FRM Exam Part II · Volatility Smiles and Volatility Surfaces

Implied Volatility and Black-Scholes Pricing Assumptions

Updated 11 October 2026 · Fact-checked

Implied volatility is the volatility that, put into the Black-Scholes-Merton formula, returns the observed option price. You find it numerically because the formula cannot be inverted. A call and a put with the same strike and maturity share one implied volatility. It varies by strike and maturity, which shows constant volatility is wrong.

Understand Implied Volatility and Black-Scholes Pricing Assumptions

The Black-Scholes-Merton (BSM) model prices a European option from five inputs: spot price, strike, time to maturity, interest rate and volatility. Four are observable. Volatility is not. So the market quotes the price and you work backwards to the volatility that makes the model match it. That number is the implied volatility. It is a way of quoting price, not a forecast you can read directly.

The formula cannot be rearranged to give volatility. You solve it by trial: guess a volatility, compute the BSM price, compare it with the market price, and adjust. Option price rises steadily with volatility, so there is only one answer. Newton-Raphson uses vega (the price change per unit of volatility) to make each adjustment. Bisection and interpolation also work.

Historical volatility is the standard deviation of past returns, annualised. Implied volatility is forward-looking and comes from today's option prices. It also includes a premium for risk, so it often sits above later realised volatility. Do not treat the two as the same thing.

BSM assumes the asset price follows geometric Brownian motion with constant volatility, so log returns are normal. It also assumes a constant risk-free rate, no arbitrage, continuous trading, no transaction costs, and (in the basic form) no dividends. If these held, every option on the same asset would give the same implied volatility. In real markets they do not. Implied volatility changes with strike (the smile or skew) and with maturity (the term structure). Real returns have fat tails, jumps and volatility that moves, and traders pay more for protection against large falls. The pattern of implied volatilities across strikes and maturities is the volatility surface.

Put-call parity is model-free for European options: c + K e^(−rT) = p + S0 e^(−qT). It holds for the same strike and maturity. So a call and a put at that strike must have the same implied volatility. If they differ, there is an arbitrage or a data problem, such as stale quotes, early-exercise features or wrong dividend inputs.

Key formulas to remember

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.

How to solve Implied Volatility and Black-Scholes Pricing Assumptions questions

Use this order for any question on implied volatility or the BSM assumptions.

1Identify what is given: option type, S0, K, T, r, q and the market price or the quoted volatility. Check that the option is European.
2Decide what is unknown. If it is price, compute d1, d2 and the BSM value. If it is volatility, you are inverting the formula.
3For volatility, bracket the answer. Price a low and a high volatility, then interpolate or apply Newton-Raphson using vega.
4If the question gives a call price and asks about the put (or the reverse), use put-call parity to find the other price. At the same strike and maturity, the implied volatility is the same.
5Compare with historical volatility if asked. Implied is forward-looking and includes a risk premium. Historical is backward-looking.
6State the interpretation. If implied volatility differs by strike or maturity, constant volatility fails, and you should name the smile, skew or surface.
7Check units: volatility annualised, T in years, r and q continuously compounded, price in the correct currency.

Quickest way: Bracket and interpolate

When to use it: Use when the question gives model prices at two volatilities and a market price between them, or asks you to compare implied volatilities.

If the market price lies between the two given model prices, interpolate linearly on volatility. Near the money, price is almost linear in volatility, so this is accurate.
If the question mentions the same strike and maturity for a call and a put, answer with equal implied volatility. No calculation is needed.
If the question gives a call price and asks for the put, use p = c + K e^(−rT) − S0 e^(−qT) and discount the strike.
For conceptual options, eliminate any answer that says implied volatility is constant across strikes or equals historical volatility.

