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Volatility smile

The volatility smile is a U-shaped pattern in the graph of implied volatility against strike prices for a set of European options on the same underlying asset and with identical expiration dates, characterized by higher implied volatilities for deep in-the-money and deep out-of-the-money options compared to at-the-money options.^[1] This phenomenon reveals that market participants price options as if the underlying asset's future volatility is not constant but varies depending on the strike price relative to the current asset price.^[1] In contrast to the Black-Scholes-Merton model, which assumes constant volatility across all strike prices and thus predicts a flat implied volatility curve, the volatility smile highlights the model's limitations in capturing real-world market dynamics, such as fat-tailed return distributions and sudden price jumps.^[2] The pattern became prominently observed in equity options markets following the 1987 stock market crash, after which implied volatilities for low-strike (out-of-the-money put) options rose sharply, often resulting in an asymmetric variant known as the volatility skew or smirk.^[2] Prior to the crash, implied volatility surfaces were relatively flat, aligning more closely with Black-Scholes assumptions.^[2] The volatility smile arises primarily from investor demand for protection against extreme market moves, leading to higher premiums (and thus implied volatilities) for tail-risk options, as well as from stochastic volatility processes where future volatility itself is uncertain and correlated with asset returns.^[1] It is more pronounced for shorter-term options and tends to flatten for longer maturities, reflecting the diminishing impact of short-term jumps over time.^[1] In practice, the smile informs advanced pricing models like stochastic volatility or jump-diffusion frameworks, aids in extracting risk-neutral probability distributions from option prices, and serves as a gauge of market sentiment, with steeper smiles indicating heightened fear or uncertainty.^[1]

Fundamentals of Implied Volatility

Definition and Calculation

Implied volatility represents the market's forward-looking estimate of an asset's price variability, derived as the specific value of the volatility parameter \sigma that equates the Black-Scholes model's theoretical option price to the observed market price.^[3] The Black-Scholes formula for the price c of a European call option on a non-dividend-paying stock is given by

c = S N(d_1) - K e^{-r \tau} N(d_2),

where

d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)\tau}{\sigma \sqrt{\tau}}, \quad d_2 = d_1 - \sigma \sqrt{\tau},

S is the current asset price, K is the strike price, r is the risk-free interest rate, \tau is the time to maturity, and N(\cdot) is the cumulative distribution function of the standard normal distribution.^[3] A put-call parity relation provides the corresponding formula for European put options.^[3] To calculate implied volatility, the equation c = BS(S, K, \tau, r, \sigma) (or the put equivalent) is solved numerically for \sigma, as no closed-form expression exists for this inverse problem.^[4] Iterative techniques, such as the Newton-Raphson method, are commonly employed; this approach starts with an initial guess for \sigma and refines it using the option's vega \partial c / \partial \sigma until convergence to the market price is achieved, typically within a few iterations for well-behaved surfaces.^[4] The Black-Scholes model presupposes constant volatility \sigma across all strike prices and maturities, along with assumptions of geometric Brownian motion for the underlying asset, continuous trading, no dividends, and frictionless markets.^[3] In practice, computing and plotting implied volatilities for options sharing the same maturity but varying strike prices K against those strikes reveals a non-constant pattern known as the volatility smile, underscoring the model's inadequacy in capturing real-market dynamics like fat-tailed return distributions or volatility clustering.^[5] This smile typically manifests as a U-shaped curve in foreign currency options, where implied volatilities are elevated for both low-strike (in-the-money calls or out-of-the-money puts) and high-strike (out-of-the-money calls) options relative to at-the-money strikes, reflecting symmetric concerns over large exchange rate movements.^[6] In contrast, equity index options prior to the 1987 crash exhibited milder U-shapes, but post-crash patterns shifted to a negatively skewed "smirk," with pronounced higher implied volatilities for low strikes due to heightened demand for protective puts amid crash fears.^[7]^[5]

Comparison with Historical Volatility

Historical volatility serves as a backward-looking measure of an asset's price fluctuations, calculated as the annualized standard deviation of its logarithmic returns over a specified past period. For a 21-day rolling window using daily data, it is given by

\sigma_h = \sqrt{252 \times \frac{1}{20} \sum_{i=1}^{21} \left( \ln \left( \frac{S_t}{S_{t-1}} \right) - \bar{r} \right)^2},

where S_t denotes the asset price at time t, \bar{r} is the mean log return over the period, and 252 approximates the number of trading days in a year.^[8] This contrasts with implied volatility, which is forward-looking and derived from current option prices by inverting the Black-Scholes model to match observed market premiums. Empirical analyses reveal frequent divergences between the two metrics, especially during heightened uncertainty, where implied volatility rises to reflect anticipated risks not evident in past data. For instance, in the 2008 financial crisis, implied volatility for S&P 500 options—as captured by the VIX index—surged to 80.06% on October 27, 2008, well above contemporaneous historical volatility levels typically around 20-30%.^[9]^[10] Such discrepancies underscore implied volatility's role as a market consensus on prospective volatility, incorporating expectations of future events like earnings announcements or economic shifts, whereas historical volatility quantifies only realized past movements. Studies confirm that implied volatility often overestimates future realized volatility but provides superior information content over historical measures during volatile regimes.^[9]^[11] In trading applications, historical volatility aids in calibrating baseline models for option pricing and risk assessment, offering a stable reference from past behavior. However, practitioners adjust using implied volatility to incorporate the volatility smile's effects, ensuring strategies account for current market pricing dynamics and potential divergences, such as selling overpriced options when implied exceeds historical levels amid uncertainty.^[10]

