FRM Part I · FRM Exam Part I

Option Sensitivity Measures: The "Greeks": formula sheet

Full chapter guide

Key formulas

Definition of delta: Δ = ∂V ÷ ∂S; Sensitivity of option or portfolio value to the underlying price.
d1 (no dividends): d1 = [ln(S ÷ K) + (r + σ²÷2)·T] ÷ (σ·√T); r is the continuously compounded risk-free rate; σ is annual volatility; T is in years.
d2: d2 = d1 − σ·√T; Used for the price, not for delta.
European call delta: Δcall = N(d1); Between 0 and 1 for a non-dividend stock.
European put delta: Δput = N(d1) − 1 = −N(−d1); Between −1 and 0.
Delta with continuous yield q: Δcall = e^(−qT)·N(d1); Δput = e^(−qT)·[N(d1) − 1]; d1 uses (r − q + σ²÷2). Applies to indices and, with q = foreign rate, to currencies.
Delta of forward and futures: Forward on non-dividend stock: Δ = 1 per unit; futures: Δ = e^(rT) with respect to spot; Delta with respect to the futures price itself is 1.
Portfolio delta: ΔP = Σ (nᵢ × Δᵢ); Use negative quantity for short positions.
Delta-neutral hedge: Shares to hold = −(portfolio delta); Short 1 call with delta 0.6 needs 0.6 shares long per option unit.
Put-call delta relation: Δcall − Δput = 1 (no dividends); Follows from put-call parity.
Delta of an option position: Position delta = N × Δ_option (N is signed: positive long, negative short); Use the contract multiplier if options cover many shares, such as 100 per contract.
Delta-neutral condition: Σ (Nᵢ × Δᵢ) = 0; Sum across all options and the underlying. The underlying has Δ = 1 per unit.
Shares needed to hedge: Shares to trade = -(Total option position delta); A positive option delta means sell shares short. A negative option delta means buy shares.
Black-Scholes-Merton delta (no dividend): Call Δ = N(d₁); Put Δ = N(d₁) - 1; With a continuous dividend yield q, call Δ = e^(-qT) N(d₁) and put Δ = e^(-qT)[N(d₁) - 1].
Delta of a forward-like exposure: Call Δ - Put Δ = 1 (same strike, same maturity, no dividends); This follows from put-call parity.
Delta-gamma approximation of value change: ΔV ≈ Δ × ΔS + ½ × Γ × (ΔS)²; For a delta-neutral position, the first term is zero and gamma drives the P&L.
Gamma to neutralize with another option: N_T = -Γ_P ÷ Γ_T; Trading option T changes delta, so you then re-hedge delta with the underlying.
Gamma definition: Γ = ∂Δ/∂S = ∂²V/∂S²; Second derivative of option value V with respect to underlying price S.
Black-Scholes-Merton gamma (no dividends): Γ = N'(d1) ÷ (S × σ × √T); N'(d1) is the standard normal density at d1. Same for calls and puts with the same strike and expiry.
Gamma with a continuous yield q: Γ = e^(−qT) × N'(d1) ÷ (S × σ × √T); Use for index or currency options.
Delta change approximation: ΔΔ ≈ Γ × ΔS; New delta ≈ old delta + Γ × ΔS.
Taylor approximation of option value: ΔV ≈ Δ × ΔS + ½ × Γ × (ΔS)²; Ignores theta and vega. The gamma term is always positive for a long option.
Gamma-neutral hedge: w_T = −Γ_P ÷ Γ_T; Number of traded options w_T to add to a portfolio with gamma Γ_P, where Γ_T is the traded option's gamma.
Delta after gamma hedge: Δ_new = Δ_P + w_T × Δ_T; underlying position = −Δ_new; Re-neutralize delta with the underlying, which has zero gamma.
Vega of a European option (no dividends): Vega = S0 × N'(d1) × √T; Same for calls and puts. N'(x) = (1 ÷ √(2π)) × e^(−x²÷2). Result is per 1.00 change in volatility; divide by 100 for a 1 percentage point change.
Vega with continuous dividend yield q: Vega = S0 × e^(−qT) × N'(d1) × √T; Use for index and currency options (in the currency case, q is the foreign risk-free rate).
Definition: Vega = ∂V ÷ ∂σ; Change in option value per unit change in volatility.
