FRM Part II · FRM Exam Part II

Volatility Smiles and Volatility Surfaces: formula sheet

Full chapter guide

Key formulas

BSM European call price: c = S0 e^(−qT) N(d1) − K e^(−rT) N(d2); q is the continuous dividend yield. Set q = 0 for a non-dividend stock. N(·) is the standard normal cumulative distribution.
BSM European put price: p = K e^(−rT) N(−d2) − S0 e^(−qT) N(−d1); Uses the same d1 and d2 as the call.
d1 and d2: d1 = [ln(S0 ÷ K) + (r − q + σ²÷2)T] ÷ (σ√T); d2 = d1 − σ√T; σ is the unknown when you solve for implied volatility.
Put-call parity (European): c + K e^(−rT) = p + S0 e^(−qT); Same strike and maturity. Model-free, so it does not depend on the volatility assumption. Implies equal implied volatility for the call and the put.
Vega: Vega = S0 e^(−qT) √T N'(d1); Same for a call and a put. Always positive. Used in Newton-Raphson updates: σ(new) = σ(old) + (market price − model price) ÷ vega.
BSM volatility assumption: dS ÷ S = (μ − q)dt + σ dz, with σ constant; Gives lognormal prices. The volatility smile and term structure show this is not what the market uses.
Lognormal assumption: dS = (r_d − r_f) S dt + σ S dz; Black-Scholes-type model for exchange rate S under risk-neutral measure. σ is constant, so implied volatility would be flat across strikes.
Smile shape for currencies: Implied volatility is lowest at-the-money and rises for both low and high strikes; Roughly symmetric U shape. State this directly in answers.
Implied vs lognormal distribution: Implied distribution: higher peak, fatter left and right tails than lognormal with the same mean and standard deviation; Both tails are heavier, so deep OTM calls and puts are priced above Black-Scholes at at-the-money volatility.
Reasons for fat tails: Fat tails = stochastic volatility + jumps; Effect of both is that the smile exists. Jumps matter more for short maturities; stochastic volatility has a relatively greater effect on longer-dated options. The smile generally becomes less pronounced as maturity increases.
Put-call parity for implied volatility: Implied volatility of a European call = implied volatility of a European put with the same strike and maturity; Use this to read one side of the smile from the other. It holds only at the same strike. A 25-delta call and a 25-delta put have different strikes, so this equality does not apply between them.
Skew direction (equity): σ_implied falls as K rises; Downward-sloping. Low strikes have the highest implied volatility, high strikes the lowest.
Implied distribution vs lognormal: Left tail: heavier than lognormal; right tail: thinner than lognormal; Compared with a lognormal distribution with the same mean and standard deviation.
Flat Black-Scholes benchmark: σ_implied(K) = σ for all K; A constant volatility gives a flat line. Any slope or curve signals a departure from lognormality.
Leverage effect: Equity value ↓ ⇒ D/E ↑ ⇒ equity volatility ↑; Negative relationship between stock price and volatility.
Put-call parity consequence: c − p = S − K·e^(−rT) (no dividends); A European call and put with the same strike and maturity must have the same implied volatility, so the skew is the same from either side.
Forward price: F0 = S0 × e^((r − q) × T); r is the risk-free rate and q the dividend or foreign-currency yield, both continuous. For FX, q is the foreign rate.
Spot moneyness: K/S0; Equals 1 when the strike equals spot. Not the same as at-the-money-forward unless r = q.
Forward moneyness: K/F0; Equals 1 when the strike equals the forward. This is the at-the-money-forward point.
d1 in Black-Scholes-Merton: d1 = [ln(S0/K) + (r − q + σ²/2) × T] ÷ (σ × √T); If K = F0, d1 = σ√T ÷ 2.
Call and put delta: Call delta = e^(−qT) × N(d1); Put delta = e^(−qT) × [N(d1) − 1]; Put delta = call delta − e^(−qT). With q = 0, put delta = call delta − 1.