Characteristics of the Volatility Smile

Shape and Patterns

The volatility smile typically manifests as a non-linear pattern when implied volatility is plotted against strike prices (or moneyness, defined as the ratio of strike price to spot price, K/S) for options with the same maturity, deviating from the flat surface predicted by the Black-Scholes model.^[12] This pattern arises because market participants price in varying expectations of future volatility across different strike levels, often reflecting asymmetric risk perceptions. In foreign exchange (FX) options markets, the smile commonly exhibits a symmetric U-shape, where implied volatility is lowest for at-the-money (ATM) options and rises symmetrically for both deep out-of-the-money (OTM) calls (high strikes) and OTM puts (low strikes).^[12] This symmetry stems from the balanced risk of large upward or downward movements in currency pairs, as exchange rates are ratios without inherent directional bias like crashes.^[12] In contrast, equity index options, particularly after the 1987 stock market crash, display a negatively skewed "smirk," characterized by significantly higher implied volatilities for low-strike OTM puts compared to high-strike OTM calls, creating a downward-sloping curve.^[13] The 1987 crash led to a permanent steepening of this equity smirk, with the spread between OTM put and ATM implied volatilities increasing markedly, driven by heightened crash fears that inflate demand for downside protection.^[13] Commodity options often feature a forward skew, where implied volatility increases with higher strike prices, reversing the equity pattern and showing elevated levels for OTM calls relative to OTM puts.^[12] This forward skew reflects market expectations of positive correlations between commodity prices and volatility, such as sudden supply disruptions driving prices upward.^[12] Across these markets, patterns are influenced by moneyness: low K/S levels (OTM puts) command higher volatilities in equities due to crash aversion, while in FX, extremes on both sides elevate volatility symmetrically.^[12] The slope of the volatility smile, known as the skew, quantifies this asymmetry and is commonly measured by the 25-delta risk reversal, which is the difference in implied volatility between the 25-delta call (OTM call with delta ≈ 0.25) and the 25-delta put (OTM put with delta ≈ -0.25).^[14] A negative 25-delta risk reversal indicates a steeper equity-like smirk, while a positive value signals forward skew in commodities; in FX, it tends toward zero due to symmetry.^[14]

Volatility Surface

The volatility surface extends the volatility smile observed at individual maturities into a three-dimensional structure, plotting implied volatilities against strike prices along one axis and time to maturity along the other, thereby capturing the joint variation in these dimensions. This representation reveals how the smile evolves over time, with implied volatilities typically derived from market prices of European options using numerical inversion of the Black-Scholes formula.^[15]^[16] Construction of the surface involves interpolating implied volatilities across strikes and maturities to form a continuous, arbitrage-free grid, often employing parametric approaches like the SABR model or non-parametric techniques such as cubic splines. The SABR model, developed by Hagan et al. in 2002, parameterizes the surface through stochastic volatility dynamics that effectively reproduce observed smiles, with parameters calibrated via least-squares fitting to market data for each maturity slice. Spline methods, including piecewise convex polynomials, complement these by ensuring monotonicity and no butterfly arbitrage while smoothing between sparse market quotes. Empirical studies confirm SABR's superior performance in estimating equity surfaces, particularly for short maturities, with interpolation errors minimized to within 1-2 basis points in pricing accuracy.^[17]^[18] The term structure embedded in the surface shows that short-term smiles are generally steeper, driven by near-term events such as earnings releases or economic data, leading to elevated out-of-the-money volatilities that reflect acute market uncertainty. In contrast, smiles at longer horizons tend to flatten, as forward-looking expectations incorporate mean reversion and reduced event-specific risks, often converging toward a constant volatility level. This pattern is evident in equity options data, where inverted term structures occasionally emerge during stress periods, with short-dated implied volatilities exceeding long-dated ones by up to 10-20 percentage points.^[2]^[16]^[17] Visualization of the surface highlights anomalies like twists—where skew direction reverses across maturities—or localized bumps signaling impending risks, such as geopolitical events, which distort the otherwise smooth curvature. These features aid traders in identifying mispricings or hedging opportunities beyond single-maturity analysis.^[19]^[18] Practical computation relies on standardized quoting conventions that parameterize the surface efficiently: at-the-money (ATM) volatility sets the central level, 25-delta risk reversals quantify the skew by differencing call and put volatilities at 25% delta, and butterfly spreads measure curvature via symmetric wing options around ATM. These inputs, quoted for key maturities (e.g., 1 month, 3 months, 1 year), enable rapid surface bootstrapping, with SABR often used to extrapolate to unquoted strikes while preserving no-arbitrage constraints like calendar spread positivity. Market data from exchanges like CBOE typically provide these for major indices, ensuring the surface reflects real-time dynamics.^[17]^[18]^[20]

Dynamics and Evolution

Sticky Strike Rule

The sticky strike rule describes a dynamic in the implied volatility surface where the implied volatility \sigma(K, T) for a given absolute strike price K and time to maturity T remains constant as the underlying spot price S changes.^[21] This assumption implies parallel shifts in the volatility smile when expressed in terms of moneyness K/S, as the fixed strikes become relatively more or less in-the-money with spot movements.^[21] Empirical observations indicate that the sticky strike rule holds particularly in calm market environments or periods of slow mean-reversion in volatility, where implied volatilities for specific strikes exhibit stability despite minor spot fluctuations.^[21] For example, examinations of S&P 500 index options during stable trading range regimes, such as those in late 1997 and mid-1998, reveal adherence to this rule, with volatilities persisting at fixed strike levels amid low overall market turbulence.^[21] In terms of pricing implications, the sticky strike rule results in decreasing implied volatilities for options at fixed moneyness levels as the spot price rises, since strikes that were previously at-the-money become out-of-the-money relative to the new spot.^[21] This leads to systematically lower volatility quotes for at-the-money options following upward spot moves, influencing delta hedging strategies and the valuation of option portfolios under Black-Scholes frameworks adapted to the smile.^[21] Historically, the sticky strike rule served as a foundational assumption in option quoting practices shortly after the 1987 market crash, when the volatility smile first emerged prominently, prior to the wider adoption of the sticky delta alternative.^[21] This usage reflected early practitioner efforts to model the persistent skew in equity index options amid heightened crash awareness.^[21]

