Stochastic Processes and Financial Mathematics
(part two)

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15.3 The Black-Scholes equation

We now look at how a replicating portfolio can be found explicitly, for a given contingent claim $\Phi (S_T)$. We will, from now on, assume that our model is free of arbitrage.

In the case of the binomial model we found ourselves solving pairs of linear equations for each time-step; now, in continuous time, we will instead find ourselves solving a partial differential equation. This is natural – our linear equations told us how things changed across a single time-step, and PDEs can describe the changing state of a system in continuous time.

Let $\Phi (S_T)$ be a contingent claim in the Black-Scholes market (with parameters $r,\mu ,\sigma $), and suppose that $F(t,s)$ is a (suitably differentiable) function such that $F(t,S_t)$ denotes the value of the contingent claim $\Phi (S_T)$ at time $t\in [0,T]$. Then, as we will show in Theorem 15.3.1, for all $s> 0$ and $t\in [0,T]$ we have

\begin{align} \frac {\p F}{\p t}(t,s)+rs\frac {\p F}{\p s}(t,s)+\frac 12s^2\sigma ^2\frac {\p ^2 F}{\p s^2}(t,s)-rF(t,s)&=0,\label {eq:bs_eq}\\ F(T,s)&=\Phi (s)\label {eq:bs_boundary}. \end{align} This is known as the Black-Scholes equation. It dates from a now famous research article by Fischer Black and Myron Scholes¹, published in 1973. The rigorous mathematical basis for the model was provided, also in 1973, by Robert C. Merton² In 1997 Merton and Scholes received the Nobel prize in economics, in recognition of their contributions (Black died in 1995, and Nobel prizes are only awarded to the living).

¹ Black, Fischer; Myron Scholes (1973). ”The Pricing of Options and Corporate Liabilities”. Journal of Political Economy. 81 (3): 637–654.

² Merton, Robert C. (1973). ”Theory of Rational Option Pricing”. Bell Journal of Economics and Management Science. The RAND Corporation.

Theorem 15.3.1 Let $\Phi (S_T)$ be a contingent claim such that $\E ^\Q [\Phi (S_T)]<\infty $. Then a replicating portfolio for $\Phi (S_T)$ is given by
$\seteqnumber{0}{15.}{11}$
\begin{align*} x_t&=\frac {1}{rC_t}\l (\frac {\p F}{\p t}(t,S_t)+\frac {1}{2}\sigma ^2S_t^2\frac {\p ^2F}{\p s^2}(t,S_t)\r )\\ y_t&=\frac {\p F}{\p s}(t,S_t) \end{align*} where $F(t,s)$ is the solution to (15.10),(15.11). The value of this portfolio at time $t$ is equal
$\seteqnumber{0}{15.}{11}$
\begin{equation} \label {eq:bs_rnv_pre} F(t,S_t)=e^{-r(T-t)}\E ^\Q [\Phi (S_T)\|\mc {F}_t], \end{equation}

and in particular its value at time $0$ is
$\seteqnumber{0}{15.}{12}$
\begin{equation} \label {eq:bs_rnv} e^{-rT}\E ^\Q [\Phi (S_T)]. \end{equation}

The formulae given for $x_t,y_t$ are a big improvement on Theorem 15.2.5, because a computer can simulate solutions to (15.10), as well as their partial derivatives, without (much) difficulty. This allows computation of the replicating portfolio in real-time. You are not expected to memorize these formulae for $h_t=(x_t,y_t)$. The other formula in Theorem 15.3.1 can be found on the formula sheet, in Appendix E.

Of course, (15.13) is known as the risk-neutral valuation formula, in direct analogy to the discreet time version we found in Proposition 5.2.6.

We will most of this section proving the claims in Theorem 15.3.1.

Proof: First let us deduce that (15.11). Equation (15.11) is known as the boundary condition, because it relates to the exercise time $T$. It holds because, by definition of $F(t,S_t)$, at the exercise time $T$ the value of $\Phi (S_T)$ must be equal $F(T,S_T)$ i.e. $\Phi (S_T)=F(T,S_T)$. Since $S_T$ may take any positive value we simply replace $S_T$ with a general $s>0$.

Note that we don’t need to worry about $s<0$ because $S_t$ is a geometric Brownian motion, which (from Section 13.2) is always positive.

We now work towards proving that (15.10) holds. From Theorem 15.2.5 we know that $\Phi (S_T)$ can be replicated by a self-financing portfolio strategy $h_t=(x_t,y_t)$. Therefore, because we assume that our model is free of arbitrage, the value of this portfolio strategy at time $t$ must be equal to $F(t,S_t)$:

\begin{equation} \label {eq:deriv_replicate} F(t,S_t)=V^h_t=x_tC_t+y_tS_t \end{equation}

In particular, this implies that $dF(t,S_t)=dV^h_t$. We plan to calculate both these stochastic differentials, written out in full, and then use Lemma 12.4.5 to equate the $dt$ and $dB_t$ coefficients. It will turn out that this leads to precisely (15.10), and along the way we will discover formulae for $x_t$ and $y_t$.

