Delta

The solution to the above problem will come from the portfolio, made from the stock and the riskless bond, that replicates the option. The delta of any portfolio is defined as the rate of change of the portfolio value with respect to the price of the underlying asset

$$\Delta _{\Pi }=\frac{\partial \Pi }{\partial S}$$.

If one manages to create a portfolio that has $\Delta _{\Pi }=0$, called delta neutral, then its value will not be affected when the underlying asset price changes [during the next instant]. Apparently, the value of $\Delta$ will depend on the asset price itself, therefore it will change over time. In order to maintain a delta neutral portfolio, one has to rebalance it in a continuous fashion, a strategy called dynamic delta hedging. This is carried out after we observe that

• The underlying asset has delta equal to one
  $$\Delta _{S}=\frac{dS}{dS}=1$$, and

• If we have two portfolios with values $\Pi _{1}$ and $\Pi _{2}$, then the composite portfolio $\Pi =\Pi _{1}+\Pi _{2}$ will have delta equal to the sum of the individual deltas
  $$\Delta _{\Pi }=\frac{\partial \Pi }{\partial S}=\frac{\partial \l... ...\Pi _{2}\right) }{\partial S}=\Delta _{\Pi _{1}}+\Delta _{\Pi _{2}}%% \text{.}$$

Now suppose that one starts with a portfolio $\Pi _{1}$, with delta $\Delta _{\Pi _{1}}$, and wants to take a position in shares, $\Pi _{2}=w_{S}S$, to make the composite position delta neutral. Apparently, the delta of the position in shares will be $\Delta _{\Pi _{2}}=w_{S}\Delta _{S}=w_{S}$.

Delta neutrality

The above observations will imply that the composite portfolio has to be augmented by selling $\Delta _{\Pi _{1}}$ shares, $w_{S}=-\Delta _{\Pi _{1}}$. Then,

$$\Delta _{\Pi }=\Delta _{\Pi _{1}}+\Delta _{\Pi _{2}}=0$$.

We now turn to examine the deltas of the two major derivative contracts: the forwards and the options. We do not consider the case of a futures contract, since its value is always zero. Say that a forward maturing at time $\tau$ is written at time 0, and the delivery price is $F\left( 0,\tau \right)$. At any point $t\in \left[ 0,T\right]$ the value of the contract will be $%% F\left( t,T\right) =S\left( t\right) -F\left( 0,\tau \right) e^{-r\left( \tau -t\right) }$. If a portfolio consists of just one forward contract is considered, the value of $\Delta _{F}$ would be

$$\Delta _{F}=\frac{dF\left( t,\tau \right) }{dS}=1$$.

In this case the portfolio does not need rebalancing: a short forward can be perfectly hedged by buying one share, while a long forward can be perfectly hedged by selling one share. Rebalancing is not needed since $\Delta _{F}=1$, no matter what the value of the forward contract is; such schemes are called hedge-and-forget schemes.

Now consider an option which is at-the-money, meaning that the relationship $%% S=K_{ATM}e^{-rT}=S^{\star }$ holds. In addition, make the assumption that the BS formula holds. In this case, the value of the call is given by

$$C_{ATM}=S\mathcal{N}\left( d_{1}\right) -S^{\star }\mathcal{N}\left( -d_{1}\right)$$   ,

where $d_{1}=\frac{\sigma _{ATM}}{2}\sqrt{T}$. In this case one can compute the delta of the call option as

$$\Delta _{C}=\frac{dC_{ATM}}{dS}=\mathcal{N}\left( d_{1}\right)$$   .

For option which are not at-the-money, the above expression is used as an approximation, now setting $d_{1}$ to its normal BS value,

$$d_{1}=\frac{\ln \left( S/K\right) +\left( r+\sigma ^{2}/2\right) T}{\sigma \sqrt{T}}\text{.}$$

The values of delta for the above example, considering different spot prices and different maturities is displayed in figure 8.2.

Figure 8.2: Behavior of a Call option Delta. Part (a) gives the behavior of the delta on a three options with specifications $(K,\tau,r,\sigma)$ equal to $(1,0.2,0,0.5)$ for the solid line, $(1,0.2,0,1)$ for the thin line, and $(1,0.5,0,0.5)$ for the dashed line. Part (b) gives the behavior of the delta for a contract which is in-the-money [solid line], at-the-money [thin line], and out-of-the-money [dashed line].