Volatility Smiles and Surfaces

Trading desks and risk managers use the volatility surface to value options at strikes with no market quote and to calculate risks such as delta.

A volatility surface set by strike and expiry

If the assumptions of the Black-Scholes model held, options on the same underlying asset would have the same implied volatility regardless of strike price and expiry. In actual markets, the value differs by strike and expiry, and its shape changes over time¹. How the model's assumptions differ from the market is covered in The Black-Scholes Model.

Let $C_{\text{mkt}}(K, T)$ be the market price of a call with strike $K$ and expiry $T$. The volatility surface $\sigma(K, T)$ is the value that satisfies the following equation¹.

$$C_{\text{mkt}}(K, T) = C_{\text{BS}}(S, K, T, r, \sigma(K, T))$$

$C_{\text{BS}}$ is the call price from the Black-Scholes formula, $S$ is the price of the underlying, and $r$ is the interest rate. For European options, put-call parity gives a put with the same strike and expiry the same value, so calls and puts share one surface². A slice at a single expiry is the volatility smile, and a slice at a single strike shows how the value changes with expiry (the term structure).

Consider a hypothetical example in which a stock index has a price of 100 and an interest rate of 1%, and consider its surface. The worked examples that follow use the values in the table below.

At every expiry, the value is higher at lower strikes. The gap between strike 80 and strike 120 is 16.0 percentage points at 3 months, 13.0 at 1 year, and 9.0 at 2 years. The shorter the expiry, the steeper the slope, and the smile flattens toward 2 years.

Figure description

The figure connects the 1-year row of the table above with a line, with strike on the horizontal axis and implied volatility (%) on the vertical axis. It slopes downward from 28% at strike 80 to 15% at strike 120. The vertical dashed line marks the underlying price of 100. If the Black-Scholes assumptions held, the implied volatility would be a horizontal line at 20% for every strike. The two lines cross at strike 100: at lower strikes the smile is above the flat value, and at higher strikes it is below. This downward-sloping shape is called a skew (covered in the next section).

Smile shapes by asset and differences by expiry

For options on stocks and stock indices, the smile at each expiry usually slopes downward (a skew): the higher the strike, the lower the implied volatility. The slope is steeper at shorter expiries³. Before the Wall Street crash of 1987, there was little or no skew in the market⁴. As expiry lengthens, the skew flattens, and a rule of thumb says the slope shrinks roughly in inverse proportion to the square root of the expiry. No-arbitrage conditions, however, keep this rule of thumb from holding all the way out to long expiries⁵.

Currency options are quoted by delta, not by strike. For each expiry, the market describes the smile with the ATM volatility and with risk reversals and butterflies (strangles in some sources) at deltas such as 25-delta. The risk reversal shows the slope, and the butterfly shows the curvature⁶. The butterfly here is a volatility used in quoting, and is different from the butterfly (a combination of options) in the section "No-arbitrage constraints." The sign of the risk reversal determines whether the call side or the put side is higher.

Interest rate options also have smiles. In an example of yen swaptions in September 2021, implied volatility was low near ATM and rose further out of the money⁷. A swaption has a third axis besides strike and expiry: the term (tenor) of the underlying interest rate swap. The values arranged along these three axes are called a volatility cube⁷.

Looking along the expiry direction at a single strike, implied volatility can rise or fall with expiry. According to lecture notes from Columbia University (2013), it approaches a constant value as expiry lengthens, and at short expiries it is often downward sloping (inverted), with shorter expiries having much higher values. This is especially so when markets are unstable⁸.

When volatility depends only on time and not on the price of the underlying, the square of the implied volatility $\sigma(T)$ for expiry $T$ multiplied by $T$ (the total variance) equals the accumulated square of the volatility $\sigma(u)$ up to expiry⁹.

$$\sigma(T)^2 \, T = \int_0^T \sigma(u)^2 \, du$$

The implied volatility of a long-dated option includes the part up to a shorter expiry. The volatility between two expiries is found from the difference between their total variances. With the ATM values in the table (20.0% at 1 year, 20.5% at 2 years), the volatility from 1 year to 2 years out is $\sqrt{0.205^2 \times 2 - 0.20^2 \times 1} = \sqrt{0.04405} \approx 20.99\%$¹⁰.

Building the surface from market points

Market prices are visible only for options at some strikes. To draw the surface, you therefore interpolate between the points¹¹. Some market points have a wide gap between the bid and offer quotes (the bid-ask spread), and some have quotes that are slow to update. Such illiquid options are sometimes left out when building the surface¹².

