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Social Mysteries of Prices of Assets and Derivatives

J. Michael SteeleThe Wharton SchoolUniversity 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 David Ricardo (1772-

1823) made a fortune speculating on British bonds before the battle of Waterloo.

Irving Fisher (1867-1947) invented the “rolodex,” made a fortune, and lost it all speculating in 1929.

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MSO: Can You Find the Day When Martha Was Convicted?

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A Three Part Plan with a Bonus 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

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 ModelArbitrage Price: St P+ - K e-rt P-

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 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 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 risk-free 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 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 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 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 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 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’

time

Ot

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What Do We Do? What Do We Get ? 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

LogicHere we have a price process which contains both the BS model and one that is highly predictable.

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 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: “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… 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” 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…

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 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: 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 ofRevisionism… Examples and “Reasons” 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 well- established standards, and our many of our streams have become polluted.

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“Take Aways” and a … Trailer 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 “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!