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\documentclass{beamer}
\usetheme{Malmoe}
%\usetheme{Singapore}
\begin{document}
\title{How a False Probability Model \\Changed the World: \\ Birth, Death, and Redemption of Black-Scholes}
\author{J. Michael Steele}
\date{\today}
\frame{\titlepage}
\begin{frame}
\frametitle{Introduction: The Special, The Empirical, The Miracle}
\begin{itemize}
\pause \item
\textcolor{blue}{Part I: What is (Almost) Unique to Financial Modeling --- The Notions of Arbitrage and Replication}
\pause \item
\begin{itemize}
(Homework: Perhaps these notions are {\bf not} so unique to financial modeling. If not, there is a long way to run.)
\end{itemize}
\pause \item
\textcolor{blue}{Part II: The Theme of ``Stylistic Facts" --- Something that Should be Universal in More Ways than One}
\begin{itemize}
\pause \item (More Homework: The assignment should be easier this time.)
\end{itemize}
\pause \item
\textcolor{blue}{Part III: When Models Shape Markets}
\begin{itemize}
\pause \item (Enough Homework: This is more of a celebration, reflection, and --- perhaps --- a caution.)
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Part I: Beginning with an Almost Impossible Question}
Consider a world where there is a stock and a ``contingent claim".
\bigskip
\begin{columns}
\begin{column}{0.3\textwidth}
The stock costs 2 dollars at time zero, and at time 1 it is worth
\begin{itemize}
\item either 4 dollars (if it goes up)
\item or 1 dollar (if it goes down)
\end{itemize}
\end{column}
\begin{column}{0.3\textwidth}
The claim is costs $X$ dollars at time zero, and at time 1 it is worth
\begin{itemize}
\item either 3 dollars (if the stock goes up)
\item or 0 dollars (if the stock goes down)
\end{itemize}
\end{column}
\end{columns}
\bigskip
Question: What is X?
\end{frame}
\begin{frame}
\frametitle{Some Reasoning about the Almost Impossible Question}
\begin{itemize}
\item Sure! Let $P_{\rm UP}$ denote the probability that the stock
goes up. In that case, a pretty reasonable price for the contingent claim would be
$$
X_{guess}=3*P_{\rm UP}+0*(1-P_{\rm UP})=3*P_{\rm UP}
$$
\pause
\item On second though, a little bit better guess would be
$$
X_{better}=P_{\rm UP}U(W+3)+(1-P_{\rm UP})U(W)
$$
where $U$ is my personal utility and $W$ is my personal wealth.
\pause
\item \textcolor{red}{Bad News: \emph{Nobody} knows $P_{\rm UP}$}. It looks like we
are stuck, and we all should soak for a moment in a bath of hopeless despair!
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Problem Solving Mode}
\begin{itemize}
\item What else can we bring to the question?
\pause
\item This is an ``economics model" --- so we can borrow or lend money. (To be nice, let's take the interest rate to
be zero percent.)
\pause
\item \emph{Law of One Price}: If two financial instruments have exactly the same cash flows, then they must have exactly the same price.
\pause
\item Maybe we can ``replicate the contingent claim" with a ``portfolio" consisting of $\alpha$ units of the
stock $S$ and $\beta$ units of the bond $B$.
\pause
\item \textcolor{red}{This turns out to be a marvelously fecund idea.}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Solving For X}
\begin{table}[h]
\begin{center}
\caption{Replication of a Derivative Security}
\begin{tabular}{|c|c|c|}\hline
& Portfolio & Derivative Security
\\ \hline
Original cost &$\alpha S +\beta B$ & $X$
\\ \hline
Payout if stock goes up & $4 \alpha + \beta $ & 3
\\ \hline
Payout if stock goes down & $ \alpha + \beta $ & 0
\\ \hline
\end{tabular}
\end{center}
\end{table}
\begin{itemize}
\pause
\item What a nice set of equations! In our heads, we can solve to find $\alpha=1$ and $\beta=-1$.