Implied volatility: σ_imp solves: Market price = BSM(S, K, T, r, q, σ_imp); Each strike and maturity has its own σ_imp. The BSM price is a quoting tool, not a belief in constant volatility.
Moneyness: K ÷ S (or K ÷ F for forward moneyness); Surfaces are often indexed by moneyness so that the grid stays relevant as the spot price moves.
Linear interpolation in strike: σ(K) = σ₁ + (K − K₁) ÷ (K₂ − K₁) × (σ₂ − σ₁), for K₁ ≤ K ≤ K₂; Use it for a strike between two quoted strikes at the same maturity.
Interpolation in maturity (total variance): σ²(T) × T = linear in T between T₁ and T₂, using σ₁²T₁ and σ₂²T₂; Interpolating total variance is a common refinement. Simple linear interpolation of σ is also accepted in basic exam questions. Follow the method the question states.
Term structure shape rule: σ(long) > σ(short): upward sloping. σ(long) < σ(short): downward sloping (inverted); Usually: low current volatility gives upward slope, high current volatility gives downward slope.
Smile-adjusted delta: Δ_smile = ∂f_BS/∂S + (∂f_BS/∂σ_imp) × (∂σ_imp/∂S) = Δ_BS + Vega × ∂σ_imp/∂S; Total derivative of the option price with respect to S, allowing implied volatility to change with S.
Minimum variance delta: Δ_MV = ∂f_BS/∂S + (∂f_BS/∂σ_imp) × E[Δσ_imp] ÷ ΔS; Hull-White form. It minimises the variance of the hedged portfolio's change. The expected volatility change comes from the model or from observed data, so the result is empirical and model-dependent. For equity calls, the observed negative price-volatility relationship typically puts it below BS delta.
Sticky strike: σ_imp(K, T) unchanged when S changes, so ∂σ_imp/∂S = 0; Smile-adjusted delta equals BS delta.
Sticky delta (sticky moneyness): σ_imp = g(K ÷ S), so ∂σ_imp/∂S = −(K ÷ S²) × g′(K ÷ S); The smile moves with the asset price. For a downward-sloping equity skew, g′ < 0, so this term is positive and raises the adjusted delta of a call. For an upward-sloping smile, g′ > 0 and the term is negative. Do not confuse this with the empirical minimum variance result.
Black-Scholes benchmark: dS = μ S dt + σ S dz, with σ constant; Gives a lognormal price and a flat implied volatility across strikes.
Jump-diffusion process: dS ÷ S = (μ − λk) dt + σ dz + dp; λ is the average number of jumps per year, k is the average proportional jump size, dp is the jump component. The −λk term compensates for the expected effect of jumps on the drift.
Stochastic volatility process (Heston-type form): dS ÷ S = μ dt + √V dz₁ ; dV = a(V_L − V) dt + ξ V^α dz₂; a is the mean reversion rate, V_L the long-run variance, ξ the volatility of variance. The correlation ρ between dz₁ and dz₂ drives skew.
Local volatility: dS ÷ S = (r − q) dt + σ(S, t) dz; Volatility is a deterministic function of price and time, so no extra random source is added.
Correlation and skew rule: ρ < 0: downward skew; ρ = 0: symmetric smile; ρ > 0: upward skew; Describes the direction of the skew under stochastic volatility.
Risk-neutral density from call prices: g(K) = e^(rT) × ∂²c/∂K²; c is the European call price as a function of strike K, for one maturity T. g(K) is the risk-neutral density of the price at T.
Finite-difference estimate: g(K) ≈ e^(rT) × [c(K − δ) + c(K + δ) − 2c(K)] ÷ δ²; Strikes must be equally spaced by δ. The bracket is the cost of the butterfly spread.
Butterfly price and density: Butterfly price ≈ e^(−rT) × g(K) × δ²; The payoff is δ at S_T = K and falls to zero at K ± δ. This is the same relation rearranged.
Approximate probability in a band: P(K − δ/2 < S_T < K + δ/2) ≈ g(K) × δ; Use for a rough risk-neutral probability around K. It is an approximation.
No-arbitrage check: c(K − δ) + c(K + δ) − 2c(K) ≥ 0; If negative, the implied density is negative and a butterfly arbitrage exists.
Smile shape versus lognormal: Equity: heavier left tail, thinner right tail. Currency: both tails heavier, higher peak.; Heavier tail means higher implied volatility for strikes in that tail.