d1: d1 = [ln(S0 ÷ K) + (r + σ²÷2) × T] ÷ (σ × √T); Needed to compute N'(d1) in vega.
Approximate P&L from a volatility move: ΔV ≈ Vega × Δσ; Keep Δσ in the same units as the vega quote.
Vega-neutral hedge position: Number of hedge options = − Vega of portfolio ÷ Vega of one hedge option; A negative number means sell hedge options. Then rebalance delta with the underlying.
Definition of theta: Θ = ∂f / ∂t; Change in option value per unit of time, other inputs fixed. Quoted per year unless told otherwise; divide by 365 for a per-day figure.
Black-Scholes-Merton equation (no dividends): Θ + r S Δ + ½ σ² S² Γ = r Π; Holds for any portfolio Π of derivatives on the same stock. Π is the portfolio value.
Delta-neutral case: Θ + ½ σ² S² Γ = r Π; If also Π = 0 (or r is negligible), Θ = −½ σ² S² Γ.
With continuous dividend yield q: Θ + (r − q) S Δ + ½ σ² S² Γ = r Π; Use for indices and currencies, where q is the dividend yield or foreign rate.
Theta of a European call (no dividends): Θ(call) = −S₀ N′(d₁) σ ÷ (2√T) − r K e^(−rT) N(d₂); Always negative for a non-dividend stock. N′(x) = e^(−x²÷2) ÷ √(2π).
Theta of a European put (no dividends): Θ(put) = −S₀ N′(d₁) σ ÷ (2√T) + r K e^(−rT) N(−d₂); Can be positive when deep in the money.
d₁ and d₂: d₁ = [ln(S₀ ÷ K) + (r + σ²÷2)T] ÷ (σ√T); d₂ = d₁ − σ√T; Needed for both theta formulas.
Rho of a European call (no dividends, Black-Scholes-Merton): ρ_call = K × T × e^(−rT) × N(d2); Positive. d2 = d1 − σ√T. Units: change in price per 1.00 (100%) change in r.
Rho of a European put (no dividends): ρ_put = −K × T × e^(−rT) × N(−d2); Negative. N(−d2) = 1 − N(d2).
Rho difference from put-call parity: ρ_call − ρ_put = K × T × e^(−rT); Holds for European options on a non-dividend-paying asset. Useful to check or to get one rho from the other.
Price change from a rate move: Δ(option price) ≈ ρ × Δr; Use the same unit for ρ and Δr. If ρ is per 1.00, a 1% move is Δr = 0.01. If ρ is quoted per 1%, use Δr = 1.
d2: d2 = [ln(S/K) + (r − σ²/2) T] ÷ (σ√T); Needed for N(d2). With continuous compounding r.
Portfolio Greek: Portfolio Greek = Σ n_i × Greek_i; n_i is positive for long and negative for short. Multiply by the contract size when the Greek is quoted per unit of the underlying.
Delta-gamma-theta approximation: ΔV ≈ δ ΔS + ½ Γ (ΔS)² + Θ Δt; Use the portfolio δ, Γ and Θ. Θ and Δt must use the same time unit, for example Θ per year with Δt in years.
Full Taylor with vega and rho: ΔV ≈ δ ΔS + ½ Γ (ΔS)² + Θ Δt + ν Δσ + ρ Δr; Add the vega and rho terms only if volatility or the interest rate changes. Δσ and Δr must be in the units that vega and rho use.
Delta-neutral portfolio: ΔV ≈ ½ Γ (ΔS)² + Θ Δt when δ = 0; Delta-neutral protects only against small moves. Gamma and theta still drive the result.
Gamma-neutralising trade: n_T = − Γ_P ÷ Γ_T; n_T is the number of traded options to add. Then change the underlying position by − (δ_P + n_T × δ_T) to restore delta neutrality.
Options on an index or currency: Call delta = e^(−qT) N(d1); Put delta = e^(−qT) [N(d1) − 1]; Gamma = e^(−qT) N′(d1) ÷ (S σ √T); q is the index dividend yield, or the foreign risk-free rate for a currency option. Call and put have the same gamma for the same strike and maturity.
Options on futures (Black's model): Call delta = e^(−rT) N(d1); Put delta = e^(−rT) [N(d1) − 1]; Gamma = e^(−rT) N′(d1) ÷ (F σ √T); Same as the yield formulas with q = r and S replaced by the futures price F. Delta is relative to the futures price.
Black-Scholes-Merton differential equation (no income): Θ + r S δ + ½ σ² S² Γ = r Π; Π is the portfolio value. For a delta-neutral portfolio, Θ + ½σ²S²Γ = rΠ, so theta and gamma have opposite signs when Π is small.