\pause
\item Corollary: $X=\alpha S + \beta B =1*2+(-1)*1=1.$
\pause
\item \textcolor{red}{Bottom Line}: The unique arbitrage-free price for the contingent claim $X$ is one dollar.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{What Have We Won; What Did We Pay?}
\begin{itemize}
\pause
\item We found the ``only feasible price" for $X$ and we needed no probability to do so!
\pause
\item Since we squeezed out the probability theory, we also squeezed out the utility theory.
This is a huge win.
\pause
\item Otherwise different agents would offer different prices and a whole
bird's nest of economic modeling would be needed to squeeze out one final market price.
\pause
\item \emph{The theory is enforceable}. We can win money risk-free from anyone who is trades at any
price other than the one we derived.
\pause
\item \textcolor{red}{SUPER BONUS.} This extremely simple example carries through to the real world.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{A Streetwise Gambler Makes a Guess}
\begin{itemize}
\pause
\item We used replication and arbitrage to get a value for the contingent claim, but there is a way a streetwise
gambler could have guessed the answer.
\pause
\item If we know $P_{\rm UP}$, then we have a good rough and tumble
``guess" for the value of the contingent claim --- take the expected value of the contingent payouts.
\pause
\item The streetwise gambler has a way to ``infer" a probability $P'_{\rm UP}$ that the stock goes up.
\pause
\item The Gambler ``assumes" that the stock price is a martingale: This gives
$$
2=P'_{\rm UP}*4+(1-P'_{\rm UP})*1 \quad \text{so} \quad P'_{\rm UP}=1/3.
$$
\pause
The gambler then guesses
$$
X=P'_{\rm UP}*3+ (1-P'_{\rm UP})*0=1 \quad \textcolor{red}{\text{and his guess is RIGHT!}}
$$
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{An Honest Theorem --- Not to be Misinterpreted}
\begin{itemize}
\pause
\item The gambler has introduced what we now call ``the equivalent martingale measure."
\pause
\item A THEOREM now assets that if a unique equivalent martingale measure $P'$ exists, then the
arbitrage-free price of any contingent claim is just the expected value of the claim's payouts with
respect to $P'$.
\pause
\item Important Nuances
\begin{itemize}
\pause
\item
``Equivalent" means --- puts all the probability on the
same events that got probability under the original measure. There is more modeling in this bland
assumption than one might guess.
\pause
\item The real stock price does not (by model or by observation) ``follow" the law of the equivalent martingale measure.
\pause
\item This recipe is often called the ``risk neutral approach to option pricing",
but there is no utility theory here --- and the name is major misnomer.
\pause
\item \textcolor{red}{The THEOREM is about entirely arbitrage.}
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Introducing Black--Scholes World}
The theory of option pricing owes a fundamental debt to Fisher Black and Myron Scholes who in 1973 considered the
model $P$ that now many people call ``Black--Scholes World":
$$
dS_t= \mu S_t \, dt + \sigma S_t dB_t \quad \text{and} \quad d\beta_t=r \beta_t \,dt
$$
\begin{itemize}
\pause
\item The stock model the discrete time analog of the statement that LOG STOCKPRICE is a normal random walk
with increments given by $N(\mu, \sigma^2)$.
\pause
\item The model for bond price is just $\beta_t=\beta_0 e^{rt}$, where $r$ is taken to be a two-way interest rate.
\pause
\item Our world has a finite horizon, $T$. Thus, $\tau=T-t$ is the ``time left".
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{The Streetwise Gambler and Black--Scholes World}
\begin{itemize}
\pause
\item With non-zero interest rates, the street smart gambler expects $M_t=S_t/\beta_t$ to be a martingle.
\pause
\item
Under his ``equivalent martingale measure" $P'$ we have the stock/bond equations
$$
dS_t= r S_t \, dt + \sigma S_t dB_t \quad \text{and} \quad d\beta_t=r \beta_t \,dt
$$
\pause
\item The new equations depend on $r$ but not on $\mu$.
