last updated: October 16, 2024

Stochastic Processes and Financial Mathematics
(part two)

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16.2 The Greeks

As usual, let \(F(t,s)\) be a differentiable function such that \(F(t,S_t)\) denote the value, at time \(t\), of a portfolio that replicates the contingent claim \(\Phi (S_T)\). We adopt one key idea from the previous section: we allow this portfolio strategy to include options as well as cash and stock.

In this section we explore the sensitivity of replicating portfolios to changes in:

  • 1. The price of the underlying stock \(S_t\).

  • 2. The model parameters \(r,\mu \) and \(\sigma \).

In the first case, we are interested to know, at a given time, how exposed our current portfolio is to changes in the asset price. That is, if the stock price were to quickly fall/rise, how much value are we likely to gain/lose?

In the second case, our concern is that the model parameters we’re using may not be a good match for reality (or that reality may change so as new parameters are needed). This is a serious issue, since in practice the values used for \(r,\mu ,\sigma \) are obtained by statistical inference, and it is not easy process to obtain them.

Various derivatives of \(F\) are used to assess the sensitivity of the associated portfolio, and they are known collectively as the Greeks. They are

\[ \begin {alignedat}{2} \Delta &=\frac {\p F}{\p s} \hspace {5pc} &&(Delta)\\ \Gamma &=\frac {\p ^2 F}{\p s^2} &&(Gamma)\\ \Theta &=\frac {\p F}{\p t} &&(Theta)\\ \rho &=\frac {\p F}{\p r} &&(rho)\\ \Vega &=\frac {\p F}{\p \sigma } &&(Vega) \end {alignedat} \]

all of which are evaluated at \((t,S_t)\). Note that, for \(\rho \) and \(\Vega \), we regard \(r\) and \(\sigma \) as variables (instead of constants) and differentiate with respect to them. There is no point in having a derivative with respect to \(\mu \) because, as we have seen in Theorem 15.3.1, \(\mu \) does not affect the risk-neutral world and consequently its value has no effect on arbitrage free prices: \(\frac {\p F}{\p \mu }=0\).

If we have an explicit formula for \(F\), such as the Black-Scholes formula (15.23) for European call options, then we can differentiate to find explicit formulae for the Greeks. This can involve some quite messy calculations, so we don’t focus on this aspect of the Greeks in this course (but, see exercise 16.12 if you like doing messy calculations). In general, they can be estimated numerically.

For us, \(\Delta \) and \(\Gamma \) are most important. In the next section we study hedging strategies based on \(\Delta \) and \(\Gamma \).

  • Remark 16.2.1 In fact, \(\Vega \) is not a letter of the Greek alphabet, it is a calligraphic Latin \(V\), but the terminology ‘the Greeks’ is used anyway.