The interpolation axis follows the way the market quotes. Currency options are quoted by delta, so first, for each expiry, you find the volatility and strike at ATM, 10-delta, and 25-delta. One implementation then interpolates linearly in strike, and beyond the outermost strikes extends the end values flat¹³.

For swaptions, you can connect the points with straight lines. Interpolating the smile with the SABR model, a stochastic volatility model (covered later in "Models that handle the smile"), is widely used as the industry's general-purpose tool. Using a model gives a better fit to market values and avoids interpolation that creates obvious arbitrage opportunities¹¹.

The slope of the parts extended beyond both ends (the wings) also has an upper bound that comes from no-arbitrage. Even as the strike moves far away, implied volatility cannot grow faster than a certain rate. So the functions used to extend the ends cannot take forms that grow faster than this¹⁴. The formula for the bound and its link to the tails of the distribution are covered later in the deep dive in "The smile and the risk-neutral distribution."

No-arbitrage constraints

The shape of the surface cannot be chosen freely. The prices of options at the same expiry with different strikes, and of options at the same strike with different expiries, must be ordered so that no arbitrage opportunity arises¹⁷. Market surfaces respect these constraints, but interpolated surfaces¹¹ and surfaces shifted for risk management¹⁷ can break them. There are three constraints.

The first is the ordering of prices by strike. For strikes $K_1 < K_2$, call and put prices are ordered as follows¹⁸.

$$C(K_1) \ge C(K_2), \quad P(K_1) \le P(K_2)$$

This is because the payoff of the put with the higher strike is never below the payoff of the put with the lower strike. If the skew is too steep, the put with the lower strike becomes more expensive, and this ordering breaks¹⁷.

The second is the butterfly. The combination of buying one call each at strikes $K - \Delta K$ and $K + \Delta K$ and selling two calls at $K$ (a butterfly) has a payoff at expiry that is never negative¹⁹. Its price is therefore 0 or more.

$$C(K - \Delta K) - 2C(K) + C(K + \Delta K) \ge 0$$

The third is the calendar spread. When interest rates and dividends are 0, the price of a call at a given strike does not fall as expiry lengthens¹⁷. In this case, the Black-Scholes call price is higher the larger $\sigma\sqrt{T}$ is, so the constraint is the same as saying that the total variance $\sigma(K, T)^2 \, T$ does not decrease with expiry²⁰. Even an inverted term structure is allowed, but if it is too inverted, this constraint breaks.

Worked example: checking the constraints on the 1-year smile

Using the 1-year smile in the table (underlying price 100, interest rate 1%), find the call and put prices²¹. For comparison, the table also shows the prices for a flat case with a volatility of 20% at every strike.

The put prices rise with the strike, at 2.8104, 4.6009, 7.4383, 12.3981, and 19.8219, so the first constraint holds. The butterflies centered on strikes 90, 100, and 110 ($\Delta K = 10$) are 1.0468, 2.1225, and 2.4639, all positive. For example, the one centered on 100, calculated from the rounded values in the table, is 15.4965 − 2 × 8.4333 + 3.4927 = 2.1226, and from the unrounded values it is 2.1225²².

Now check a case where the slope is too steep. Keep the volatility at strike 100 at 20% and raise the volatility at 90 to 35%. The 90 put becomes 8.3864, above the 100 put at 7.4383. You can then buy the 100 put and sell the 90 put, receive 0.9481, and still have a payoff at expiry that is never negative. The 90 put has the same price as the 100 put when the volatility at 90 is 32.29%, and any higher value breaks the first constraint²³.

Check the calendar spread with an example at a zero interest rate. With an underlying price of 100 and a strike of 100, if the 6-month volatility is 25% and the 1-year volatility is 17%, the 6-month call is 7.0432 and the 1-year call is 6.7739, so the longer expiry is cheaper. The total variance also falls, from 0.25² × 0.5 = 0.03125 at 6 months to 0.17² × 1 = 0.0289 at 1 year. If the 1-year volatility is 17.68% (the square root of 0.03125) or higher, this constraint holds²⁴.

The smile and the risk-neutral distribution

The price of a butterfly at a given expiry is proportional to the risk-neutral probability that the underlying price at expiry ends up near $K$. Let $f(K, T)$ be the probability density of the underlying price at expiry $T$. The butterfly price gives the following approximation, and as $\Delta K$ approaches 0, it becomes the second formula¹⁹.

$$f(K, T) \approx e^{rT} \frac{C(K - \Delta K) - 2C(K) + C(K + \Delta K)}{(\Delta K)^2}$$ $$f(K, T) = e^{rT} \frac{\partial^2 C}{\partial K^2}$$

Once you know the surface, the distribution of the underlying price at each expiry is determined. The second constraint (a butterfly is 0 or more) corresponds to this probability density never being negative.

