Chapter 6 The Black–Scholes formula

We have seen how to price contingent claims using the binomial model in terms of the given parameters \(u,d,r\), etc. But this is only of practical use of the binomial model gives a reasonable approximation to real stock price behaviours. In this chapter we look at how we can try to match the binomial model to observed data, as well as some particular choices of \(u,d, p_u\) that give rise to the Black–Scholes formula for pricing a European call option.

Our first step is to move “towards” working in continuous time, by dividing our interval \([0,T]\) into a larger, but still finite, number of segments. In reality, the time delay between potential trading times is only limited by the processing speed of our computers, and we usually view this as a continuous process. Nonetheless, we will consider a market in which trades can happen at times \(t \in \{ 0, \frac{T}{n}, \frac{2T}{n}, \dots, T\}\). We set \(h=\frac{T}{n}\) as the time difference between each pair of consecutive trading times.

6.1 Non-binomial prices

One approach we can take (following that in Stochastic Finance, by Turner and Zeindler) is to consider a more general form for the way our share prices evolve. It seems sensible to focus on the relative change in the share prices at each time step, rather than the absolute change, on the basis that a change of £1 is more significant if the current share price is £0.5 than if it is £500.

Recall that in Chapter \(\ref{Chapter3}\), we defined the share prices via \[ S_t(\omega) = Z_t(\omega) S_{t-1}(\omega),\] where \(Z_t\) is a random variable which takes value \(u\) if \(\omega_t = 1\), or \(d\) if \(\omega_t=0\).

Let’s consider what will happen if we replace \(Z_t(\omega)\) with a random variable with a different distribution, say \(\xi_t\).

At time \(t = \frac{kT}{n}\), we have \[ S_k = \xi_k S_{k-1} = S_0 \prod_{j=1}^k \xi_j.\] Taking logs of both sides, we can write this as \[ \log S_k - \log S_0 = \sum_{j=1}^k \log \xi_j.\]

Now we have a sum of random variables, which we can reasonably assume to be independent from each other (if we suppose that the market forces affecting share prices at one time are independent of those at another time), and identically distributed (if we suppose that these market forces are not meaningfully different from one moment to another). By the Central Limit Theorem, this sum is approximately Normally distributed, with mean \(k \mathbb{E}[\log \xi_1]\) and variance \(k \text{Var}(\log \xi_1)\).

The approximation is exact when the \(\log \xi_j\)s are Normally distributed. Deciding to model the \(\log\xi_j\)s as a sequence of i.i.d Normal random variables is a reasonable choice: the market forces consist of many small components, contributing to one “large-scale” change in the price. If we take \(\xi_j = e^{I_j}\), where \(I_j \sim \mathcal{N}(a,b^2)\), then \[ \log S_T - \log S_0 \sim \mathcal{N} ( n a, n b^2). \tag{6.1} \]

How do we decide what values of \(a\) and \(b\) make sense? We work in terms of the drift, usually denoted \(\mu\), and the volatility, which we denote by \(\sigma^2\). These are properties of the share price which we can observe. The drift rate tells us about the direction in which the expected share price is moving, and how quickly, and the volatility tells us about how much the share price fluctuates around these values. Mathematically, we have \[ \mathbb{E}[S_t] = e^{\mu t} \\ \Var( \log S_t) = \sigma^2 t. \]

Using (6.1), we should set \[ n b^2 = \sigma^2 T,\] or in other words, \(b^2 = \sigma^2 \frac{T}{n} = \sigma^2 h\). Then, recalling the moment generating function of a Normal distribution, we can write \[ \mathbb{E}[\xi_j] = \mathbb{E}[e^{I_j}] = \exp\left\{ a + \frac{1}{2} b^2\right\},\] to show that the correct choice for \(a\) is \[ a = (\mu - \frac{1}{2} \sigma^2) \frac{T}{n} = (\mu - \frac{1}{2}\sigma^2) h.\]

Conclusion: When we model the relative change in the share price at each time step via \[ \xi_j = e^{I_j},\] with \[I_j \sim \mathcal{N} \left( \big(\mu - \frac{1}{\sigma^2} \big) h, \sigma^2 h \right),\] then the share prices are in the form \[ S_t = S_0 \prod e^{ I_j},\] and have the correct drift rate and volatility.

In a bit of foreshadowing for next term, we can rearrange this formula a bit. Set \[ I_j = \left( \mu - \frac{1}{2} \sigma^2\right) h + \sigma N_j,\] where \(N_j\) is a Normally-distributed random variable with mean 0 and variance \(h\). (Convince yourself that this still gives us the same expectation and variance as before.) Then, if we write \[ W_t^{(n)} = \sum_{j \leq nt} N_j,\] we can write \[ S_t = S_0 e^{\left( \mu - \frac{1}{2} \sigma^2\right) t} \exp \{ \sigma W_t^{(n)}\}.\] This useful version of the formula allows us to pull out all of the deterministic parts of \(S_t\) (the first two terms), and deal with the random part (the final term) in its own right.

As \(n \to \infty\), the distribution of \(\exp \{ \sigma W_t^{(n)}\}\) converges to that of the stochastic process geometric Brownian motion, evaluated at time \(t\). To prove that this convergence holds at a single point is fairly straightforward, but proving that as a function, \(t \mapsto \exp\{ \sigma W_t^{(n)}\}\) converges in distribution to \(t \mapsto GBM_t\) is much more complicated.