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\section*{Rating} %one of 
% Everyone: contains no mathematics.
% Student: contains scenes of mild algebra or calculus that may require guidance.
Mathematically Mature: may contain mathematics beyond calculus with proofs.
% Mathematicians Only: prolonged scenes of intense rigor.

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\visual{QuestionofDay}{../../../../CommonInformation/Lessons/question_mark.jpeg}
\section*{Question of the Day}
What is the most important idea in the derivation of the binomial
option pricing model?

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\section*{Key Concepts}

\begin{enumerate}
    \item
        The derivation of the Black-Scholes equation uses
        \begin{enumerate}
            \item
                tools from calculus,
            \item
                the quadratic variation of geometric Brownian motion,
            \item
                the no-arbitrage condition to evaluate growth of
                non-risky portfolios,
            \item
                and a simple but profound insight
                to eliminate the randomness or risk.
        \end{enumerate}
\end{enumerate}

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\visual{Vocabulary}{../../../../CommonInformation/Lessons/vocabulary.jpeg}
\section*{Vocabulary}
\begin{enumerate}
    \item
        A \defn{backward parabolic PDE} is  a partial differential
        equation of the form $ V_t + D V_{xx} + \ldots = 0 $ 
        with highest derivative terms in $ t $ of order 1 and highest
        derivative terms $ x $ of order $2$ respectively 
        \defn{Terminal values} $ V(S, T) $ at an end time $t = T$ 
        must be satisfied in
        contrast to the initial values  at $ t = 0 $ required by many
        problems in physics and engineering.
    \item
        A \defn{terminal condition} for a backward parabolic equation
        is the specification of a function at the end time of the
        interval of consideration to uniquely determine the
        solution.  It is analogous to an initial condition for an
        ordinary differential equation, except that it occurs at the end
        of the time interval, instead of the beginning.
\end{enumerate}

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\section*{Mathematical Ideas}

\subsection*{Explicit Assumptions Made for Modeling and Derivation}

For mathematical modeling of a market for a risky 
security we will ideally assume
\begin{enumerate}
    \item
        that a very large number of identical, rational traders always
        have complete information about all assets they are trading,
    \item
        changes in prices are given by a continuous random variable with some
        probability distribution,
    \item
        that trading transactions take negligible time,
    \item
        purchases and sales can be made in any amounts, that is, the
        stock and bond are divisible, we can buy them in any amounts
        including negative amounts (which are short positions),
    \item
        the risky security issues no dividends.
\end{enumerate}

The first assumption is the essence of what economists call the \defn{efficient
market hypothesis}, which leads to the second assumption as a
conclusion, called the \defn{random walk hypothesis.} Another version of
the random walk hypothesis is that traders cannot predict the direction
of the market or the magnitude of the change in a stock so the best
predictor of the market value of a stock is the current price.  
We will make the second assumption stronger and more precise
by specifying the probability distribution of the changes with a
stochastic differential equation.  The remaining hypotheses are
simplifying assumptions which can be relaxed at the expense of more
difficult mathematical modeling.

Mathematically, we assume
\begin{enumerate}
    \item
        the price of the underlying security for which we are
        considering a derivative financial instrument follows the
        stochastic differential equation
        $$
            dS = r S \; dt + \sigma S \; dW
        $$
        or equivalently that $ S(t) $ is a Geometric Brownian motion
        with parameters $ r-\sigma^2/2 $ and $ \sigma $.
    \item
        The risk free interest rate $ r $ and $ \sigma $ are constants,
    \item
        the value $ V $ of the derivative depends only on the current
        value of the underlying security $ S $ and the time $ t $, so we
        can write $ V(S,t) $,
    \item
        All variables are real-valued, and all functions are
        sufficiently smooth to justify necessary calculus operations.
\end{enumerate}

The first assumption is a mathematical translation of a strong
form of the efficient market hypothesis from economics.  It is a
reasonable modeling assumption but finer analysis strongly
suggests that security prices have a higher probability of large price
swings than Geometric Brownian Motion predicts.  Therefore the first
assumption  is not absolutely supported by data.  However, it is
useful since we have good analytic understanding of Geometric Brownian
Motion.  