From the butterflies in the worked example in the previous section, find approximate probability densities, using $e^{0.01} = 1.010050$²⁵.

With the skew in this example, compared with the flat case, the probability density is lower near 90 and higher near 100 and 110. On the other hand, the put with a strike of 80 is 2.8104, 2.6 times the 1.0673 of the flat case. The put price is the discounted expected amount by which the price ends below 80 at expiry, so this shows that the contribution from large falls well below 80 is bigger than in the flat case. Conversely, the call with a strike of 120 is 1.0159, lower than the 2.3406 of the flat case, and the contribution from rises above 120 is smaller.

Deep diveThe slope of the wings and the tails of the distribution (the moment formula)

Write the position of the strike as $x = \ln(K / (S e^{rT}))$. For a fixed expiry, where $|x|$ is large enough, implied volatility satisfies the following upper bound¹⁴.

$$\sigma^2(x, T) \le \frac{2|x|}{T}$$

Let $\beta_R$ be the upper limit of the value that $\sigma^2 T / |x|$ approaches at the right end, and $\beta_L$ the value at the left end. Both lie between 0 and 2. For the underlying price at expiry $S_T$, the values are determined only by the upper limit $\tilde p$ of the $p$ for which $E[S_T^{1+p}]$ is finite (right side), and the upper limit $\tilde q$ of the $q$ for which $E[S_T^{-q}]$ is finite (left side)¹⁴.

$$\beta_R = 2 - 4\left(\sqrt{\tilde p^2 + \tilde p} - \tilde p\right)$$

The smaller $\tilde p$ is, the larger $\beta_R$ is: $\beta_R$ is 2 at $\tilde p = 0$ and approaches 0 as $\tilde p$ grows²⁶. The fatter the tail of the distribution, and the fewer higher moments that are finite, the steeper the slope of the wing. The same relationship holds between $\beta_L$ and $\tilde q$ on the left side.

The surface determines only the distribution of the underlying price at each expiry. It does not determine how prices at several points in time combine (the joint distribution)¹⁹. So the price of a digital option, whose payoff depends only on the price at expiry, is determined from the surface as $-\partial C / \partial K$, the derivative of the call price with respect to the strike, and the slope of the skew affects its price²⁷. The price of a barrier option, whose payoff changes with prices before expiry, is not determined by the surface alone²⁸.

How the surface moves when the market moves

When the price of the underlying moves, the shape of the surface moves too. There are two views of how the surface moves. Sticky strike assumes that the implied volatility at each strike stays the same. Sticky delta assumes that the value at each ratio of the strike to the underlying price (moneyness) stays the same²⁹³⁰. According to lecture notes from Columbia University (2013), equity markets often use sticky strike when calculating delta, while foreign exchange markets often use sticky delta²⁹.

Take the 1-year smile example, and suppose the underlying price rises from 100 to 105 on the same day (the time to expiry stays at 1 year). Under sticky strike, the volatility of the call with a strike of 100 stays at 20%, and its price is 11.4698. Under sticky delta, the strike of 100 corresponds to the position 100 × 100 ÷ 105 = 95.24 on the surface before the move. Interpolating linearly between 90 and 100 gives 24 − 4 × 0.5238 = 21.90%, and the price is 12.2087. The price rises by 3.0365 (8.4333 → 11.4698) under sticky strike and by 3.7754 (8.4333 → 12.2087) under sticky delta. The choice of view makes a difference of about 0.74, and under sticky delta the price reacts more strongly to the underlying price³¹.

The delta calculated under sticky strike is the Black-Scholes delta with the volatility held fixed. If you assume the volatility moves with the underlying price, delta gains an extra term through vega $\nu$²⁹.

$$\Delta = \frac{\partial C_{\text{BS}}}{\partial S} + \nu \, \frac{\partial \sigma(K, T)}{\partial S}$$

With a downward-sloping skew like the one in this example, under sticky delta, the volatility at strike 100 rises when the underlying price rises ($\partial \sigma / \partial S > 0$). The second term is therefore positive, and delta is larger than under sticky strike³².

The actual daily moves of the surface are more complex than these two views. A study of data on options on US and UK stock indices explained most of the daily changes with three moves: a shift up or down in the overall level, a change in slope in which the low-strike and high-strike sides move in opposite directions, and a change in curvature³³.

Models that handle the smile

To describe the smile, the Black-Scholes model has to be extended. The extensions fall into three kinds: making volatility a function of the underlying price and other variables, letting volatility itself move randomly, and adding price jumps³⁴.