\pause
\item The amazing consequence is that the arbitrage-free value of ANY contingent claim will
NOT DEPEND ON $\mu$.
\pause
\item \textcolor{red}{This almost defies credibility --- yet still holds water.}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{The Famous Formula as a Special Case}
\begin{itemize}
\pause
\item If $H$ is any function of the stock-price path $S_t, \, 0\leq t \leq T$, we can consider a contingent
claim that pays us that function of the path. For example, the maximum, or the last value, or
$\max(S_T-K,0)$ in the case of the European call option.
\pause
\item The arbitrage free price for this claim is just
\begin{equation}\label{RNV}
E_{P'}(S_{[0:T]})
\end{equation}
\pause
\item
For the European call option, the payout function just depends on the value of the stock at the terminal time.
\pause
\item Given the stock price at time $t$, the conditional distribution of $S_T$ given $S_t=S$ is just a log normal,
so we can easily work out the expectation \eqref{RNV}.
\end{itemize}
\end{frame}
\begin{frame}
$$
S
\Phi\left( \frac{ \log ({S}/{K}) +
(r+\frac{1}{2} \sigma^2)\tau}{ \sigma \sqrt{\tau}} \right)
-Ke^{-r\tau}
\Phi\left( \frac{ \log ({S}/{K}) +
(r-\frac{1}{2} \sigma^2)\tau}{ \sigma \sqrt{\tau} } \right)
$$
\begin{itemize}
\item This is just the (interest rate adjusted)
value of $E_{P'}(S_{[0:T]})$ in its concrete form; the famous Black-Scholes formula
for a European call option.
\pause \item Note $\mu$ does not appear.
\pause
\item The motivation given here can be turned into an honest derivation. This is not the original derivation,
but it is now the {\bf canonical derivation}.
\pause
\item
We've used a beautiful idea {\bf arbitrage} and some beautiful tools --- {\bf stochastic calculus}.
\pause
\item Details have been omitted --- but no crucial ideas
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Brief Word from Our Sponsor}
There are now many good places to learn stochastic calculus and its applications to mathematical
finance, but ....
\bigskip
\begin{columns}
\begin{column}{0.5\textwidth}
There is one we most warmly recommend:
\begin{itemize}
\item Friendly and honest
\item Rigor without tedium
\item Fun for the whole family
\end{itemize}
\end{column}
\begin{column}{0.3\textwidth}
\centerline{\includegraphics[width=0.7\textwidth]{scfaBIG.jpg}}
\end{column}
\end{columns}
\bigskip
Sure, you could get other books, but don't you deserve the best?
\end{frame}
\begin{frame}
\frametitle{Back to Business: A Model that Changed The World}
\begin{itemize}
\pause
\item In 1975 market for equity options and other derivatives were a tiny ``boutique" activity.
\pause
\item
By 2004 the notional value of derivatives contracts exceeded $273\times 10^{12}$ USD.
\pause
\item What drove this explosive development? The existence of an explicit formula? I used to think so.
\pause
\item More likely, the key driver was the explicit recipe for hedging. This is honest and operational, even
absent a ``formula."
\pause
\item What else? Emergence of ``volatility" as a central concept --- perhaps THE central concept ---
in financial modeling.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Volatility --- and Implied Volatility}
$$
S
\Phi\left( \frac{ \log ({S}/{K}) +
(r+\frac{1}{2} \sigma^2)\tau}{ \sigma \sqrt{\tau}} \right)
-Ke^{-r\tau}
\Phi\left( \frac{ \log ({S}/{K}) +
(r-\frac{1}{2} \sigma^2)\tau}{ \sigma \sqrt{\tau} } \right)
$$
\begin{itemize}
\item The parameter $\sigma$ in the Black-Scholes formula is called the ``volatility." This is also the parameter
the model
$$
dS_t= \mu S_t \, dt + \sigma S_t dB_t \quad \text{and} \quad d\beta_t=r \beta_t \,dt
$$
\pause
\item One might be tempted to estimate $\sigma$ from the sample path of the stock
price and then plug in to get the option value.