Sticky Delta Rule

The sticky delta rule, also known as the sticky moneyness rule, posits that the implied volatility for options of a given maturity remains constant when expressed as a function of the option's delta or moneyness, such as \sigma(\delta, T) or \sigma(\log(K/S), T), where \delta is the delta, K is the strike price, S is the underlying spot price, and T is the time to maturity.^[21] Under this rule, as the underlying price S changes, the strikes corresponding to fixed deltas or moneyness levels shift accordingly, causing the volatility smile to roll horizontally along the moneyness axis while preserving the shape of the smile relative to the spot.^[6] This contrasts with the sticky strike rule, which assumes volatilities remain fixed at absolute strike levels.^[22] Empirical evidence from equity markets, particularly S&P 500 options data spanning 1998 to 2002, demonstrates that the sticky delta rule provides a superior fit to observed implied volatility surfaces compared to the sticky strike rule, achieving an R^2 of 94.93% and RMSE of 0.0073 versus much poorer performance for the alternative.^[6] This rule has been observed to prevail during trending market periods post-1990s, effectively capturing the dynamics of volatility skews in indices where sustained movements in the underlying are common.^[21] The implications of the sticky delta rule include adjustments in implied volatilities that result in higher levels for out-of-the-money (OTM) puts during underlying declines, as the smile shifts to maintain constant volatility at fixed moneyness levels, thereby reflecting heightened demand for crash protection in negatively skewed markets.^[22] This behavior underscores the rule's alignment with risk premia that intensify during downturns. Market conventions exhibited a transition from the sticky strike rule toward the sticky delta rule around the 2000s, driven by its superior performance in hedging strategies amid trending equity environments, as evidenced by improved empirical fits and stability in at-the-money volatility preservation.^[6]^[21]

Theoretical Implications

Risk-Neutral Distributions

In financial mathematics, option prices reflect expectations of future asset prices under the risk-neutral measure, denoted as the Q-measure, where the discounted asset price is a martingale. Under this measure, the price of a European call option with strike K and maturity T is given by the expected value of the payoff discounted at the risk-free rate r, integrated against the risk-neutral probability density function q(S_T) of the terminal asset price S_T. This framework allows market-implied option prices to directly encode the risk-neutral distribution without requiring assumptions about the physical measure or investor risk preferences. The Breeden-Litzenberger theorem provides a foundational method to extract this risk-neutral density from observed option prices. Specifically, the second derivative of the call option price C with respect to the strike K yields the discounted risk-neutral density:

\frac{\partial^2 C}{\partial K^2} = e^{-rT} q(K)

This relation holds under no-arbitrage conditions and assumes twice-differentiable option prices across strikes. In practice, the density is approximated by numerically differentiating interpolated call prices or using parametric fits to the implied volatility surface as input.^[23] The presence of a volatility smile, characterized by elevated implied volatilities for out-of-the-money (OTM) puts relative to at-the-money options, implies deviations from the lognormal distribution assumed in the Black-Scholes model. Higher OTM put implied volatilities indicate a higher market-implied probability of large downward moves, resulting in fatter left tails in the risk-neutral density, with negative skewness (typically less than 0) and excess kurtosis (greater than 3). These features reflect anticipated crash risk, as evidenced in equity index options where the smile's steep left skew corresponds to leptokurtic densities. A flat volatility smile corresponds to a lognormal risk-neutral density, consistent with Black-Scholes assumptions of constant volatility and no jumps. In contrast, the post-1987 equity volatility smile produces skewed densities with kurtosis values often exceeding 10 and skewness around -0.5 to -1.0 for S&P 500 options, aligning with empirical observations of market crashes and tail events. Such deviations quantify the market's pricing of non-lognormal risks, with the smile's slope and curvature directly mapping to moments of the implied distribution.

Forward Probability Insights

The risk-neutral density q(K), derived from option prices embedded in the volatility smile, enables the computation of forward probabilities by integrating the density function. Specifically, the risk-neutral probability that the underlying asset price S_T exceeds a strike K at expiration T is given by P(S_T > K) = \int_K^\infty q(s) \, ds, while tail risks, such as crash probabilities, can be assessed by integrating over extreme lower tails.^[23] This approach, building on the Breeden-Litzenberger theorem that links option prices to the second derivative of the call pricing function, provides a market-implied view of future price distributions under the risk-neutral measure. In practice, the left skew of the volatility smile—characterized by higher implied volatilities for out-of-the-money put options—translates to elevated risk-neutral probabilities of significant downside moves, allowing traders to quantify crash risks. Conversely, the right tail of the smile informs upside potential, such as the likelihood of substantial rallies in the underlying asset. These probabilistic insights are particularly valuable for risk management and portfolio hedging, as they reflect collective market expectations of non-lognormal price dynamics.^[24] Following the 2008 financial crisis, equity index volatility smiles exhibited pronounced left skews, implying downside probabilities far exceeding those under a lognormal distribution during the height of market turmoil. A key limitation is that these forward probabilities are expressed under the risk-neutral measure, which incorporates investor risk premiums and thus overstates downside risks compared to physical (real-world) probabilities; converting to physical measures requires estimating and adjusting for time-varying risk aversion, often via historical data or equilibrium models.

