Wing Asymptotics and Moment Constraints¶

Introduction¶

The wings of the implied volatility smile refer to the behavior of $\sigma_{\text{IV}}(K, T)$ as the strike $K$ moves far from the current spot $S_0$ or forward $F$. Understanding wing asymptotics is crucial for: - Ensuring arbitrage-free extrapolation beyond traded strikes - Characterizing tail risk and extreme events - Constraining the moments of the risk-neutral distribution - Pricing deep out-of-the-money options and variance swaps

This section develops the complete theory of wing asymptotics and establishes fundamental connections to moment constraints.

Lee's Moment Formula¶

1. Statement of the Theorem¶

Theorem 4.4.7 (Lee, 2004)
Let $q(S_T)$ be the risk-neutral probability density at maturity $T$. Define the growth rate of total implied variance in the wings:

\[ p_+ := \lim_{y \to +\infty} \frac{\sigma_{\text{IV}}^2(y, T) \cdot T}{y} \]

\[ p_- := \lim_{y \to -\infty} \frac{\sigma_{\text{IV}}^2(y, T) \cdot T}{|y|} \]

where $y = \ln(K/F)$ is log-moneyness.

Then the maximum finite moment of the distribution is characterized by:

\[ p_{\pm} = \frac{2}{m_{\pm}} \]

where $m_+$ is the largest $p$ such that $\mathbb{E}^\mathbb{Q}[S_T^p] < \infty$ (right tail), and $m_-$ is the largest $p$ such that $\mathbb{E}^\mathbb{Q}[S_T^{-p}] < \infty$ (left tail).

Interpretation: The slope of the implied variance in the wings is inversely proportional to the maximum finite moment.

2. Proof Sketch¶

The call option price for large strikes admits the tail expansion:

\[ C(K, T) \sim e^{-rT} \mathbb{E}^\mathbb{Q}[(S_T - K)^+] \sim e^{-rT} \int_K^\infty (S - K) q(S) dS \]

For large $K$, if the density tail behaves as $q(S) \sim S^{-\alpha}$, then:

\[ C(K, T) \sim K^{1 - \alpha} \]

Using the Black-Scholes formula approximation for deep OTM:

\[ C(K, T) \approx e^{-rT} S_0 e^{-\frac{1}{2}d_1^2} \sim \exp\left(-\frac{y^2}{2\sigma_{\text{IV}}^2 T}\right) \]

Matching the power law decay $K^{1-\alpha}$ with the Gaussian tail requires:

\[ \sigma_{\text{IV}}^2(y, T) T \sim \frac{2y}{p} \]

where $p = \alpha - 1$ is the moment exponent. □

3. Special Cases¶

Finite variance ($p_+ = p_- = 2$):

\[ \lim_{|y| \to \infty} \frac{\sigma_{\text{IV}}^2(y, T) T}{|y|} = 2 \]

The wings grow exactly linearly in log-moneyness.

Infinite variance ($p_+ < 2$ or $p_- < 2$):

Wings grow slower than linear—flatter smile indicates heavier tails and infinite variance.

Finite fourth moment ($p_+ = p_- = 4$):

\[ \lim_{|y| \to \infty} \frac{\sigma_{\text{IV}}^2(y, T) T}{|y|} = \frac{1}{2} \]

Very steep wings, thin tails.

Right Wing: Large-Strike Asymptotics¶

1. Power Law Tails¶

Assume the risk-neutral density has a power law right tail:

\[ q(S) \sim C_+ S^{-\alpha} \quad \text{as } S \to \infty \]

for some $\alpha > 1$ (to ensure normalization).

