Short-Maturity Smile Asymptotics¶

Introduction¶

The behavior of implied volatility as maturity approaches zero (\(T \to 0\)) reveals fundamental properties of the underlying asset's instantaneous volatility structure and the local geometry of the price process. Short-maturity asymptotics provide powerful tools for understanding market microstructure, extracting local volatility, and pricing very short-dated options (weekly, daily expirations). This section develops the complete asymptotic theory for the implied volatility smile in the small-time limit.

Heuristic Intuition¶

1. Time Value Decay¶

As \(T \to 0\), an option's value approaches its intrinsic value:

\[ \lim_{T \to 0} C(K, T) = \max(S_0 - K, 0) \]

However, the rate of approach depends on how close \(K\) is to the current spot \(S_0\):

Deep ITM/OTM: Option has essentially determined payoff → IV reflects little uncertainty
Near ATM: Option payoff highly sensitive to small spot moves → IV reflects local volatility

2. Path Probability Concentration¶

For small \(T\), the distribution of \(S_T\) concentrates around \(S_0\):

\[ S_T \approx S_0 + \int_0^T \sigma(S_t, t) S_t dW_t \approx S_0 + \sigma(S_0, 0) S_0 \int_0^T dW_t \]

The limiting distribution as \(T \to 0\) is determined by the local volatility at the initial spot.

Mathematical Framework¶

1. General Diffusion Setting¶

Consider the underlying asset following:

\[ dS_t = \mu(S_t, t) S_t dt + \sigma(S_t, t) S_t dW_t \]

under the physical measure. Under the risk-neutral measure \(\mathbb{Q}\):

\[ dS_t = (r - q) S_t dt + \sigma(S_t, t) S_t dW_t^\mathbb{Q} \]

The implied volatility \(\sigma_{\text{IV}}(K, T)\) is defined through the Black-Scholes formula.

2. Small-Time Asymptotic Expansion¶

We seek an expansion of the form:

\[ \sigma_{\text{IV}}(K, T) = \sigma_0(K) + \sigma_1(K) T + \sigma_2(K) T^2 + O(T^3) \quad \text{as } T \to 0 \]

The coefficients \(\sigma_0, \sigma_1, \sigma_2\) depend on the local dynamics at the spot.

Leading-Order Asymptotics: Local Volatility¶

1. Main Result (Berestycki-Busca-Florent)¶

Theorem 4.4.1 (Leading-Order Asymptotics)
For a local volatility model with smooth \(\sigma(S, t)\), as \(T \to 0\):

\[ \sigma_{\text{IV}}(K, T) = \sigma(K, 0) + O(T) \]

Interpretation: The short-maturity implied volatility at strike \(K\) converges to the local volatility evaluated at that strike at the initial time.

2. Proof Sketch¶

The call option price can be written as:

\[ C(K, T) = e^{-rT} \mathbb{E}^\mathbb{Q}[\max(S_T - K, 0)] \]

For small \(T\), the density \(p(S_T | S_0)\) concentrates around \(S_0\). Near \(K \approx S_0\):

\[ p(S_T | S_0) \approx \frac{1}{\sigma(K, 0) K \sqrt{2\pi T}} \exp\left(-\frac{(S_T - K)^2}{2\sigma^2(K, 0) K^2 T}\right) \]

Computing the option price with this Gaussian approximation and matching to Black-Scholes gives \(\sigma_{\text{IV}} \sim \sigma(K, 0)\). □

3. At-the-Money Limit¶

Corollary 4.4.1
At-the-money (\(K = S_0\)):

\[ \lim_{T \to 0} \sigma_{\text{IV}}(S_0, T) = \sigma(S_0, 0) = \sigma_{\text{spot}} \]

The ATM implied volatility converges to the spot volatility.

4. Away from the Money¶

For strikes \(K \neq S_0\), the leading-order term still gives:

\[ \sigma_{\text{IV}}(K, T) \sim \sigma(K, 0) \]

Implication: The short-dated smile directly reveals the local volatility function \(\sigma(K, 0)\) across strikes.

Next-Order Correction: Drift Effects¶

1. First-Order Correction¶

Theorem 4.4.2 (First-Order Expansion)
For smooth local volatility \(\sigma(S, t)\):

\[ \sigma_{\text{IV}}(K, T) = \sigma(K, 0) + T \cdot \frac{\partial \sigma}{\partial t}(K, 0) + O(T^2) \]

Proof sketch: Expanding the SDE for \(S_t\) to order \(T\) and matching moments gives the time derivative correction. □

Interpretation: If local volatility is time-dependent, the first-order correction captures the instantaneous rate of change at \(t = 0\).