Common mistakes in Implied Volatility and Black-Scholes Pricing Assumptions

Treating implied volatility as a forecast of realised volatility.
It is called a volatility, so it looks like a prediction.
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.
Students recall c − p = S0 − K from memory.
Fix: Write c − p = S0 e^(−qT) − K e^(−rT). Only with r = 0 and q = 0 does the discounting disappear.
Saying a call and a put at the same strike have different implied volatilities because their prices differ.
Prices differ, so students assume the volatilities do.
Fix: Parity ties the two prices together. For European options with the same strike and maturity, the implied volatilities are equal.
Saying BSM implies the same implied volatility for all strikes, so the smile is a model flaw in the data.
Mixing up what the model assumes with what the market shows.
Fix: BSM assumes constant volatility. The market shows a smile or skew. The market prices differ from BSM because returns have fat tails and volatility changes.
Mixing up vega signs or applying it to puts differently.
Students link put values with falling prices and assume the opposite sign.
Fix: Vega is positive for both calls and puts and is the same at the same strike and maturity. Higher volatility raises both prices.
Using the wrong time unit or compounding.
Quotes come in days or months, and rates may be quoted with other compounding.
Fix: Convert T to years and use continuously compounded r and q in the BSM formulas.

Worked examples

Example 1

A European call and a European put on a non-dividend stock have the same strike of $100 and one year to maturity. The spot is $100 and the continuously compounded risk-free rate is 5%. The call trades at $10.45, which corresponds to an implied volatility of 20%. What is the put price, and what is its implied volatility?

Show the solution

Use put-call parity with q = 0: p = c + K e^(−rT) − S0.
Discount the strike: 100 × e^(−0.05) = 100 × 0.951229 = 95.1229.
Compute p = 10.45 + 95.1229 − 100 = 5.5729, about $5.57.
Parity ties the put price to the call price at the same strike and maturity. A put priced at $5.57 therefore has the same implied volatility as the call.

Answer: The put is worth about $5.57 and its implied volatility is 20%, the same as the call.

Example 2

For a one-year at-the-money European call (S0 = K = $100, r = 5%, no dividends), the BSM price is $10.45 at σ = 20% and $14.23 at σ = 30%. The market price is $12.34. Estimate the implied volatility, and say what it tells you if the stock's historical volatility is 18%.

Show the solution

The market price of $12.34 lies between $10.45 and $14.23, so the implied volatility lies between 20% and 30%.
Interpolate linearly: fraction = (12.34 − 10.45) ÷ (14.23 − 10.45) = 1.89 ÷ 3.78 = 0.50.
Implied volatility ≈ 20% + 0.50 × 10% = 25%.
Check: at σ = 25%, d1 = (0.05 + 0.03125) ÷ 0.25 = 0.325 and d2 = 0.075. N(d1) ≈ 0.6274 and N(d2) ≈ 0.5299, so c ≈ 62.74 − 95.1229 × 0.5299 ≈ 12.34. The estimate holds.
Compare with historical volatility of 18%. The market is pricing about 25%, above past realised volatility. Implied volatility is forward-looking and includes a premium for uncertainty and for demand for protection, so a gap like this is normal.

Answer: Implied volatility is about 25%. It is above the 18% historical figure, so the option is priced for higher future volatility than the recent past, plus a risk premium.

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.

Practice questions from Volatility Smiles and Volatility Surfaces

Implied Volatility and Black-Scholes Pricing Assumptions in other exams

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

Implied Volatility and Black-Scholes Pricing Assumptions: frequently asked questions

How do you calculate implied volatility from an option price?

You cannot solve the BSM formula for volatility directly. You guess a volatility, compute the BSM price, and adjust until the model price equals the market price. Newton-Raphson using vega, bisection and interpolation all work. On the exam you will usually bracket and interpolate.

What is the difference between implied volatility and historical volatility?

Historical volatility is the annualised standard deviation of past returns. Implied volatility is backed out of current option prices and is forward-looking. Implied volatility also reflects a risk premium, so it often sits above realised volatility.

Do a call and a put at the same strike have the same implied volatility?

Yes, for European options with the same strike and maturity on the same underlying. Put-call parity links the two prices without any assumption on volatility, so they give one implied volatility. A difference points to an arbitrage opportunity or bad inputs.

Why does constant volatility fail to match market prices?

If BSM held, every strike and maturity would give the same implied volatility. Market implied volatilities vary by strike and maturity, forming a smile or surface. Real returns have fat tails, jumps and stochastic volatility, and the market prices these features.