Delta-Hedging P&L

Ask a desk trader what an option is really a bet on and the answer is not direction, a delta-hedged option is a bet on volatility. This section derives the single most important equation in practical options trading: the P&L of a delta-hedged option equals its gamma weighted by the difference between realized and implied variance. It explains why you buy options when you think the world will be more volatile than the price implies, and it turns the abstract Greeks into cash.

Setup

You are long one option, priced and hedged using Black-Scholes at implied volatility $\sigma_{\text{imp}}$, but the stock actually realizes volatility $\sigma_{\text{real}}$. You continuously short $\Delta = V_S$ shares. What is the total P&L over the life of the trade?

One-step P&L

Over a small step $\delta t$ the position value changes by the option's move minus the hedge's move minus financing. Ignoring rates for clarity and using a Taylor expansion of the option value $V(S,t)$:

$\delta V = \Theta\,\delta t + \Delta\,\delta S + \tfrac12\Gamma\,(\delta S)^2 + \dots$

The delta hedge contributes $-\Delta\,\delta S$, cancelling the first-order spot term. The hedged P&L over the step is therefore

$\delta\Pi = \delta V - \Delta\,\delta S = \Theta\,\delta t + \tfrac12\Gamma\,(\delta S)^2.$

Now use the key fact that the option was priced at implied vol, so its theta satisfies the Black-Scholes relation $\Theta = -\tfrac12\sigma_{\text{imp}}^2 S^2\Gamma$ (delta-hedged, zero rates). Substitute:

$\boxed{\,\delta\Pi = \tfrac12\Gamma\Big[(\delta S)^2 - \sigma_{\text{imp}}^2 S^2\,\delta t\Big] = \tfrac12\Gamma S^2\Big[\Big(\tfrac{\delta S}{S}\Big)^2 - \sigma_{\text{imp}}^2\,\delta t\Big].\,}$

This is the fundamental hedging P&L equation. Each period you collect $\tfrac12\Gamma S^2$ multiplied by the gap between the realized squared return $(\delta S/S)^2$ and the implied variance budget $\sigma_{\text{imp}}^2\,\delta t$ the theta charged you. Being long an option is being long a stream of "realized minus implied variance," dollar-weighted by gamma.

Aggregating: the P&L over the whole trade

Summing over all steps, the expected realized squared return is $\mathbb{E}[(\delta S/S)^2] = \sigma_{\text{real}}^2\,\delta t$, so in expectation each step earns $\tfrac12\Gamma S^2(\sigma_{\text{real}}^2 - \sigma_{\text{imp}}^2)\,\delta t$. Over the life of the option the total (dollar-gamma weighted) P&L is

$\Pi = \frac12\int_0^T \Gamma_t\,S_t^2\big(\sigma_{\text{real},t}^2 - \sigma_{\text{imp}}^2\big)\,dt.$

If realized vol exceeds implied every day, the long option makes money continuously; if it lags, the position bleeds. The quantity $\Gamma_t S_t^2$ is dollar gamma, the P&L weight, and it is largest for at-the-money, short-dated options. Crucially, the sign of the total P&L depends only on realized versus implied variance, not on the direction of the stock: a delta hedge has stripped out direction and left a pure vol bet. This is exactly the intuition that variance swaps make into a clean, gamma-independent instrument.

Path dependence: the catch

The P&L is not simply $\propto(\sigma_{\text{real}}^2 - \sigma_{\text{imp}}^2)$ because the weight $\Gamma_t S_t^2$ is itself random and path-dependent. You earn the vol spread most on days the option sits near the money (high gamma) and least when it drifts deep in or out of the money (low gamma). Two paths with identical realized vol but different trajectories through strike give different P&L. So even a correct vol forecast can lose if the realized moves arrive when your gamma is low, the P&L is dollar-gamma-weighted realized variance, not plain realized variance.

Discrete-hedging error

The clean equation assumed continuous rebalancing. Hedging only every $\delta t$ leaves a residual. The per-step hedging error has mean zero (you are unbiased) but a variance that scales with the rebalancing interval. A standard result: the total hedging error from discrete rebalancing has standard deviation of order

$\text{sd(hedge error)} \sim \sqrt{\frac{\pi}{4}}\,\cdot\,(\text{option vega-scale})\cdot\sqrt{\delta t},$

i.e. it shrinks like $\sqrt{\delta t}$, halve the hedging interval and you cut the error by $\sqrt 2$, but you double transaction costs. This tension defines the optimal-hedging problem (Leland's model adds transaction costs and effectively bumps the hedging volatility).

Worked example

You buy a 30-day ATM straddle priced at $\sigma_{\text{imp}} = 16\%$ (so implied daily move $\approx 1\%$) and delta-hedge daily. On a day the stock moves 2%:

$\delta\Pi = \tfrac12\Gamma S^2\big[(0.02)^2 - (0.01)^2\big] = \tfrac12\Gamma S^2(0.0004 - 0.0001) = \tfrac12\Gamma S^2(0.0003) > 0.$

You made money, realized (2%) beat implied (1%). On a quiet day with a 0.3% move, $(0.003)^2 = 0.000009 < 0.0001$, and you lose theta. Over the month, if the stock realizes 20% annualized against 16% implied, the long straddle profits regardless of where the stock ends up.

What breaks in practice

• Gamma is short in disguise. Selling options to earn theta is selling insurance: you collect small daily gains and lose big on the rare large move. The P&L equation shows the short is hurt precisely when $(\delta S)^2 \gg \sigma_{\text{imp}}^2 S^2\delta t$, a gap. Short-gamma blowups (LTCM-flavored) are this equation biting.
• Implied vol is not constant. With a skew, the vol you hedge at changes as spot moves; the naive constant-$\sigma$ theta is wrong and a vanna/volga correction enters.
• Transaction costs. Continuous hedging is infinitely expensive. Real desks hedge on bands or fixed intervals, accepting the $\sqrt{\delta t}$ error to keep costs sane.
• Jumps. No delta hedge catches a discontinuous move; the Taylor expansion fails and the loss is first-order in the jump size.

In interviews

The signature question: "you're long a delta-hedged call and the stock ends where it started, did you make money?" The answer is it depends on realized vs implied variance, via $\delta\Pi = \tfrac12\Gamma S^2[(\delta S/S)^2 - \sigma_{\text{imp}}^2\delta t]$. Be able to derive that from a Taylor expansion plus $\Theta = -\tfrac12\sigma_{\text{imp}}^2 S^2\Gamma$. Follow-ups: "does direction matter?" (no, the hedge removed it), "why might a correct vol forecast still lose?" (dollar-gamma weighting is path-dependent), and "how does hedging error scale with frequency?" ($\sqrt{\delta t}$). This equation is the bridge from the Greeks to variance swaps.