When asked what God was doing before the Creation, J. E. Littlewood remarked, “Millions of words must have been written on this; but he was doing Pure Mathematics and thought it would be a pleasant change to do some Applied.” (Littlewood’s Miscellany, p. 135)
Thus, in the ‘Mathematical Bible’, we find Genesis 1:3 rendered as follows:
And God said: \[ “\,\nabla\boldsymbol{\cdot}\mathbf {E} = \frac{\rho}{\epsilon_0},\quad \nabla\times\mathbf {E} =-\frac{\partial \mathbf {B}}{\partial t},\quad \nabla\boldsymbol{\cdot} \mathbf {B} = 0,\quad \nabla \times \mathbf {B} =\frac{1}{c^2}\left(\frac{\mathbf {j}}{\epsilon_0} + \frac{\partial \mathbf {E}}{\partial t}\right).” \] And there was light.
Euclid demonstrated (Elements, Book IX, Prop. 20) that the primes are ‘unending’, or, in his words: “More prime numbers may be given than any multitude whatsoever of prime numbers propounded.” This proposition implies there are infinitely many primes, and the proof can be condensed into one line:
“Given \(p_{1}\!,\,\ldots,\,p_{n}\) prime, \(1+p_{1}\cdots\, p_{n}\) is not divisible by any \(p_{m}.\)”
We consider a derivative’s price at a general time \(t\) (not at time \(0\)). If \(T\) is the maturity date, the time to maturity is \(T-t\).
The stock price process we are assuming is \begin{equation} dS=\mu S\,dt+\sigma S\,dz, \end{equation}
where \(S\) is the stock price at time \(t\), \(\mu\) is the stock’s expected rate of return, \(\sigma\) is the volatility of the stock price, and \(dz\) is a Wiener process. Suppose that \(f\) is the price of a call option or other derivative contingent on \(S\). The variable \(f\) must be some function of \(S\) and \(t\). Hence, \begin{equation} df=\biggl(\frac{\partial f}{\partial S}\mu S+\frac{\partial f}{\partial t}+ {\textstyle\frac{1}{2}}\frac{\partial^2f}{\partial S^{2}}\sigma^2 S^{2}\biggr)\, dt+ \frac{\partial f}{\partial S}\sigma S\,dz. \end{equation} The discrete versions of equations (1) and (2) are \begin{equation} \Delta S =\mu S\,\Delta t+\sigma S\,\Delta z \end{equation} and \begin{equation}\Delta f =\biggl(\frac{\partial f}{\partial S}\mu S+\frac{\partial f}{\partial t}+{\textstyle\frac{1}{2}}\frac{\partial^2f}{\partial S^{2}}\sigma^{2}S^{2}\biggr )\,\Delta t+\frac{\partial f}{\partial S}\sigma S\,\Delta z, \end{equation} where \(\Delta f\) and \(\Delta S\) are the changes in \(f\) and \(S\) in a small time interval \(\Delta t\). The Wiener processes underlying \(f\) and \(S\) are the same. In other words, the \(\Delta z\) (\({}= \epsilon\sqrt{\!\Delta t}\,)\) in equations (3) and (4) are the same. It follows that a portfolio of the stock and the derivative can be constructed so that the Wiener process is eliminated. This portfolio is short one derivative and long an amount \(\partial f/\partial S\) of shares. Define \(\Pi\) as the value of the portfolio. By definition, \begin{equation} \Pi=-f+\frac{\partial f}{\partial S}S. \end{equation} The change \(\Delta \Pi\) in the value of the portfolio in the time interval \(\Delta t\) is given by \begin{equation} \Delta \Pi=-\Delta f+\frac{\partial f}{\partial S}\,\Delta S. \end{equation} Substituting equations (3) and (4) into equation (6) yields \begin{equation} \Delta \Pi=\biggl (-\frac{\partial f}{\partial t}-{\textstyle\frac{1}{2}}\frac{\partial^2f}{\partial S^{2}}\sigma^{2}S^{2}\biggr )\,\Delta t. \end{equation} Since this equation does not involve \(\Delta z\), the portfolio must be riskless during time \(\Delta t\). The assumptions of Black–Scholes–Merton imply that the portfolio must instantaneously earn the same rate of return as other short-term risk-free securities. If it earned more than this return, arbitrageurs could make a riskless profit by borrowing money to buy the portfolio; if it earned less, they could make a riskless profit by shorting the portfolio and buying risk-free securities. It follows that \begin{equation} \Delta \Pi=r\Pi \,\Delta t, \end{equation} where \(r\) is the risk-free interest rate. Substituting from equations (5) and (7) into (8), we obtain \begin{equation} \biggl (\frac{\partial f}{\partial t}+{\textstyle\frac{1}{2}}\frac{\partial^2f}{\partial S^{2}}\sigma^{2}S^{2}\biggr )\Delta t=r \biggl (f-\frac{\partial f}{\partial S}S\biggr )\, \Delta t, \end{equation} so that \begin{equation} \frac{\partial f}{\partial t}+rS\frac{\partial f}{\partial S}+{\textstyle\frac{1}{2}}\sigma^{2}S^{ 2}\frac{\partial^2f}{\partial S^{2}}=rf. \end{equation} Equation (10) is the Black–Scholes–Merton differential equation. It has many solutions, corresponding to all the different derivatives that can be defined with \(S\) as the underlying variable. The particular derivative that is obtained when the equation is solved depends on the boundary conditions that are used. These specify the values of the derivative at the boundaries of possible values of \(S\) and \(t\). In the case of a European call option, the key boundary condition is \begin{equation*} f=\max(S-K,\,0) \quad \hbox{when }t=T. \end{equation*} In the case of a European put option, it is \begin{equation*} f=\max(K-S,\,0)\quad \hbox{when } t=T. \end{equation*} One point that should be emphasized about the portfolio used in the derivation of equation (10) is that it is not permanently riskless. It is riskless only for an infinitesimally short period of time. As \(S\) and \(t\) change, \(\partial f/\partial S\) also changes. To keep the portfolio riskless, it is therefore necessary to frequently change the relative proportions of the derivative and the stock in the portfolio.