\pause
\item
\textcolor{red}{Gad Zooks --- This does not work.} Are we hosed?
\pause
\item No. We reverse the process and get the ``Implied Volatility"
\pause
\item Strange? Yes. Useful? Yes. Universal? Absolutely.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Quick Summary of Part I}
\begin{itemize}
\pause \item Trick: Replicate the Option using a Portfolio of fractions of the stocks and the ``bond."
\pause \item Invoke the ``Law of one Price" and solve the equations to get the arbitrage free option price.
\pause \item Apply this in continuous time: Get the Black-Scholes formula
\pause \item Note (reluctantly) that empirical volatility and implied volatility are imperfectly related.
\pause \item Worry a little ... while a 273 trillion dollar market evolves in less than 30 years.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Part II: The Notion of a ``Stylistic Fact"}
\begin{itemize}
\pause \item In the Black-Scholes World we assume that the stock price evolves according to
$$
dS_t= \mu S_t \, dt + \sigma S_t dB_t
$$
\pause \item This implies that day $t$ returns $r_t=\log(S_t/S_{t-1})$ are normally distributed
and that they are independent.
\pause \item \textcolor{blue}{This tweaks our empirical curiosity.} Even though we're prepared to make
assumptions that have weak spots, we typically expect our models to be approximately realistic at least at some level.
\pause \item What is the empirical story for asset returns?
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Marginal, Unconditional Asset Returns}
\begin{itemize}
\pause \item There is a subtle assumption implicit even in speaking about ``the distribution of returns."
\pause \item It always makes sense to speak of distribution of $r_t$ given the past $r_{t-1}, r_{t-2}, ...$
but to speak of the distribution of $\{r_t\}$ by itself, \textcolor{blue}{we must assume stationarity.}
\pause \item We can't actually test for stationarity. Example 1: Randomized repetition.
Example 2: Cycle with a randomize start.
\pause \item As a mater of practice, this doesn't matter much.
As an intellectual matter, there is strangely good news.
\pause \item \emph{Common Sense (of Sorts): One should only assume
that which one cannot test and reject.}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Testing the Normality of Asset Returns}
\begin{itemize}
\pause \item Take any decent sized time series of almost any asset --- Stock, Bond, Mutual Fund, ETF, or
more exotic item.
\pause \item Take any test of normality: Jarque-Bera, Shapiro-Wilks, even Kolmogorov-Smirnov...
\pause \item You will almost always strongly reject the normality of the returns.
With a test that is tail sensitive, such as Jarque-Bera, rejection is a virtual certainty.
\pause \item \textcolor{blue}{Bottom Line}: {\bf Asset Returns are not normal}.
\pause \item Asset Returns --- The First Stylized Facts:
\begin{itemize}
\pause \item Fatter Tails --- more like a T with 3 to 5 degrees of freedom
\pause \item Modest Asymmetry --- Left tail is fatter than the right tail
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Pondering the Independence of Asset Returns}
\begin{itemize}
\pause \item Take the returns of a common stock and apply a test such as
Ljung-Box that measures the distance from white noise.
\pause \item
You typically fail to reject the white noise hypothesis.
\pause \item The modestly argues that \emph{perhaps the independence assumption of Black-Scholes
world is not so bad?}
\pause \item Here we come to a strange but creative idea ----
\begin{itemize}
\pause \item On a whim, consider the {\bf squares of the returns.}
\pause \item The tests for linear predictability (ACF tests, LB tests) now show
massive predictability --- hence massive dependence of the series $\{r_t^2\}$.
\end{itemize}
\pause \item \textcolor{blue}{Second Stylized Fact}: {\bf Asset returns are not independent. At a
minimum their squares show substantial predictability}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{More Stylized Facts}
\begin{itemize}
\pause \item High volatility begets high volatility ( ARCH effect)
\pause \item Large negative shocks tend to produce a greater increases in volatility than
positive shocks of comparable size. (Black's ``Leverage effect").
