Lesson 6.3: Implied volatility and the smile
Learn to compute implied volatility from an option price, why it varies by strike (smile and skew), what the volatility surface is and what traders do with it.
By the end of this lesson you will be able to compute an implied volatility from an option price with a root-finding method, estimate it quickly from a straddle, explain why options at different strikes have different implied volatilities, read a smile and a skew, describe the volatility surface and its no-arbitrage checks, and say what the variance risk premium and event volatility mean for a trader. You will show that a six-month at-the-money call quoted at 7.00 has an implied volatility of 19.42 per cent, that a flat 19 per cent volatility would price a put 20 per cent below the money at 0.14 when a skewed market charges 0.62, that the straddle rule of thumb gives an implied volatility of 20.1 per cent against a true 20 per cent, and that a volatility fall of 8 points after an event takes about 2.2 off a call even if the share price does not move. You should have read Lesson 6.2. The mathematics is the Black–Scholes formula, its derivative with respect to volatility and a little iteration. This course is education about how markets and research work, not investment advice.
Lesson 6.2 gave a formula that turns five inputs, the price, the strike, the time, the interest rate and the volatility, into an option price. Four of the inputs can be seen on a screen. The fifth, the volatility of the share over the life of the option, is the future, and it cannot be seen. A trader's view of that number is what the option market trades. So the market runs the formula in reverse: it observes the price and asks which volatility makes the formula agree. That number is the implied volatility, and it is the most watched quantity in the options business.
Implied volatility: running the formula backwards
Given a market price for an option, the implied volatility is the value of σ for which the Black–Scholes price equals the market price. There is no formula for σ in terms of the price, so it is found numerically. The price rises smoothly with σ, so a root-finding method works well. The standard one is Newton's method, which uses the slope of the price with respect to volatility, which is vega, to improve a guess:
σnew = σold − (Black–Scholes price at σold − market price) ÷ vega
Worked example 1. A six-month at-the-money call on a share at 100 with a strike of 100 and 6 per cent interest is quoted at 7.00. Start with a guess of 20 per cent.
Step · Guess for σ · Model price · Vega (per 1.00 of volatility) · Next guess0 · 20.0000% · 7.1559 · 27.103 · 19.4248%
1 · 19.4248% · 7.0001 · 27.071 · 19.4245%
2 · 19.4245% · 7.0000 · 27.071 · stops
Two steps are enough. The implied volatility is 19.42 per cent. Newton's method converges so fast here because the price is nearly a straight line in σ near the answer. It can fail for deep out-of-the-money options, whose vega is tiny, and for prices below the intrinsic value, for which no volatility exists, so careful code limits the step and falls back on a bracketing method such as bisection.
A rule of thumb from the straddle
You can estimate the implied volatility of an at-the-money option without a computer. For an at-the-money option, the price of a straddle, a call and a put at the money, is approximately
straddle ≈ 0.8 × S × σ × √T
with 0.8 standing in for √(2 ÷ π) = 0.7979. Turn it round to get σ ≈ straddle ÷ (0.8 × S × √T). In the example, the call is 7.16 and the put 4.20 at a volatility of 20 per cent, so the straddle is 11.36, and the rule gives 11.36 ÷ (0.7979 × 100 × 0.7071) = 20.1 per cent, within a tenth of a point of the truth. The straddle price, as a share of S, is also a quick guide to the market's expected move: the volatility times the square root of time, scaled, is the typical size of the move over the life of the option. Here the market expects the share to move by about 0.2 × 0.707 = 14 per cent either way, one standard deviation, over six months.
The smile: implied volatility is not the same at every strike
If Black–Scholes were true, with one constant volatility, then the implied volatilities of all options on the same asset and expiry would be equal. They are not. Plot implied volatility against strike and you do not get a flat line. For equity indices the shape is a skew or a smirk: implied volatility is highest for low strikes, the out-of-the-money puts, and falls as the strike rises. For some currencies and commodities the plot has two raised ends, which gives the smile that names the whole subject.
