FRM Part II · FRM Exam Part II
Volatility Smiles and Volatility Surfaces for FRM Part II
A volatility smile is the pattern you see when implied volatility, backed out of Black-Scholes prices, changes with strike price instead of staying flat. A volatility surface adds maturity. To solve questions, identify the asset, read the shape, link it to the distribution of returns, and state the pricing effect.
What this chapter covers
This chapter starts from a simple fact. Black-Scholes assumes one constant volatility for an asset. Market option prices do not agree. When you invert the formula to get implied volatility, the number changes with strike and with maturity. That pattern is the smile, skew or surface.
The chapter then does three jobs. It describes the shapes: a smile for foreign currency options, a downward skew for equity and index options, and a term structure across maturities. It explains why they exist, using fatter tails, skewed return distributions, jumps and stochastic volatility. It also shows how traders use them: to measure Greeks, to build implied distributions and to spot possible mispricing.
In Part II this sits inside Market Risk Measurement and Management. It connects to option risk measurement, valuation and risk models, and stress testing. Questions are applied. You are given a quoted shape or a set of implied volatilities and asked what it says about tails, hedging or pricing.
This chapter is short on formulas and heavy on interpretation, which suits an 80-question applied exam. Every question counts equally, so a topic that you can answer by reasoning is good value. If you can link a shape to a return distribution and then to a pricing or hedging consequence, you can handle most question styles. The ideas also support later work on option risk, model risk and tail events, so the effort pays off beyond this chapter.
Volatility Smiles and Volatility Surfaces: topics in the order to study them
- 1Implied Volatility and Black-Scholes Pricing AssumptionsEverything else depends on knowing what the model assumes and what implied volatility means, so you start here.
- 2Volatility Smiles for Foreign Currency OptionsThe currency smile is the cleanest example: a symmetric shape tied to fat tails, so it builds your intuition.
- 3Volatility Skew for Equity and Index OptionsOnce you know the symmetric smile, the downward skew is a contrast that you can explain with leverage and crash fear.
- 4Alternative Ways to Characterize the Volatility SmileThis gives you other ways to describe the same shape, such as by moneyness or delta, which questions often use.
- 5The Volatility Term Structure and Volatility SurfacesYou now add maturity to strike, which turns the smile into a surface.
- 6Greek Letters and the Volatility SmileWith the surface understood, you can see how it changes delta and vega and why hedges shift.
- 7Models for Smiles: Jumps, Stochastic Volatility and AlternativesThis explains why smiles arise and how models try to reproduce them, so it comes after you know the shapes.
- 8Determining Implied Distributions and Detecting MispricingIt is the most advanced use, linking prices to the risk-neutral distribution, so you finish with it.
How to prepare Volatility Smiles and Volatility Surfaces
Aim to read a shape and explain it in two sentences. Practise that more than memorising definitions.
- Write out the Black-Scholes assumptions in your own words, and note which one the smile breaks: constant volatility and lognormal returns.
- Sketch each shape from memory: currency smile, equity skew, and a term structure that is upward or downward sloping. Beside each, write the return distribution it implies.
- For each shape, note the effect on out-of-the-money puts and calls compared with Black-Scholes using a flat volatility. Be clear on which side is priced higher.
- Work through how the smile affects delta, gamma and vega. Focus on the direction of the change rather than exact numbers.
- Learn the models by their main idea: jumps give fat tails and short-dated smiles, stochastic volatility links volatility to the asset price and affects longer maturities.
- Do applied multiple-choice questions under time pressure. For each wrong answer, write down whether you misread the shape, the distribution or the pricing effect.
Common mistakes in Volatility Smiles and Volatility Surfaces
Mixing up the currency smile and the equity skew
Fix: Tie each shape to its asset: currency is roughly symmetric with two fat tails; equity is downward sloping with a heavy left tail.
Saying that the Black-Scholes price is wrong because the smile exists
Fix: Explain that implied volatility is a way to quote prices. The smile shows the model's assumptions fail, not that the formula gives arithmetic errors.
Getting the direction of tail pricing backwards
Fix: Remember that higher implied volatility means a higher option price. Ask which strikes have higher implied volatility, then say which tail is heavier.
Ignoring maturity when reading a surface
Fix: Always check the expiry. Note that the smile is usually steeper at short maturities and flattens at longer ones.
Assigning the wrong model to the wrong feature
Fix: Link jumps to sudden price moves and short-dated smiles, and stochastic volatility to changing volatility and longer-dated effects.
Treating a mispricing signal as a sure profit
Fix: State that it may reflect model error, liquidity or transaction costs. Only a violation of no-arbitrage conditions is a pure arbitrage.
Last-day revision: Volatility Smiles and Volatility Surfaces
- 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.
Volatility Smiles and Volatility Surfaces practice questions
- A bank's Black-Scholes model uses a flat 20% volatility to price a 1-year European call with a strike far above the current spot, while the …
- A risk manager uses a simple practitioner approach in which implied volatility is modeled as a quadratic function of strike, fitted to marke…
- A desk holds a long position in a call option and uses a sticky-strike assumption, where each strike's implied volatility is fixed regardles…
- A trader holds a long position in a 3-month out-of-the-money index put struck at 90% of spot. The trader prices it with the at-the-money imp…
- A bank's equity options desk is short a portfolio of out-of-the-money puts and calls on a stock. Black-Scholes vega for the whole book is co…
- A risk analyst observes that implied volatilities for equity index options decline steadily as the strike price rises from well below to wel…
- A desk fits a smooth implied volatility curve across strikes and uses it to derive an implied risk-neutral distribution for a currency. The …
- A trader uses a volatility surface to price a European option with a 9-month maturity. Quoted implied volatilities at the same strike are 22…
Volatility Smiles and Volatility Surfaces in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Volatility Smiles and Volatility Surfaces: frequently asked questions
What is the difference between a volatility smile and a volatility surface?
A smile shows implied volatility against strike for one maturity. A surface shows implied volatility against both strike and maturity. The surface is what traders use to price options at any strike and expiry.
Why do equity options show a skew rather than a symmetric smile?
Equity returns are seen as having a heavier left tail, as markets fall faster than they rise. Low-strike options therefore cost more and have higher implied volatility. Leverage effects and demand for downside protection are common explanations.
Do I need to calculate implied volatility by hand for FRM Part II?
No. The exam tests interpretation, not numerical solving. You should know what implied volatility is, how it differs across strikes and maturities, and what that says about the market's view of the distribution.
How much time should I give this chapter?
It is conceptual and shorter than many chapters, so a focused block is usually enough. Spend most of it on sketching shapes and doing applied questions, and revisit it near the exam with the quick revision list.