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Social Mysteries of Prices
of Assets and Derivatives
J. Michael Steele
The Wharton School
University of Pennsylvania
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First: Some Ambidextrous Attitudes
Toward Speculation
“While London's financial men
toiled many weary hours in
crowded offices, he played
the market from his bed for
half an hour each morning.
This leisurely method of
investing earned him
several million pounds for
his account and a tenfold
increase in the market value
of the endowment of his
college, King's College,
Cambridge.” (B. Malkiel)
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The Spirit of Speculation Has Been Part
of a Long Tradition in Economics
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David Ricardo (17721823) made a fortune
speculating on British
bonds before the battle
of Waterloo.
Irving Fisher (18671947) invented the
“rolodex,” made a
fortune, and lost it all
speculating in 1929.
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G.W. Bush in ’04: Futures Contacts
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Pays 100 points if GWB
wins in November
TradeSports.com
Contract Hi~75, Lo~57
Point=Dime (1/10 USD)
Bid-Ask Gap~.8 point
Exchange Fee= 4 cents
each way.
One contract: $6 bet,
open contracts ~ 218K
4
MSO: Can You Find the Day When
Martha Was Convicted?
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A Three Part Plan with a Bonus
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Introductory Observations that Illustrate
“anomalous” price processes (that’s done!)
Reflection on the recent past: BS as a social
event, as applied mathematics, and as a
science paradigm. (I know you know; I’ll be
quick --- but maybe you don’t know.)
Facing Empirical Realities --- The Main Point.
… ah, yes, the Bonus.
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The Samuelson Model
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Stock Model: dSt = µ St dt + σ St dWt
Bond Model: dβt = r βt dt
Some Features of note:
(1) the “volatility” σ is constant, and
(2) the model is Markovian.
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Without it we would not be here today:
Pricing of European Call Option
Under the Black-Scholes Model
Arbitrage Price: St P+ - K e-rt P
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P+/-=Φ(d+/-/{σ sqrt(T-t)} )
d+= log (St/K)+(r +σ2/2)
d-= log (St/K)+(r -σ2/2)
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Examination of the Social Epistemology
of “Black-Scholes” :The Technical Side.
Black and Scholes give
two arguments for their
pricing formula.
One of these is widely
repeated and uses the
Ito “analog” of
(f/g)’=f’/g.
The other argument has
“not been seen again.”
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The Famous Delta Hedge Argument
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In 1973 Black and Scholes follow a lead
from Beat the Market by Thorp and Kassouf.
Linearizing through the origin they consider
the portfolio:
Xt = St - f( t, St ) / fx( t, St )
Ito’s Formula with the odd (φ/ψ)’= φ’/ψ twist
Yields the Black-Scholes PDE
Economic vs Mathematical Reasoning
Motivation for a “PDE Model”
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The Less Famous CAPM Argument
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Return on any asset will (in theory) be equal
to the risk-free rate plus a multiple of the
“return of the market” in excess of the riskfree rate.
The multiplier is just the covariance of the
asset return and the market return, divided by
the variance of the market return (Beta).
Apply this to St and f(t, St) to get two
equations. Clear the market, get the BS-PDE
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Two Questions with (Partial) Answers
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What if you don’t use (φ/ψ)’= φ’/ψ in the delta
hedge argument? What do you get?
Answer: You get a nonlinear PDE which
must be in some sense approximated by the
Black-Scholes PDE, but no one seems to
have pursued this program.
Why did the CAPM argument just disappear?
Answer: Because it was pure flim-flam. You
can replace CAPM with a cubic or quadratic
and the “argument” goes through.
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Where the Arguments Took Us First
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Empirical performance is not particularly
good --- not then, not now.
The Delta Hedge idea had serious impact on
the practical world of finance.
The two motivational arguments of Black and
Scholes have been supplemented by more
satisfying arguments by Merton and
especially by Harrison and Kreps.
Martingale theory now almost completely
eclipses the PDE theory.
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Reexamination of the Fundamentals
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Here we have made assumptions about both
the underlying price process and the logic of
arbitrage.
Much is known about the drawbacks of GBM
as a model for price --- though we will soon
review some new findings.
It is harder, but still possible, to question the
logic of arbitrage pricing.
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One Way to Examine the Logic:
A “New” Textbook Example
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Simple, with a decent story
Explicitly solvable with the tools at hand
Suggesting simple inferences that are at
odds with intuition
Resolved by seeing that these confusions
with us all along…
And revived by suggesting that those
confusions may not be silly after all.
