Theta-Gamma Relationship and Delta-Hedged P&L

Worked Solution

How to Think About It: When you delta-hedge an option, you remove exposure to the stock's direction. What is left? Two competing forces: theta (time decay eating away at you) and gamma (the convexity payoff when the stock actually moves). Every day, you are paying rent via theta and collecting windfalls via gamma whenever the stock moves. The daily scorecard is literally realized variance versus implied variance. If the stock is moving more than the option's price assumed, you come out ahead. If not, theta grinds you down. This is the most important relationship in options trading.

Quick Sanity Checks: Before deriving anything, check what should be true. A long gamma position should profit from large moves in either direction (gamma is non-directional). The breakeven should be where realized vol equals implied vol -- that is when theta and gamma exactly offset. And the P&L should scale with $\Gamma S^2$, which is the dollar gamma -- the actual dollar exposure to variance.

Derivation:

Start with the Taylor expansion of the option value $V(S, t)$ over a small interval $\Delta t$:

$$\Delta V \approx \frac{\partial V}{\partial t}\Delta t + \frac{\partial V}{\partial S}\Delta S + \frac{1}{2}\frac{\partial^2 V}{\partial S^2}(\Delta S)^2 = \Theta\,\Delta t + \Delta_{\text{opt}}\,\Delta S + \frac{1}{2}\Gamma(\Delta S)^2$$

Your hedge is short $\Delta_{\text{opt}}$ shares. The P&L of the hedged portfolio is:

$$\Delta\Pi = \Delta V - \Delta_{\text{opt}}\,\Delta S = \Theta\,\Delta t + \frac{1}{2}\Gamma(\Delta S)^2$$

The delta term cancels -- that is the whole point of delta hedging. You are left with theta plus the gamma P&L from the realized move.

Now use the Black-Scholes PDE to eliminate $\Theta$. Under BSM, the option value satisfies:

$$\Theta + \frac{1}{2}\sigma_{\text{imp}}^2 S^2 \Gamma + rS\Delta_{\text{opt}} = rV$$

The $rS\Delta_{\text{opt}}$ and $rV$ terms are small over short intervals (they are $O(r\,\Delta t)$ effects). Dropping them:

$$\Theta \approx -\frac{1}{2}\sigma_{\text{imp}}^2 S^2 \Gamma$$

This is the theta-gamma trade-off in its purest form: theta is the cost you pay for owning convexity.

Substitute back into the P&L expression:

$$\Delta\Pi \approx -\frac{1}{2}\sigma_{\text{imp}}^2 S^2 \Gamma\,\Delta t + \frac{1}{2}\Gamma(\Delta S)^2$$

Factor out $\frac{1}{2}\Gamma S^2$:

$$\boxed{\Delta\Pi \approx \frac{1}{2}\Gamma S^2 \left[\left(\frac{\Delta S}{S}\right)^2 - \sigma_{\text{imp}}^2 \Delta t\right]}$$

Practical Interpretation:

The term $(\Delta S / S)^2$ is the realized variance over the interval. The term $\sigma_{\text{imp}}^2 \Delta t$ is the implied variance "budget" for that interval. The P&L is proportional to their difference, scaled by dollar gamma $\frac{1}{2}\Gamma S^2$.

• Long gamma ($\Gamma > 0$): you profit when realized moves exceed implied. Every day the stock moves more than "expected," you collect. Every quiet day, you bleed theta.
• Short gamma ($\Gamma < 0$): the mirror image. You collect theta every day and pray the stock does not move too much.
• Breakeven: realized vol $=$ implied vol. Over a long enough horizon, if realized vol matches implied, the gamma gains and theta losses exactly cancel.

This is why options market makers obsess over realized versus implied volatility. When you buy an option at 20 vol and the stock realizes 25 vol, you make money not from any directional bet, but from the gamma scalping -- each rebalance of the delta hedge locks in a small profit. Conversely, if you buy at 20 vol and the stock only realizes 15, theta eats you alive.

Answer: The delta-hedged P&L is $\frac{1}{2}\Gamma S^2[(\Delta S/S)^2 - \sigma_{\text{imp}}^2 \Delta t]$, which equals dollar gamma times the difference between realized and implied variance. Long gamma profits when realized vol exceeds implied vol because the gamma gains from actual stock moves outweigh the theta cost baked into the option price.