Here are illustrative implied volatilities for six-month options on a share at 100, with 6 per cent interest.
Strike · 80 · 90 · 100 · 110 · 120Implied volatility · 26.0% · 22.0% · 19.0% · 17.5% · 17.0%
What does the skew mean in rupees? Price the 80 put, which is 20 per cent below the money, in two ways. With a flat volatility of 19 per cent, Black–Scholes values it at 0.14. With the market's 26 per cent, it is worth 0.62, 4.6 times as much. The 90 put is 1.03 at the flat volatility and 1.54 at the skew. Out-of-the-money puts are expensive relative to the model because the market charges more for protection against large falls than a constant-volatility, normal-return model allows.
Why the skew exists
Several explanations work together, and none is complete.
Fat tails and crash risk. As Lesson 2.2 showed, large falls are far more likely than a normal distribution allows. A model that knows about them has to charge more for low-strike puts, and the market's skew is the way it does so.
The leverage effect. When a share falls, a company's debt is a larger share of its value and its equity becomes riskier, so volatility tends to rise as prices fall. The negative link between price and volatility makes low prices go with high volatility, which tilts the distribution towards big falls.
Demand for protection. Fund managers buy puts to insure portfolios, and few investors buy upside calls on indices. Persistent demand for puts pushes their prices up, in the same way as any shortage of supply.
Supply and risk. The sellers of puts are short gamma in a falling market, the worst place to be, so they ask a premium.
Jumps. A price that can jump has fatter tails than the model, and the option market prices the possibility.
The skew is therefore not an error to arbitrage away. It is the market's price for a risk the model leaves out.
The volatility surface
Options trade at many strikes and many expiries. The grid of implied volatilities by strike and expiry is the volatility surface. Two further features:
The term structure is the pattern across expiries at a fixed strike. In calm markets, longer-dated options usually have higher implied volatility than short-dated ones, since the long run holds more uncertainty. After a shock the curve inverts: near-term volatility jumps above long-term, because the market expects the turbulence to fade.
Strikes are usually measured not in rupees but as moneyness, such as K ÷ F or the logarithm of K ÷ F, or by the option's delta, so that surfaces from different days are comparable.
Traders summarise the shape of the skew with two numbers. A risk reversal is the implied volatility of an out-of-the-money call minus that of an out-of-the-money put at a symmetric distance: with the wings at 110 and 90 in the example, 17.5 − 22.0 = −4.5 volatility points. The negative sign is the skew. A butterfly is the average of the two wings minus the at-the-money volatility: (22.0 + 17.5) ÷ 2 − 19.0 = 0.75, a measure of how much the wings rise above the centre, which is the smile. Market quotes often come in this form, as an at-the-money volatility, a risk reversal and a butterfly at a given delta.
Large markets publish a volatility index. In India, India VIX, computed by the NSE from prices of Nifty options, shows the volatility the market expects over the next 30 days. It tends to rise when the index falls sharply, the leverage effect at index level, and traders watch it as a gauge of fear. Check the exchange's current methodology note for the details.
The surface must obey no-arbitrage rules
A surface fitted to market quotes is not free to take any shape, because some shapes would imply riskless profits. Two checks are standard.
Strike convexity (butterfly arbitrage). The call price must be a convex function of the strike: buying the calls at 90 and 110 and selling two at 100 must never cost a negative amount, since its payoff is never negative. With the skewed prices of the example, 14.199 − 2 × 6.885 + 2.419 = 2.85, which is positive, so the surface passes. A negative number would signal a bad quote or a bad fit.
Calendar spreads. Total variance, σ² × T, must not fall as the expiry lengthens at a fixed strike: a longer option should not be cheaper than a shorter one.
Surfaces are built from a smooth parametric form, such as the SVI family, or from a model, such as the local and stochastic volatility models of Lesson 8.4, and checked against these rules.