The second assumption is a reasonable assumption for a modeling
attempt although good evidence indicates neither interest rates nor
the volatility are constant.  On reasonably short times scales, say a
period of three months for the lifetime of most options, the interest
rate and the volatility are approximately constant.  The third and
fourth assumptions are mathematical translations of the assumptions
that securities can be bought and sold in any amount and that trading
times are negligible, so that standard tools of mathematical analysis
can be applied.

\subsection*{Derivation of the Black-Scholes equation}

This discussion is drawn from ``Financial Derivatives and Partial
Differential Equations'' by Robert Almgren, in \emph{American
Mathematical Monthly}, Volume 109, January, 2002, pages 1--11.

We consider a simple derivative instrument, an option written on an
underlying asset, say a stock that trades in the market at price $
S(t) $.  A payoff function $ \Lambda(S) $ determines the value of the
option at expiration time $ T $.  For $ t < T $, the option value
should depend on the underlying price $ S $ and the time $ t $.  We
denote the price as $ V(S,t) $.  So far all we know is the value at
the final time $ V(S,T) = \Lambda(S) $.  We would like to know the
value $V(S, 0)$ so that we know an appropriate buying or selling price.

As time passes, the value of the option changes, both because the
expiration date approaches and because the stock price changes.  We
assume the option price changes smoothly in both the security
price  and the time.
Across a short time interval $ \delta t $ we can write by the Taylor
series expansion of $ V $ that:
$$
    \delta V = V_t \delta t + V_s \delta S + \frac{1}{2} V_{SS} (\delta
    S)^2 + \ldots
$$
The neglected terms are of order $ (\delta t)^2 $, $ \delta S \delta t $,
and $ (\delta S)^3 $ and higher.  We rely on our intuition of random
walks and Brownian motion to explain why we keep the terms of order $ (\delta
S)^2 $ but neglect the other terms.  More about this later.

To determine the price, we construct a replicating portfolio.  This will
be a specific investment strategy involving only the stock and a cash
account that will yield exactly the same eventual payoff as the option
in all possible future scenarios.  Its present value must therefore be
the same as the present value of the option and if we can determine one
we can determine the other.  We thus define a portfolio $ \Pi $
consisting of $ \phi(t) $ shares of stock and $ \psi(t) $ units of the
cash account $ B(t) $.  The portfolio must be constantly changing in
value as the security price changes randomly and the cash account
accumulates interest.

In a short time interval, we can take the changes in the portfolio
to be
$$
    \delta \Pi = \phi(t) \delta S + \psi(t) r B(t) \delta t
$$
since $ \delta B(t) \approx r B(t)\delta t $, where $ r $ is the
interest rate.  This says that short-time changes in the portfolio value
are due only to changes in the security price, and the interest growth
of the cash account, and not to additions or subtraction of the
portfolio amounts.  Any additions or subtractions are due to subsequent
reallocations financed through the changes in the components
themselves.  The reallocations 

The difference in value between the two portfolios changes by
$$
    \delta (V - \Pi) = (V_t - \psi(t) r B(t) ) \delta t + (V_S - \phi(t)
    ) \delta S + \frac{1}{2} V_{SS} (\delta S)^2 + \ldots
$$

Next come a couple of linked insights:  As an initial insight we will
eliminate the first order dependence on $ S $ by taking $ \phi(t) = V_S $.
Note that this means the rate of change of the derivative value with
respect to the security value determines a policy for $ \phi(t) $.
Looking carefully, we see that this policy removes the ``randomness''
from the equation for the difference in values! (What looks like a
little ``trick'' right here hides a world of probability theory.  This is
really a Radon-Nikodym derivative that defines a change of measure that
transforms a diffusion, i.e.\  a transformed Brownian motion with drift,
to a standard Wiener measure.)

