What is Theta (Options Time Decay)?
Theta is the rate at which an option loses value with the passage of time, holding everything else constant. It is the "rent" you pay to be long an option.
Definition
Theta (Θ) is the derivative of option value with respect to calendar time. By convention it is reported as a negative number for long option positions (you bleed) and positive for short positions (you collect). Theta is highest for at-the-money short-dated options — the closer to the strike and the closer to expiry, the faster the time value evaporates. Deep ITM and deep OTM options have small thetas because they have little time value left to lose. Theta is the counterpart of gamma: a long-gamma position pays gamma P&L through hedging and bleeds theta through the clock. The whole game is whether realised volatility delivers enough gamma P&L to pay your theta bill.
Why it matters & how it's calculated
For a European call under a standard pricing model, Θ = −(S · φ(d₁) · σ) / (2√T) − r·K·e^(−rT)·N(d₂). The dominant term is the first one for short-dated near-the-money options. Theta accelerates non-linearly into expiry — a 30-DTE ATM option loses far less today than the same option 5 days from expiry loses tomorrow. On a desk, the relationship Θ ≈ −½ · Γ · σ² · S² (in dollar terms) ties the two Greeks together: every long-gamma book has an implied "fair" theta bill assuming vol is at the level you paid for it. If you are paying more theta than that, you overpaid vol; if you are paying less, you got a deal. Theta also has a weekend component: brokers typically book theta over calendar days, not trading days, which is why Friday-close-to-Monday-open premium decay is harshest.
Formula
Θ ≈ −½ · Γ · σ² · S² | P&L_realised ≈ Γ × ½ × (ΔS)² + Θ × Δt
Worked example
You're long 100 SPX ATM straddles 7 days from expiry. Each straddle has theta around −$120/day. Total daily bleed = $12,000. If SPX moves 30 points one direction by tomorrow, your gamma P&L is roughly ½ × Γ × 30² × 100. With Γ ≈ 0.025 per option, that's ½ × 100 × 100 × 0.025 × 900 ≈ $11,250 — almost exactly your theta. That break-even spot move IS the daily implied vol the market sold you.
Related concepts
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