Volatility skew, implied probabilities and risk reversals

How to read the options market beyond delta

Preface                                                                                         

In the Black–Scholes model implied volatility is a constant: for a given underlying and expiry, every option should therefore quote the same volatility. Empirical evidence shows instead a structure across strikes and maturities: the volatility surface.

Skew is the cross-sectional component of that surface — the way implied volatility changes with strike or, more usefully on a desk, with delta. It is not a pricing curiosity. It summarises the relative cost of distributional tails, the demand for protection, the Street’s willingness to sell convexity, and the risk premium embedded in option prices.

In equity markets the classic case is put skew: out-of-the-money puts generally quote higher implied volatility than out-of-the-money calls. In markets characterised by structural demand for upside — certain commodities, individual growth names, occasionally FX — a call skew can emerge instead.

The essential point is this: skew is not automatically a directional forecast. It is, first and foremost, the relative price the market assigns to different regions of the underlying’s future distribution.

1         From constant volatility to the volatility surface                            

Black–Scholes assumes a lognormal distribution of future prices, constant volatility and frictionless markets. In that setting, a call and a put of the same expiry, with strikes consistent with put-call parity, should be recoverable from a single volatility.

In practice, inverting the model on observed prices produces different implied volatilities:

σ_impl = f (K, T ) where K is the strike and T is time to expiry.

The surface has three operational dimensions:

• Level: overall implied volatility, often approximated by at-the-money IV.
• Skew: the slope of the volatility curve as strike or delta changes.
• Curvature: the extent to which volatility rises in the tails — a pronounced smile or wing.

A market with constant IV would be “flat”. A typical equity market instead shows a curve that declines from low strikes to high strikes: OTM puts carry higher IV, at-the-money options

quote an intermediate level, and OTM calls tend to be relatively cheaper in implied-volatility terms.

Comparing IVs is not the same as comparing premia. Implied volatility is the Black–Scholes parameter inverted from price: equal differences in volatility points do not, in general, correspond to equal differences in premium, particularly away from the money and at short tenors.

2  Put skew: why OTM puts cost more                                           

Put skew is observed when OTM puts quote higher implied volatility than at-the-money options and than delta-matched OTM calls.

The configuration reflects at least two components.

2.1      Left-tail risk

Equity returns are not well described by a symmetric normal. Drawdowns tend to be faster, more correlated and accompanied by a rise in realised volatility. In other words, the actual distribution often has a heavier left tail than a simple lognormal would imply.

An OTM put does not merely hedge an “ordinary” decline: it delivers convex exposure precisely in the states where liquidity deteriorates, correlations go to one and volatility rises. Its economic value is therefore higher than a constant-volatility distribution would suggest.

2.2      Structural demand for protection

Foundations, pension funds, long-only managers, multi-asset funds and institutional investors are often structurally long equity risk. Buying OTM puts is a form of insurance against severe drawdowns.

Recurring demand for protection tends to lift the price of OTM puts. Because implied volatility is the parameter the market backs out of the option price, richer puts imply higher IV at low strikes.

The result is a downward-sloping curve:

σput OTM > σATM > σcall OTM

This relationship should not be read as “the market necessarily expects a decline”. It can instead mean that the market assigns a high price to protection against non-linear, expensive-to-hedge downside.

3         Call skew: when upside becomes expensive                                  

Call skew is the opposite configuration: OTM calls quote higher IV than comparable OTM puts. It is common in markets where upside is perceived as discontinuous or squeeze-prone: com-modities with physical scarcity risk, names with heavy speculative demand, digital assets, and

certain FX markets in phases of strong demand for a particular currency.

Typical dynamics include:

• aggressive demand for OTM calls;
• short-squeeze risk for call sellers;

• a perceived distribution with a more expensive right tail;
• market-maker dynamic hedging that can amplify upside moves. The inverse relationship then holds:

σcall OTM > σATM > σput OTM

Here too the signal is not a directional certainty. Call skew can embed both an expectation of upside and a risk premium attached to extreme moves higher.

4         Delta and probability: two different quantities                               

One of the most common errors is to treat a call’s delta as the probability that the option expires in-the-money.

In the Black–Scholes model, for a European call with no dividends:

∆_call = N (d₁) With a continuous dividend yield q, delta becomes

∆call = e−qT N (d1).