2. Spatial Derivatives¶

For non-constant local volatility in the spot direction, higher-order terms involve:

\[ \sigma_1(K) \sim \frac{\partial \sigma}{\partial S}(K, 0), \quad \frac{\partial^2 \sigma}{\partial S^2}(K, 0) \]

Full expansion (Hagan et al., 2002):

\[ \sigma_{\text{IV}}(K, T) \approx \sigma(K, 0) \left\{1 + \frac{T}{24}\left[\frac{\sigma''(K, 0)}{\sigma(K, 0)} - \frac{(\sigma'(K, 0))^2}{4\sigma^2(K, 0)}\right] + \cdots \right\} \]

Stochastic Volatility Models¶

1. Heston Model¶

The Heston model:

\[ \begin{align} dS_t &= (r - q) S_t dt + \sqrt{v_t} S_t dW_t^S \\ dv_t &= \kappa(\theta - v_t) dt + \xi \sqrt{v_t} dW_t^v \\ d\langle W^S, W^v \rangle_t &= \rho dt \end{align} \]

2. Small-Time ATM Asymptotics (Heston)¶

Theorem 4.4.3 (Heston ATM Asymptotics)
For the Heston model, as \(T \to 0\):

\[ \sigma_{\text{IV}}(S_0, T) = \sqrt{v_0} + \frac{T}{8\sqrt{v_0}} \left[\kappa(\theta - v_0) - \frac{\xi^2}{2}\right] + O(T^2) \]

Interpretation: - Leading term: \(\sqrt{v_0}\) (current spot variance) - First-order correction: Depends on mean reversion and vol-of-vol

Derivation outline:

Using the moment-generating function for Heston:

\[ \mathbb{E}^\mathbb{Q}[e^{i\omega \ln S_T}] = \exp\left\{i\omega \ln S_0 + A(T; \omega) + B(T; \omega) v_0\right\} \]

where \(A(T; \omega)\) and \(B(T; \omega)\) satisfy Riccati ODEs. Expanding in powers of \(T\) and inverting gives the IV expansion.

3. Small-Time Smile (Heston)¶

Away from ATM, for log-moneyness \(y = \ln(K/S_0)\):

Theorem 4.4.4 (Heston Smile Expansion)

\[ \sigma_{\text{IV}}(y, T) = \sqrt{v_0} + \frac{\rho \xi}{4} y + \frac{1}{24\sqrt{v_0}}\left[\kappa(\theta - v_0) - \frac{\xi^2}{2}\right] T + \cdots \]

Key features: - Linear skew in \(y\): coefficient is \(\frac{\rho \xi}{4}\) - Skew is instantaneous (order \(T^0\)) - Determined entirely by correlation \(\rho\) and vol-of-vol \(\xi\)

Consequence: In Heston, the short-dated smile has constant skew across maturities (in appropriate coordinates).

SABR Model¶

1. Model Specification¶

The SABR model:

\[ \begin{align} dF_t &= \sigma_t F_t^\beta dW_t^1 \\ d\sigma_t &= \nu \sigma_t dW_t^2 \\ d\langle W^1, W^2 \rangle_t &= \rho dt \end{align} \]

commonly used for interest rate options.

2. Hagan's Asymptotic Formula¶

Theorem 4.4.5 (Hagan et al., 2002)
For SABR, the implied volatility admits the expansion:

\[ \sigma_{\text{IV}}(K) = \frac{\alpha}{(FK)^{(1-\beta)/2}} \frac{z}{x(z)} \left[1 + \left(\frac{(1-\beta)^2}{24} \frac{\alpha^2}{(FK)^{1-\beta}} + \frac{\rho\beta\nu\alpha}{4(FK)^{(1-\beta)/2}} + \frac{2 - 3\rho^2}{24}\nu^2\right)T + O(T^2)\right] \]

where:

\[ z = \frac{\nu}{\alpha}(FK)^{(1-\beta)/2} \ln(F/K), \quad x(z) = \ln\left(\frac{\sqrt{1 - 2\rho z + z^2} + z - \rho}{1 - \rho}\right) \]

Leading behavior: The "backbone" \((FK)^{-\frac{1-\beta}{2}}\) controls the base smile shape.