A local volatility model makes volatility a fixed function $\sigma(S, t)$ of the underlying price and time. The function that fits all of today's option prices on the surface can be found from the derivatives of the call price with respect to strike and expiry (Dupire's formula)³⁵. In a stochastic volatility model, volatility moves with a separate source of randomness, not only with the underlying price and time³⁶. The Heston model is one of them. The SABR model used for swaptions is a stochastic volatility model. Its approximation formula computes implied volatility quickly, and its relationship to the market-standard Black model is clear³⁷. In practice, models that combine the two (stochastic local volatility models) are also used³⁴.

How a model is fitted to the surface is covered in Model Calibration. Differences between models remain even after they are fitted to the surface. As shown in "The smile and the risk-neutral distribution," the surface determines only the distribution at each expiry. Two models that fit the same surface can give different prices for a barrier option if they link prices across points in time differently³⁸.

Notes

1. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, 2013), equation (14) in Section 3 "The Volatility Surface" (page 4). ↩ ↩²

2. Martin Haugh, "Black-Scholes and the Volatility Surface," note 5 on page 4. On page 5, the source also says that for options on individual stocks, many of which are American options, calls and puts have different surfaces. ↩

3. Martin Haugh, "Black-Scholes and the Volatility Surface," page 4 (the Eurostoxx 50 index surface on November 28, 2007) and "Why is there a Skew?" on page 5. ↩

4. Martin Haugh, "Black-Scholes and the Volatility Surface," page 6. ↩

5. Roger W. Lee, "Implied Volatility: Statics, Dynamics, and Probabilistic Interpretation," Section 3 (page 11) and Remark 3.1 in Section 3.1.1 (page 13). ↩

6. OpenGamma, "Vanilla Forex Options: Garman-Kohlhagen and Risk Reversal/Strangle" (May 2012, Version 1.5), Section 3 "Smile" (pages 2–3). The source calls the butterfly on this page a strangle. ↩

7. Takahiro Hattori, "スワップション入門（モデル編）―ノーマル（ブラック）・モデルおよびSABRモデルについて―" (Introduction to swaptions (models): the normal (Black) model and the SABR model), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), "４．４ ボラティリティ・サーフィス（キューブ）" (4.4 Volatility surface (cube)), pages 64–65. The source's figure shows the yen interest rate swaption surface in September 2021, and writes "surface" as "サーフィス." ↩ ↩²

8. Martin Haugh, "Black-Scholes and the Volatility Surface," page 4 (for a fixed strike, implied volatility can rise or fall with expiry, approaches a constant value as expiry lengthens, and is often inverted at short expiries, especially in times of market stress). ↩

9. Lee, "Implied Volatility," Section 2.1 (pages 4–5). ↩

10. From the formula in the text, $\sigma_{1,2}^2 \times (2 - 1) = 0.205^2 \times 2 - 0.20^2 \times 1 = 0.08405 - 0.04 = 0.04405$, and its square root is 0.209881. ↩

11. Takahiro Hattori, "スワップション入門（モデル編）―ノーマル（ブラック）・モデルおよびSABRモデルについて―" (Introduction to swaptions (models): the normal (Black) model and the SABR model), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), Section 4.1 (pages 61–62). The source covers the need for interpolation, SABR as a general-purpose tool, linear interpolation as an option, and the advantages of a model. ↩ ↩² ↩³

12. Martin Haugh, "Black-Scholes and the Volatility Surface," the note to Exercise 8 on page 20. The source says that options with wide bid-ask spreads carry little information, that quotes on illiquid options may be stale, and that it is often better to leave them out when building the surface. ↩

13. OpenGamma, "Vanilla Forex Options: Garman-Kohlhagen and Risk Reversal/Strangle," Section 3 "Smile" (page 3). The source says that its implementation calculates the strike for each delta, interpolates linearly in strike, and extrapolates flat beyond the outermost strikes. ↩

14. Lee, "Implied Volatility," equation (3.5) and Application 3.2 in Section 3.1.2 (page 13), and the moment formula (page 14). The source writes implied volatility as $I$. Application 3.2 advises against fitting the wings with functions that grow faster than $x^{1/2}$. ↩ ↩² ↩³

15. OpenGamma, "Vanilla Forex Options: Garman-Kohlhagen and Risk Reversal/Strangle," Section 5 "Implementation" (page 4). The source calls the butterfly on this page a strangle. ↩

16. Takahiro Hattori, "スワップション入門（モデル編）―ノーマル（ブラック）・モデルおよびSABRモデルについて―" (Introduction to swaptions (models): the normal (Black) model and the SABR model), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), formula (6) in Section 4.3 (page 63), and page 64 (broker quotes are used as market prices). ↩ ↩²

