FRM Exam Part II · Volatility Smiles and Volatility Surfaces

Greek Letters and the Volatility Smile: Minimum Variance Delta

Updated 11 October 2026 · Fact-checked

When implied volatility depends on strike and the asset price, Black-Scholes delta is incomplete. The smile-consistent delta adds a vega term: delta = BS delta + vega × ∂σ_imp/∂S. The minimum variance delta picks the hedge ratio that minimises the variance of the hedged position's value change, given the expected volatility move.

Understand Greek Letters and the Volatility Smile

Black-Scholes assumes one constant volatility. In the market, implied volatility changes with strike and maturity. This is the smile or skew. It means the implied volatility an option trades at also moves when the asset price moves.

The standard delta is the partial derivative of the Black-Scholes price with respect to the asset price, holding volatility fixed. If implied volatility shifts as the price moves, holding volatility fixed is wrong. The true price change has two parts: the direct effect of S, and the indirect effect through the change in implied volatility.

This gives the smile-adjusted delta: BS delta + vega × (expected change in implied volatility per unit change in S). The sign of the adjustment depends on how the smile moves. In observed equity data, volatility usually rises when the price falls. The correlation between price and volatility is negative, so the empirically estimated adjustment is usually negative for both calls and puts, because vega is positive for both. A call's delta falls. A put's delta becomes more negative, so it is larger in absolute size.

How the smile moves is called smile dynamics. Under sticky strike, the implied volatility of each strike stays the same when S moves, so the adjustment is zero and BS delta is unchanged. Under sticky delta (also called sticky moneyness), volatility depends on K ÷ S, so the whole smile slides with the asset price. Then the volatility of a fixed strike changes when S moves. Write σ = g(K ÷ S). Then ∂σ/∂S = −(K ÷ S²) × g′. With a downward-sloping equity skew, g′ is negative, so a rise in S lowers K ÷ S and raises the volatility at a fixed strike. The vega term is then positive, and the adjusted delta of a call is above BS delta. With an upward-sloping smile in K ÷ S, the term is negative.

The minimum variance delta (Hull and White) is a practical version. It chooses the hedge ratio that minimises the variance of the change in the hedged portfolio, using the observed relationship between price and volatility changes. For a skewed equity market this empirical relationship is negative, so the minimum variance delta is typically below the Black-Scholes delta for calls. This is an empirical, model-dependent result. It is not the same as the sticky-delta result above. Vega hedging also changes: a model with a smile gives vega relative to the parameters of that model, such as a stochastic volatility factor, not just a single flat volatility.

Key formulas to remember

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.

How to solve Greek Letters and the Volatility Smile questions

Use this method for any question on how the smile affects Greeks or hedges.

1Identify the option, the BS delta and the BS vega given in the question.
2Identify the smile dynamics: sticky strike, sticky delta, or an estimated relationship between volatility and price.
3Work out the expected change in implied volatility per unit move in S, including its sign.
4Compute the adjustment: vega × ∂σ_imp/∂S, in matching units (vega per 1 percentage point versus per 1.00).
5Add the adjustment to BS delta to get the smile-consistent or minimum variance delta.
6Convert to a hedge: shares to hold = −(adjusted delta) × number of options.
7Check the direction. Sticky strike means no change. Equity skew with volatility rising as price falls usually pushes the call's minimum variance delta below BS delta.
8State what stays unhedged, mainly vega and the volatility-of-volatility risk.

Quickest way: Three-question shortcut

When to use it: Use for conceptual or comparison MCQs under time pressure.

Ask: does implied volatility of this strike move when S moves? If no, it is sticky strike and delta equals BS delta.
If yes, find the sign of ∂σ_imp/∂S. Equity: price down, volatility up, so it is negative.
Multiply vega by that slope and add to BS delta. A positive vega with a negative slope lowers delta.