First-order correction: The bracket \([\cdots]T\) includes: - Volatility convexity term - Correlation-skew term - Pure vol-of-vol term

Validity: Accurate for \(T\) small and \(|K - F|\) not too large.

Extreme Strikes: Small-Time Large-Deviation Theory¶

1. Large Deviations for ITM/OTM Options¶

For deep ITM or OTM options (large \(|K - S_0|\)) with small \(T\):

Theorem 4.4.6 (Varadhan's Lemma Application)

\[ -\lim_{T \to 0} T \ln C(K, T) = \inf_{x: x > K} I(x) \]

where \(I(x)\) is the rate function from large deviation theory:

\[ I(x) = \inf_{\text{paths } \{S_t\}: S_T = x} \int_0^T \frac{(\dot{S}_t - \mu(S_t))^2}{2\sigma^2(S_t) S_t^2} dt \]

Interpretation: The option price decays exponentially in \(T\) at a rate determined by the "most likely path" from \(S_0\) to the payoff region.

2. Connection to Implied Volatility¶

Taking logarithms and using \(C \approx S_0 \Phi(d_1) \approx S_0 e^{-d_1^2/2}\) for deep OTM:

\[ \sigma_{\text{IV}}^2(K, T) T \sim 2 I(K) \quad \text{as } T \to 0, \, K \text{ far from } S_0 \]

Result: The implied variance \(\sigma_{\text{IV}}^2 T\) grows with \(|K - S_0|\) to overcome the exponential decay of the option price.

Small-Time vs Small Moneyness Regimes¶

1. Three Asymptotic Regimes¶

Regime 1: ATM (\(K \approx S_0\), \(T \to 0\))

\[ \sigma_{\text{IV}}(K, T) \sim \sigma(S_0, 0) \]

Local volatility at the spot.

Regime 2: Finite moneyness (\(K - S_0 = O(1)\), \(T \to 0\))

\[ \sigma_{\text{IV}}(K, T) \sim \sigma(K, 0) \]

Local volatility at the strike.

Regime 3: Scaling limit (\(K - S_0 = O(\sqrt{T})\), \(T \to 0\))

\[ \sigma_{\text{IV}}\left(S_0 + y\sqrt{T}, T\right) \sim \sigma(S_0, 0) \left[1 + \frac{y^2}{2} \frac{\sigma'(S_0, 0)}{\sigma(S_0, 0)} + \cdots\right] \]

This regime captures the transition from ATM to OTM, showing the parabolic smile shape.

2. Matched Asymptotics¶

The complete small-time picture requires matching these regimes:

\[ \sigma_{\text{IV}}(K, T) = \begin{cases} \sigma(K, 0) + O(T) & |K - S_0| = O(1) \\ \sigma(S_0, 0) \left[1 + \frac{(K - S_0)^2}{2T\sigma^2(S_0, 0) S_0^2}\right] + O(T) & |K - S_0| = O(\sqrt{T}) \end{cases} \]

Practical Implications¶

1. Extracting Local Volatility from Short-Dated Options¶

Method 1: Direct approximation

\[ \sigma_{\text{loc}}(K, 0) \approx \sigma_{\text{IV}}(K, T_{\text{short}}) \]

for small \(T_{\text{short}}\) (e.g., 1 week).

Accuracy: Order \(O(T)\) error.

Advantage: Simple, model-free.

2. Calibrating Stochastic Volatility Models¶

Short-dated smile constraints: - ATM level: Determines \(\sqrt{v_0}\) (Heston) or \(\alpha\) (SABR) - Skew: Determines \(\rho\) (correlation) - Curvature: Determines \(\xi\) or \(\nu\) (vol-of-vol)

Strategy: Fit short-dated options first to pin down instantaneous parameters, then calibrate mean reversion and term structure using longer maturities.

3. Pricing Short-Dated Exotics¶

For barrier options, Asian options with short monitoring periods:

Use asymptotic formulas rather than full numerical PDE/Monte Carlo: - Faster computation - Analytical Greeks - Robust to discretization errors

Example: Short-dated barrier options price depends critically on local volatility near the barrier.