The risk-neutral probability of finishing above the strike — under the measure Q with the money-market account as numeraire — is instead

Q(S_T > K) = N (d₂)

with

2

√                   ln(F/K) + 1 σ²T

d₁ = d₂ + σ

For a call, since d₁ > d₂,

T,          d₁ =

σ√T

,          F = S0e(r−q)T .

N (d₁) > N (d₂).

As a result, call delta tends to overstate the risk-neutral probability of finishing ITM. For a put the opposite holds: the absolute value of delta tends to be smaller than the risk-neutral probability of exercise,

-∆_put– = e^−qT N (−d₁) < N (−d₂).

The economic reason is straightforward. Delta measures the sensitivity of the option price to a marginal move in the underlying. That sensitivity places more weight on states in which a call finishes deep ITM, because the payoff is large in those states. The ITM probability, by contrast, assigns the same weight to every outcome above the strike: finishing a tick above K and finishing well above K count equally.

In numeraire terms: N (d₂) is the probability under Q; N (d₁) is the probability under the

share measure (the underlying as numeraire). This is not an empirical adjustment. It is a change of measure.

FX adds a further layer of convention: spot delta, forward delta and premium-adjusted delta.

A “25-delta” is not a unique object across platforms.

4.1      Trading implication

A 25-delta call does not necessarily correspond to a 25% probability of finishing ITM. The gap widens with:

• higher volatility;
  • longer time to expiry;
  • greater distance between strike and forward;
  • asymmetry and skew in the surface.

At short tenors and moderate volatility the difference can be modest, and using delta as a proxy may be acceptable. At long tenors, for options far from the money, or on a steeply skewed surface, the approximation becomes fragile.

5         Probability from the price of a spread                                          

A more direct way to extract an implied probability is to look at the price of a very tight vertical spread.

Consider a long call spread

C(K₁) − C(K₂)

with K₂ > K₁. At expiry the payoff is:

• zero if S_T≤K₁;
  • linearly increasing between K₁ and K₂;
  • equal to K₂ − K₁ if S_T ≥ K₂.

As the distance between strikes tends to zero, the normalised payoff tends to a digital. For

European calls, Breeden–Litzenberger implies

∂C          −rT

−∂K = e        Q(S_T > K).

Hence, with discrete strikes,

∆K

T

C(K) − C(K + ∆K) ≈ e−rT Q(S  > K)

in the sense of an average of Q over the interval [K, K + ∆K]. The ratio of spread price to spread width is the price of a cash-or-nothing digital — that is, the discounted risk-neutral probability, not raw Q.

If a 117/118 call spread costs 0.19 and the width is 1, the market is quoting about 0.19 for a maximum payoff of 1 across that strike band. At a short remaining tenor the factor e^−rT is negligible and a “19%” reading is an acceptable working approximation. At longer tenors, and with non-zero rates, the figure must be restated as a probability after allowing for discounting, dividends and the fact that the spread is not infinitesimal.

The reading is useful, but it requires discipline.

5.2      Conditions for a reliable estimate

• A spread tight enough to approximate the derivative, but not so tight that it is only bid-ask.
  • Liquid strikes and credible bid-ask quotes.
  • Mid prices used only when the market is actually tradable around that level.
  • A discounting adjustment, especially at longer tenors.
  • Care with dividends, rates, early exercise on American options, and liquidity gaps.

In particular, a theoretical price built from the mids of two illiquid options can produce an apparent probability that cannot actually be traded. In that case one is measuring the market-maker’s spread, not the implied distribution.

The risk-neutral density, as distinct from the cumulative probability, comes from the second derivative: e^rT ∂²C/∂K². Operationally, that is a normalised butterfly.

6         Delta-matched risk reversal: the standard measure of skew           

A risk reversal compares the implied volatility of an OTM call and an OTM put, normally at the same absolute delta and the same expiry.

A common convention, aligned with the FX market, is

RR25∆ = σcall 25∆ − σput 25∆.

The interpretation is immediate:

• RR_25∆< 0: the 25-delta put is richer than the 25-delta call — typical put skew;
  • RR_25∆ > 0: the 25-delta call is richer than the 25-delta put — typical call skew;
  • RR_25∆ ≈ 0: a relatively symmetric structure at that delta.

The sign is not universal. Some equity reports and platforms publish put minus call. A risk-reversal number without a stated convention is not interpretable.

In FX, the 25-delta risk reversal is a standard quote. In equities and indices the same logic is highly useful, subject to different delta conventions, American exercise and the impact of dividends.

The corresponding structure, under this convention, is:

• long the OTM 25-delta call;
  • short the OTM 25-delta put.