Historical Development

Pre-1987 Observations

Prior to the establishment of organized options trading, financial markets featured limited and over-the-counter option contracts, primarily on equities and commodities, with no standardized exchange until the Chicago Board Options Exchange (CBOE) launched in April 1973. This development coincided with the publication of the Black-Scholes model, which assumed constant volatility across strike prices and maturities, leading practitioners to initially price options under a flat implied volatility surface. In the early years following the CBOE's inception, empirical observations of equity options revealed implied volatilities that were largely flat across different strike prices, consistent with the Black-Scholes premise, though subtle deviations occasionally appeared. Mark Rubinstein's 1985 analysis of trading data from 30 heavily traded NYSE stocks between August 1976 and August 1978 found that implied volatilities for calls and puts were approximately constant with respect to moneyness, with minor variations—such as slightly higher volatilities for deep out-of-the-money options—attributed to market microstructure noise like bid-ask spreads rather than systematic patterns. These deviations were dismissed as insignificant, reflecting a prevailing view that the constant volatility assumption held adequately for equity options. For equities, the general flatness stemmed from low awareness of tail risks like market crashes, resulting in the relative underpricing of out-of-the-money puts, as market participants underestimated extreme downside probabilities. In contrast, currency options traded on the Philadelphia Stock Exchange (PHLX) since December 1982 exhibited mild volatility smiles even before 1987, with implied volatilities increasing modestly for out-of-the-money strikes. James Bodurtha and Georges Courtadon's 1987 study of PHLX data from 1982 to 1984 documented this U-shaped pattern, linking it to perceived risks from central bank interventions that could induce jumps in exchange rates, though the smiles remained shallow compared to post-1987 equity patterns.^[25]

Post-1987 Crash Emergence

The 1987 stock market crash, known as Black Monday, occurred on October 19, 1987, when the Dow Jones Industrial Average plummeted by 22.6% in a single trading session, marking the largest one-day percentage decline in its history.^[26] This event dramatically exposed the underpricing of out-of-the-money (OTM) put options on equity indices, as the severe downside move revealed that market participants had previously underestimated tail risks under the prevailing Black-Scholes framework, which assumed constant volatility and lognormal distributions.^[27] In the immediate aftermath, implied volatilities for low-strike options—corresponding to deep OTM puts—surged dramatically, with spreads relative to at-the-money (ATM) implied volatilities spiking above 10% for 10% OTM puts with one month to maturity, compared to pre-crash averages of just 1.83%. This shift transformed the previously relatively flat implied volatility patterns observed in equity index options into a pronounced "smirk," an asymmetric volatility smile characterized by steeply higher implied volatilities for low strikes and relatively flatter levels for high strikes, reflecting heightened crash fears.^[27] Post-crash, these deep OTM put implied volatilities averaged 8.21% higher than ATM levels, establishing a permanent tilt in the volatility curve. Market participants and regulators responded swiftly to the turmoil. The Brady Commission investigated the crash and recommended market reforms, including circuit breakers to halt trading during extreme volatility. Options exchanges such as the Chicago Board Options Exchange (CBOE) implemented higher margin requirements for broad-based index options, while circuit breakers were introduced across major exchanges in 1988 to curb excessive leverage and volatility in derivatives trading.^[28] Academics began formalizing the documentation of this phenomenon, notably Mark Rubinstein in his 1994 paper "Implied Binomial Trees," which introduced methods to infer risk-neutral distributions from observed option prices and highlighted the smile's implications for non-lognormal asset return assumptions.^[29] Over the longer term, the volatility smile became a standard feature of equity options markets by the early 1990s, prompting a shift in industry quoting conventions from absolute strike prices or raw implied volatilities to delta-based terms, such as 25-delta risk reversals, to better capture the skewed dynamics across moneyness levels. This evolution underscored the crash's role in embedding crash risk premia into option pricing, contrasting sharply with the subtler, more symmetric patterns seen prior to 1987.^[27]

Explanations and Causes

Fat Tails and Crash Risk

The concept of fat tails in asset returns refers to the observation that empirical distributions exhibit excess kurtosis greater than 3, implying a higher probability of extreme events compared to the normal distribution's thin tails.^[30] This leptokurtosis is a stylized fact across equity markets, where daily stock returns typically display kurtosis values ranging from 10 to 50, leading to more frequent large deviations than predicted by Gaussian assumptions. Such fat-tailed behavior contributes to the volatility smile by necessitating higher implied volatilities for out-of-the-money (OTM) options to account for the elevated risk of tail events. A key driver of the volatility smile's leftward skew is the crash risk premium embedded in option prices, where investors pay a premium for OTM puts as protection against sudden downward jumps, such as market drops exceeding 20% in a single day.^[31] This demand elevates implied volatilities for low-strike puts, reflecting the market's pricing of asymmetric downside risk under risk-neutral measures, distinct from the symmetric volatility assumed in standard models. The 1987 stock market crash served as a pivotal example, with pre-event option prices already incorporating expectations of such jumps.^[31] Empirical evidence links the equity volatility skew to broader market fear, as post-crash periods show a strong positive correlation between the skew (measured by the difference in implied volatilities between OTM puts and calls) and spikes in the VIX index, which captures near-term volatility expectations.^[32] For instance, during the 2008 financial crisis, this correlation intensified, with skew steepening alongside VIX surges above 80, underscoring the smile's role in pricing tail risks.^[32] More recently, amid 2024-2025 fears of an AI-driven equity bubble, similar left skew patterns emerged in technology-heavy indices, driven by heightened demand for protective puts amid concerns over overvaluation and potential corrections.^[33] The leverage effect further amplifies this downside skew through a negative correlation between asset returns and future volatility, where declining stock prices increase financial leverage and thereby heighten equity volatility.^[34] This dynamic, empirically observed in stock markets with correlation coefficients around -0.2 to -0.5, exacerbates the pricing of crash risks by making negative return shocks more volatility-inducing than positive ones, contributing to the persistent left tilt in the volatility smile.^[35]

