Straddle Delta
Consider a straddle on an underlying asset \(S\) with a strike price \(K\) and expiry \(T\). You also have a put option on \(S\) with the same strike \(K\) and expiry \(T\). This put option has a delta \(\Delta\) of \(-0.31\). Assuming Black–Scholes dynamics, what is the delta \(\Delta\) of the straddle?
Answer
Under Black–Scholes, \(\Delta_{\rm call}=1+\Delta_{\rm put}\). With \(\Delta_{\rm put}=-0.31\),
\[
\Delta_{\rm call}=1-0.31=0.69,
\quad
\Delta_{\rm straddle}=\Delta_{\rm call}+\Delta_{\rm put}=2\times0.69-1=0.38.
\]
Notes and comments
Comment 1: The delta of a straddle (which is almost always ATM) mostly depends on time to maturity \(T\). The higher the maturity the higher the delta of the call and hence the delta of the Straddle. With that said, in a general setting, most likely it will be close to \(0\).