Implications: - Finite $p$-th moment: $\mathbb{E}[S_T^p] < \infty$ iff $p < \alpha - 1$ - Maximum finite moment: $m_+ = \alpha - 1$

2. Implied Volatility Asymptotics¶

Theorem 4.4.8 (Right Wing Asymptotics)
If $q(S) \sim C_+ S^{-\alpha}$ as $S \to \infty$, then:

\[ \sigma_{\text{IV}}^2(y, T) T \sim \frac{2y}{\alpha - 1} \quad \text{as } y \to +\infty \]

Equivalently:

\[ \sigma_{\text{IV}}(K, T) \sim \sqrt{\frac{2 \ln(K/F)}{T(\alpha - 1)}} \]

Proof: From Lee's formula with $m_+ = \alpha - 1$, we have $p_+ = \frac{2}{\alpha - 1}$. The result follows by definition of $p_+$. □

3. Exponential Tails¶

If instead the density has exponential decay:

\[ q(S) \sim C_+ e^{-\lambda S} \quad \text{as } S \to \infty \]

Implications: - All moments are finite: $\mathbb{E}[S_T^p] < \infty$ for all $p$ - Lee's formula: $p_+ = +\infty$

Wing behavior:

\[ \sigma_{\text{IV}}^2(y, T) T \gg y \quad \text{as } y \to +\infty \]

The implied variance grows faster than linear (steep wings).

Example (Gaussian tail):
For lognormal distribution:

\[ q(S) \sim \frac{1}{S\sigma\sqrt{2\pi T}} e^{-\frac{(\ln S - \mu)^2}{2\sigma^2 T}} \]

The IV wings grow as:

\[ \sigma_{\text{IV}}^2(y, T) T \sim 2|y| + \text{const} \]

(Linear growth with a different constant than power law.)

Left Wing: Small-Strike Asymptotics¶

1. Put-Call Symmetry¶

By put-call parity:

\[ P(K, T) = C(K, T) - S_0 e^{-qT} + K e^{-rT} \]

The implied volatility of a put equals that of a call:

\[ \sigma_{\text{IV}}^{\text{put}}(K, T) = \sigma_{\text{IV}}^{\text{call}}(K, T) \]

Thus, left wing analysis proceeds analogously using puts.

2. Small-Strike Behavior¶

Define the left wing slope:

\[ p_- = \lim_{y \to -\infty} \frac{\sigma_{\text{IV}}^2(y, T) T}{|y|} \]

Theorem 4.4.9 (Left Wing Asymptotics)
If the density tail behaves as:

\[ q(S) \sim C_- S^{\beta} \quad \text{as } S \to 0 \]

for some $\beta > -1$ (to ensure integrability), then:

\[ p_- = \frac{2}{\beta + 1} \]

Special case (Black-Scholes):
Lognormal density has $q(S) \sim S^{-1}$ as $S \to 0$, giving $\beta = -1$. However, this is boundary case—technically infinite $p_-$ due to exponential correction.

3. Absorbing Barrier at Zero¶

If there is probability mass at $S_T = 0$ (complete default):

\[ \mathbb{P}^\mathbb{Q}(S_T = 0) = p_0 > 0 \]

Consequence: The put price has a floor:

\[ P(K, T) \geq K e^{-rT} p_0 \]

This creates a kink in the left wing:

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

(Infinite IV needed to match the put price floor.)

Symmetric vs Asymmetric Wings¶

1. Symmetric Distribution¶

If the risk-neutral density is symmetric around $F$:

\[ q(F + x) = q(F - x) \quad \text{for all } x \]

Then:

\[ p_+ = p_- = 2 \]

Both wings have finite variance tails.

Markets: Foreign exchange options often exhibit near-symmetric wings.

2. Asymmetric Distribution (Skewed)¶

Equity markets: Typically exhibit:

\[ p_- < p_+ \quad (\text{or } p_- < 2 < p_+) \]

Interpretation: Left tail is heavier (fatter) than right tail, reflecting crash risk.

Example:
- $p_- = 1.5$ → Inverse cubic tail ($\alpha_- = 2.5$) - $p_+ = 2.0$ → Quadratic tail (finite variance)

Left wing flatter (higher IV) than right wing.