Supply-Demand Dynamics

The volatility smile arises in part from imbalances in the supply and demand for options among heterogeneous market participants, including hedgers, speculators, and market makers, which distort implied volatilities across strike prices. Hedgers, such as institutional investors protecting equity portfolios against downturns, disproportionately demand out-of-the-money (OTM) put options, elevating implied volatilities for low strikes and contributing to the downward skew observed in equity index options.^[36] In contrast, speculators often pursue upside potential by buying OTM call options, though this demand is typically less intense than protective put buying, resulting in relatively higher implied volatilities for high strikes as well, forming the "smile" shape.^[37] These dynamics reflect trader-specific needs rather than solely underlying asset characteristics. On the supply side, market makers provide liquidity by quoting bid-ask spreads but face inventory risks and hedging costs, particularly for illiquid or high-risk strikes like deep OTM puts, leading them to widen spreads and embed premia into prices that perpetuate the smile. When demand surges for certain strikes, dealers cannot perfectly arbitrage away mispricings due to limits like capital constraints, allowing supply-demand pressures to influence implied volatilities directly.^[38] Behavioral factors exacerbate these imbalances; loss aversion, where investors overvalue downside protection due to the asymmetric pain of losses relative to gains, drives excessive demand for OTM puts and inflates their prices beyond risk-neutral expectations.^[39] Institutional mandates, such as regulatory requirements for downside hedging in pension funds and insurance portfolios, further amplify this skew by channeling concentrated buying into protective options. While fat tails in return distributions motivate such hedging needs, trader behaviors intensify the resulting volatility patterns. Empirical evidence supports these supply-demand explanations. Coval and Shumway (2001) demonstrated that the negative expected returns on index options, particularly puts, stem from persistent overpricing driven by hedging demand, consistent with a demand-induced skew. More recently, in 2025, heightened policy uncertainty from U.S. tariff announcements spurred demand for protective put options amid market turbulence, steepening the volatility smile in S&P 500 options as traders bet on trade disruptions.^[40]

Modeling Approaches

Black-Scholes Limitations

The Black-Scholes model, developed in 1973, assumes that the volatility of the underlying asset is constant over the life of the option.^[41] This assumption is embedded in the model's partial differential equation, which derives option prices under geometric Brownian motion with fixed volatility σ. Applying Itô's lemma to the option price C(S, t, σ) yields the decomposition:

dC = \left( \Delta \frac{\partial C}{\partial S} + \theta \frac{\partial C}{\partial t} + \frac{1}{2} \Gamma \frac{\partial^2 C}{\partial S^2} \sigma^2 S^2 \right) dt + \vega \frac{\partial C}{\partial \sigma} d\sigma,

where Δ, θ, Γ, and vega are the Greeks representing delta, theta, gamma, and vega, respectively. The model sets dσ = 0, implying no volatility risk and enabling perfect dynamic replication through delta hedging alone.^[41]^[42] Even prior to the 1987 crash, the Black-Scholes model slightly underpriced deep OTM options, though implied volatility surfaces were relatively flat, aligning more closely with its assumptions.^[27] Post-1987, the emergence of the volatility smile exacerbated this mismatch: using a constant σ calibrated to at-the-money options systematically underprices OTM puts and overprices OTM calls, or vice versa, leading to inconsistent pricing across strikes.^[2] The model's hedging framework also breaks down under varying implied volatilities. In the presence of a volatility smile, delta hedging—designed under the constant σ assumption—becomes imperfect because changes in the underlying price alter the effective volatility (via vanna and volga exposures), introducing unhedged vega risk and increasing replication costs.^[42]^[6] Specifically, the Black-Scholes delta understates the hedge ratio for low-strike options and overstates it for high-strike options in a smiling volatility surface, resulting in residual risks that the original replication strategy cannot eliminate.^[6] The volatility smile directly challenges the Black-Scholes lognormal assumption by implying that the risk-neutral volatility σ depends on both the strike price K and time to maturity T, denoted as σ(K, T). This strike and maturity dependence reflects a non-lognormal risk-neutral distribution with fatter tails than predicted by the model, necessitating alternative frameworks to capture market-implied dynamics.^[2]^[43]

