Stochastic Processes and Financial Mathematics
(part two)

16.3 Delta and Gamma Hedging

As in the previous section, consider a portfolio consisting of both options, stock and cash, whose value at time \(t\) is given by \(F(t,S_t)\). In this section we will make heavy use of the first two Greeks, \(\Delta (t,S_t)=\frac {\p F}{\p s}(t,S_t)\) and \(\Gamma (t,S_t)=\frac {\p ^2 F}{\p s^2}(t,S_t)\).

In this section we will need to consider several portfolios at once. If \(F(t,S_t)\) is the value of a portfolio at time \(t\) then we say that \(F\) is the price function of this portfolio, and we write the corresponding \(\Delta \) and \(\Gamma \) as

\[\Delta _F=\frac {\p F}{\p s},\hspace {5pc}\Gamma _F=\frac {\p ^2 F}{\p s^2}.\]

We focus first on \(\Delta _F\).

Let us think about delta neutrality for a moment. In words, \(\Delta _F(t,S_t)=0\) says that the derivative, with respect to the stock price, of the value of the replicating portfolio \(F\), is zero. This means that if \(S_t\) where to vary (slightly), we would not expect the value of \(F\) to vary much. In other words, to some extent, the value of \(F\) is not exposed to changes in the price of the underlying stock; which is a good thing, since it means the holder of the portfolio takes less risk. With this motivation, we will now look at a hedging strategy that tries to keep \(\Delta \approx 0\).

For a contingent claim \(\Phi (S_T)\), let \(F(t,S_t)\) be the value of the ‘usual’ hedging portfolio \(h_t=(x_t,y_t)\), consisting of just cash and stocks, that is provided by Theorem 15.3.1. Such a portfolio is typically not delta neutral. We consider including an amount \(z_t\) of some derivative (say, a call option) with itself has hedging portfolio with value \(Z(t,S_t)\). The value of our new portfolio is therefore

\[V(t,S_t)=F(t,S_t)+z_tZ(t,S_t).\]

We would like this new portfolio to be delta neutral, say at time \(t\). That is, we would like \(\Delta _V=\frac {\p V}{\p s}=0\), which gives us the equation

\begin{equation} \label {eq:zt_delta_hedge} \frac {\p F}{\p s}+z_t\frac {\p Z}{\p s}=0 \end{equation}

and solving for \(z_t\) we see that we should hold

\[z_t=-\frac {\Delta _F}{\Delta _Z}\]

units of the derivative. Adding this amount of the deriative into our usual replicating portfolio, with the aim of having \(\Delta _V=0\), is known as a delta hedge.

There is a spanner in the works here. The value of \(z_t\) changes with time. If, at time \(t\), we delta hedge using \(z_t\), then at time \(t+\epsilon \) we will discover that \(z_{t+\epsilon }\) is slightly different from \(z_t\) and our delta hedge is no longer working. On the other hand, if we continually adapt our portfolio to precisely match \(z_t\) then we will suffer high transaction costs.

To handle this issue, there is a procedure known as discrete rebalanced delta hedge. We explain it in the setting of Example 16.3.2, where we have sold one unit of an option with price function \(P\), and wish to delta hedge the sale.

First, we fix some \(\epsilon >0\). Then:

• • At time \(t=0\), sell one unit of an option with price \(P\).

• • Compute \(z_t=\Delta _P(t,S_t)\) (using Theorem 15.3.1) at \(t=0\) and buy (or sell) this many units of stock.

• • Wait for time \(\epsilon \). Recompute \(z_t=\Delta _P(t,S_t)\) at time \(t=\epsilon \), then buy/sell stock to re-balance the amount of ‘extra’ stock that we hold, to match this new amount.

• • Repeat the rebalancing at each time \(t=\epsilon ,2\epsilon ,3\epsilon ,4\epsilon ,\ldots \) and so on.

Of course, a smaller \(\epsilon \) results in closer approximation of \(z_t\approx \Delta _P(t,S_t)\) and (consequently) a more effective delta hedge, but with higher transaction costs; a larger \(\epsilon \) results in less effective delta hedge but lower transaction costs. This is natural – we can’t expect to reduce risk for free.

Of course, there is no need for all of our rebalancing time intervals to have length \(\epsilon \). In fact, if \(\Delta _P\) is changing rapidly then we will need to rebalance frequently in order to keep \(z_t\approx \Delta _P(t,S_t)\), but if \(\Delta _P\) is relatively stable then we’ll want to rebalance infrequently and spend less on transaction costs. This observation leads us on to the idea of \(\Gamma \) neutrality.

The key idea is that \(\Gamma _P(t,S_t)=\frac {\p }{\p s}\Delta _P(t,S_t)\) measures how quickly \(\Delta _P\) changes in response to changes in the underlying stock price \(S_t\). When \(\Gamma _P\approx 0\), we have that \(\Delta _P\) does not change quickly in response to small changes in the stock price. For this reason, it is advantageous to hold portfolios which are delta neutral and gamma neutral. How to achieve this?

We now find ourselves wanting to augment a replicating portfolio (that, recall, has price function \(F(t,S_t)\)) into a portfolio with price function \(V(t,S_t)\) in such a way as both

\begin{equation} \label {eq:delta_gamma_hedge} \Delta _V=\frac {\p V}{\p s}=0,\hspace {3pc}\Gamma _V=\frac {\p ^2 V}{\p s^2}=0. \end{equation}

It should be intuitively clear that, because we now have two conditions to satisfy, we’ll need to consider adding in two extra quantities in order to achieve this. So, consider adding in \(w_t\) of some derivative with price function \(W(t,S_t)\) and \(z_t\) of some other derivative with price function \(Z(t,S_t)\). Then \(V(t,S_t)=F(t,S_t)+w_tW(t,S_t)+z_tZ(t,S_t)\) and to satisfy (16.3) we need that both

\begin{align} \Delta _F+w_t\Delta _W+z_t\Delta _Z&=0 \notag \\ \Gamma _F+w_t\Gamma _W+z_t\Gamma _Z&=0. \label {eq:gamma_eqs} \end{align} We could solve this pair of linear equations to find formulae for \(w_t\) and \(z_t\), in terms of the \(\Delta \)s and \(\Gamma \)s. Since the formulae themselves are not particularly interesting to see, we won’t bother. Including the resulting amounts \(w_t\) of \(W\), plus \(z_t\) of \(Z\), into a portfolio, in order to achieve (16.3) is known as a gamma hedge.