Connections to Variance Swaps¶

1. Variance Swap Pricing¶

The fair strike for a variance swap is:

\[ K_{\text{var}} = \frac{2 e^{rT}}{T} \left(\int_0^F \frac{P(K)}{K^2} dK + \int_F^\infty \frac{C(K)}{K^2} dK\right) \]

2. Wing Contribution¶

The integral is dominated by the wings:

\[ \int_F^\infty \frac{C(K)}{K^2} dK \sim \int_{y=0}^\infty C(Fe^y) e^{-2y} F dy \]

For large $y$, using the tail behavior $C(K) \sim K^{1 - \alpha}$:

\[ \int_{y_{\text{large}}}^\infty e^{y(1 - \alpha)} e^{-2y} dy \sim \int_{y_{\text{large}}}^\infty e^{-y(\alpha + 1)} dy \]

Convergence condition: $\alpha + 1 > 1$, i.e., $\alpha > 0$ (finite first moment).

Connection to Lee's formula:
If $\alpha = m_+ + 1 < 3$ (infinite variance), the variance swap integral diverges.

Practical implication: Variance swaps are not well-defined for distributions with infinite variance, requiring careful wing truncation.

Arbitrage Constraints on Wings¶

1. Minimum Wing Slope¶

For the density to be a valid probability measure:

Lower bound:

\[ p_{\pm} > 0 \]

Upper bound (finite first moment):

\[ p_{\pm} < \infty \quad \Rightarrow \quad \mathbb{E}[S_T] < \infty \]

The forward price $F = \mathbb{E}^\mathbb{Q}[S_T]$ is finite by no-arbitrage, requiring:

\[ p_+ \geq 2/m_+ \quad \text{with } m_+ \geq 1 \]

Thus:

\[ p_+ \leq 2 \]

Similarly for the left wing.

2. Calibration Constraints¶

When fitting parametric smile models (SVI, SSVI), the wing slopes must satisfy:

\[ 0 < p_- \leq 2, \quad 0 < p_+ \leq 2 \]

Violations indicate: - Mispriced options - Model over-fitting - Extrapolation error

Best practice: Explicitly constrain wing slopes during calibration.

Empirical Observations¶

1. Equity Index Wings¶

S&P 500 typical values: - $p_- \approx 1.0$ to $1.5$ (fat left tail, crash risk) - $p_+ \approx 2.0$ to $2.5$ (thin right tail)

Interpretation: Market prices put insurance heavily (left wing flat), while call side is closer to Gaussian.

2. FX Wings¶

Major currency pairs (EUR/USD, USD/JPY): - $p_- \approx 2.0$ - $p_+ \approx 2.0$

Interpretation: Symmetric finite-variance distribution, two-sided jump risk.

3. Commodity Wings¶

Crude oil: - $p_- \approx 1.5$ (supply disruption) - $p_+ \approx 1.8$ (demand collapse)

Natural gas: - Highly asymmetric depending on season and storage levels

Wing Approximations in Practice¶

1. Polynomial Extrapolation¶

For strikes beyond the last traded option, fit:

\[ \sigma_{\text{IV}}^2(y, T) T = a + b|y| + c y^2 + \cdots \]

Constraint: The linear coefficient $b$ should match $p_{\pm}$ from Lee's formula.

2. Power Law Tails (SVI)¶

The SVI parametrization:

\[ w(y) = a + b\left(\rho(y - m) + \sqrt{(y - m)^2 + \sigma^2}\right) \]

Wing slopes:

\[ p_+ = 1 + \rho, \quad p_- = 1 - \rho \]

Constraint: $\rho \in [-1, 1]$ ensures $0 < p_{\pm} < 2$ (finite variance).

Limitation: SVI cannot model infinite variance distributions ($p < 2$ arbitrary).

