No option gives you one Greek on its own. Buy an at-the-money straddle and you get gamma, vega, negative theta and almost no delta in one package, and the package changes shape as the stock moves and the days pass. This lesson puts the five together, because that is how positions actually behave.
Gamma and theta are one trade
The theta lesson called decay the rent for gamma. It is more precise than a metaphor. For a hedged option, one day's profit is roughly half of gamma times the day's move squared, minus theta. Set the two equal and you get the day's break-even move: the square root of two times theta divided by gamma. Take a straddle with a gamma of 0.08 and a theta of minus 0.12 per share. The break-even move is the square root of 2 times 0.12 over 0.08, about $1.73. A day the stock moves more than that pays; a day it moves less costs.
That break-even is not arbitrary. Option prices come from implied volatility, and the one-day move that volatility implies is almost exactly this break-even. Buying options is a bet that the stock will move more than implied volatility says. Selling them is a bet that it will move less. The chart below runs the bet day by day.
A simulated stock path for a $100 stock. Each day the straddle is re-hedged to delta zero. Gamma earned against theta paid gives the net. Set realized equal to implied and try several new paths: the net ends near zero on average.
A position's Greeks change shape
A long straddle's delta is near zero at the strike. Let the stock rally and the call's delta grows while the put's shrinks, so the straddle becomes long; let it fall and it becomes short. Its gamma, the reason for that, is a hump centered on the strike, and its theta is a mirror-image dip in the same place.
Pick a Greek. Delta runs from minus one to one across the strike; gamma and vega peak at it; theta is deepest there. The 7-day lines show everything sharpening into expiry.
Switch the chart between the Greeks and you see the pattern that holds for every long option position: gamma and vega are positive and peak at the strike, theta is negative and deepest at the strike. The seller's chart is the same, flipped. The next chart is the same straddle as a profit and loss, now and at expiry.
Bought with 30 days left at 30% volatility. Pull the days slider down to watch theta drag the curve toward the expiry V; raise volatility to watch vega lift it.
Spreads cancel some Greeks on purpose
A call spread, long one strike and short a higher one, keeps some delta but hands back much of the gamma, theta and vega, because the short call carries its own. That is the point of a spread: you choose which Greeks to keep. It also makes the Greeks flip sign across the strikes. Near the lower strike the long call dominates and the spread is long gamma and vega; near the upper strike the short call dominates and it is short both.
Gamma and vega flip sign between the strikes: long volatility near the lower strike, short it near the upper. Pick theta to see the mirror image.
A long calendar, short the near expiry and long a later one at the same strike, turns the usual relationship around. The short-dated leg has the bigger gamma and theta, the long-dated leg the bigger vega, so the calendar is short gamma, collects theta and is long vega. Module 8 builds on exactly this.
The maths: the break-even move and implied volatility
For a delta-hedged option over a short interval, P&L ≈ ½·Γ·dS² + θ·dt. With no rates or dividends the pricing equation gives θ = -½·v²·S²·Γ, so P&L ≈ ½·Γ·S²·[(dS/S)² - v²·dt]. The position wins when the squared return over the interval is bigger than the implied variance for that interval, v²·dt, so the break-even daily move is S·v·√(1/365), the one-day implied move.
Over a whole holding period the hedged P&L adds up these terms, each weighted by gamma at that moment. That is why realized volatility near the strike counts for more than realized volatility far from it, and why two stocks with the same realized volatility can pay a straddle owner very differently.
Your turn
Open the MSFT chain on the monthly expiration. Add the call and put gamma at the money, then the call and put theta, and work out the straddle's daily break-even move. Then build the same straddle in Strategy Lab and check its net Greeks against your sums.