently, that there are no
asset-price bubbles.
Summarizing, we have proven the following fact:
Standard option pricing theory is valid if and only if
the market is efficient with respect to an information set F, or
equivalently, there are no asset-price bubbles.
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This observation is not well known. It implies
that if one believes an asset-price bubble exists or the
market is inefficient with respect to the information set
F, then risk-neutral valuation does not work for pricing
or hedging derivatives. Also, the market need not have
any arbitrage opportunities.
Alternatively stated, if you believe that there are
positive alpha trading strategies based on F, then RNV
can not be used to price or hedge derivatives. Or, if you
believe the stock price exhibits a bubble, then risk-neutral
valuation cannot be used to price or hedge derivatives.
Under these conditions, using the standard risk-neutral
valuation approach will indicate false arbitrage opportunities. The methodology also leads to poor hedging,
because the stock price process is falsely assumed to be
a martingale. This implies that the assumed stock-price
process is incorrect. That is, those using this process are
hedging the wrong stochastic evolution.
CONCLUSION
The previous insights increase our understanding
of option-pricing theory and its relation to an efficient
market. We see that an efficient market is a necessary and
sufficient condition for the valid use of pricing derivatives via risk-neutral valuation. As such, we see that the
two concepts are not independent, but interrelated. No
arbitrage and market completeness are not all that is
needed to price derivatives. We also need the existence
of risk-neutral probabilities such that the normalized
stock price is a martingale, which is equivalent to market
efficiency.
But do these new insights generate prudent adjustments to standard market practice for either testing market
efficiency or pricing options? The answer is yes.
In testing for market efficiency, it provides a new
methodology that avoids the curse of the joint-model
hypothesis. To reject an efficient market, we just need
to find dominated assets or arbitrage opportunities. This
rejects an efficient market without assuming a joint
model. To accept market efficiency, we need to use a
conditional joint model. First assume a stochastic process
for the underlying asset’s price. Estimate the stochastic
process’ parameters and perform a goodness-of-fit test
for the model’s validity. If the evolution is valid, we can
test for efficiency by checking the estimated parameters
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to see if the process admits a martingale measure or
not. If a martingale, then the market is efficient. If not,
then the market is inefficient. The test for whether the
stochastic process is a martingale or not, in this context,
is equivalent to the test for bubbles. (See Jarrow, Kchia,
and Protter [2011].) If the evolution is rejected, we need
to find an alternative stochastic process and repeat the
procedure. Given the plethora of stochastic processes,
a suitable candidate that provides a good fit can always
be obtained.
With respect to option-pricing practice, if we
believe the underlying asset is exhibiting a price
bubble—the market is inefficient—then the standard
methodology should not be used for either pricing or
hedging. It should not be used for pricing because riskneutral valuation doesn’t hold. It should not be used for
hedging because the wrong stochastic process has been
assumed. Alternative pricing methods must be employed
in this circumstance.
These new insights also imply that more emphasis
should be placed on testing for bubbles before using an
option-pricing model. This can easily be accomplished
by performing goodness-of-fit tests using historical data
for the assumed stochastic process for the underlying
assets. Once a historically valid evolution is obtained,
we test to see if it exhibits a price bubble. If there is no
price bubble, it can be used to price options, using the
standard approach. If not, then the market is inefficient
and the model should not be used to price options. In
this circumstance, however, because the correct evolution is obtained, hedging is still possible. Although there
will be dominated assets in the market, given a valid
stochastic process, we can still determine the minimalcost method of hedging any portfolio.
There is an important caveat. After fitting a stochastic evolution for the underlying assets, based on
historical data, it is still possible for the economy to
have structural shifts. A process that had no bubbles in
the past could suddenly exhibit a bubble today. This
possibility suggests that any option-pricing procedure
should monitor for such an event by constantly re-estimating the model’s parameters to ensure that no bubble
is present. If a bubble appears, the methodology should
not be employed. Adding a prudent check will avoid
costly misuses of the standard theory in practice.
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ENDNOTES
1
These conditions are frictionless and competitive markets and situations in which investors are risk averse with
preferences that are increasing in wealth. Investors know the
information set F and choose their stock holdings to maximize their preferences. An equilibrium price is one at which
aggregate supply equals aggregate demand and all investors
are at their optimal holdings.
2
Formally, the price process is measurable with respect
to F.
3
Formally, no arbitrage is in the sense of NFLVR,
which is defined in the next section of the article.
4
Formally, the stochastic process Xt is a local martingale
if there exists a sequence of stopping times τn → ∞ a.s., such
that the stopped process X min(τn, t) is a martingale. (See Protter
[2005]).
5
Find the definition of a self-financing trading strategy
in Jarrow and Larsson [2012].
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To order reprints of this article, please contact Dewey Palmieri
at dpalmieri@ iijournals.com or 212-224-3675.
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