3. Rational Function Extrapolation¶

For more flexible wing behavior:

\[ w(y) \sim a|y| + b \log|y| + c \quad \text{as } |y| \to \infty \]

This allows for: - Leading linear term (controls moment) - Logarithmic correction (finer tail structure)

Moment-Constrained Calibration¶

1. Imposing Known Moments¶

If external information provides moment bounds:

\[ \mathbb{E}^\mathbb{Q}[S_T^2] = M_2, \quad \mathbb{E}^\mathbb{Q}[S_T^3] = M_3, \quad \ldots \]

Calibration strategy:

Use Lee's formula to determine $p_{\pm}$ from moment constraints
Fit smile model with wing slopes constrained to $p_{\pm}$
Verify that extracted density has correct moments

Example: If variance is known from variance swap:

\[ \mathbb{E}[S_T^2] = F^2 + \text{Var}(S_T) = F^2 + \sigma_{\text{var}}^2 F^2 T \]

This constrains the wings to decay at least as fast as $p_+ = p_- = 2$.

Extreme Wing Behavior: Pathological Cases¶

1. Flat Wings (p → 0)¶

If $\sigma_{\text{IV}}(y, T)$ is constant for large $|y|$:

\[ \sigma_{\text{IV}}(y) = \sigma_\infty \quad \text{for } |y| > y_{\text{large}} \]

Consequence (Lee): All moments are infinite.

Interpretation: Extremely heavy tails—unrealistic for most assets.

2. Steep Wings (p → ∞)¶

If IV grows faster than $\sqrt{|y|/T}$:

\[ \sigma_{\text{IV}}^2(y, T) T \sim |y|^\gamma \quad \text{with } \gamma > 1 \]

Consequence: Density has compact support (zero probability beyond some threshold).

Interpretation: Asset price is bounded—possible for commodities (price can't be negative, may have physical upper bound).

Relationship to Greeks¶

1. Wing Vega¶

The vega in the wings is:

\[ \mathcal{V}(K) = S_0 e^{-qT} \phi(d_1) \sqrt{T} \]

For large $K$, $d_1 \to -\infty$, so $\phi(d_1) \to 0$ exponentially.

Implication: Deep OTM options have very low vega—changes in wing IV have limited P&L impact unless positions are large.

2. Wing Delta¶

Deep OTM call delta:

\[ \Delta_{\text{call}} = e^{-qT} \Phi(d_1) \to 0 \quad \text{as } K \to \infty \]

Deep OTM put delta:

\[ \Delta_{\text{put}} = -e^{-qT} \Phi(-d_1) \to -e^{-qT} \quad \text{as } K \to 0 \]

Wing hedging: Minimal sensitivity to spot moves, but gamma risk remains.

Computational Aspects¶

1. Numerical Stability¶

Computing $\sigma_{\text{IV}}$ for deep OTM options:

Challenge: Option price $C(K) \approx 10^{-6}$ or smaller → numerical precision issues

Solution: - Use higher-precision arithmetic - Work in log-space: compute $\log C$ directly - Extrapolate from traded strikes using wing asymptotics

2. Wing Interpolation Algorithms¶

Algorithm:

Fit SVI or SSVI to liquid strikes (90-110% of spot)
Determine wing slopes $p_{\pm}$ from fit
Extend IV linearly in $\sqrt{|y|}$ for $|y| > y_{\text{liquid}}$:

$$ \sigma_{\text{IV}}^2(y, T) T = w_{\text{last}} + p_{\pm} (|y| - y_{\text{liquid}}) $$

Guarantee: Asymptotically correct behavior, arbitrage-free.

Summary¶

Wing asymptotics reveal:

1. Lee's moment formula:¶

\[ p_{\pm} = \frac{2}{m_{\pm}} \]

Wing slope inversely proportional to maximum finite moment.

2. Arbitrage constraints:¶

\[ 0 < p_- \leq 2, \quad 0 < p_+ \leq 2 \]

Finite variance requires wings grow at least linearly in $|y|$.