What traders do with implied volatility
Compare it with realised volatility
Implied volatility is the market's forecast, and realised volatility is what the share actually did. On average across many markets, index options' implied volatility has been higher than the volatility subsequently realised. The difference is called the variance risk premium, and it is a documented feature of index option markets, though its size varies by market and period. It is usually explained as insurance: sellers of options are paid to take the risk of large moves, a risk that buyers will pay to remove. A strategy that systematically sells options to collect it earns the premium most of the time and suffers rare large losses, which is the shape that Lessons 4.2 and 5.2 warned about, and Lesson 6.4 shows what it does in simulation.
Trade the shape
Skew trades buy and sell options at different strikes, for example buying a put and selling a call (a risk reversal), to bet that the skew will steepen or flatten.
Term-structure trades buy a short-dated option and sell a long-dated one, or the reverse, to bet on the shape of the curve.
Relative value. Compare the implied volatilities of related assets, such as a stock and its index, or two similar stocks.
Event volatility
Scheduled events, such as quarterly results, a central bank decision, an election or a Union Budget, add uncertainty that the options expiring after the event must price. Implied volatility rises ahead of the event and falls sharply after it, when the uncertainty is resolved. This volatility crush can hurt the buyer of an option even if the price moves in their favour.
Worked example 2. A six-month at-the-money call is priced with an implied volatility of 28 per cent before an event and is worth 9.34. The event passes, the share price does not change, and implied volatility returns to 20 per cent. The call is now worth 7.16, a loss of 2.18, 23 per cent of its value. The linear estimate from vega gives 8 × 0.271 = 2.17, almost the same. The buyer needed the share to move enough to make up for that loss, and a move that is smaller than the market expected loses even in the right direction.
Use the simulator below to estimate the effect of a change in implied volatility on an option you hold.
The next calculator turns the wings of a skew into the two summary numbers that the market quotes.
Beyond Black–Scholes: models that produce a smile
The skew is the signal that Black–Scholes is incomplete, and a family of models is built to fit it. Local volatility models, due to Bruno Dupire, let volatility depend on the price and the time so as to match every option on the surface exactly. Stochastic volatility models, such as Heston's, let volatility follow its own random process, correlated with the price, and produce skews and term structures naturally. Jump-diffusion models add random jumps. Each model has to be calibrated, which means choosing its parameters to match the market's prices, and each makes its own prediction about how the surface will move, which can differ from the data. Lesson 8.4 covers the main ones, calibration and model risk.
The code below finds an implied volatility with Newton's method, with a bracketing fallback, and computes the straddle approximation, the smile prices, the butterfly check and the volatility crush.
python
import numpy as np
from math import log, sqrt, exp
from scipy.stats import norm
from scipy.optimize import brentq
def bscall(S, K, r, sigma, T):
d1 = (log(S / K) + (r + 0.5 sigma 2) T) / (sigma sqrt(T))
return S norm.cdf(d1) - K exp(-r T) norm.cdf(d1 - sigma sqrt(T))
def bsput(S, K, r, sigma, T):
return bscall(S, K, r, sigma, T) - S + K exp(-r T)
def vega(S, K, r, sigma, T):
d1 = (log(S / K) + (r + 0.5 sigma 2) T) / (sigma sqrt(T))
return S norm.pdf(d1) sqrt(T) # per 1.00 of volatility
def impliedvol(price, S, K, r, T, guess=0.2):
intrinsic = max(S - K exp(-r T), 0)
if price <= intrinsic or price >= S:
return float("nan") # no volatility can produce this price
sigma = guess
for step in range(20):
diff = bscall(S, K, r, sigma, T) - price
v = vega(S, K, r, sigma, T)
print(f" step {step}: sigma {sigma:.6f} model price {price + diff:.5f} vega {v:.3f}") if step < 3 else None
if abs(diff) < 1e-8:
return sigma
if v < 1e-6: # vega too small for Newton: bracket instead
return brentq(lambda x: bscall(S, K, r, x, T) - price, 1e-4, 5.0)
sigma -= diff / v
return sigma
S, K, r, T = 100, 100, 0.06, 0.5
print("Newton's method for a call quoted at 7.00:")
iv = impliedvol(7.00, S, K, r, T)
print("implied volatility", round(iv 100, 2), "%")
the straddle rule of thumb
straddle = bscall(S, K, r, 0.20, T) + bsput(S, K, r, 0.20, T)
print("straddle at 20%:", round(straddle, 3), "| rule of thumb volatility", round(straddle / (0.7979 S sqrt(T)) 100, 2), "%")
a skew: flat against skewed prices
strikes = [80, 90, 100, 110, 120]
smile = [0.26, 0.22, 0.19, 0.175, 0.17]
for k, v in zip(strikes, smile):
print(f"strike {k}: put at the skew {bsput(S, k, r, v, T):7.3f} | put at flat 19% {bsput(S, k, r, 0.19, T):7.3f}"
f" | call at the skew {bscall(S, k, r, v, T):7.3f} | call at flat 19% {bscall(S, k, r, 0.19, T):7.3f}")
print("80 put: skew price is", round(bsput(S, 80, r, 0.26, T) / bsput(S, 80, r, 0.19, T), 1), "times the flat price")
no-arbitrage check: butterfly on the skewed call prices
c90, c100, c110 = (bscall(S, k, r, v, T) for k, v in ((90, 0.22), (100, 0.19), (110, 0.175)))
print("butterfly 90/100/110 on skewed call prices:", round(c90 - 2 c100 + c110, 3), "(must be >= 0)")
print("risk reversal", round(0.175 - 0.22, 3), "| butterfly", round((0.175 + 0.22) / 2 - 0.19, 4))
volatility crush
before, after = bscall(S, K, r, 0.28, T), bscall(S, K, r, 0.20, T)
print("call at 28%:", round(before, 2), "at 20%:", round(after, 2), "loss", round(before - after, 2),
"| linear estimate", round(8 vega(S, K, r, 0.20, T) / 100, 2))
Common mistakes
Quoting an option price without its implied volatility. The volatility is what lets you compare options across strikes, expiries and days.
Treating implied volatility as a forecast that is right. It is a price, which includes a risk premium, and it differs from the volatility that will be realised.
Using one volatility for all strikes. The skew means the wings are priced at their own volatilities, and a flat volatility misprices them badly.
Solving for implied volatility with a method that can fail. Use a bracketing fallback, and reject prices below intrinsic value.
Using in-the-money options to read volatility. Out-of-the-money options carry the information, and in-the-money prices are mostly intrinsic value with a tiny volatility component.
Buying options before an event and ignoring the crush. The price already includes the event, so a modest move loses.
Treating the skew as an arbitrage. It is the market's price of crash risk and of demand for protection.
Fitting a surface that violates no-arbitrage rules. Check convexity in strike and the calendar condition.
Ignoring liquidity. Quotes for far strikes are wide and thin, and the implied volatility there may not be tradable.
Key takeaways
Implied volatility is the volatility that makes the Black–Scholes price match the market price. Newton's method, σnew = σold − (model price − market price) ÷ vega, finds it in two steps for a call quoted at 7.00: 19.42 per cent.
For an at-the-money option, straddle ≈ 0.8 S σ √T, so σ ≈ straddle ÷ (0.8 S √T). A straddle of 11.36 on a share at 100 with six months to go implies about 20.1 per cent.
Implied volatility varies with strike. Equity indices show a skew, with out-of-the-money puts at the highest volatilities. A flat 19 per cent prices the 80 put at 0.14, and a 26 per cent skew prices it at 0.62, 4.6 times more.
The skew reflects fat tails, the leverage effect, demand for protection, hedgers' risk and jumps. It is the market's price for risks that Black–Scholes leaves out.
