MR 17 – Volatility Smiles and Volatility Surfaces

Option traders quote and value European options with the Black-Scholes-Merton formula, yet they do not feed every option the same volatility. If the model were literally true, the asset price at any future date would be lognormal, one volatility would describe that distribution, and every option on the asset would back out the same implied volatility. Real quotes do not: inverting each market price through the formula gives an implied volatility that changes with strike and with maturity.

Traders accept this and build it into their pricing. For options of one maturity, the curve that plots implied volatility against strike price is called a volatility smile. Stacking the smiles for many maturities gives a three-dimensional object, with implied volatility depending jointly on strike and maturity, known as the volatility surface. On a trading desk, a new option is priced by reading its volatility off the surface.

The shape of a smile is not arbitrary. Each smile corresponds to a particular risk-neutral probability distribution for the future asset price, and the gap between that distribution and the lognormal one is what bends the curve. A smile that rises at both ends signals heavier tails on both sides than the lognormal allows, while a curve that falls steadily from low strikes to high strikes, known as a volatility skew, signals a heavy left tail and a light right tail.

A European call and a European put written on the same asset, with the same strike K and the same maturity T, are tied together by put-call parity. With a continuous dividend yield q on the asset, a risk-free rate r for maturity T, and S₀ the asset price today, the relation is as follows.

Parity rests only on the absence of arbitrage and assumes nothing about how the asset price is distributed, so it holds whether or not that distribution is lognormal.

The dollar pricing error is the same for both options

Pick any volatility and compute the model values p_BS and c_BS. Because the Black-Scholes-Merton formulas satisfy parity, the first line below holds. Because market prices must also respect parity when there is no arbitrage, the second line holds for p_mkt and c_mkt.

Taking one line from the other cancels the asset and strike terms and leaves a striking equality.

So whatever dollar error the formula makes on the put at a chosen volatility, it makes exactly the same error on the matching call. Suppose the put has an implied volatility of 22%. At 22% the model price of the put equals its market price, the left side of Equation 17.2 is zero, and the right side must be zero as well. The call is then also priced correctly at 22%, so its implied volatility is 22% too.

The consequence is very convenient. For a given strike and maturity there is one implied volatility, shared by the European call and the European put, so a smile or a surface can be drawn without saying which type of option it refers to. American options obey the same result closely, though not exactly.

Currency option traders work with a smile that is close to a symmetric U. Implied volatility is lowest for at-the-money options and climbs steadily as the strike moves away from the current exchange rate in either direction, so both deep in-the-money and deep out-of-the-money options carry higher volatilities. The curve is usually plotted against K/S₀, the strike divided by the current exchange rate, with the minimum near 1.0.

What the smile says about the distribution

The probability distribution of the future asset price that is consistent with the quoted smile is called the implied distribution. It is a risk-neutral distribution, and the final section shows how it is extracted from option prices. Compared with a lognormal curve that matches its mean and standard deviation, the implied distribution for an exchange rate has heavier tails on both sides. It is also more sharply peaked, which means very small moves and very large moves are both more likely than the lognormal suggests, while intermediate moves are less likely. This combination is what statisticians call excess kurtosis.

The link between the two pictures runs through deep-out-of-the-money options. Take a call with a high strike K₂, with K₂/S₀ well above 1.0. It pays off only if the exchange rate ends above K₂, and the implied distribution assigns more probability to that region than the lognormal does. The option is worth more under the implied distribution, and since a higher price can only be produced by a higher volatility input, its implied volatility is high. A put with a low strike K₁, well below the current rate, works the same way through the left tail: more probability below K₁ gives a higher price and a higher implied volatility. Both wings of the smile lift, which is the U shape traders use.

The currency smile embodies a belief that the lognormal model understates extreme moves. Hull tests it with daily data for ten exchange rates against the US dollar over the 10 years from 2005 to 2015, including the euro, yen, pound and Swiss franc. For each rate the standard deviation of daily percentage changes is estimated first. The next step counts how often the actual daily change was larger than 1, 2, up to 6 standard deviations, and the last step compares those frequencies with what a normal distribution of daily changes would produce. Over one day the lognormal model makes percentage changes almost exactly normal.

Source: Hull, Table 17.1. The last column is derived.

The pattern is unmistakable. Moves beyond 3 standard deviations happened on 1.30% of days against a model figure of 0.27%, close to five times as often. Moves beyond 4, 5 and 6 standard deviations, which the model treats as practically impossible, turned up on 0.49%, 0.24% and 0.13% of days. At the other end, moves of more than 1 standard deviation were less common than the model predicts, which matches the sharper peak described in the previous section.