3. Asymptotic behavior:¶

Power law tail: $q(S) \sim S^{-\alpha} \Rightarrow \sigma_{\text{IV}}^2 T \sim \frac{2|y|}{\alpha - 1}$

Exponential tail: $q(S) \sim e^{-\lambda S} \Rightarrow \sigma_{\text{IV}}^2 T \gg |y|$ (steep)

4. Practical applications:¶

Extrapolation: Use power law wings beyond traded strikes
Calibration: Constrain wing slopes to ensure valid density
Variance swaps: Require finite variance ($p_{\pm} = 2$)
Moment inference: Extract tail behavior from observed wings

5. Empirical patterns:¶

Equity: Asymmetric ($p_- < p_+$), fat left tail
FX: Symmetric ($p_- \approx p_+ \approx 2$)
Commodities: Variable, depends on market structure

Wing asymptotics provide a rigorous framework for understanding tail risk, constraining models, and ensuring arbitrage-free pricing across the entire strike range.

Exercises¶

Exercise 1. Using Lee's moment formula $p_{\pm} = 2/m_{\pm}$, compute the wing slopes $p_+$ and $p_-$ for a risk-neutral distribution with maximum finite right moment $m_+ = 3$ and maximum finite left moment $m_- = 1.5$. Which wing is steeper? What does this say about the relative heaviness of the tails?

Solution to Exercise 1

Using Lee's moment formula $p_{\pm} = 2/m_{\pm}$:

Right wing: $m_+ = 3$, so

\[ p_+ = \frac{2}{3} \approx 0.667 \]

Left wing: $m_- = 1.5$, so

\[ p_- = \frac{2}{1.5} = \frac{4}{3} \approx 1.333 \]

Since $p_- > p_+$, the left wing is steeper (the implied variance $\sigma_{\text{IV}}^2 T$ grows faster with $|y|$ on the left side). By Lee's formula, a steeper wing corresponds to a smaller maximum finite moment, meaning the corresponding tail is lighter. Indeed, $m_- = 1.5 < m_+ = 3$, so fewer left-tail moments are finite.

Equivalently, the right tail is heavier than the left tail: the distribution has a fat right tail (large upward moves are more probable), while the left tail decays faster. This is the opposite of the typical equity pattern, where the left tail is fatter due to crash risk.

Exercise 2. Consider a risk-neutral density with power law right tail $q(S) \sim C_+ S^{-4}$ as $S \to \infty$. (a) Determine the maximum finite moment $m_+$. (b) Compute the right wing slope $p_+$. (c) Write the asymptotic formula for $\sigma_{\text{IV}}^2(y, T) T$ as $y \to +\infty$. (d) Is the variance of $S_T$ finite?

Solution to Exercise 2

Given: $q(S) \sim C_+ S^{-4}$ as $S \to \infty$, so $\alpha = 4$.

(a) Maximum finite moment:

\[ m_+ = \alpha - 1 = 4 - 1 = 3 \]

This means $\mathbb{E}[S_T^p] < \infty$ for all $p < 3$, and $\mathbb{E}[S_T^3] = \infty$.

(b) Right wing slope:

\[ p_+ = \frac{2}{m_+} = \frac{2}{3} \approx 0.667 \]

(c) Asymptotic formula: By Theorem 4.4.8:

\[ \sigma_{\text{IV}}^2(y, T) T \sim \frac{2y}{\alpha - 1} = \frac{2y}{3} \quad \text{as } y \to +\infty \]

Equivalently:

\[ \sigma_{\text{IV}}(K, T) \sim \sqrt{\frac{2\ln(K/F)}{3T}} \]

(d) Variance of $S_T$: The variance requires $\mathbb{E}[S_T^2] < \infty$. Since $m_+ = 3 > 2$, the second moment is finite, so yes, the variance of $S_T$ is finite. In fact, all moments up to (but not including) the third are finite.

Exercise 3. The SVI parametrization has wing slopes $p_+ = 1 + \rho$ and $p_- = 1 - \rho$. (a) For $\rho = -0.4$, compute both wing slopes and verify $0 < p_{\pm} < 2$. (b) What value of $\rho$ produces symmetric wings? (c) Explain why $|\rho| > 1$ would violate arbitrage constraints.

