Put the last two lessons together and a hedged long option has one source of income, gamma, and one cost, theta. Over a holding period of weeks the day-by-day scalps average out, and the result comes down to a single comparison. The option was priced on an assumption about how much the stock would move, the implied volatility. The stock then moved by some actual amount, the realized volatility. A hedged long option earns when realized beats implied and loses when it does not. That is the whole trade.
Why it comes down to two numbers
Theta is not arbitrary. It is the daily gamma profit the option would earn if the stock moved exactly one implied standard deviation each day. If the stock moves more than that, gamma earns more than theta costs, and the hedged position makes money. If it moves less, theta wins. Direction never enters, because the hedge removed it.
Use the straddle from last lesson: gamma 0.08, theta $12 a day, priced at 32% implied. Suppose the stock instead realizes 40%. Gamma profit scales with the square of volatility, so the daily earnings average 12 × (40 ÷ 32)², about $19, against $12 of theta: roughly $7 a day in the trader's favor. Suppose it realizes 24% instead. Now the earnings average 12 × (24 ÷ 32)², about $7, against the same $12: a loss of about $5 a day. Multiply by the days in the trade and that is the profit or loss, before trading costs.
Bought at 30% implied, and the simulated stock also moves at 30%. Press New price path several times: the net finishes above zero on some paths and below on others, scattered around zero.
When realized equals implied, the expected result is zero, but any one path is not. The hedge removed direction, not luck: a stock can realize 30% in a pattern that happens to suit the position, or one that does not.
Bought at 30% implied on a stock that moves at 18%. Theta runs ahead of the gamma banked on nearly every path. Slide realized back up toward 30% and watch the gap close.
Pull realized below implied and the picture changes. Theta's red line falls steadily, the gamma line cannot keep up, and the net drifts down on almost every path you try.
Where the stock moves matters
There is a catch in the two-number summary. Each day's scalp is weighted by that day's gamma, and gamma is large near the strike and small far from it. If the stock whipsaws around the strike for three weeks, its movement lands where the position has the most gamma, and the straddle earns more than the average realized volatility suggests. If the stock drifts 15% away early and does its moving out there, the same realized volatility lands where gamma is small and earns much less. Timing matters too: at the money, gamma grows into expiry, so a big move in the last week counts for more than the same move in the first.
So the precise rule is this. A hedged option earns the gap between realized and implied variance, which is simply volatility squared, weighted day by day by its gamma. Two stocks with identical realized volatility can pay a straddle owner very differently.
One more piece moves the mark: if implied volatility changes while you hold, vega moves the price before a single scalp. Held to expiration, that part washes out.
IV against HV
So the trade is a view that the stock will move more, or less, than its options are pricing. Two figures let you check where that has stood recently. IV 30d is the implied volatility of options about 30 days out, the vol you would be paying today. HV 20d is the historical, or realized, volatility of the stock over its last 20 trading days, the vol it has actually been delivering. Both are annualized percentages, so they compare directly.
If IV 30d is 28 and HV 20d is 20, options are priced for far more movement than the stock has produced lately. A hedged buyer would have been paying $12 of theta to earn about $6. The seller, hedged the other way, would have been collecting the difference. If IV 30d is 22 and HV 20d is 30, the sign flips and the buyer has had the edge.
Two cautions. Realized looks backward and implied looks forward, so the gap shows who had the edge lately, not who will next month. And a one-off earnings gap sits inside the 20-day window, inflating HV until it drops out.
The premium, most days
Look at a year of the two lines on almost any large stock and implied sits above realized more often than not. That persistent gap is called the volatility risk premium, VRP for short. Measured properly, each day's implied is set against the realized vol that followed it; the Terminal's VRP card sets today's implied against the last 20 days, the backward-looking version, so read it as a guide to the premium rather than the premium itself. Either way the pattern holds: option buyers pay a little extra for protection, and hedged sellers collect it, most of the time. The exception is the point. When realized jumps above implied it usually does so fast and by a lot, and those are the weeks that hand the sellers' earnings back. Reading IV against HV is how you tell which regime you are in today.
The maths: the hedged P&L is a gamma-weighted variance gap
Start from the one-day formula, dΠ ≈ ½·Γ·dS² + θ·dt. With no rates or dividends the pricing model fixes theta: θ = -½·Γ·S²·σ_I² per year, where σ_I is the implied volatility. Write the day's move as a return, dS = S·r, and substitute: dΠ ≈ ½·Γ·S²·(r² - σ_I²·dt). The squared return r² is that day's realized variance, σ_R²·dt. Add up the days and the hedged P&L is Σ ½·Γ·S²·(σ_R² - σ_I²)·dt: the gap between realized and implied variance, weighted each day by ½·Γ·S², the position's gamma in dollar terms.
Worked example, per share: a $100 straddle with gamma 0.08, so ½ × 0.08 × 100² = 400. Implied is 32%, realized 40%. The variance gap is 0.40² - 0.32² = 0.16 - 0.1024 = 0.0576 a year, or 0.0576 ÷ 365 = 0.000158 a day. Times 400 is $0.063 a share, $6.31 a straddle each day. The same formula puts theta at 400 × 0.1024 ÷ 365 = $0.112 a share, $11.22 a straddle; the main text rounded it to $12, which is why it showed about $7. Note that variances subtract, not volatilities: 40% against 32% is an 8-point gap in volatility but a 56% bigger variance, which is why gains grow faster than the gap as realized pulls away.
Your turn
Open MSFT. Read IV 30d from the Vol Snapshot and HV 20d from Price Context, decide which side has had the edge, then open the VRP tab and see how long that has been true.