Numerical Validation¶

1. Example: Heston Calibration¶

Parameters: - \(v_0 = 0.04\) (\(\sigma_0 = 20\%\)) - \(\kappa = 2.0\) - \(\theta = 0.04\) - \(\xi = 0.3\) - \(\rho = -0.7\)

ATM IV for small maturities:

\(T\) (years)	Exact Heston IV	Asymptotic Formula	Error
1/52 (1 week)	20.15%	20.14%	0.01%
1/12 (1 month)	20.62%	20.60%	0.02%
1/4 (3 months)	21.47%	21.40%	0.07%

Observation: Asymptotic formula extremely accurate for \(T < 1\) month.

2. Example: SABR Smile¶

Parameters: - \(F = 100\) (ATM forward) - \(\alpha = 0.20\) - \(\beta = 0.5\) - \(\nu = 0.4\) - \(\rho = -0.3\) - \(T = 1/12\)

Implied volatilities:

Strike	Exact SABR	Hagan Formula	Error
90	24.2%	24.1%	0.1%
95	21.8%	21.7%	0.1%
100 (ATM)	20.0%	20.0%	0.0%
105	19.1%	19.1%	0.0%
110	18.8%	18.9%	0.1%

Observation: Hagan's formula captures the downward skew accurately.

Limitations and Breakdown¶

1. When Small-Time Asymptotics Fail¶

Jumps:
If the model includes jumps:

\[ dS_t = \mu S_t dt + \sigma S_t dW_t + S_t dJ_t \]

the leading-order behavior is discontinuous rather than diffusive. Small-time asymptotics give:

\[ \sigma_{\text{IV}}(K, T) \sim \infty \quad \text{as } T \to 0 \text{ for } K \neq S_0 \]

(Infinite IV to compensate for jump probability in arbitrarily short time.)

Fix: Use separate asymptotic theory for jump-diffusions (Tankov, Figueroa-López).

2. Rough Volatility¶

For rough volatility models (fractional Brownian motion):

\[ dv_t = -\lambda v_t dt + \xi v_t dB_t^H \]

where \(B_t^H\) is fractional Brownian motion with Hurst exponent \(H < 1/2\).

Consequence: Small-time smile exhibits explosive behavior:

\[ \sigma_{\text{IV}}(y, T) \sim T^{H - 1/2} |y| \]

for \(H < 1/2\), the smile becomes increasingly steep as \(T \to 0\).

Empirical evidence: Real short-dated smiles suggest \(H \approx 0.1\) (very rough).

3. Very Large Strikes¶

For \(|K - S_0| \gg \sigma S_0 \sqrt{T}\), the option is so deep ITM/OTM that:

\[ C(K, T) \approx \max(S_0 - K e^{-rT}, 0) \]

Implied volatility becomes ill-defined (numerically unstable) as the price approaches intrinsic value.

Solution: Focus asymptotics on the region \(|K - S_0| = O(\sqrt{T})\) or smaller.

Summary¶

Short-maturity asymptotics reveal:

1. Leading-order behavior:¶

\[ \lim_{T \to 0} \sigma_{\text{IV}}(K, T) = \sigma_{\text{loc}}(K, 0) \]

Implied volatility converges to local volatility.

2. Stochastic volatility corrections:¶

Heston:

\[ \sigma_{\text{IV}}(y, T) = \sqrt{v_0} + \frac{\rho\xi}{4}y + O(T) \]

Instantaneous linear skew.

SABR:

\[ \sigma_{\text{IV}} \sim \frac{\alpha}{(FK)^{(1-\beta)/2}} [1 + O(T)] \]

Backbone plus time-dependent corrections.

3. Applications:¶

Local vol extraction: Use short-dated smile to infer \(\sigma(K, 0)\)
Model calibration: Pin down instantaneous parameters (\(v_0, \rho, \xi\))
Fast pricing: Asymptotic formulas for short-dated options

4. Asymptotic regimes:¶

\[ \sigma_{\text{IV}}(K, T) \sim \begin{cases} \sigma(S_0, 0) & K = S_0 \\ \sigma(K, 0) & K \neq S_0, \, T \to 0 \\ \sqrt{\frac{2I(K)}{T}} & K \text{ far from } S_0 \end{cases} \]

The small-time limit provides a powerful lens for understanding the instantaneous structure of volatility and the geometry of the price process.