Solution to Exercise 3

SVI wing slopes: $p_+ = 1 + \rho$ and $p_- = 1 - \rho$.

(a) For $\rho = -0.4$:

\[ p_+ = 1 + (-0.4) = 0.6 \]

\[ p_- = 1 - (-0.4) = 1.4 \]

Verification: $0 < p_+ = 0.6 < 2$ and $0 < p_- = 1.4 < 2$. Both satisfy the arbitrage constraint.

Note that $p_- > p_+$, so the left wing is steeper than the right wing. This means the right tail is heavier (consistent with $\rho < 0$ producing an equity-like skew where the left wing IV is higher but flatter in total variance space).

(b) Symmetric wings require $p_+ = p_-$:

\[ 1 + \rho = 1 - \rho \implies 2\rho = 0 \implies \rho = 0 \]

When $\rho = 0$, both wing slopes equal $1$, giving symmetric behavior.

(c) If $|\rho| > 1$, say $\rho = 1.2$, then $p_+ = 2.2 > 2$ and $p_- = -0.2 < 0$. A negative $p_-$ would mean implied variance decreases with $|y|$ in the left wing, which is impossible for a valid probability density (it would imply that deep OTM puts are underpriced relative to any non-degenerate distribution). Similarly, $p_+ > 2$ would imply the first moment $\mathbb{E}[S_T]$ is infinite, violating the no-arbitrage condition $\mathbb{E}^{\mathbb{Q}}[S_T] = F < \infty$. Therefore $|\rho| > 1$ leads to arbitrage violations.

Exercise 4. An equity index has empirical wing slopes $p_- \approx 1.2$ and $p_+ \approx 2.0$. Using Lee's formula, determine the maximum finite moments $m_-$ and $m_+$. Explain the economic interpretation: why does the left tail exhibit a lower maximum finite moment?

Solution to Exercise 4

Using Lee's formula $p_{\pm} = 2/m_{\pm}$ to find the maximum finite moments:

Left tail: $p_- = 1.2$, so

\[ m_- = \frac{2}{p_-} = \frac{2}{1.2} = \frac{5}{3} \approx 1.667 \]

Right tail: $p_+ = 2.0$, so

\[ m_+ = \frac{2}{p_+} = \frac{2}{2.0} = 1.0 \]

Interpretation: The left tail has maximum finite moment $m_- \approx 1.667$, meaning $\mathbb{E}[S_T^{-p}] < \infty$ for $p < 1.667$. The right tail has $m_+ = 1.0$, meaning $\mathbb{E}[S_T^p] < \infty$ for $p < 1$ (borderline — the first moment $\mathbb{E}[S_T]$ is just barely finite).

Economic interpretation: The lower maximum finite moment on the left ($m_- < m_+$ in the wing slope sense, but actually $m_- > m_+$ in the moment exponent sense here — let us clarify). With $p_- = 1.2 < p_+ = 2.0$, the left wing is flatter in total variance, meaning it requires less implied variance to grow. This corresponds to a fatter left tail: the risk-neutral distribution assigns substantial probability to large downward moves (market crashes). This is characteristic of equity indices, where investors aggressively bid up OTM put prices as crash protection, making the left tail of the risk-neutral distribution much heavier than the right tail. The market is pricing significant downside risk relative to upside potential.

Exercise 5. Suppose a variance swap has fair strike $K_{\text{var}} = 0.04$ (corresponding to $\sigma_{\text{var}} = 20\%$). The variance swap formula involves

\[ K_{\text{var}} = \frac{2 e^{rT}}{T} \left(\int_0^F \frac{P(K)}{K^2} dK + \int_F^\infty \frac{C(K)}{K^2} dK\right) \]

Explain why this integral diverges if the implied volatility wings are flat (constant $\sigma_{\text{IV}}$ for large $|y|$), and connect this to the condition that all moments must be infinite when $p_{\pm} = 0$.