Whether the structure opens for a credit or a debit depends on the relative richness of the two wings of the smile. If the OTM put is much richer in IV, selling the put and buying the call can generate a credit, but the structure remains exposed to severe downside. It should never be read as a simple “zero-cost” bet.

7         Why a matched-delta comparison is cleaner                                  

Comparing options at equal absolute distance from spot can look natural, but it does not isolate skew rigorously.

Suppose spot is 100 and the two strikes are 89 and 117. The 89 put and the 117 call appear equally “far” from the money, but they are not necessarily comparable in risk-neutral probability, position relative to the forward, or contribution to the distribution.

There are three reasons.

7.1      The centre of the distribution is the forward

In the presence of rates and dividends, the balancing point is not spot but the forward:

F0 = S0e(r−q)T

where r is the risk-free rate and q is the continuous dividend yield. Strikes equidistant from spot are therefore not, in general, equidistant from the economic centre of the distribution.

7.2      Lognormality is already asymmetric

The underlying cannot fall below zero, while upside is unbounded. At the same cash distance, the call and the put intercept non-equivalent regions of the distribution.

The right tail can extend indefinitely; the left tail is compressed by the zero bound. That affects both payoffs and the mapping among delta, probability and strike.

7.3      Put-call parity creates mechanical differences

The relation

C − P = S₀e^−qT − Ke^−rT = e^−rT (F − K)

ties calls and puts deterministically. Comparing premia, or the debit/credit of structures struck equidistant from spot, embeds the forward, carry, lognormal asymmetry and put-call parity. The result is not a “pure” measure of implied-volatility asymmetry.

That is why a matched-delta structure is preferable: it compares options that sit in more similar locations on the risk-neutral distribution and that the market itself tends to quote as corresponding points on the surface.

8  Risk reversals and market expectations                                         

Saying that a negative risk reversal means “the market thinks the underlying will fall” is a simplification that is often wrong.

Skew can be decomposed, conceptually, into two elements.

8.1      An asymmetric risk-neutral distribution

The distribution implied by prices may assign more weight to extreme declines than a return-symmetric lognormal would. This component is tied to the shape of the risk-neutral density extracted from options.

8.2      A crash-risk premium

Investors may be willing to pay more than the risk-neutral — or even the physical — expected value to hold protection in adverse states. A put can therefore be expensive not only because a

crash is viewed as likely, but because its payoff is particularly valuable precisely when marginal wealth and liquidity are scarce.

Equity skew is therefore often the price of insurance against a systemic loss, not a pure subjective probability of a decline. Moving from the measure Q to the physical measure P requires a price of risk. Skew alone does not identify it.

9  Strategies that use put skew                                                       

“Using the skew” does not mean simply selling what looks rich and buying what looks cheap. Every structure transfers a different mix of direction, convexity, vega, gamma and tail risk.

9.1      Long put spread

When OTM puts are expensive, buying a put and financing part of the premium by selling a further OTM put reduces the cost of protection.

The structure retains downside protection over a defined range, but gives up convexity beyond the short strike. It suits a hedge against a significant but bounded correction, not a fear of an open-ended collapse.

9.2      Short call spread just out of the money

If OTM calls are relatively cheap in IV, a short call spread can harvest premium with a defined maximum loss. The idea still requires attention to gamma dynamics: in a fast rally the short call can pick up delta very quickly.

9.3      Long call spread farther out of the money

A distant call spread can be relatively efficient when the upper part of the IV curve is cheaper than OTM puts. The risk is that the required upside is unlikely, or that the position remains too sensitive to time decay.

9.4      Risk reversal

A long-call / short-put risk reversal benefits from a normalisation of put skew and from an upside move in the underlying, but it is exposed to material losses on the downside. It is not a neutral strategy. It is bullish exposure financed by selling protection.

9.5      Jade lizard

The combination of a short put and a short call spread can be used to collect premium where put skew makes the put relatively rich, while capping upside risk through the call spread. The construction condition is that net credit is at least equal to the width of the short call spread: only then does an upside move produce no loss at expiry. Downside risk remains potentially large. Favourable skew does not turn a short put into a risk-free trade.

10   Strategies in a call-skew market                                                  

When OTM calls are relatively richer, the logic reverses.

• Naked puts look relatively less expensive, while retaining the inherent risk of a short position.