Profiting from heavy tails before the market priced them

This knowledge once paid. If everyone else prices currency options with a flat Black-Scholes-Merton volatility, deep-out-of-the-money calls and puts look cheap to someone who knows the tails are heavy, because more of them will finish in the money than the lognormal implies. Buying a spread of such options across many currencies and holding them earns payoffs whose present value, on average, comfortably exceeds the premiums. A handful of traders did exactly this in the mid-1980s. By the late 1980s the whole market priced currency options with a smile and the opportunity was gone.

Why exchange rates are not lognormal

A lognormal price requires two things: a constant volatility, and a price path that moves smoothly with no jumps. Exchange rates satisfy neither. Their volatility drifts, and they jump, sometimes on central bank action. Both a nonconstant volatility and jumps make extreme outcomes more likely, which fattens the tails.

Maturity changes the strength of each effect. As maturity lengthens, nonconstant volatility has a growing percentage effect on option prices but usually a shrinking percentage effect on implied volatility. Jumps lose influence on both prices and implied volatility, because over a long horizon they tend to be averaged out and the resulting distribution looks much like one produced by smooth moves. So the currency smile flattens as maturity increases.

For equity options, on single stocks and on stock indices alike, the smile slopes downward rather than curving up at both ends, which is why it is usually called a volatility skew. A low strike carries a much higher implied volatility than a high strike. In option language, deep-out-of-the-money puts and deep-in-the-money calls (both low strikes) are priced with high volatilities, while deep-in-the-money puts and deep-out-of-the-money calls (both high strikes) are priced with low ones. Before the crash of October 1987 this pattern was not marked; it has been a steady feature ever since.

The implied distribution behind the skew

For an equity price the implied distribution has more mass in its left tail and less in its right tail than a lognormal curve matched on mean and standard deviation. The deep-out-of-the-money option argument from the currency case carries over directly. A call with a high strike K₂ needs the stock to finish above K₂; the implied distribution gives that region less probability than the lognormal, so the call is cheaper and its implied volatility is low. A put with a low strike K₁ needs the stock to finish below K₁; the implied distribution gives that region more probability, so the put is dearer and its implied volatility is high. Volatility therefore falls as the strike rises, which is the skew in Figure 1.

Why equity prices and volatility move in opposite directions

The skew traces back to the way equity prices and volatility tend to move against each other: when prices fall, volatility tends to rise, and when they rise, it tends to fall. Three explanations are offered.

Leverage is the first. A drop in the equity value of a firm raises its leverage, and a more highly geared equity claim is more volatile, while a rise in the equity price does the reverse. The volatility feedback effect runs the causation the other way. When volatility increases for some outside reason, investors demand a higher return to hold the stock, and the only way to deliver a higher expected return is for the price to fall now; a fall in volatility lifts the price.

The third explanation is crashophobia, a term proposed by Mark Rubinstein. Because the steep skew appeared only after the 1987 crash, he suggested that traders fear a repeat and price low-strike options to reflect it. Some evidence supports this view: when the S&P 500 falls, the skew tends to steepen, and when the index rises, the skew tends to flatten.

Whatever the mix of causes, a falling stock price brings higher volatility, which makes further large falls more likely and stretches the left tail. A rising price brings lower volatility, which makes further large gains less likely and trims the right tail.

The most obvious horizontal axis for a smile is the strike price K, but a smile drawn against K moves whenever the asset price moves. As the asset rises, the at-the-money strike rises with it and the whole curve shifts to the right; as the asset falls, the curve shifts left. To get a picture that stays put, practitioners usually plot implied volatility against K/S₀, the strike as a fraction of the current asset price, which is the axis used in Figure 1. Derman has found that for exchange-traded options this adjustment is sometimes sticky, meaning the smile does not always shift fully with the price.

A refinement replaces the spot price with the forward price, plotting volatility against K/F₀, where F₀ is the forward price for a contract maturing on the same date as the options. On the same reasoning, many traders define an at-the-money option as one with K = F₀ rather than K = S₀. The justification is that F₀, not S₀, is the expected asset price at maturity in a risk-neutral world.

A third approach uses the delta of the option as the horizontal axis. This sometimes lets a smile be applied to options other than standard European and American calls and puts. Under this convention the at-the-money options are the call whose delta is 0.5 and the put whose delta is -0.5, called 50-delta options.

When smiles for different maturities are compared, the smile for longer options is typically flatter. Some financial engineers remove much of that maturity effect by measuring the horizontal axis as the following quantity, where T is the time to maturity.

On this axis smiles for different maturities lie much closer together.

Implied volatility also depends on maturity, and the pattern across maturities is the volatility term structure. When short-dated volatility is low by historical standards, the market expects it to rise, so implied volatility increases with maturity. When short-dated volatility is unusually high, the market expects it to fall back, so implied volatility decreases with maturity.

