Stochastic Processes and Financial Mathematics
(part two)

15.2 Completeness

We’ll begin our analysis of the Black-Scholes model by showing that, in the Black-Scholes market, we can replicate any contingent claim that has a well-defined expectation. This is, essentially, what is meant by completeness in the Black-Scholes model.

In the binomial model we discovered the importance of the so-called risk-neutral world, \(\Q \). We will see that the corresponding concept is equally important in continuous time.

The key point here is that, compared to the ‘real’ dynamics of \(S_t\), given in (15.1), we have replaced \(\mu \) with \(r\).

The next lemma, at first glance, appears rather odd. It gives an awkward looking condition under which we can replicate a contingent claim. The point, as we will see immediately after, is that it turns out we can always satisfy this condition.

For reasons that will become clear in Section 17.2, in this section we’ll tend to write \(X\) (instead of our usual \(\Phi (S_T)\)) for contingent claims.

Proof: We are looking for a portfolio strategy \(h_t=(x_t,y_t)\) such that both

\begin{align} V^h_T&=X\hspace {6pc},\label {eq:repl_rep}\\ dV^h_t&=x_tdC_t+y_tdS_t.\label {eq:repl_sf} \end{align} Here, the first equation is ‘replication’ and the second is ‘self-financing’. The portfolio strategy we will use is

\begin{align*} x_t&=\l (M_t-e^{-rt}f_tS_t\r )\\ y_t&=f_t. \end{align*} It is immediate that \(x_t\) and \(y_t\) are continuous and adapted, so \(h_t=(x_t,y_t)\) is a portfolio strategy and we must check (15.6) and (15.7) hold. Recalling that \(C_t=e^{rt}\), we note that

\begin{align} V^h_t &=x_tC_t+y_tS_t\notag \\ &=e^{rt}M_t-f_tS_t+f_tS_t\notag \\ &=e^{rt}M_t.\label {eq:Vht_ito} \end{align} Hence \(V^h_T=e^{rT}M_T=e^{rT}e^{-rT}\E ^\Q [X\|\mc {F}_T]=X\), because \(X\in \mc {F}_T\). This checks (15.6) i.e. \((h_t)\) replicates \(X\).

Now, for (15.7). We need to apply Ito’s formula to obtain \(dV^h_t\), and we’ll do it using (15.8). This means that we need to calculate \(dM_t\), which we can do using (15.5), meaning that we’ll need to start by finding an expression for \(dZ_t\).

Using Ito’s formula on \(Z_t=e^{rt}S_t\) we obtain that

\begin{align*} dZ_t&=\l (-rS_te^{-rt}+\mu S_t e^{-rt}+0\r )\,dt+\sigma S_t e^{-rt}\,dB_t\\ &=e^{-rt}S_t(\mu -r)\,dt+\sigma e^{-rt}S_t\,dB_t. \end{align*} This represents \(Z\) as an Ito process. Hence, using (15.5),

\[dM_t=e^{-rt}f_tS_t(\mu -r)\,dt+\sigma e^{-rt}f_tS_t\,dB_t\]

and we are now ready to apply Ito’s formula to \(V^h_t=e^{rt}M_t\). We obtain

\begin{align*} dV^h_t &=\l (re^{rt}M_t+e^{-rt}f_tS_t(\mu -r)e^{rt}+0\r )\,dt+\sigma e^{-rt}f_tS_te^{rt}\,dB_t\\ &=\l (re^{rt}M_t-rf_tS_t\r )\,dt+f_t\l [\mu S_t\,dt+\sigma S_t\,dB_t\r ]\\ &=\l (M_t-e^{-rt}f_tS_t\r )\l [re^{rt}\,dt\r ]+f_t\l [\mu S_t\,dt+\sigma S_t\,dB_t\r ].\\ &=\l (M_t-e^{-rt}f_tS_t\r )\,dC_t+f_t\,dS_t. \end{align*} The second and third lines are collecting terms, so that in the final line we can use the definitions of \(dC_t\) and \(dS_t\) from (15.2) and (15.1). Recalling the definitions of \(x_t\) and \(y_t\) we now have

\[dV^h_t=x_t\,dC_t+y_t\,dS_t\]

and, as required, we have checked (15.7) i.e. \((h_t)\) is self-financing.   ∎

Proof: We define \(M_t=e^{-rT}\E ^\Q \l [X\|\mc {F}_t\r ]\) and look to apply Lemma 15.2.4. Note that since \(\E ^\Q [|X|]<\infty \) this conditional expectation is well defined. From Example 3.3.9 (which is easily adapted to continuous time) we have that \(\E [X\|\mc {F}_t]\) is a martingale. Hence, since \(e^{-rT}\) is just a constant, we know that \(M_t\) is a martingale.

The key step is the next one: use the martingale representation theorem (Theorem 13.4.3), in the risk neutral world \(\Q \), to say that under \(\Q \) there exists a process \(g_t\) such that

\begin{equation} \label {eq:dMt_gtBt} dM_t=g_t\,dB_t. \end{equation}

Let \(Z_t=e^{-rt}S_t\), as in Lemma 15.2.4. Under \(\Q \), the dynamics of \(S\) are that \(dS_t=rS_t\,dt+\sigma S_t\,dB_t\), so in world \(\Q \) when we apply Ito’s formula to \(Z_t\) we obtain

\begin{align*} dZ_t&=\l (-re^{-rt}S_t+rS_te^{-rt}+0\r )\,dt+\sigma e^{-rt}S_t\,dB_t\\ &=\sigma Z_t\,dB_t. \end{align*} Combining with (15.9) we obtain

\[dM_t=\frac {g_t}{\sigma Z_t}\sigma Z_t\,dB_t=\frac {g_t}{\sigma Z_t}\,dZ_t.\]

This shows that (15.5) holds, with \(f_t=\frac {g_t}{\sigma Z_t}\). Finally, we apply Lemma 15.2.4 and deduce that there exists a replicating portfolio strategy for \(X\).   ∎

In some ways, Theorem 15.2.5 is very unsatisfactory. It asserts that (essentially, all) contingent claims can be replicated, but it doesn’t tell us how to find a replicating portfolio. The root cause of this issue is that we used the martingale representation theorem – which told us the process \(g\) existed, but couldn’t give us an explicit formula for \(g\). Without calculating \(g\), we can’t calculate \(x_t,y_t\) either.

Of course, to trade in a real market, we would need a replicating portfolio \(h_t=(x_t,y_t)\), or at the very least a way to numerically estimate one. With this in mind, in the next section, we will show how to find a replicating portfolio explicitly. The argument we will use to do so relies on already knowing that a replicating portfolio exists; for this reason, we needed to prove Theorem 15.2.5 first.

Another issue is that we have not addressed is to ask if our replicating portfolio is unique. Potentially, two different replicating portfolios could exist. We will show in the next section that the replicating portfolio is, in fact, unique.