Common mistakes in Greek Letters and the Volatility Smile

Saying BS delta is always correct if the Black-Scholes price matches the market.
Matching the price feels like matching the hedge.
Fix: Remember that BS delta holds volatility fixed. If volatility moves with S, the price sensitivity differs, so the hedge differs.
Adding the vega term under sticky strike.
Students apply the formula mechanically.
Fix: Under sticky strike, ∂σ_imp/∂S = 0, so the adjustment is zero.
Getting the sign of the adjustment wrong for equities.
They forget that volatility tends to rise when prices fall.
Fix: Equity skew gives a negative price-volatility relationship. Positive vega times a negative slope reduces the delta.
Mixing units of vega.
Vega is often quoted per 1 percentage point of volatility.
Fix: Match the units. If vega is per 1 point, the volatility change must also be in points.
Confusing minimum variance delta with an exact, riskless hedge.
The word 'minimum' suggests no risk.
Fix: It only minimises variance. Residual risk remains, and vega hedging is still needed.

Worked examples

Example 1

A call has S = 100, K = 100, a Black-Scholes delta of 0.60 and a vega of 30 per 1.00 change in volatility. The market follows sticky delta, so σ = g(K ÷ S). The smile slopes upward in K ÷ S: g′ = +0.05, meaning implied volatility rises by 0.05 for each 1.00 increase in K ÷ S. Find the smile-adjusted delta.

Show the solution

Sticky delta: ∂σ/∂S = −(K ÷ S²) × g′.
K ÷ S² = 100 ÷ 10,000 = 0.01.
∂σ/∂S = −0.01 × 0.05 = −0.0005 per unit of S.
Vega is per 1.00 of volatility and the slope is in decimal volatility, so the units match.
Adjustment = 30 × (−0.0005) = −0.015.
Adjusted delta = 0.60 + (−0.015) = 0.585.

Answer: The smile-adjusted delta is 0.585, slightly below the BS delta of 0.60. The sign is negative because the smile slopes upward in K ÷ S. A downward-sloping equity skew under sticky delta would give a positive adjustment.

Example 2

A call on an equity index has a BS delta of 0.55 and a BS vega of 30 per 1.00 change in volatility. The index is at 1,000. Past data show that when the index falls by 10 points, implied volatility rises by 0.5 percentage points, so the slope is −0.0005 per index point. Find the minimum variance delta and the number of index units to hold to hedge a short position in 10,000 calls.

Show the solution

Convert the volatility move: 0.5 percentage points = 0.005 in decimal volatility.
When the index falls 10 points, volatility rises 0.005, so ∂σ/∂S = −0.005 ÷ 10 = −0.0005 per point.
Vega adjustment = 30 × (−0.0005) = −0.015.
Minimum variance delta = 0.55 + (−0.015) = 0.535.
You are short 10,000 calls, so you hold index units long: 0.535 × 10,000 = 5,350 units.

Answer: Minimum variance delta = 0.535. Hold 5,350 index units long, versus 5,500 under plain BS delta.

Exam tips

Know the sign. For equity indexes, falling price and rising volatility makes the adjustment negative for options with positive vega.
Sticky strike means delta equals BS delta. Sticky delta means the smile moves with S.
Expect conceptual questions comparing minimum variance delta with BS delta, not heavy computation.
Watch the units of vega and the volatility change before you add the terms.
Remember the aim: minimum variance delta reduces hedge variance, it does not remove risk.

Practice questions from Volatility Smiles and Volatility Surfaces

Greek Letters and the Volatility Smile in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Greek Letters and the Volatility Smile: frequently asked questions

What is minimum variance delta?

It is the hedge ratio that minimises the variance of the change in a hedged option position. It equals BS delta plus vega times the expected change in implied volatility per unit change in the asset price.

How does the volatility smile affect delta hedging?

Implied volatility moves with the asset price, so BS delta, which holds volatility fixed, is not the true sensitivity. The hedge needs a vega-based adjustment that depends on smile dynamics.

What is the difference between sticky strike and sticky delta?

Under sticky strike, each strike keeps its implied volatility when S moves. Under sticky delta, volatility depends on moneyness K ÷ S, so the smile shifts with the asset price.

Why is the minimum variance delta usually lower than BS delta for equity calls?

It is an empirical result, not a general rule. In equity markets, prices and implied volatility have tended to move in opposite directions. The expected volatility change per unit of price is then negative, and with positive vega this lowers the call's minimum variance delta below BS delta. The size depends on the model and the data used.