Social Mysteries of Prices of Assets and Derivatives

1
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

• 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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G.W. Bush in 04 Futures Contacts

• Pays 100 points if GWB wins in November
• TradeSports.com
• Contract Hi75, Lo57
• PointDime (1/10 USD)
• Bid-Ask Gap.8 point
• Exchange Fee 4 cents each way.
• One contract 6 bet, open contracts 218K

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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 (thats done!)
• Reflection on the recent past BS as a social
  event, as applied mathematics, and as a science
  paradigm. (I know you know Ill be quick --- but
  maybe you dont know.)
• Facing Empirical Realities --- The Main Point.
• ah, yes, the Bonus.

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The Samuelson Model

• Stock Model dSt µ St dt s St dWt
• Bond Model dßt r ßt dt
• Some Features of note
• (1) the volatility s is constant, and
• (2) the model is Markovian.

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Without it we would not be here todayPricing of
European Call Option Under the Black-Scholes
Model

• Arbitrage Price St P - K e-rt P-
• P/-F(d/-/s sqrt(T-t) )
• d log (St/K)(r s2/2)
• d- log (St/K)(r -s2/2)

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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 )
• Itos Formula with the odd (f/?) f/? 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 dont use (f/?) f/? 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 CandidateThe Observational Model
for Stock Price

• BM with Drift dXt µ dt s dWt
• Model for Wobble dOt -a Ot dt e 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 -aOt dt edWt
Ot
time
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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 s 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.
• 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
  (Im 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 doesnt matter --- but can
  we trust what we have taught them?

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John von Neumann once said

• In mathematics, you dont 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 cant say I didnt
  trymaybe another day. Anyway, lets 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, well 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 Confusingbut it is What Weve 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 dont. We
  collectively agree that we dont have a handle on
  µt.
• We know s is not constant, but there is
  (probably) no point in pretending that Log(st) 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 dont 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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