# Breaking even on a delta-hedged portfolio

When an options portfolio is delta-neutral, its P&L over a small time change is:

\begin{equation*}
P\&L = -\Theta \delta t + \frac{1}{2}\Gamma(\delta S)^2
\end{equation*}

When you own options you're long gamma, so when the underlying moves it makes you money. Your option then loses some extrinsic value every day through theta.

Black-Scholes says that when the underlying moves by one standard deviation, your gamma profits should offset your theta losses. This known as the break even move.

You can use the current underlying price plus the implied volatility to work out the expected daily standard deviation. This is given by \(S \sigma \sqrt{\delta t}\)
.

But, as your gamma and theta change everyday, this breakeven move can also be expressed by rearranging the equation at the top. This calculates how much the underlying needs to move *that day* to break even.

\begin{equation*}
\delta S = \sqrt{\frac{2\Theta}{\Gamma}}
\end{equation*}

## Example

If we have a 1 year ATM call option on a stock where the underlying's price is currently $100.00 and the volatility is 20%.

Today the option has a gamma of \(0.028138\)
and a theta of \(-0.01543\)
, which makes our break even move

\begin{equation*}
\delta S = \sqrt{\frac{2\Theta}{\Gamma}} = \sqrt{\frac{2 * 0.01543}{0.028138}} = 1.047
\end{equation*}