Combining the smile with the term structure produces the volatility surface, a grid of volatilities from which an option of any strike and maturity can be priced. The table below is an example of the kind of surface that might be used for currency options, with K/S₀ across the top and maturity down the side. Some cells come straight from options with reliable market quotes, whose implied volatilities are computed from their prices; the remaining cells are usually filled in by interpolation.

Source: Hull, Table 17.2. The last column is derived.

The derived column makes the flattening visible: the premium of the 0.90 wing over the at-the-money volatility shrinks from 2.2 points at 1 month to 0.4 points at 5 years. The at-the-money volatility itself rises from 12.0% to 14.4%, an upward-sloping term structure.

The textbook formulas for delta and the other Greek letters hold implied volatility fixed while the asset price moves. Traders rarely expect that to happen. For a stock or stock index option, the negative correlation between price and volatility means that a rise in the price normally comes with a fall in implied volatility. A delta that allows for this joint move is called the minimum variance delta. Writing f_BSM for the Black-Scholes-Merton option price, σ_imp for its implied volatility and E(σ_imp) for the expected implied volatility as a function of the asset price S, it is defined as follows.

Here Δ_BSM and ν_BSM are the delta and vega from the constant-volatility model. Vega is always positive, and for equities the expected change in implied volatility per unit rise in the price is negative. So the second term is negative and the minimum variance delta is smaller than the Black-Scholes-Merton delta.

If every option is allowed its own volatility, it is fair to ask how much the pricing model still matters. One view is that Black-Scholes-Merton has become a sophisticated interpolation tool: the desk uses it to keep a new option in line with the prices of options that trade actively. A switch to another plausible model would reshape the smile and the surface, because the same prices would be inverted through a different formula, yet quoted dollar prices would arguably barely move.

The model matters much more for hedging, because Greek letters depend on it and an unrealistic model tends to produce poor hedges. Models also have their biggest effect on price when no similar derivative trades actively. Many nonstandard exotic options fall into this group, and their values are model-dependent because there are no liquid neighbours to anchor them.

An unusual smile can appear when a single big event is coming. Suppose a stock trades at $50 and an announcement due in a few days, perhaps the result of a takeover bid or a court ruling, is expected to move the price up by $8 or down by $8. The distribution of the price in 1 month is then a mixture of two lognormal distributions, one centred on good news and one on bad news. The true distribution is bimodal, nothing like a single lognormal with the same mean and standard deviation.

The cleanest way to see the effect on the smile is to push this to the extreme, where only two prices are possible. In 1 month the stock will be either $42 or $58, and money earns 12% a year risk-free. A one-step binomial tree then values every option exactly.

A call with strike K pays 58 – K with risk-neutral probability 0.5314, and a put pays K – 42 with probability 0.4686; each is discounted for one month at 12%. Inverting each call price through the formula gives the implied volatility, which is the same for the put of that strike.

Source: Hull, Table 17.3, columns reordered. The last column is derived and matches 50 minus the discounted strike to within a cent of rounding, as parity requires.

The pattern is a volatility frown. Because the distribution has no tails at all beyond $42 and $58, options far from the money are cheap relative to the lognormal, and implied volatility drops on both sides of the money. A practical consequence is that the volatility implied by the at-the-money option with a strike of 50 would overprice options struck at 44 or 56.

The printed volatility of 49.0% for the $56 strike does not reconcile with its own call price: inverting $1.05 over one month at 12% gives about 49.9%, while every other row reproduces to within 0.1 point. The frown conclusion is unaffected.

The price of a European call with strike K and maturity T is the discounted risk-neutral expectation of its payoff. With a constant interest rate r, S_T the asset price at T and g the risk-neutral probability density function of S_T, the price can be differentiated twice with respect to the strike.

This result, due to Breeden and Litzenberger (1978), means the density can be read from the curvature of call prices across strikes, and since a smile delivers a price for every strike, it delivers an implied distribution. In practice the derivative is approximated with three calls of prices c₁, c₂ and c₃ struck at K – δ, K and K + δ, with δ small.

The butterfly spread interpretation

The numerator is the cost of a butterfly spread: long one call at K – δ, long one call at K + δ, short two calls at K. Its payoff is a triangle that is zero outside the range K – δ to K + δ and peaks at δ when S_T = K. The triangle has base 2δ and height δ, so its area is 0.5 × 2δ × δ = δ². With δ small, g is nearly constant across the triangle, so the value of the spread is e^-rTg(K)δ², and setting this equal to c₁ + c₃ – 2c₂ gives Equation 17A.2.