How do I decompose P&L from a delta-hedged options position?

Question

AcadiFi · Accepted Answer

The daily P&L of a delta-hedged option position decomposes via a Taylor expansion: dP = Theta*dt + 0.5*Gamma*(dS)^2 + Vega*dSigma + 0.5*Vomma*(dSigma)^2 + Vanna*dS*dSigma + residual. In a Black-Scholes world with constant vol, the three theta-gamma-vega terms dominate. The classic decomposition for long gamma, delta-hedged: Gamma P&L = 0.5 * Gamma * (dS)^2 (always positive), Theta P&L = Theta * dt (always negative for long options), Vega P&L = Vega * dSigma (sign depends on vol move). If realized vol > implied vol, gamma scalping gains exceed theta decay and the trade is profitable before vega. For Cascade's long 100 ATM calls, Gamma = 0.05 per $1 move per contract, Theta = -$120/day total, Vega = $450 per 1vol, position delta-hedged daily. On a day with S moving $2, sigma rising 0.3vol: Gamma P&L = 0.5 * 0.05 * 100 * 100 * 4 = $1,000. Theta P&L = -$120. Vega P&L = $450 * 0.3 = $135. Total = $1,015. ```mermaid flowchart LR G[Gamma
P&L] --> T[Total
P&L] TH[Theta
P&L] --> T V[Vega
P&L] --> T R[Residual] --> T ``` The breakeven move size satisfies 0.5*Gamma*dS^2 = -Theta*dt, giving dS = sqrt(-2*Theta*dt/Gamma) = sigma_implied*S*sqrt(dt). If realized daily move exceeds this, long gamma makes money. Cascade should report each component separately plus residual (which captures smile dynamics and higher-order Greeks).

How do I decompose P&L from a delta-hedged options position?

Master Level III with our CFA Course

Related Questions

Practice Questions