Since $(h_t)$ is self-financing we have

\begin{equation*} dV^h_t=x_t\,dC_t+y_t\,dS_t. \end{equation*}

Substituting in (15.1) and (15.2), this becomes

\begin{equation} \label {eq:deriv_self_fin_2} dV^h_t=(x_t r C_t +y_t\mu S_t) \,dt+y_t\sigma S_t \,dB_t. \end{equation}

Next: recalling that $dS_t=\mu S_t \,dt+\sigma S_t \,dB_t$, by Ito’s formula we have

\begin{equation} \label {eq:deriv_bs} dF(t,S_t)=\l (\frac {\p F}{\p t}+\mu S_t\frac {\p F}{\p s}+\frac 12\sigma ^2S_t^2\frac {\p ^2 F}{\p s^2}\r )\,dt + \sigma S_t \frac {\p F}{\p s}\,dB_t \end{equation}

where we have suppressed the $(t,S_t)$ arguments of the partial derivatives of $F$.

Remark 15.3.2 Here, we assume, without justifying ourselves, that $F(t,s)$ is differentiable (once in $t$ and twice in $s$). This is a minor issue that can be dealt with using appropriate results from analysis, but it is beyond the scope of our course.

Equating the $dB_t$ coefficients between (15.15) and (15.16) gives us that

\[y_t=\frac {\p F}{\p s}.\]

With this in hand, equating the $dt$ coefficients gives us

\begin{align} \frac {\p F}{\p t}+\mu S_t\frac {\p F}{\p s}+\frac 12\sigma ^2S_t^2\frac {\p ^2 F}{\p s^2} &=x_t r C_t + \frac {\p F}{\p s}\mu S_t\label {eq:deriv_self_fin_3} \end{align} and the terms $\mu S_t\frac {\p F}{\p s}$ cancel giving

\[\frac {\p F}{\p t}+\frac 12\sigma ^2S_t^2\frac {\p ^2 F}{\p s^2}=x_t r C_t.\]

so as

\[x_t=\frac {\frac {\p F}{\p t}+\frac 12\sigma ^2S_t^2\frac {\p ^2 F}{\p s^2}}{rC_t}.\]

Using (15.14) to substitute in for $C_t$ in (15.17), we obtain

\begin{align*} \frac {\p F}{\p t}+\frac 12\sigma ^2S_t^2\frac {\p ^2 F}{\p s^2} &=x_t r \l (\frac {F-y_tS_t}{x_t}\r )\\ &=rF-rS_t\frac {\p F}{\p s}. \end{align*} This is equation (15.10), but with $S_t$ in place of $s$. Equation (15.10) follows, because we can see from (13.9) that $S_t$ takes values on the whole of the positive reals $(0,\infty )$. Note that we have also discovered formulae for $x_t,y_t$ along the way, which must satisfy $F(t,S_t)=V^h_t$ because they were derived to satisfy (15.14).

Next we use Lemma 14.1.3, which tells us that the solution to be Black-Scholes equation can be written as

\[F(t,s)=e^{-r(T-t)}\E _{t,s}^\Q [\Phi (S_T)].\]

Note that here we take expectation in the risk neutral world $\Q $, in which $S_t$ follows $dS_t=rS_t\,dt+\sigma S_t\,dB_t$ (in the notation of Lemma 14.1.3 we have $\alpha (t,s)=rs$ and $\beta (t,s)=\sigma s$). We deduce that the value at time $t$ of the replicating portfolio is

\begin{align*} F(t,S_t) &=e^{-r(T-t)}\E ^\Q _{t,S_t}[\Phi (S_T)] \end{align*}

To finish the proof, we use that $S_t$ is a Markov process. By the Markov property at time $t$, from Lemma 14.2.1, we have

\[\E ^\Q _{t,S_t}[\Phi (S_T)]=\E ^\Q [\Phi (S_T)\|\mc {F}_t].\]

Setting $t=0$, and recalling that $\mc {F}_0$ is the trivial $\sigma $-field (containing no information) we have

\[\E ^\Q [\Phi (S_T)\|\mc {F}_0]=\E ^\Q [\Phi (S_T)],\]

as required. ∎

Corollary 15.3.3 The replicating portfolio given in Theorem 15.3.1 for $\Phi (S_T)$ is unique.

Proof: The calculations in the proof of Theorem 15.3.1 showed that any replicating portfolio (consisting solely of stocks and cash) could be written in terms of the solution to the Black-Scholes equation, as in the statement of Theorem 15.3.1. It is known from the world of PDEs that the Black-Scholes equation has a unique solution, so the replicating portfolio is also unique. ∎

Remark 15.3.4 We have now seen that the Feynman-Kac formula is important to mathematical finance; it is the key to establishing the risk-neutral valuation formula (15.13). This formula links arbitrage free pricing theory to Brownian motion, through the stochastic differential equation $dS_t=\mu S_t\,dt+\sigma S_t\, dB_t$.

The Feynman-Kac formula has much wider applications too, alongside other ways of connecting stochastic processes to PDEs. We have already mentioned heat diffusion and movements of particles within fluids, which is an example of a different type of connection known as ‘duality’ that we won’t discuss in detail here. We list two further examples here:
- • The Keynmann-Kac formula was originally developed to describe solutions to the Schrödinger equation, a PDE which is the equivalent in quantum mechanics of Newton’s second law (i.e. $F=ma$, the ‘law of motion’).
- • The method of duality is used, in a variety of SDE based models, by mathematicians trying to model evolution, to analyse the positions and/or proportions of genes within a population.
There are many others – Brownian motion sits right at the heart of the physical world.

Stochastic Processes and Financial Mathematics (part two)

15.3 The Black-Scholes equation

Stochastic Processes and Financial Mathematics
(part two)