Stochastic Processes and Financial Mathematics
(part two)
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Chapter 17 Further topics on the Black-Scholes model \(\offsyl \)
In this section we briefly survey a number of further ways in which the standard Black-Scholes model is often extended. We include some (perhaps surprising) information about what the Black-Scholes model is used
for, in practice. This chapter mostly contains discussions rather than mathematical arguments. As in Chapter 16 we
follow a strategy of keeping things simple by studying each extension in isolation. Note that the whole of Chapter 17 is
off-syllabus, and marked with \(\offsyl \)s.
17.1 Time inhomogeneity \(\offsyl \)
We’ve kept the model parameters \(r,\mu \) and \(\sigma \) as fixed, deterministic constants throughout our analysis of the Black-Scholes model. In fact, there is no need to do so. It is common to allow \(\mu \)
and \(\sigma \) to depend on both \(t\) and \(S_t\), written \(\mu (t,S_t)\) and \(\sigma (t,S_t)\).
The situation for \(r\) is similar. It is common to allow \(r\) to depend on \(t\), but not on \(S_t\), written simply \(r(t)\). The for this choice is simply that reason interest rates are generally not thought to
be dependent on stock prices. Moreover, interest rates tend to vary much more slowly than stock prices, and it is not unusual to assume that \(r\) is constant.
Allowing \(r\), \(\mu \) and \(\sigma \) to vary makes the problem of parameter inference much more difficult, but it is easily absorbed (without major changes) into the pricing theory that we’ve studied in this
course. Essentially, the reason that no major changes occur is that our use of Ito’s formula did not ever require us to differentiate \(r\), \(\mu \) or \(\sigma \).
In the situation where \(r=r(t)\), \(\mu =\mu (t,S_t)\) and \(\sigma =\sigma (t,S_t)\), it turns out that the risk neutral valuation formula for the contingent claim \(\Phi (S_T)\) turns into
\[\Pi _t=e^{-\int _t^T r(u)\,du}\,\E _{t,S_t}^\Q \l [\Phi (S_T)\r ],\]
where the risk neutral world \(\Q \) has the dynamics of \(S\) as
\[dS_t=r(t)S_t\,dt+\sigma (t,S_t)S_t\,dB_t.\]
It is easily seen that this generalizes the version of risk-neutral valuation that we proved in Theorem 15.3.1: when \(r\) is constant we have \(\int
_t^Tr\,du=r(T-t)\). Note, though, that in this case we don’t have the explicit formula (15.20) for \(S_T\) in
terms of \(S_t\).