Stochastic Volatility Models

Stochastic volatility models introduce a random process for the asset's volatility, typically mean-reverting, to endogenously produce volatility smiles without relying on deterministic volatility functions. These models capture the empirical observation that volatility is not constant but fluctuates stochastically, with key features like the leverage effect—negative correlation between asset returns and volatility changes—driving the skew in implied volatilities. The volatility of volatility further shapes the smile's curvature, allowing for more realistic pricing of options across strikes and maturities.^[44] A seminal example is the Heston model, which posits the following stochastic differential equations for the asset price S_t and its variance v_t:

dS_t = \mu S_t \, dt + \sqrt{v_t} S_t \, dW_t^{(1)}

dv_t = \kappa (\theta - v_t) \, dt + \xi \sqrt{v_t} \, dW_t^{(2)}

where \kappa > 0 is the mean-reversion speed, \theta > 0 is the long-term variance, \xi > 0 is the volatility of volatility, and the Brownian motions satisfy \langle dW_t^{(1)}, dW_t^{(2)} \rangle = \rho \, dt with \rho < 0 to reflect the negative correlation observed in equity markets.^[44] In this setup, the negative \rho generates a pronounced downside skew in the volatility smile by linking volatility increases to asset price declines, while \xi controls the smile's overall slope and the degree of kurtosis in the risk-neutral distribution.^[44] Calibration of the Heston model to the implied volatility surface involves minimizing the difference between model and market prices, often using efficient techniques like Fourier inversion of the characteristic function for European option pricing or Monte Carlo simulations for broader instrument sets. These methods enable precise fitting to observed smiles, and the model's stochastic nature provides superior forward volatility dynamics, such as realistic smile evolution over time, relative to models with fixed volatility paths.^[45] Variants of stochastic volatility models extend the Heston framework for specific applications, such as the SABR model, which models the forward price F_t and its volatility \alpha_t via:

dF_t = \alpha_t F_t^\beta \, dW_t^{(1)}, \quad d\alpha_t = \nu \alpha_t \, dW_t^{(2)}

with correlation \rho and \beta \in [0,1] to adjust the diffusion's behavior near zero.^[46] The SABR model excels in capturing short-term volatility smiles, particularly in interest rate derivatives, by approximating implied volatilities through asymptotic expansions that match market skews and curvatures effectively.^[46]

Local and Jump Diffusion Models

Local volatility models address the limitations of the constant volatility assumption in the Black-Scholes framework by positing that volatility is a deterministic function of the underlying asset price and time, σ_loc(S_t, t).^[47] This approach, pioneered by Bruno Dupire in 1994, derives the local volatility surface directly from observed market prices of European options using a forward partial differential equation (PDE) that ensures exact replication of the implied volatility smile at any given maturity.^[48] Specifically, the local volatility σ_loc(K, T) is calibrated such that σ_loc(K, T) = σ(S = K, t = T), where K is the strike and T is the maturity, allowing the model to fit the entire volatility surface without arbitrage. However, while arbitrage-free by construction, these models often produce unrealistic forward volatility dynamics, such as explosive skew evolution, because the volatility is perfectly correlated with the asset price path.^[49] Jump diffusion models extend the Black-Scholes-Merton framework by incorporating discontinuous price movements via a compound Poisson process superimposed on the continuous diffusion component, enabling better capture of the fat tails observed in the volatility smile. Robert Merton introduced this paradigm in 1976, where jumps occur with intensity λ and have lognormal size distributions with mean μ_J, fattening the tails of the risk-neutral density to explain out-of-the-money option pricing discrepancies. To address the symmetry limitations of Merton's model, Steven Kou developed a double exponential jump diffusion model in 2002, employing asymmetric jump sizes with exponential distributions for upward and downward moves (parameters η_1 for positive jumps and η_2 for negative jumps), which more accurately reproduces the skewness in empirical volatility smiles.^[50] Hybrid models combine local volatility with jump diffusions to improve fit across the full smile surface, particularly in the tails, by allowing state-dependent diffusion alongside discontinuous jumps.^[51] For instance, extensions of Dupire's implied diffusion approach to Poisson jumps, as explored by Andersen and Andreasen in 1999, enable numerical calibration that matches market-observed smiles while preserving arbitrage-free conditions.^[51] These hybrids offer enhanced flexibility over pure local or jump models but introduce calibration challenges due to the increased parameter space.^[52] A key advantage of local volatility models is their ability to perfectly hedge vanilla options within a complete market framework, akin to Black-Scholes, since volatility is deterministic.^[53] In contrast, jump diffusion models excel at modeling crash risk and leptokurtosis but complicate hedging due to incomplete markets and require risk premia assumptions for jumps.^[54] Calibration in jump models is often computationally intensive, involving Fourier transforms or numerical PDE solutions, though they provide superior tail risk pricing compared to diffusion-only approaches.^[54] Stochastic volatility models serve as a complementary random volatility framework, but local and jump diffusions emphasize state-dependence and discontinuities for static smile fitting.^[52]

Modern Developments

Since the early 2010s, modeling approaches have evolved to incorporate more empirical features of volatility dynamics. Rough volatility models, introduced around 2018 by Gatheral, Jaisson, and Rosenbaum, treat volatility paths as rough fractional Brownian motion with Hurst parameter H ≈ 0.1, providing superior fits to short-dated volatility smiles and surfaces observed in equity and FX markets. These models capture the explosive skew behavior and realistic forward smile dynamics better than classical stochastic volatility frameworks.^[55] Additionally, machine learning techniques have gained prominence for volatility smile modeling as of 2025. Methods such as meta-learning neural processes reconstruct implied volatility surfaces from sparse data, outperforming traditional parametric models like SABR in accuracy and speed, particularly for long maturities. These data-driven approaches integrate structural priors from classical models and enable real-time calibration, addressing limitations in high-dimensional fitting.^[56]