Solution to Exercise 5

If the implied volatility wings are flat ($\sigma_{\text{IV}}(y, T) = \sigma_\infty$ for large $|y|$), then the total implied variance is:

\[ w(y) = \sigma_{\text{IV}}^2(y, T) \cdot T = \sigma_\infty^2 T = \text{constant} \]

for large $|y|$, which means $p_{\pm} = \lim_{|y| \to \infty} w(y)/|y| = 0$.

By Lee's formula, $p_{\pm} = 2/m_{\pm}$, so $p_{\pm} = 0$ implies $m_{\pm} = \infty$. But $m_+ = \infty$ means that $\mathbb{E}[S_T^p] = \infty$ for all $p > 0$ — even the first moment is infinite. This contradicts the no-arbitrage requirement $\mathbb{E}^{\mathbb{Q}}[S_T] = F < \infty$.

Why the variance swap integral diverges: Consider the right wing contribution:

\[ \int_F^\infty \frac{C(K)}{K^2} dK \]

For large $K$, the Black-Scholes call price with constant IV $\sigma_\infty$ behaves as:

\[ C(K) \approx S_0 \Phi(d_1) - K e^{-rT}\Phi(d_2) \sim S_0 \frac{\phi(d_1)}{|d_1|} \]

where $d_1 = \frac{-\ln(K/F) + \sigma_\infty^2 T/2}{\sigma_\infty \sqrt{T}}$. For large $K$, $d_1 \to -\infty$ and:

\[ C(K) \sim \frac{S_0}{\sqrt{2\pi} |d_1|} \exp\left(-\frac{(\ln(K/F))^2}{2\sigma_\infty^2 T}\right) \]

The ratio $C(K)/K^2$ still decays, but only as a Gaussian in $\ln K$, not as a power law. Converting the integral to log-moneyness $y = \ln(K/F)$:

\[ \int_{y_0}^\infty \frac{C(Fe^y)}{F^2 e^{2y}} F e^y dy \sim \int_{y_0}^\infty e^{-y} \exp\left(-\frac{y^2}{2\sigma_\infty^2 T}\right) \frac{dy}{y} \]

This integral converges for any fixed $\sigma_\infty$, but the point is that the flat-wing model implies unrealistic tail behavior. Specifically, the density implied by flat wings has all moments infinite, meaning the variance $\text{Var}(S_T) = \mathbb{E}[S_T^2] - F^2 = \infty$. Since the variance swap strike equals $\text{Var}(\ln(S_T/F))/T$ (which relates to the full integral), and the underlying distribution has pathological tails, the replicating portfolio for the variance swap would require infinite notional in the wings — making the variance swap ill-defined in practice.

Exercise 6. Consider an absorbing barrier model where $\mathbb{P}^{\mathbb{Q}}(S_T = 0) = p_0 = 0.02$. (a) Compute the minimum put price $P(K, T) \geq K e^{-rT} p_0$ for $K = 50$ with $r = 0.05$ and $T = 1$. (b) Explain why the left wing implied volatility must satisfy $\sigma_{\text{IV}}(K, T) \to \infty$ as $K \to 0$. (c) In which asset classes might this model be realistic?

Solution to Exercise 6

(a) Minimum put price:

\[ P(K, T) \geq K e^{-rT} p_0 = 50 \times e^{-0.05 \times 1} \times 0.02 = 50 \times 0.9512 \times 0.02 = 0.9512 \]

So the put price is at least $\$0.95$ regardless of how far OTM it is.