17. Martin Haugh, "Black-Scholes and the Volatility Surface," "Arbitrage Constraints on the Volatility Surface" on page 5. The source says that market surfaces do not violate these constraints, but they are hard to enforce when the surface is moved for risk management. ↩ ↩² ↩³ ↩⁴

18. Lee, "Implied Volatility," equation (3.1) in Section 3.1.1 "Slope bounds" (page 12). ↩

19. Martin Haugh, "Black-Scholes and the Volatility Surface," equation (18) and the text after it in "What the Volatility Surface Tells Us" on page 6. ↩ ↩² ↩³

20. When interest rates and dividends are 0, the call price increases in $\sigma\sqrt{T}$, so not decreasing in expiry is the same as $\sigma^2 T$ not decreasing. ↩

21. Calculated with blackScholesPrice in src/lib/simulators/black-scholes.ts and rounded to four decimal places. ↩

22. Calculated from the unrounded values, it is 2.122471. The value of the expression in the text is calculated from the rounded values in the table, so the fourth decimal place differs. ↩

23. The prices are calculated with blackScholesPrice in src/lib/simulators/black-scholes.ts (8.386415 for the 90 put and 7.438302 for the 100 put). The 32.29% is the volatility at which the 90 put has that price, found by bisection. ↩

24. The prices are calculated with blackScholesPrice in src/lib/simulators/black-scholes.ts, with interest rates and dividends set to 0 (7.043198 and 6.773861). The square root of 0.03125 is 0.176777. ↩

25. The butterflies are calculated from the unrounded prices behind the table, and the probability densities are $1.010050 \times \text{butterfly} \div 10^2$ rounded to four decimal places (for the example centered on 100, 1.010050 × 2.122471 ÷ 100 = 0.021438). ↩

26. At $\tilde p = 0$, $\beta_R = 2$, and as $\tilde p$ grows, $\beta_R$ approaches 0. ↩

27. Martin Haugh, "Black-Scholes and the Volatility Surface," "Pricing a Digital Option" under Section 6 "Pricing Exotics" (page 14), and Example 3. ↩

28. Martin Haugh, "Black-Scholes and the Volatility Surface," "Pricing Other Exotics" on page 16. The source says that because the surface determines only the marginal distributions of the stock price, the price of a barrier option cannot be determined from the surface. ↩

29. Martin Haugh, "Black-Scholes and the Volatility Surface," page 8 in Section 4 (the paragraph after the formula for delta). The source writes sticky-by-strike and sticky-by-delta. ↩ ↩² ↩³

30. Lee, "Implied Volatility," Remark 4.3 in Section 4.2 (page 24). ↩

31. The prices are calculated with blackScholesPrice in src/lib/simulators/black-scholes.ts, with an underlying price of 105, a strike of 100, 1 year, and an interest rate of 1% (11.469827 and 12.208697). The interpolated value is calculated at 21.90%, and the price before the move is 8.433319. ↩

32. This is the direction of the second term in the formula for $\Delta$ in the text. Under sticky delta, as the underlying price rises, the position of strike 100 moves toward the low-strike side (the side with higher volatility), so $\partial \sigma / \partial S > 0$. The vega of a call is positive, so the second term is positive. ↩

33. Lee, "Implied Volatility," Section 4.2 (pages 24–25). The source covers Cont and da Fonseca's analysis of S&P 500 and FTSE data, use in risk-management simulation (Application 4.4), and that the model is not intended to specify no-arbitrage drifts for pricing. ↩ ↩² ↩³

34. Takahiro Hattori, "スワップション入門（モデル編）―ノーマル（ブラック）・モデルおよびSABRモデルについて―" (Introduction to swaptions (models): the normal (Black) model and the SABR model), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), Section 4.2 (page 62) and note 17. ↩ ↩²

35. Lee, "Implied Volatility," Section 2.2 (pages 5–6), equations (2.1), (2.3), and (2.4). ↩

36. Lee, "Implied Volatility," Section 2.3 (page 8). ↩

37. Takahiro Hattori, "スワップション入門（モデル編）―ノーマル（ブラック）・モデルおよびSABRモデルについて―" (Introduction to swaptions (models): the normal (Black) model and the SABR model), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), Sections 4.1 and 4.2 (page 62). ↩

38. Martin Haugh, "Black-Scholes and the Volatility Surface," Example 2 on page 7. In the form of a question, the source shows that two securities with the same marginal distributions but different joint distributions have the same surface, but that the prices of a knock-in put need not be the same. ↩