References

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[PDF] Implied Volatility, Volatility Smile/Skew/Smirk, and Risk
If one is willing to accept the smile, one can still use the Black-Scholes model to evaluate options or predict future asset prices. More Realistic Model. Does ...Missing: explanation | Show results with:explanation
[2]
[PDF] Black-Scholes and the Volatility Surface
The principal features of the volatility surface is that options with lower strikes tend to have higher implied volatilities. For a given maturity, T, this ...
[3]
The Pricing of Options and Corporate Liabilities
A theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options.
[4]
A review on implied volatility calculation - ScienceDirect.com
Chance (1996) [22] starts with the implied volatility σ ∗ = σ B S of an at-the-money-call C ∗ option obtained with the Brenner–Subrahmanyam approximation ...
[5]
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[6]
[PDF] Post-'87 Crash Fears in S&P 500 Futures Options
In essence, an asymmetric "volatility smirk" pattern alternated with a more symmetric "volatility smile" pattern over 1983-87, with patterns persisting ...
[7]
Estimating volatility-of-volatility: A comparative analysis
Historical variance H V t is calculated as: H V t = 1 n − 1 ∑ i = 1 n r i , t − r ̄ t 2 , where r i , t represents the log-returns of the underlying asset, and ...
[8]
[PDF] Implied Volatility and Historical Volatility - DiVA portal
Empirical findings of encompassing regression tests imply that the implied volatility index does not surpass historical volatility in terms of forecasting ...
[9]
[PDF] Option Volatility & Arbitrage Opportunities - LSU Scholarly Repository
To best estimate the volatility of an option, traders use the historical volatility and the implied volatility given by the marketplace. The volatility factor ...
[10]
Empirical Comparison of Alternative Implied Volatility Measures of ...
Feb 9, 2012 · According to the results, Heston's (1993) implied volatility dominated other implied volatility measures as well as historical volatility in ...
[11]
[PDF] Volatility Surfaces: Theory, Rules of Thumb, and Empirical Evidence
A U-shaped volatility smile is commonly observed for options on a foreign currency. ... Rubinstein (1994). When the level of the S&P 500 decreases ...
[12]
Explaining asset pricing puzzles associated with the 1987 market ...
The 1987 market crash was associated with a dramatic and permanent steepening of the implied volatility curve for equity index options.
[13]
Introduction to CVOL Skew - CME Group
Traders use the risk reversal to express a view of the cost of downside protection versus upside protection. For example, the 25-delta risk reversal is ...
[14]
Understanding the Volatility Surface in Options Trading - Investopedia
The volatility surface is a three-dimensional chart that shows how implied volatility varies across different strike prices and expiration dates for options ...Missing: steep | Show results with:steep
[15]
Volatility Smiles | FRM Part 2 Study Notes - AnalystPrep
Jun 2, 2019 · Volatility smiles are implied volatility patterns that arise in pricing financial options. When the implied volatility of options – with the same expiration ...
[16]
[PDF] Construction Methodologies for Implied Volatility Surfaces
Apr 29, 2016 · We will see that the SABR model is the most appropriate construction methodology with respect to both esti- mating implied volatility surfaces ...
[17]
[PDF] Implied volatility surface: construction methodologies and ... - arXiv
Jul 10, 2011 · The most commonly considered stochastic volatility models are Heston and SABR and their extensions. (such as time dependent parameters, etc) and ...
[18]
Reading the Volatility Surface - Menthor Q
This article explains how the volatility surface shapes gamma scalping efficiency by influencing option pricing, skew, term structure, and profitability.
[19]
https://menthorq.com/guide/reading-the-volatility-surface/
[20]
[PDF] Quantitative Strategies Research Notes
Volatility behavior using the sticky-strike rule. The Sticky-Delta Rule. The sticky-delta rule is a more subtle view of what quantity remains invariant as ...Missing: post- | Show results with:post-
[21]
[PDF] Volatility Surfaces: Theory, Rules of Thumb, and Empirical Evidence
For foreign currencies this skew becomes a “smile” (that is, the implied volatility is a U-shaped function of strike price). For both types of assets, the ...
[22]
[PDF] Patterns of Volatility Change - Copyright Emanuel Derman 2008
Copyright Emanuel Derman 2008 ... Table 1 summarizes the behavior of volatilities under the sticky-strike rule. TABLE 1. Volatility behavior using the sticky- ...
[23]
[PDF] A Simple and Reliable Way to Compute Option-Based Risk-Neutral ...
This paper describes a method for computing risk-neutral density functions based on the option-implied volatility smile. Its aim is to reduce complexity and ...
[24]
[PDF] Forecasting Crashes with a Smile - LSE
Martin (2017) shows how to use option prices to calculate the probability of a crash in the market from the perspective of a log investor who holds the market.
[25]
Analysing Implied Volatility Smirk to Predict the US Stock Market ...
Aug 10, 2025 · ... 2008. Finally, volatility of returns was unusually high in 2008, obviously due to the prevailing global financial crisis. View full-text.
[26]
Gold Silver: Major Factors That Could Impact Implied Volatility and ...
Dec 3, 2024 · There are major volatility risks for gold and silver next year amid the potential for tariffs against trading partners.Missing: fears | Show results with:fears
[27]
Tests of an American Option Pricing Model on the Foreign Currency ...
Specifically, we consider a cur? rency option written on a currency with a 10.00-percent annual volatility. ... Bodurtha and Courtadon 159 the previous day, but ...
[28]
Stock Market Crash of 1987 | Federal Reserve History
In the wake of the Crash of 1987, option volatility surfaces changed and the probabilities of fat tail (kurtosis)/skew distributions increased, leading to ...