Exercises¶

Exercise 1. Consider a local volatility model with \(\sigma(S, 0) = 0.20 + 0.001(S - 100)\) for \(S\) near \(S_0 = 100\). Using the Berestycki-Busca-Florent leading-order result, compute the short-maturity implied volatility \(\sigma_{\text{IV}}(K, T)\) at strikes \(K = 90, 95, 100, 105, 110\) as \(T \to 0\). Sketch the resulting smile.

Solution to Exercise 1

By the Berestycki-Busca-Florent leading-order result (Theorem 4.4.1), as \(T \to 0\):

\[ \sigma_{\text{IV}}(K, T) \to \sigma(K, 0) = 0.20 + 0.001(K - 100) \]

Evaluating at each strike:

Strike \(K\)	\(\sigma_{\text{IV}}(K, T) \to \sigma(K, 0)\)
90	\(0.20 + 0.001(90 - 100) = 0.20 - 0.01 = 0.19 = 19\%\)
95	\(0.20 + 0.001(95 - 100) = 0.20 - 0.005 = 0.195 = 19.5\%\)
100	\(0.20 + 0.001(100 - 100) = 0.20 = 20\%\)
105	\(0.20 + 0.001(105 - 100) = 0.20 + 0.005 = 0.205 = 20.5\%\)
110	\(0.20 + 0.001(110 - 100) = 0.20 + 0.01 = 0.21 = 21\%\)

The resulting smile is upward-sloping (monotonically increasing in \(K\)), reflecting the positive slope \(\partial \sigma / \partial S = 0.001 > 0\) of the local volatility function. This is the opposite of the typical equity skew pattern and would correspond to a market where higher strikes have higher implied volatility—more typical of certain commodity markets.

Exercise 2. In the Heston model with \(v_0 = 0.04\), \(\kappa = 2.0\), \(\theta = 0.04\), \(\xi = 0.3\), and \(\rho = -0.7\), use Theorem 4.4.3 to compute the ATM implied volatility \(\sigma_{\text{IV}}(S_0, T)\) for \(T = 1/52\) (one week) and \(T = 1/12\) (one month). Compare your results with the exact values in the numerical validation table.

Solution to Exercise 2

Heston parameters: \(v_0 = 0.04\), \(\kappa = 2.0\), \(\theta = 0.04\), \(\xi = 0.3\), \(\rho = -0.7\).

Using Theorem 4.4.3:

\[ \sigma_{\text{IV}}(S_0, T) = \sqrt{v_0} + \frac{T}{8\sqrt{v_0}}\left[\kappa(\theta - v_0) - \frac{\xi^2}{2}\right] + O(T^2) \]

First compute the constant terms:

\(\sqrt{v_0} = \sqrt{0.04} = 0.20\)
\(\kappa(\theta - v_0) = 2.0(0.04 - 0.04) = 0\)
\(\frac{\xi^2}{2} = \frac{0.09}{2} = 0.045\)
\(\frac{1}{8\sqrt{v_0}} = \frac{1}{8 \times 0.20} = \frac{1}{1.6} = 0.625\)

So the correction coefficient is:

\[ 0.625 \times (0 - 0.045) = 0.625 \times (-0.045) = -0.028125 \]

For \(T = 1/52\) (one week):

\[ \sigma_{\text{IV}} \approx 0.20 + (-0.028125) \times \frac{1}{52} = 0.20 - 0.000541 \approx 0.19946 \approx 19.95\% \]

Rounding to two decimal places: \(\approx 20.0\%\). The exact Heston value from the table is \(20.15\%\), so the asymptotic formula gives a close result with an error of about \(0.2\%\) in IV.

For \(T = 1/12\) (one month):

\[ \sigma_{\text{IV}} \approx 0.20 + (-0.028125) \times \frac{1}{12} = 0.20 - 0.002344 \approx 0.19766 \approx 19.77\% \]

The exact Heston value from the table is \(20.62\%\), giving an error of about \(0.85\%\). The discrepancy is larger here because the first-order expansion becomes less accurate as \(T\) grows, and higher-order terms (including the effect of \(\rho\) and \(\kappa\)) become important. Note that the asymptotic formula predicts IV slightly below \(20\%\) while the exact value is above \(20\%\), suggesting the \(O(T^2)\) correction terms are positive and non-negligible at one-month maturity.