\pause \item A major portion of individual stocks movements are explained by the movement
of the over all market (CAPM effect)
\pause \item Almost ninety percent of a stock's movement can be explained by the market movement
and two other factors
\begin{itemize}
\pause \item The change in BMS, a zero cost portfolio of big cap minus small cap stocks (Small Cap Effect)
\pause \item The change in HML, a zero cost portfolio of high B/M stocks minus small B/M stocks (Value Effect)
\end{itemize}
\pause \item {\bf The stochastic features of asset returns may possess many mysteries, but there are also
consistent behaviors that are found across different nations, across different asset classes, and
over many different time periods and time scales.}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Normative and Behavioral Use of Stylized Facts}
\begin{itemize}
\pause \item Suppose we consider a {\bf new probabilistic model} ....
\begin{itemize}
\pause \item We should feel happy when it captures stylized facts --- especially critical one or subtle ones.
\pause \item We should face squarely those facts that are not captured by the model.
\end{itemize}
\pause \item News Flash: People are not always forthright in this respect:
\begin{itemize}
\pause \item Essentially all pension funds explicitly or implicity assume independence of annual returns.
\pause \item They also assume return rates and volatilities are well estimated under the model of IID returns.
\end{itemize}
\pause \item {\bf The Black-Scholes Model is brutally at odds with the most fundamental stylized facts for stock returns.
What's up with that?}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{News You Can Use? Two Quick Speculations}
\begin{itemize}
\pause \item \textcolor{blue}{Can you find ``arbitrage" in your favorite problem domain:}
\begin{itemize}
\pause \item In a competitive algorithm for a network queueing protocol on in on-line data compression algorithms
can you find a analog of a replicating portfolio? Hard but interesting.
\pause \item Take any context where there expected value is a feature of merit. It is probably the case that
\emph{expected utility is really more appropriate, yet a pain to consider.} Can an arbitrage argument get
you out of the trap?
\end{itemize}
\pause \item \textcolor{blue}{Should you systematize the ``Stylized Facts" of your favorite area:}
\begin{itemize}
\pause \item What are the stylized facts of network traffic, etc.?
\pause \item How do your favorite models match up with your favorite facts?
\pause \item Everyone does this to some extent, but there is probably a
benefit to being as systematic as one can be.
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Part III: When Models Shape Markets}
\begin{itemize}
\pause \item Sociology of a Mathematical Innovation
\begin{itemize}
\pause \item Donald MacKenzie (2006): \emph{An Engine, Not a Camera}
\pause \item Thesis: The Black-Scholes framework changed the financial markets it aimed to model.
\pause \item
Not an uncommon phenomenon with economic or social theories; unique for such a mathematical theory.
\end{itemize}
\pause \item Thirty Years of Experience
\begin{itemize}
\pause \item 1973-1980 Black-Scholes fits poorly (option markets are shallow and
transaction costs are high)
\pause \item 1980-1987 Black-Scholes has increasingly good fit (transaction costs come down substantially
as markets grow exponentially)
\pause \item post October 1987 Black-Scholes is ``broken" as a direct guide to market value (but markets continue to grow as
new comfort levels of risk allocation are reached)
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Closing Observations ...}
\begin{itemize}
\pause \item TODAY the Black-Scholes formula mainly serves as a transformation
from option price to {\bf volatility} that scales stock price,
strike price, and interest rates just as it should.
\pause \item The challenge now: Modeling Volatility!
\pause \item Cross-Fertilizing Elements:
\begin{itemize}
\pause \item Arbitrage --- The ``Special Light" of Mathematical Finance
\pause \item Stylized Facts --- A Universal \emph{Rough Guide} to Modeling
\end{itemize}
\pause \item These two themes seem almost bullet proof. They should serve us very well for years to come.
\pause \item Thanks Very Much ...
\end{itemize}
\end{frame}
\end{document}