(b) Why $\sigma_{\text{IV}} \to \infty$ as $K \to 0$: As $K \to 0$, a standard (no-default) model would give $P(K, T) \to 0$ exponentially fast. But with positive default probability, $P(K, T) \geq K e^{-rT} p_0 > 0$ for all $K > 0$. In the Black-Scholes formula, matching a non-vanishing put price at small $K$ requires:

\[ P_{\text{BS}}(K, T, \sigma) \geq K e^{-rT} p_0 \]

For small $K$, the Black-Scholes put price is approximately $K e^{-rT}\Phi(-d_2)$ where $d_2 = \frac{\ln(S_0/K) + (r - \sigma^2/2)T}{\sigma\sqrt{T}}$. As $K \to 0$, $d_2 \to +\infty$ and $\Phi(-d_2) \to 0$ for any finite $\sigma$. To keep $\Phi(-d_2) \geq p_0 > 0$, we need $d_2$ to remain bounded, which requires $\sigma \to \infty$ to absorb the growing $\ln(S_0/K)$ term. Therefore $\sigma_{\text{IV}}(K, T) \to \infty$ as $K \to 0$.

(c) Realistic asset classes: An absorbing barrier at zero is realistic for:

Corporate bonds/credit: Individual firms can default, with recovery at or near zero
Sovereign credit: Countries can default (e.g., Argentina, Greece)
Distressed equities: Companies near bankruptcy have non-negligible probability of stock price going to zero
Cryptocurrency: Some tokens have gone to zero
Structured products: Tranched credit products (e.g., equity tranches of CDOs) can be completely wiped out

Exercise 7. A parametric smile model produces wing behavior $\sigma_{\text{IV}}^2(y, T) T = 0.05 + 0.8|y| + 0.1 y^2$ for large $|y|$. (a) Identify the leading-order behavior and determine the effective wing slope. (b) Does this satisfy the arbitrage constraint $0 < p_{\pm} \leq 2$? (c) Compute the implied maximum finite moments $m_{\pm}$ using Lee's formula.

Solution to Exercise 7

(a) Leading-order behavior and effective wing slope:

The total implied variance is $w(y) = \sigma_{\text{IV}}^2(y, T) T = 0.05 + 0.8|y| + 0.1 y^2$. For large $|y|$, the quadratic term $0.1 y^2$ dominates, so:

\[ \frac{w(y)}{|y|} = \frac{0.05 + 0.8|y| + 0.1 y^2}{|y|} = \frac{0.05}{|y|} + 0.8 + 0.1|y| \to \infty \]

as $|y| \to \infty$. The effective wing slope in Lee's sense is:

\[ p_{\pm} = \lim_{|y| \to \infty} \frac{w(y)}{|y|} = +\infty \]

The leading-order behavior is quadratic ($w(y) \sim 0.1 y^2$), which grows faster than linear.

(b) Arbitrage constraint check: The constraint requires $0 < p_{\pm} \leq 2$. Here $p_{\pm} = +\infty > 2$, so this violates the arbitrage constraint. A wing slope exceeding $2$ implies that the first moment $\mathbb{E}[S_T]$ would be infinite (the forward price cannot be finite), which contradicts no-arbitrage.

More precisely, $p_{\pm} = \infty$ means the distribution has compact support (all moments are finite). While compact support is not itself an arbitrage violation, the super-linear growth of $w(y)$ creates issues: the implied density may become negative for large strikes, which is a butterfly arbitrage violation. In practice, any parametric model with faster-than-linear growth of total variance in the wings must be carefully checked for arbitrage.

(c) Maximum finite moments: Since $p_{\pm} = \infty$, Lee's formula gives:

\[ m_{\pm} = \frac{2}{p_{\pm}} = \frac{2}{\infty} = 0 \]

This is paradoxical: it would suggest no finite moments at all. The resolution is that Lee's formula $p_{\pm} = 2/m_{\pm}$ applies to the linear growth rate of total variance. When $w(y)$ grows super-linearly (quadratically here), the formula's limit is infinite, and the correct interpretation is that all moments $\mathbb{E}[S_T^p]$ are finite for all $p > 0$. The distribution has lighter tails than any power law — effectively compact support or sub-exponential decay. The model should be modified to cap the wing growth at a linear rate to ensure consistency with Lee's moment constraints.