[29]
[PDF] Explaining Asset Pricing Puzzles Associated with the 1987 Market ...
Moreover, the crash triggered a permanent shift in index option prices: Prior to the crash, implied 'volatility smiles' for index options were relatively flat.
[30]
A Brief History of the 1987 Stock Market Crash with a Discussion of ...
When adjusting the margin accounts, the exchanges first made margin calls ... Failure of retail investors to meet margin calls spurred liquidations in options ...
[31]
Implied Binomial Trees - RUBINSTEIN - 1994 - The Journal of Finance
This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European ...
[32]
[PDF] Empirical properties of asset returns: stylized facts and statistical ...
The kurtosis of the increments of asset prices is far from its Gaussian value: typical values for T = 5 minutes are (see table 1): κ ≃ 74 (US$/DM exchange rate ...
[33]
Was It Expected? The Evidence from Options Markets - jstor
Options prices suggest a crash was expected before the 1987 crash, with out-of-the-money puts becoming unusually expensive.Missing: surge | Show results with:surge
[34]
[PDF] Stock Market Volatility during the 2008 Financial Crisis - NYU Stern
Apr 1, 2010 · In this report, we study the stock market volatility and the behavior of various measures of volatility before, during and after the 2008 ...Missing: probabilities | Show results with:probabilities<|separator|>
[35]
https://arxiv.org/abs/cond-mat/0101120
[36]
[PDF] The Leverage Effect Puzzle: Disentangling Sources of Bias at High ...
The leverage effect refers to the observed tendency of an asset's volatility to be negatively correlated with the asset's returns. Typically, rising asset ...
[37]
[cond-mat/0101120] The leverage effect in financial markets - arXiv
Jan 9, 2001 · We investigate quantitatively the so-called leverage effect, which corresponds to a negative correlation between past returns and future volatility.
[38]
Does Net Buying Pressure Affect the Shape of Implied Volatility ...
Mar 25, 2004 · This paper examines the relation between net buying pressure and the shape of the implied volatility function (IVF) for index and individual stock options.
[39]
Investors' net buying pressure and implied volatility dynamics
Bollen and Whaley (2004) suggest that options' NBP affects the IV dynamics because of limitations on arbitrage trading, in favor of the limits-to-arbitrage ...
[40]
Does Net Buying Pressure Affect the Shape of Implied Volatility ...
This paper examines the relation between net buying pressure and the shape of the implied volatility function (IVF) of S&P 500 index options and options on ...
[41]
https://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/black_scholes73.pdf
[42]
Volatility Traders Revel in S&P 500's Wild Tariff-Driven Swings
Mar 9, 2025 · Options traders who thrive on volatility are loving the market turbulence sparked by President Donald Trump's trade fights.Missing: demand smile
[43]
[PDF] Fischer Black and Myron Scholes Source: The Journal of Political Eco
Author(s): Fischer Black and Myron Scholes. Source: The Journal of Political Economy, Vol. 81, No. 3 (May - Jun., 1973), pp. 637-654. Published by: The ...
[44]
[PDF] Vega risk and the smile - CiteSeerX
1.2 Limitations of the Black-Scholes model The Black-Scholes model assumes that asset returns follow a random walk with a constant volatility. The Black- ...
[45]
[PDF] AN EMPIRICAL-DISTRIBUTION-BASED OPTION PRICING MODEL
Duan, J.C , 1999, “Conditionally Fat Tailed Distribution and the Volatility Smile in. Options,” working paper, University of Toronto. Dumas, B., J. Fleming ...
[46]
[PDF] Stochastic Volatility, Smile & Asymptotics - Mathematics
May 17, 1998 · Figure 1: The implied volatility smile curves become spikier as t → T. ... Rubinstein. Nonparametric Tests of Alternative Option Pricing ...
[47]
[PDF] A Closed-Form Solution for Options with Stochastic Volatility with ...
This paper derives a closed-form solution for European call options with stochastic volatility, allowing correlation between volatility and spot-asset returns, ...
[48]
The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo ...
Jun 13, 2013 · This hybrid model combines the main advantages of the Heston model and the local volatility model introduced by Dupire (1994) and Derman & Kani ...
[49]
[PDF] Smile Risk - Deriscope
To resolve this problem, we derive the SABR model, a stochastic volatility model in which the asset price and volatility are correlated. Singular perturbation ...
[50]
http://www.columbia.edu/~sk75/KouWangMS.pdf
[51]
[PDF] Local Volatility Pricing Models for Long-dated FX Derivatives - arXiv
Apr 3, 2012 · In this section we derive the expression of the local volatility function by using the same technique as. Dupire [Dupire, 1994] and Derman and ...
[52]
[PDF] Option Pricing under Hybrid Stochastic and Local Volatility
Second, the geometric struc- ture of the implied volatilities shows a smile fitting the market data better than the CEV model. Consequently, the underlying ...
[53]
[PDF] Option pricing under a double exponential jump diffusion model
The aim of this paper is to extend the analytical tractability of the Black-Scholes model to alternative models with jumps. We demon- strate that a double ...
[54]
[PDF] Jump-Diffusion Processes: Volatility Smile Fitting and ...
May 6, 1999 · This paper discusses extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps.
[55]
https://www.siam.org/publications/siam-news/articles/rough-volatility-in-financial-mathematics/
[56]
[PDF] Stochastic Volatility: Modeling and Asymptotic Approaches to Option ...
One advantage of local volatility models is that markets remain complete, meaning that deriva- tives written on S can be hedged perfectly – just as in the Black ...
[57]
[PDF] A Jump-Diffusion Model for Option Pricing - Columbia University
Merton. (1976) was the first to consider a jump-diffusion model similar to (1) and (3). In Merton's paper. Ys are normally distributed. ... Jump Diffusion Model ...