Second, since the difference portfolio is now \emph{non-risky}, it must
grow in value at exactly the same rate as any risk-free bank account:
$$
    \delta( V - \Pi) = r (V - \Pi) \delta t.
$$
This insight is actually our now familiar no-arbitrage-possibility
argument:  If $ \delta(V - \Pi) > r (V - \Pi) \delta t $, then anyone
could borrow money at rate $ r $ to acquire the portfolio $ V - \Pi $,
holding the portfolio for a time $ \delta t $, and then selling the
portfolio, with the growth in the difference portfolio more than enough
to cover the interest costs on the loan.  On the other hand if $ \delta(
V - \Pi) < r (V - \Pi) \delta t $, then short-sell the option in the
marketplace for $ V $, purchase $ \phi(t) $ shares of stock and loan the
rest of the money out at rate $ r $.  The interest growth of the money
will more than cover the repayment of the difference portfolio.  Either
way, the existence of risk-free profits to be made in the market will
drive the inequality to an equality.

So:
$$
    r (V - \Pi) \delta t = (V_t - \psi(t) r B(t) ) \delta t + \frac{1}{2}
    V_{SS} (\delta S)^2.
$$
Recall the quadratic variation of geometric Brownian motion is
deterministic, namely $ (\delta S)^2 = \sigma^2 S(t)^2 \delta t $,
$$
    r (V - \Pi) \delta t = (V_t - \psi(t) r B(t) ) \delta t + \frac{1}{2}
    \sigma^2 S^2 V_{SS} \delta t.
$$
Cancel the $ \delta t $ terms, and recall that $ V - \Pi = V
- \phi(t) S - \psi(t) B(t) $, and $ \phi(t) = V_S $, so that on the left
$ r(V - \Pi) = rV - r V_S S - r \psi(t) B(t) $.  The terms $ -r \psi(t)
B(t) $ on left and right cancel, and we are left with the Black-Scholes
equation:
$$
    V_t + \frac{1}{2} \sigma^2 S^2 V_{SS} + r S V_S - r V = 0.
$$
Note that under the assumptions made for the purposes of the
modeling the partial differential equation depends only on the constant
volatility $ \sigma $ and the constant risk-free interest rate $ r $.
This partial differential equation (PDE) must be satisfied by the value
of any derivative security depending on the asset $ S $.

Some comments about the PDE:\@
\begin{itemize}
    \item
        The PDE is linear:  Since the solution of the PDE is the worth
        of the option, then two options are worth twice as much as one
        option, and a portfolio consisting two different options has
        value equal to the sum of the individual options.
    \item
        The PDE is \defn{backwards parabolic} because of the form $ V_t
        + (1/2) \sigma^2 S^2 V_{SS} $.  Thus, \defn{terminal values} $ V
        (S, T) $ (in contrast to the initial values required by many
        problems in physics and engineering) must be specified.
        The value of an European option at expiration is
        known as a function of the security price $ S $, so we have a
        terminal value.  The PDE is solved to determine the value of the
        option at times before the expiration date.
\end{itemize}

\subsection*{Comment on the derivation:}

The derivation above follows reasonably closely the original derivation
of Black, Scholes and Merton. Option prices can also be calculated and
the Black-Scholes equation derived by more advanced probabilistic
methods.  In this equivalent formulation, the discounted price process $
\exp(-r t)S(t) $ is shifted into a ``risk-free'' measure using the
Cameron-Martin-Girsanov Theorem, so that it becomes a martingale.
The option price $ V(S,t) $ is then the discounted expected value of the
payoff $ \Lambda(t) $ in this measure, and the PDE is obtained as the
backward evolution equation for the expectation.  The derivation above
follows the classical derivation of Black and Scholes, but the
probabilistic view is more modern and can be more easily extended to
general market models.

