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 $\beta (r,s)=\sigma s$). 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. We have already mentioned heat diffusion and movements of particles within fluids, and we could re-phrase our ‘by hand’ results from Section 11.3 in terms of the Feynman-Kac formula. We list two further examples here:
- • It is used 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’).
- • It 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)