Interpreting Volatility Smile Flattening Across Option Maturities

# Volatility surfaces


# Volatility surfaces












So I'm learning about volatility smiles and volatility surfaces from Hull's Options, Futures and Other Derivatives. On p. 459, there is the following table (Table 20.2) of a volatility surface:

| Maturity | 0.90 | 0.95 | 1.00 | 1.05 | 1.10 |
| 1 month | 14.2 | 13.0 | 12.0 | 13.1 | 14.5 |
| 3 month | 14.0 | 13.0 | 12.0 | 13.1 | 14.2 |
| 6 month | 14.1 | 13.3 | 12.5 | 13.4 | 14.3 |
| 1 year | 14.7 | 14.0 | 13.5 | 14.0 | 14.8 |
| 2 year | 15.0 | 14.4 | 14.0 | 14.5 | 15.1 |
| 5 year | 14.8 | 14.6 | 14.4 | 14.7 | 15.0 |

The columns denote different values for $K/S_0$, where $K$ is the strike price and $S_0$ is the current underlying. The rows are ordered by time to maturity.

Hull also writes

> The table shows that the volatility smile becomes less pronounced as the option maturity increases.

I don't understand this. Doesn't the table show exactly the opposite? I mean, for options with 5 years time to maturity, the values all lie in the interval $[14.4, 15.0]$ of length $0.6$, but for options with 1 month time to maturity, we have to consider a larger interval $[12.0, 14.5]$, which has length $2.5$.

That said, English is not my mother tongue, so maybe I am misunderstanding the term "as maturity increases". I would have thought that increasing maturity means more mature, so less time to maturity. Is that correct?

## Answer by pandashark (score 2, accepted)

https://quant.stackexchange.com/a/85570

Your reading of the table is correct — and Hull is also correct. The confusion is purely linguistic.

"As maturity increases" means as you move from the 1-month row down to the 5-year row — i.e., longer time to expiry, not closer to expiry. In finance, "higher maturity" = more time remaining. (Hull could have said "as time to expiry increases" to avoid this ambiguity.)

The other answers have covered this terminology point well, so that answers your question. What follows is for readers who want to understand why the smile flattens — it draws on material beyond Hull's Chapter 20.

### Three reasons the smile flattens at longer maturities

#### 1. You're comparing apples to oranges (moneyness scaling)

A large part of what you see in Table 20.2 is an axis-scaling artifact. The columns use simple moneyness $K/S_0$, but the same $K/S_0$ range corresponds to very different levels of "standardised moneyness" $d = \ln(K/F)\,/\,(\sigma\sqrt{T})$ at different tenors.

Take $K/S_0 = 0.90$ with $\sigma \approx 13\%$:

| Tenor | $\sigma\sqrt{T}$ | $\ln(0.90)/(\sigma\sqrt{T})$ |
| 1M | $0.13 \times \sqrt{1/12} \approx 0.038$ | $-2.8\sigma$ |
| 5Y | $0.13 \times \sqrt{5} \approx 0.291$ | $-0.36\sigma$ |

At 1 month, the $K/S_0 = 0.90$ strike is nearly 3 standard deviations out-of-the-money. At 5 years, it is only 0.36 standard deviations — barely OTM. Naturally the implied vol at 0.36$\sigma$ is much closer to ATM vol than at 2.8$\sigma$, even if the standardised smile shape were identical.

If you re-plotted Table 20.2 on a standardised moneyness axis, the flattening would be much less dramatic. But it would still be present, because of the next two effects.

#### 2. Excess kurtosis decays as $1/T$

The smile exists because the risk-neutral return distribution has fatter tails than a lognormal (i.e., positive excess kurtosis). This kurtosis comes from jumps, stochastic volatility, and other non-BS features of real markets.

For jump-diffusion models (Merton 1976), the excess kurtosis of the log-return distribution over horizon $T$ scales as:

$$\text{excess kurtosis} \;\propto\; \frac{1}{T}$$

This is a direct consequence of cumulant scaling: the fourth cumulant grows as $T$, but kurtosis divides by variance squared, which grows as $T^2$. So kurtosis decays as $T/T^2 = 1/T$. Since smile curvature is driven by kurtosis, it decays with the same rate.

#### 3. Volatility mean-reversion concentrates integrated variance

If you model vol as mean-reverting (as in the Heston model), then over longer horizons the average realised variance concentrates around its long-run mean $\bar{v}$, by the law of large numbers applied to the vol path.

Quantitatively, the dispersion of time-averaged realised variance decays as $\sim 1/T$ for long maturities (because mean-reversion clamps the vol path around its long-run mean). Since smile curvature is driven by this dispersion, it inherits the same decay, reinforcing effect #2.