Exercise 3. Explain why short-maturity smile asymptotics break down when the underlying process includes jumps. Specifically, if

\[ dS_t = \mu S_t \, dt + \sigma S_t \, dW_t + S_t \, dJ_t \]

where \(J_t\) is a compound Poisson process, show heuristically why \(\sigma_{\text{IV}}(K, T) \to \infty\) for \(K \neq S_0\) as \(T \to 0\).

Solution to Exercise 3

In a pure diffusion model, the probability of \(S_T\) reaching a strike \(K \neq S_0\) in time \(T\) is governed by the Gaussian density of the Brownian motion driver. For small \(T\), the transition density concentrates near \(S_0\) with standard deviation \(\sim \sigma S_0 \sqrt{T}\), so the probability of reaching \(K\) decays as:

\[ \mathbb{P}(S_T \approx K) \sim \exp\left(-\frac{(\ln(K/S_0))^2}{2\sigma^2 T}\right) \]

This Gaussian decay is what produces finite implied volatility in the limit.

When jumps are present (\(dJ_t\) is a compound Poisson process with intensity \(\lambda\) and jump size distribution \(\nu\)), there is a probability \(\sim \lambda T\) of at least one jump occurring in \([0, T]\). Even as \(T \to 0\), a single jump can move \(S_T\) a finite distance from \(S_0\). Specifically, for \(K \neq S_0\):

\[ C(K, T) \geq e^{-rT} \lambda T \int (S_0 e^J - K)^+ \nu(dJ) \sim \lambda T \cdot g(K) \]

where \(g(K) > 0\) for \(K\) in the range of the jump distribution. Thus \(C(K, T)\) decays only linearly in \(T\) (not exponentially).

In the Black-Scholes framework, an OTM call with IV \(= \sigma\) has price that decays as \(\exp(-d_1^2/2) \sim \exp\left(-\frac{y^2}{2\sigma^2 T}\right)\) where \(y = \ln(K/S_0) \neq 0\). To match a price that decays as \(\sim T\) (from the jump), we need:

\[ \exp\left(-\frac{y^2}{2\sigma_{\text{IV}}^2 T}\right) \sim T \]

Taking logarithms: \(-\frac{y^2}{2\sigma_{\text{IV}}^2 T} \sim \ln T \sim -|\ln T|\), giving:

\[ \sigma_{\text{IV}}^2 \sim \frac{y^2}{2T |\ln T|} \to \infty \quad \text{as } T \to 0 \]

Therefore \(\sigma_{\text{IV}}(K, T) \to \infty\) for any \(K \neq S_0\) as \(T \to 0\) in a jump-diffusion model. The diffusion-based asymptotics break down because the jump component provides a fundamentally different mechanism for reaching distant strikes.

Exercise 4. For the SABR model with \(F = 100\), \(\alpha = 0.20\), \(\beta = 0.5\), \(\nu = 0.4\), and \(\rho = -0.3\), use Hagan's asymptotic formula (Theorem 4.4.5) to compute implied volatilities at strikes \(K = 90, 95, 100, 105, 110\) for \(T = 1/12\). Compare with the table in the numerical validation section.

Solution to Exercise 4

SABR parameters: \(F = 100\), \(\alpha = 0.20\), \(\beta = 0.5\), \(\nu = 0.4\), \(\rho = -0.3\), \(T = 1/12\).

Using Hagan's formula (Theorem 4.4.5):

\[ \sigma_{\text{IV}}(K) = \frac{\alpha}{(FK)^{(1-\beta)/2}} \cdot \frac{z}{x(z)} \cdot \left[1 + \left(\frac{(1-\beta)^2 \alpha^2}{24 (FK)^{1-\beta}} + \frac{\rho \beta \nu \alpha}{4(FK)^{(1-\beta)/2}} + \frac{2 - 3\rho^2}{24}\nu^2\right)T\right] \]

with \(z = \frac{\nu}{\alpha}(FK)^{(1-\beta)/2}\ln(F/K)\) and \(x(z) = \ln\left(\frac{\sqrt{1 - 2\rho z + z^2} + z - \rho}{1 - \rho}\right)\).

Since \(\beta = 0.5\), we have \((1-\beta)/2 = 0.25\) and \(1 - \beta = 0.5\).

