Stochastic Processes and Financial Mathematics
(part two)

15.3 The Black-Scholes equation

We now look at how a replicating portfolio can be found explicitly, for a given contingent claim $\mathrm{\Phi }(S_T)$. 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 $\mathrm{\Phi }(S_T)$ 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 $\mathrm{\Phi }(S_T)$ 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

$$\begin{array}{}\text{(15.10)}& \frac{\partial F}{\partial t}(t,s)+rs\frac{\partial F}{\partial s}(t,s)+\frac{1}{2}{s}^2{\sigma }^2\frac{{\partial }^{2}F}{\partial s^2}(t,s)-rF(t,s)& =0,\text{(15.11)}& F(T,s)& =\mathrm{\Phi }(s).\end{array}$$ 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).

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 $\mathrm{\Phi }(S_T)$ must be equal $F(T,S_T)$ i.e. $\mathrm{\Phi }(S_T)=F(T,S_T)$. 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 $\mathrm{\Phi }(S_T)$ 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 $F(t,S_t)$:

$$\begin{array}{}\text{(15.14)}& F(t,S_t)=V_t^h=x_t{C}_t+y_t{S}_t\end{array}$$

In particular, this implies that $dF(t,S_t)=d{V}_t^h$. We plan to calculate both these stochastic differentials, written out in full, and then use Lemma 12.4.5 to equate the $dt$ and $d{B}_t$ 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

$$d{V}_t^h=x_{t}d{C}_t+y_{t}d{S}_t.$$

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

$$\begin{array}{}\text{(15.15)}& d{V}_t^h=(x_{t}r{C}_t+y_t\mu S_t)dt+y_t\sigma S_{t}d{B}_t.\end{array}$$

Next: recalling that $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{B}_t$, by Ito’s formula we have

$$\begin{array}{}\text{(15.16)}& dF(t,S_t)=(\frac{\partial F}{\partial t}+\mu S_t\frac{\partial F}{\partial s}+\frac{1}{2}{\sigma }^2{S}_t^2\frac{{\partial }^{2}F}{\partial s^2})dt+\sigma S_t\frac{\partial F}{\partial s}d{B}_t\end{array}$$

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

Equating the $d{B}_t$ coefficients between (15.15) and (15.16) gives us that

$$y_t=\frac{\partial F}{\partial s}.$$

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

$$\begin{array}{}\text{(15.17)}& \frac{\partial F}{\partial t}+\mu S_t\frac{\partial F}{\partial s}+\frac{1}{2}{\sigma }^2{S}_t^2\frac{{\partial }^{2}F}{\partial s^2}& =x_{t}r{C}_t+\frac{\partial F}{\partial s}\mu S_t\end{array}$$ and the terms $\mu S_t\frac{\partial F}{\partial s}$ cancel giving

$$\frac{\partial F}{\partial t}+\frac{1}{2}{\sigma }^2{S}_t^2\frac{{\partial }^{2}F}{\partial s^2}=x_{t}r{C}_t.$$

so as

$$x_t=\frac{\frac{\partial F}{\partial t}+\frac{1}{2}{\sigma }^2{S}_t^2\frac{{\partial }^{2}F}{\partial s^2}}{r{C}_t}.$$

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

$$\begin{array}{rl}\frac{\partial F}{\partial t}+\frac{1}{2}{\sigma }^2{S}_t^2\frac{{\partial }^{2}F}{\partial s^2}& =x_{t}r\left(\frac{F-y_t{S}_t}{x_t}\right)\\ & =rF-r{S}_t\frac{\partial F}{\partial s}.\end{array}$$ 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 $(0,\mathrm{\infty })$. Note that we have also discovered formulae for $x_t,y_t$ along the way, which must satisfy $F(t,S_t)=V_t^h$ 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

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

Note that here we take expectation in the risk neutral world $\mathbb{Q}$, in which $S_t$ follows $d{S}_t=r{S}_{t}dt+\sigma S_{t}d{B}_t$ (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

$$\begin{array}{rl}F(t,S_t)& =e^{-r(T-t)}{\mathbb{E}}_{t,S_t}^{\mathbb{Q}}[\mathrm{\Phi }(S_T)]\end{array}$$

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

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

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

$${\mathbb{E}}^{\mathbb{Q}}[\mathrm{\Phi }(S_T)|{\mathcal{F}}_0]={\mathbb{E}}^{\mathbb{Q}}[\mathrm{\Phi }(S_T)],$$

as required.   ∎

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.   ∎