• Put spreads can be less efficient as protection than the call side of the surface.
• Long call spreads can be preferable to naked calls, because selling the farther wing monetises part of the elevated IV.
• Short put spreads can offer defined-risk premium collection where puts are relatively less rich.
• A reverse jade lizard can be structured when the asymmetry is concentrated on the call wing, with the same discipline on credit versus spread width.

Here too it would be a mistake to infer a bullish forecast automatically. Call skew can be driven by demand for upside, squeeze risk or covering by short-call desks. Implications for direction depend on the broader context.

11   The role of expiry                                                                    

Skew is not a still photograph. It has to be read along the term structure.

At short horizons, corporate events, macro releases, central banks or earnings can change the relationship between ATM IV and OTM IV abruptly. At long horizons, the cost of protection embeds greater cumulative uncertainty, and the gap between delta and probability tends to widen.

Because

d₁ − d₂ = σ√T,

maturity enters the difference between delta and ITM probability directly. At constant volatility, a one-year expiry produces a larger gap between d₁ and d₂ than a one-month expiry. If implied volatility is also higher, the “delta equals probability” approximation loses still more precision.

Advanced analysis should therefore consider not only:

• the level of skew;
• the slope of the curve;
• the comparison across 10-, 25- and 35-delta;
• the wing-to-ATM ratio;

but also how those quantities evolve across maturities.

12         An operating method                                                                 

A professional process for reading skew can follow this sequence.

12.1      1. Define the forward and the convention

Before comparing strikes or deltas, identify:

• spot;
  • rates;
  • expected dividends;
  • the implied forward;

• option style, European or American;
  • the delta convention used by the platform.

Without that foundation, “equidistant” or even matched-delta comparisons can mislead.

12.2      2. Observe ATM IV, the wings and the term structure

Separate the general level of volatility from the shape of the curve.

A parallel rise in IV across all strikes is not necessarily a steepening of skew. Likewise, a decline in ATM IV can coexist with OTM puts that remain very expensive.

12.3      3. Compare options at matched delta

A 25-delta put versus a 25-delta call is often an effective first summary. For a fuller reading, look at:

• the 10-delta risk reversal, more sensitive to the tails;
  • the 25-delta risk reversal, the standard measure;
  • the 25-delta butterfly, a measure of curvature versus ATM;
  • the spread between 25-delta IV and ATM IV;
  • the time path of these quantities.

12.4      4. Check tradable prices

An apparent IV anomaly can be due to:

• wide bid-ask spreads;
  • illiquid strikes;
  • stale quotes;
  • interpolation models;
  • implied-volatility errors on deep ITM options;
  • proximity to discrete events.

Before attaching meaning to skew, confirm that the structure is actually executable.

12.5      5. Map the Greeks of the strategy

A skew strategy is never only a position in relative IV. It also includes:

• directional delta;
  • gamma;
  • vega;
  • theta;
  • exposure to a shift in the surface;

• jump and gap risk;
  • liquidity under stress.

A risk reversal, for example, is simultaneously a bullish directional position, a sale of downside skew and a relative purchase of upside volatility.

13   Recurring errors                                                                        

13.1      Confusing delta with a probability

Delta can be a quick proxy, but it does not replace N (d₂) or a probability extracted from tight spreads. The approximation deteriorates with tenor and volatility, and it must be adjusted by e^−qT when dividends are not zero.

13.2      Reading skew as a pure forecast

Expensive puts can indicate demand for insurance and a risk premium, not merely a higher physical probability of a decline.

13.3      Comparing strikes equidistant from spot

A correct comparison requires at least a reference to the forward and, to isolate quoted skew, the use of comparable deltas.

13.4      Ignoring liquidity

A probability backed out of the mid price of options with a wide bid-ask can be statistically elegant and operationally irrelevant.

13.5      Selling puts because they look “too rich”

A put can look expensive precisely because it pays in the regimes where losses are largest and liquidity is most fragile. Harvesting premium on the put wing must be paired with an explicit budget for tail risk.

14    Conclusion                                                                              

The volatility surface is how the market publishes a risk-neutral distribution and a price for transferring tails. Skew is its slope; a matched-delta risk reversal is its quotable summary; a tight spread, properly discounted, is its reading in probability.

None of these quantities is a directional verdict. They are prices. They are used to choose which exposure to buy or sell — bounded protection, one-sided convexity, financing of a bias — not to replace a risk process.

Operational discipline consists in stating the convention — delta type, sign of the risk reversal, European or American — in working off executable prices, and in not confusing the Black–Scholes parameter with the quantity one is actually buying: a payoff.