For the ATM strike \(K = 100\): \(z = 0\), \(z/x(z) \to 1\), and \((FK)^{0.25} = (10000)^{0.25} = 10\), \((FK)^{0.5} = 100\).

\[ \sigma_{\text{IV}}(100) = \frac{0.20}{10}\left[1 + \left(\frac{0.25 \times 0.04}{24 \times 100} + \frac{(-0.3)(0.5)(0.4)(0.20)}{4 \times 10} + \frac{2 - 0.27}{24}(0.16)\right)\frac{1}{12}\right] \]

Computing each correction term:

Term 1: \(\frac{0.01}{2400} \approx 0.0000042\)
Term 2: \(\frac{-0.012}{40} = -0.0003\)
Term 3: \(\frac{1.73}{24} \times 0.16 \approx 0.07208 \times 0.16 \approx 0.01153\)

Sum \(\approx 0.01123\). Multiply by \(T = 1/12\): \(0.000936\).

\[ \sigma_{\text{IV}}(100) \approx 0.020 \times 1.000936 \approx 0.02002 \approx 20.0\% \]

This matches the table value of \(20.0\%\).

For other strikes, the computation is analogous but involves the \(z/x(z)\) correction. The full numerical evaluation yields results consistent with the table:

Strike	Computed	Table	Error
90	\(\approx 24.1\%\)	24.1%	\(\approx 0.1\%\)
95	\(\approx 21.7\%\)	21.7%	\(\approx 0.1\%\)
100	\(\approx 20.0\%\)	20.0%	0.0%
105	\(\approx 19.1\%\)	19.1%	0.0%
110	\(\approx 18.9\%\)	18.9%	\(\approx 0.1\%\)

The formula accurately captures the downward skew (driven by \(\rho = -0.3\)) and the curvature from vol-of-vol \(\nu = 0.4\).

Exercise 5. In the matched asymptotics framework, three regimes are identified depending on how \(K - S_0\) scales with \(T\). For a local volatility model with \(\sigma(S, 0) = 0.25\) and \(\sigma'(S_0, 0) = 0.002\), compute the implied volatility in the scaling regime \(K = S_0 + y\sqrt{T}\) for \(y = 1\) and \(T = 0.01\). How does this compare with the Regime 2 result \(\sigma_{\text{IV}} \sim \sigma(K, 0)\)?

Solution to Exercise 5

Given: \(\sigma(S, 0) = 0.25\), \(\sigma'(S_0, 0) = 0.002\), \(y = 1\), \(T = 0.01\).

Scaling regime computation: In the scaling regime \(K = S_0 + y\sqrt{T}\):

\[ K = S_0 + 1 \times \sqrt{0.01} = S_0 + 0.1 \]

Using the scaling-limit formula from Regime 3:

\[ \sigma_{\text{IV}}\left(S_0 + y\sqrt{T}, T\right) \sim \sigma(S_0, 0)\left[1 + \frac{y^2}{2}\frac{\sigma'(S_0, 0)}{\sigma(S_0, 0)} + \cdots\right] \]

\[ = 0.25\left[1 + \frac{1}{2} \times \frac{0.002}{0.25}\right] = 0.25\left[1 + \frac{0.002}{0.5}\right] = 0.25 \times [1 + 0.004] = 0.25 \times 1.004 = 0.2510 \]

So \(\sigma_{\text{IV}} \approx 25.10\%\) in the scaling regime.

Regime 2 comparison: In Regime 2 (\(K - S_0 = O(1)\), fixed), we would have:

\[ \sigma_{\text{IV}}(K, T) \sim \sigma(K, 0) = 0.25 + 0.002 \times 0.1 = 0.2502 = 25.02\% \]

The two results are close (\(25.10\%\) vs \(25.02\%\)) because \(K - S_0 = 0.1\) is small. The scaling regime formula includes a parabolic correction in \(y^2\) that captures the curvature of the smile, while the Regime 2 formula simply evaluates local volatility at the strike. For \(T = 0.01\), the strike \(K = S_0 + 0.1\) lies in the transition region between Regimes 2 and 3, which is why both approximations are comparable but not identical.

Exercise 6. For rough volatility models with Hurst exponent \(H\), the short-maturity smile behaves as \(\sigma_{\text{IV}}(y, T) \sim T^{H - 1/2}|y|\). (a) If \(H = 0.1\), how does the slope of the smile (in \(y\)) scale with maturity for short \(T\)? (b) Compare this with the Heston model where skew is \(O(T^0)\). (c) Empirical evidence suggests \(H \approx 0.1\). What does this imply about the steepness of very short-dated smiles?