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A State-Space Candidate:
The Observational Model for Stock Price
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BM with Drift: dXt = µ dt + σ dWt
Model for Wobble: dOt = -α Ot dt + ε dWt’
Model for Price: St = S0 exp (Xt +Ot )
The point is that St is essentially geometric
Brownian motion, but with a mean reverting
observational error.
Please Note: St is NOT a Markov process
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Mean Reverting Process
dOt = -αOt dt + εdWt’
Ot
time
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What Do We Do? What Do We Get ?
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Martingale Pricing Theory is up to the task.
An easy exercise gives one a formula for the
price of a European call option.
At first it may be surprising, but you get
EXACTLY the Black-Scholes formula,
Except that the old σ is replaced by a function
of the new model parameters.
The PDE approach is meaningless in this
context, but …
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“Interpolation of Models”
as a way to test our Logic
Here we have a price process which contains
both the BS model and one that is highly
predictable.
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Martingale pricing theory applies seamlessly
as we move from one extreme to the other.
Can we have appropriately priced options
under a model which makes every man a
king? You tell me (I’m sure you will!)
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The Mystery Fades Out, then Fades In
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To be fair, this “new” model may only add a
small stochastic confusion to a familiar fact
“Every baby knows” that µ does not matter in
the BS price of an option; what we see here
a variation on that old story.
They also know why µ doesn’t matter --- but
can we trust what we have taught them?
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John von Neumann once said:
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“In mathematics, you don’t understand things,
you just get used to them.”
Von Neumann had in mind such things as
the Pythagorean theorem as the basis for the
geometry of d-space, or …
Here we might ask honestly ask (after we
side-step any silly tautologies), “Is µ really
and truly irrelevant?”
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After I clean the pie off my face…
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OK, so you are unmoved. You can’t say I
didn’t try…maybe another day. Anyway, let’s
move to a less contentious issue.
Many people are willing to agree that as a
pricing model GBM is past its sell-by date.
What do we do about it? Where is our next
model coming from?
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First, Hi-Frequency Gives Us a More
Honest View of “Volatility”
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We want (squared) “volatility” to mean
something like the current growth rate of
quadratic variation.
More commonly “volatility” is use to mean the
value of some parameter in some model --so its meaning can vary from place to place.
Model-based volatility can be self-fulfilling.
If we stick with the honest definition, we need
hi-frequency data. Jonathan Weinberg has
done this,and he finds a nice story.
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What a Quick Look at the
Picture Tells Us…
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These are honest QV volatilities, but they are
not log-volatilities, we’ll get to those shortly.
We see “long-range” dependence via the
ACF, but…
The PACF tells us that 6 days, tells the tale.
Similar pictures apply for MRK, GE, etc.
To the eye, this might support the SV model,
but there is more to the story.
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What a Second Look
at the Picture Tells Us
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If one takes logs of the (QV-defined)
volatilities and fits an AR(1) model, the story
for the SV price model falls apart.
The residual time series has an ACF with
small but significant and non-decaying
coefficients.
The PACF has many significant coeffients.
Even the lame Ljung-Box is highly significant;
we reject the SV model quite handily.
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Its Maybe Confusing…but it is What
We’ve Got:
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The history of the Black-Scholes formula has
more dark alleys than is it is customary to
acknowledge.
We understand µ, or perhaps we don’t. We
collectively agree that we don’t have a handle
on µt.
We know σ is not constant, but there is
(probably) no point in pretending that Log(σt)
is an AR(1).
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We are in an interesting time of
Revisionism… Examples and “Reasons”
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More now argue that “long-range”
dependence which has had some vogue, is
perhaps just an artifact of non-stationarity of
the underlying price process.
LTCM reminds us that “in the extreme” all
markets become correlated.
Oddly enough, we don’t have solid wellestablished standards, and our many of our
streams have become polluted.
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“Take Aways” and a … Trailer
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The logic of arbitrage pricing is not yet
established beyond a question of doubt, even
if it is closed to “as good as economics gets.”
Popularity of a model should be meaningless
as far as science goes, but on a social level it
always maters more than one could imagine.
As theoreticians, we need to read the fine
print and not trust empirical work to others.
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The Promised Trailer:
The Cauchy-Schwarz Master Class
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“A three-hundred page
book about a one-line
inequality”
Coaching for problem
solving, plus all of the
classical inequalities
viewed with new eyes
The real truth about
Bunyakovsky…
Thanks!
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