Quick revision

Implied volatility is the volatility that, put into Black-Scholes, matches the market option price.
Black-Scholes assumes constant volatility and lognormally distributed returns, so it implies a flat volatility across strikes.
Currency options typically show a smile: implied volatility higher for both deep in and out of the money than at the money.
A currency smile points to fatter tails than lognormal in both directions.
Equity and index options typically show a downward skew: low strikes carry higher implied volatility than high strikes.
The equity skew implies a heavier left tail and a thinner right tail than lognormal.
The term structure of volatility shows implied volatility against maturity; the surface combines strike and maturity.
Smiles tend to be more pronounced for short maturities and flatten as maturity grows.
Jump models matter most for short-dated options; stochastic volatility has more effect on longer-dated ones.
The risk-neutral implied distribution can be inferred from option prices across strikes.
A model-based Greek must be adjusted when implied volatility changes with the asset price.

Common mistakes

Treating implied volatility as a forecast of realised volatility. Fix: Remember it is the volatility that matches the price. It contains expectations plus a risk premium, so it often exceeds later realised volatility.
Forgetting to discount the strike in put-call parity. Fix: Write c − p = S0 e^(−qT) − K e^(−rT). Only with r = 0 and q = 0 does the discounting disappear.
Saying the currency smile is a downward-sloping skew Fix: Link currencies to symmetric U shape and equities to downward skew. The reason is that equity volatility rises as prices fall, but currencies lack that link.
Claiming the implied distribution has thinner tails than lognormal Fix: Higher implied volatility at extreme strikes means the market assigns more probability to extreme moves. Tails are fatter.
Calling the equity pattern a symmetric smile. Fix: Equity is a downward slope. Currency is a smile with both tails heavy. Link equity with left tail only.
Saying the right tail is heavier because high-strike calls look cheap. Fix: Lower implied volatility at high strikes means less right-tail probability than lognormal, so a thinner right tail.
Treating K/S0 = 1 as at-the-money-forward. Fix: The forward is S0 × e^((r − q) × T). The two centres agree only when r = q.
Giving put delta as a positive number or mixing up its sign. Fix: Put delta is negative. A 25-delta put has a delta of −0.25 and equals a 75-delta call at the same strike when q = 0.
Saying an upward-sloping term structure means volatility is rising over time for sure. Fix: It shows market-implied expected average volatility for each horizon. It signals current volatility below the long-run level, not a certainty.
Thinking a high short-term volatility and an inverted term structure are unrelated. Fix: After shocks, short-dated volatility jumps above long-dated volatility because of mean reversion, so the term structure inverts.

Exam tips

Expect conceptual MCQs on why BSM fails. The standard answer is that constant volatility and lognormal returns do not match fat tails, jumps and changing volatility.
When a question says same strike and same maturity for a call and a put, think put-call parity and equal implied volatility.
Know the direction of the effects: a higher volatility raises both call and put prices, so vega is positive for both.
Read the wording. Implied volatility is backed out of prices. Historical volatility is computed from past returns. Do not swap them.
For numerical questions, check that the final price or volatility is reasonable against the given bracket before you choose an answer.
Link currency to symmetric, equity to skew. Many questions test only this contrast.
When a question gives implied volatilities by strike, compare each to the at-the-money value before saying anything about tails.
Remember the maturity link: jumps for short dated, stochastic volatility for long dated.