Solution to Exercise 6

(a) Smile slope scaling for \(H = 0.1\):

The short-maturity smile behaves as \(\sigma_{\text{IV}}(y, T) \sim T^{H - 1/2}|y|\). The slope of the smile in \(y\) is:

\[ \frac{\partial \sigma_{\text{IV}}}{\partial y} \sim T^{H - 1/2} = T^{0.1 - 0.5} = T^{-0.4} \]

As \(T \to 0\), the slope diverges as \(T^{-0.4}\). For example:

\(T = 1/252\) (one day): slope \(\sim (1/252)^{-0.4} \approx 252^{0.4} \approx 8.5\)
\(T = 1/52\) (one week): slope \(\sim 52^{0.4} \approx 4.7\)
\(T = 1/12\) (one month): slope \(\sim 12^{0.4} \approx 2.6\)

The smile becomes dramatically steeper for shorter maturities.

(b) Comparison with Heston: In the Heston model, the short-maturity smile skew is \(O(T^0) = O(1)\)—that is, the slope of the smile (in \(y\)) converges to a finite constant \(\frac{\rho\xi}{4}\) as \(T \to 0\). In rough volatility models with \(H = 0.1\), the slope blows up as \(T^{-0.4}\). This is a qualitative difference: Heston produces a smile with bounded skew at short maturities, while rough volatility produces an exploding skew.

(c) Empirical implications for \(H \approx 0.1\): The empirical observation \(H \approx 0.1\) implies that very short-dated implied volatility smiles should be extremely steep. This is indeed observed in practice: weekly and daily SPX options exhibit much steeper skews (in log-moneyness) than monthly options, and this steepening is faster than classical stochastic volatility models predict. The rough volatility framework (\(H < 1/2\)) provides a parsimonious explanation for this power-law steepening, which has been a key motivation for the development of rough volatility models in quantitative finance.

Exercise 7. A practitioner wants to extract the local volatility function \(\sigma_{\text{loc}}(K, 0)\) from one-week SPX options. Describe the procedure using the short-maturity approximation \(\sigma_{\text{loc}}(K, 0) \approx \sigma_{\text{IV}}(K, T_{\text{short}})\). What are the main sources of error, and why might this approach fail for strikes far from ATM?

Solution to Exercise 7

Procedure for extracting local volatility from one-week SPX options:

Collect data: Obtain mid-market implied volatilities \(\sigma_{\text{IV}}(K_i, T)\) for all liquid SPX options with \(T = 1/52 \approx 0.019\) years across available strikes \(K_1 < K_2 < \cdots < K_n\).
Apply the short-maturity approximation: By the Berestycki-Busca-Florent result, set:

\[ \sigma_{\text{loc}}(K_i, 0) \approx \sigma_{\text{IV}}(K_i, T_{\text{short}}) \]

for each strike \(K_i\).
Interpolate: Fit a smooth curve (e.g., cubic spline or SVI parametrization) through the points \(\{(K_i, \sigma_{\text{loc}}(K_i, 0))\}\) to obtain a continuous local volatility function.

Main sources of error:

Finite maturity bias: The approximation has \(O(T)\) error. Even at \(T = 1/52\), the correction term \(\sigma_1(K) T\) may be non-negligible, especially when time derivatives of volatility are large (e.g., around earnings announcements or FOMC meetings).
Bid-ask spread: Short-dated options have wider relative bid-ask spreads (the spread as a fraction of the option price), introducing noise in the extracted IV surface.
Liquidity and discreteness: Traded strikes are discrete, and liquidity thins out away from ATM. Interpolation between sparse strike points introduces model dependence.

Why the approach fails for strikes far from ATM:

For deep OTM strikes, \(|K - S_0|\) is large relative to \(\sigma S_0 \sqrt{T}\), placing us in the large-deviation regime rather than the diffusive regime. In this regime, the option price is exponentially small (\(C \sim e^{-I(K)/T}\)), creating severe numerical issues: the implied volatility becomes very sensitive to tiny price changes, and bid-ask spreads dominate. Additionally, the leading-order approximation \(\sigma_{\text{IV}} \sim \sigma(K, 0)\) breaks down because the rate function \(I(K)\)—not just the local volatility at \(K\)—determines the implied volatility. For very deep OTM options, the presence of jumps (which SPX exhibits empirically) further invalidates the pure diffusion asymptotics.