Quick revision

Delta = change in option price per unit change in the underlying price.
Call delta is between 0 and 1; put delta is between -1 and 0.
Position delta = number of options × delta per option, with a negative sign for short positions.
A delta-neutral position has total delta of zero, and it holds only for small price moves.
Gamma = change in delta per unit change in the underlying price; it is highest for at-the-money options near expiry.
Long options have positive gamma; short options have negative gamma.
Vega = change in option price per unit change in volatility; long calls and puts both have positive vega.
Theta = change in option value as time passes; it is usually negative for long options.
Rho = change in option price per change in the interest rate; it is generally positive for calls and negative for puts.
Delta-gamma approximation: ΔC ≈ δ × ΔS + ½ × Γ × (ΔS)².
Portfolio Greek = sum of (quantity × Greek) across all positions.
Fixing gamma or vega needs other options, because the underlying has zero gamma and zero vega.

Common mistakes

Giving a put a positive delta. Fix: Put delta = N(d1) − 1. It is always negative for a long put.
Treating delta as the probability of finishing in the money. Fix: The risk-neutral probability of exercise is N(d2) for a call. Delta is a hedge ratio, not that probability.
Hedging in the wrong direction, such as buying shares to hedge a long call. Fix: Compute the position delta with its sign first. Then trade the opposite amount in the underlying.
Forgetting the contract multiplier. Fix: Always write contracts × shares per contract × delta.
Trying to fix gamma by trading the underlying. Fix: The underlying has delta 1 and gamma 0. Only options change gamma.
Forgetting to rebalance delta after the gamma hedge. Fix: Always do gamma first, then recompute delta and neutralize it with the underlying.
Saying a put has negative vega because its price falls when the stock rises. Fix: A long put gains value when volatility rises. Long call and long put have the same positive vega for the same strike and maturity.
Believing vega is highest for deep in-the-money options. Fix: Vega follows N'(d1), the normal density, which peaks near d1 = 0. That is close to at-the-money.
Saying theta is always negative Fix: Say 'usually negative for long options'. A deep in-the-money European put can have positive theta. Short options have the opposite sign to long ones.
Forgetting the rΠ term in the Black-Scholes equation Fix: That shortcut needs Π = 0 (or r ≈ 0). Otherwise use Θ = rΠ − ½σ²S²Γ.

Exam tips

Memorise the sign and range of call and put delta; many questions are solved by elimination.
Read the underlying: dividend yield or foreign rate changes the formula.
Watch for per-share versus per-contract quantities and the multiplier.
Know the difference between delta with respect to spot and futures price.
Be ready to link delta to hedging: hedge shares equal minus portfolio delta.
Most numerical questions come down to total delta × quantity × multiplier. Do this arithmetic carefully and check the sign.
Know the delta ranges: call 0 to 1, put -1 to 0, stock 1. Use them to eliminate wrong answer choices fast.
Conceptual questions test that delta-neutral still has gamma and vega risk. Large moves produce losses for a short-gamma position.