### Quantitative demonstration with QuantLib

Here is a quick C++ program that loads Hull's exact data into a BlackVarianceSurface and computes a smile "convexity" metric (average wing vol minus ATM vol) at each tenor:

```
#include <ql/quantlib.hpp>
#include <iostream>
#include <iomanip>
#include <vector>
using namespace QuantLib;

int main() {
    Date today(3, April, 2026);
    Settings::instance().evaluationDate() = today;

    Calendar cal = NullCalendar();
    DayCounter dc = Actual365Fixed();

    // Hull Table 20.2 maturities
    std::vector<Date> expirations;
    for (auto& p : {Period(1,Months), Period(3,Months), Period(6,Months),
                     Period(1,Years),  Period(2,Years),  Period(5,Years)})
        expirations.push_back(cal.advance(today, p));

    std::vector<Real> strikes = {90, 95, 100, 105, 110};

    // Matrix(nStrikes, nDates): rows = strikes, cols = dates
    Matrix vols(5, 6);
    //             1M     3M     6M     1Y     2Y     5Y
    Real data[][6] = {
        {14.2, 14.0, 14.1, 14.7, 15.0, 14.8},  // K=90
        {13.0, 13.0, 13.3, 14.0, 14.4, 14.6},  // K=95
        {12.0, 12.0, 12.5, 13.5, 14.0, 14.4},  // K=100
        {13.1, 13.1, 13.4, 14.0, 14.5, 14.7},  // K=105
        {14.5, 14.2, 14.3, 14.8, 15.1, 15.0},  // K=110
    };
    for (Size i = 0; i < 5; ++i)
        for (Size j = 0; j < 6; ++j)
            vols[i][j] = data[i][j] / 100.0;

    BlackVarianceSurface surface(today, cal, expirations, strikes, vols, dc);
    surface.enableExtrapolation();

    std::string labels[] = {"1M","3M","6M","1Y","2Y","5Y"};
    std::cout << "Tenor   ATM Vol   Wing Avg   Convexity(bp)\n"
              << "--------------------------------------------\n";
    for (Size i = 0; i < 6; ++i) {
        Real atm  = surface.blackVol(expirations[i], 100.0);
        Real wing = (surface.blackVol(expirations[i], 90.0)
                   + surface.blackVol(expirations[i], 110.0)) / 2.0;
        std::cout << std::setw(5)  << labels[i]
                  << std::fixed << std::setprecision(2)
                  << std::setw(10) << atm*100
                  << std::setw(11) << wing*100
                  << std::setprecision(0)
                  << std::setw(14) << (wing - atm)*1e4 << "\n";
    }
}
```

Output:

```
Tenor   ATM Vol   Wing Avg   Convexity(bp)
--------------------------------------------
   1M     12.00      14.35           235
   3M     12.00      14.10           210
   6M     12.50      14.20           170
   1Y     13.50      14.75           125
   2Y     14.00      15.05           105
   5Y     14.40      14.90            50
```

The "convexity" column is a butterfly metric: (avg wing vol) $-$ ATM vol, in basis points. It drops monotonically from 235 bp at 1 month to 50 bp at 5 years — exactly what Hull describes.

TL;DR: Hull is saying the smile (curvature across strikes) gets flatter as you look at longer-dated options. Your numerical observation ($[12.0, 14.5]$ at 1M vs $[14.4, 15.0]$ at 5Y) confirms this perfectly. The confusion was just the English — "maturity increases" means longer-dated, not closer-to-expiry.

## Answer by Dimitri Vulis (score 1)

https://quant.stackexchange.com/a/85544

I am not a huge fan of late Hull's book. 11E, pages 436-437:

Given that some products underlying options also have their maturities - e.g. swaps underlying swaptions - a more common and less confusing term is "time to option's expiry🇬🇧/expiration🇺🇸", rather than "option maturity". You can have a "volatility cube", with 3 dimensions: the moneyness, the time to the expiration of the swaptions, and the time to maturity of the underling swaps. Options "expire", rather than "mature". To reduce confusion, I will use "expiration" here.

But in fairness to him, I don't think that the interval is a good way to quantify smiles. Let us define the "smile" at a given time to expiration as (IV at 90% moneyness + IV at 110% moneyness) / 2 - ATM IV. Just one possible measure of convexity, there are others. (It would be more conventional to use IVs for 25 deltas, rather than specific moneyness.)

In his table, at 1 month time to expiration: at the money (ATM) implied volatility (IV) is 12. The IV at 90% moneyness is 14.2, which is 2.2 or 18% more than ATM IV. The IV at 110% moneyness is 14.5, which is 2.5 or 21% more than ATM IV. The "smile" is 2.35=(2.2+2.5)/2.

In contrast, at 5 years time to expiration: ATM IV is 14.4. The IV at 90% moneyness is 14.8, which is only 0.4 or 3% more than ATM IV. The IV at 110% moneyness is 15, which is only .6 or 4% more than ATM IV. The "smile" is only 0.5=(0.4+0.6)/2.

Hence, this table illustrates that if you have IV quotes for the same settlement date, but different combinations of times to expiration and moneyness, the smile usually decreases as the time to expiration increases.

Likewise, if you had quotes for the same expiration date, but different settlement dates, then, with all other stuff possibly happening in the markets, you would expect the smile to decrease as the time left to expiration decreases.

You also want to measure the skew (risk reversal), which you can define as the difference between the IVs at 90% and 110% moneyness. (Again, it would be more conventional to use IVs for 25 deltas, rather than specific moneyness.)

## Answer by Jo&#227;o (score 1)

https://quant.stackexchange.com/a/85545

Adding @Dimitri's answer that explains it well, graphically it should be intuitively

## Answer by carry_and_pray (score 0)

https://quant.stackexchange.com/a/85554

You're reading of the table is not wrong.

"At time to maturity increases" means as you move from the 1-month row to the 5-year row. A vol. surface is indexed by strike and time to maturity and tenor is commonly used for remaining time until expiration. So "higher maturity" just means longer-dated option, not an option that has aged and is now closer to expiry.

The table shows what Hull states where the smile is much more curved for short maturities than for long maturities.

For example, at 1 month, the ATM vol is 12.0% while the wings are 14.2% and 14.5% so the smile depth is around 2.2% to 2.5%. At 5 years, the ATM vol is 14.4% while the wings are 14.8% and 15.0% so the smile depth is only about 0.4% to 0.6% i.e., the long-dated smile is much flatter.

The only real issue in this context is the wording. More mature can sound like closer to maturity but in market usage, higher maturity, longer maturity and higher tenor usually mean more time remaining not less.

Your intuition about the numbers is right even though the verbal phrase feels backwards at first