I elected to derive the Black-Scholes equation via the fairly intuitive
partial derivative equation approach because of the simplicity of the
derivation.  This has the value that it is
\begin{itemize}
    \item
        easily motivated and related to similar derivations of partial
        differential equations in physics and engineering,
    \item
        it avoids the burden of developing additional probability theory
        and measure theory machinery, including filtrations,
        sigma-fields, previsibility, and changes of measure including
        Radon-Nikodym derivatives and the Cameron-Martin-Girsanov
        theorem.
    \item
        it also avoids, or at least hides, martingale theory that we
        have not yet developed or exploited,
    \item
        it does depend on the stochastic process knowledge that we have
        gained already, but not more than that knowledge!
\end{itemize}
The disadvantages are that:
\begin{itemize}
    \item
        we are forced to skim certain details relying on motivation
        instead of strict mathematical rigor,
    \item
        when we are done we still have to solve the partial differential
        equation to get the price on the derivative, whereas the
        probabilistic methods deliver the solution almost automatically
        and give the partial differential equation as an auxiliary
        by-product,
    \item
        the probabilistic view is more modern and can be more easily
        extended to general market models.
\end{itemize}

\hr

\section*{Problems to Work for Understanding}
\begin{enumerate}
    \item
        Show by substitution that two exact solutions of the
        Black-Scholes equations are
        \begin{enumerate}
            \item
                $ V(S,t) = A S $, $ A $ some constant.
            \item
                $ V(S,t) = A exp(r t) $
        \end{enumerate}
        Explain in financial terms what each of these solutions
        represents.  That is, describe a simple ``claim'',
        ``derivative'' or ``option'' for which this solution to the
        Black Scholes equation gives the value of the claim at any time.

        %% \HCode{<a href="problem1.shtml">} Solution \HCode{</a>}

    \item
        Draw the expiry diagrams (that is, a graph of terminal condition
        of portfolio value versus security price $ S $) at the
        expiration time for the portfolio which is
        \begin{enumerate}
            \item
                Short one share, long two calls with exercise price $ K $.
                (This is called a \defn{ straddle }.)
            \item
                Long one call, and one put both exercise price $ K $. (This
                is also called a straddle.)
            \item
                Long one call, and two puts, all with exercise price $ K
                $. (This is called a \defn{ strip }.)
            \item
                Long one put, and two calls, all with exercise price $ K
                $. (This is called a \defn{ strap }.)
            \item
                Long one call with exercise price $ K $_1 and one put
                with exercise price $ K $_2.  Compare the three cases
                when $ K_1 > K_2 $, (known as a \defn{ strangle}), $ K_1
                = K_2 $, and $ K_1 < K_2 $.
            \item
                As before, but also short one call and one put with
                exercise price $ K $.  (When $ K_1 < K < K_2 $, this is
                called a \defn{ butterfly spread.  })
        \end{enumerate}

\end{enumerate}

\hr

\section*{Reading Suggestion:}
\begin{enumerate}
    \item
        ``Financial Derivatives and Partial Differential Equations'' by
        Robert Almgren, in \emph{American Mathematical Monthly}, Volume
        109, January, 2002, pages 1--11.
    \item
        \emph{ The Mathematics of Financial Derivatives: A Student
        Introduction } by P. Wilmott, S. Howison, J. Dewynne, Cambridge
        University Press, Cambridge, 1995, Section 3.5, pages 41--47.
    \item
        \emph{ Stochastic Calculus and Financial Applications } by J.
        Michael Steele.  Springer, New York, 2001, Section 10.1--10.2,
        pages 151--158.
    \item
        \emph{An Introduction to the Mathematics of Financial
        Derivatives } by S. N. Neftci, Academic Press, San Diego, 2000,
        Section 12.1, page 276--279.
\end{enumerate}

\hr

\section*{Outside Readings and Links:}
\begin{enumerate}
    \item
        \link{http://bradley.bradley.edu/~arr/bsm/model.html}{Bradley
        University, School of Business Administration, Finance
        Department, Kevin Rubash} A very brief description on
        the development history of option theory and the Black-Scholes
        model for calculating option value, with the notations, Greeks
        and some explanatory graphs.  Also contains a calculators for
        the option value calculation.  Submitted by Yogesh Makkar,
        November 19, 2003.
\end{enumerate}

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