The surface of implied volatilities by strike and expiry has a term structure, summarised by risk reversals (call wing minus put wing, −4.5 points in the example) and butterflies (0.75). It must satisfy convexity in strike and a calendar condition.
Implied volatility has on average exceeded realised volatility for index options, the variance risk premium, which sellers earn at the price of rare large losses. Event volatility rises before and collapses after an event: a fall from 28 to 20 per cent cost about 2.2 on a call.
Local volatility, stochastic volatility and jump models are built to reproduce the smile, and they must be calibrated and tested.
Check yourself
1. A call is quoted at 5.50 and the Black–Scholes price at 18 per cent volatility is 5.20, with a vega of 0.30 per volatility point. What is the next Newton guess, and roughly the implied volatility?
The price difference is −0.30 and vega per 1.00 of volatility is 30, so the step is +0.30 ÷ 30 = 0.01. The next guess is 19 per cent, and the implied volatility is about 19 per cent.
2. A three-month at-the-money straddle on a share at 500 costs 50. What is the approximate implied volatility?
σ ≈ 50 ÷ (0.7979 × 500 × √0.25) = 50 ÷ 199.5 = 25.1 per cent.
3. The 20 per cent out-of-the-money put has an implied volatility of 30 per cent, and the at-the-money option 20 per cent. What does this tell you, and what would a flat 20 per cent do to the put's price?
It shows a steep skew: the market prices crash risk. A flat 20 per cent would value the put far below the market price, so a trader who used it would think the put expensive and sell it, taking on a risk the skew is paying for.
4. You buy a call before earnings at an implied volatility of 35 per cent. After the announcement the share is unchanged and implied volatility is 25 per cent. The call's vega is 0.20. What is your approximate loss?
10 points × 0.20 = 2.0 per option.
5. A researcher fits a smile and finds that the butterfly of calls at 90, 100 and 110 has a negative price. What does it mean?
The fitted prices admit an arbitrage: the butterfly's payoff is never negative, so a negative price is impossible. The fit or the quotes are wrong, and the surface must be corrected.
Try this
Set aside forty-five minutes. You need Python and, if you can get them, option quotes for an index or a share for one expiry, at five to ten strikes.
1. Run the code. Check that Newton's method converges in two or three steps and that the straddle rule is close.
2. Using your quotes, compute the implied volatility at each strike, using out-of-the-money options (puts below the forward and calls above it). Plot them.
3. Compute the risk reversal and the butterfly from the wings, and describe the skew in one sentence.
4. Test the butterfly condition across three neighbouring strikes. Does your quote set pass?
5. Compute the implied volatility from the nearest straddle and compare it with the volatility of the underlying over the last 20 and 60 days. Is the variance risk premium visible?
6. Use the crush simulator to estimate what a fall of 5 volatility points would cost on one of your options.
7. Write a short note: the level and shape of the surface, how it compares with realised volatility, and what event or fear it seems to price.
A good result is a plot of implied volatility against strike with the wings labelled, an honest comparison with realised volatility and a passed arbitrage check. The next lesson turns from reading volatility to trading it: how a trader hedges an option and what a volatility strategy earns.
Read more
Emanuel Derman and Iraj Kani's 1994 paper "Riding on a Smile" and Bruno Dupire's 1994 paper "Pricing with a Smile" in Risk introduced local volatility. Jim Gatheral's The Volatility Surface: A Practitioner's Guide is the standard reference on surfaces, the smile and no-arbitrage conditions, and Steven Heston's 1993 paper "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options" in the Review of Financial Studies is the classic stochastic volatility model. Peter Carr and Liuren Wu's 2009 paper "Variance Risk Premiums" in the Review of Financial Studies studies the premium. Euan Sinclair's Volatility Trading is a practitioner's account, and for the construction of India VIX, read the NSE's methodology document.
Tags: Quant Trader Implied volatility Volatility smile Volatility surface Variance risk premium