Smile Dynamics and Hedging¶

Introduction¶

Smile dynamics describe how the implied volatility surface evolves as market conditions change. Understanding these dynamics is essential for hedging volatility risk beyond simple vega neutralization. A static view of the smile—treating implied volatility as a fixed input—leads to systematic hedging errors and unexpected P&L. This section develops the theory and practice of dynamic smile hedging.

Static vs. Dynamic Smiles¶

1. The Static Smile Assumption¶

A static smile assumes the implied volatility surface is fixed in time except for deterministic decay:

\[ \sigma_{\text{IV}}(K, T; t) = \sigma_{\text{IV}}(K, T-t; 0) \]

Under this assumption: - The smile shape remains constant - Only maturity decreases as time passes - No dependence on spot movements or volatility shocks

Implications: - Delta hedging uses Black-Scholes delta - Vega hedging is straightforward - P&L attribution is simple

Limitation: Real smiles move in complex ways, violating this assumption.

2. The Dynamic Smile Reality¶

A dynamic smile evolves with: - Spot movements: Smile shifts, steepens, or flattens when spot moves - Volatility regime shifts: Overall level changes with market conditions - Term structure evolution: Different maturities respond differently - Skew dynamics: The slope of the smile changes over time

Consequence: Delta-hedged options exhibit volatility-driven P&L even when the hedger is delta-neutral.

3. Sources of Dynamic Smile Effects¶

Source	Description	Impact on Hedging
Spot-vol correlation	Negative correlation in equities	Vanna effects
Volatility mean reversion	Vol converges to long-run level	Term structure changes
Jumps	Discrete large moves	Skew steepening
Stochastic vol-of-vol	Uncertainty in vol process	Volga effects
Event risk	Known future events	Humps in term structure

Mathematical Framework for Smile Dynamics¶

1. The Smile as a Function of State Variables¶

Model the implied volatility surface as a function of state variables:

\[ \sigma_{\text{IV}}(K, T; S_t, v_t, \ldots) = f(K, T, S_t, v_t, \ldots) \]

where $v_t$ may represent instantaneous variance, a volatility factor, or other state variables.

Taylor expansion:

\[ d\sigma_{\text{IV}} = \frac{\partial \sigma}{\partial S} dS + \frac{\partial \sigma}{\partial v} dv + \frac{\partial \sigma}{\partial t} dt + \frac{1}{2}\frac{\partial^2 \sigma}{\partial S^2}(dS)^2 + \ldots \]

2. Smile Sensitivities¶

Define the following smile sensitivities:

Spot sensitivity (at fixed strike):

\[ \Sigma_S := \frac{\partial \sigma_{\text{IV}}(K)}{\partial S} \]

Variance sensitivity:

\[ \Sigma_v := \frac{\partial \sigma_{\text{IV}}(K)}{\partial v} \]

Time decay of smile:

\[ \Sigma_t := \frac{\partial \sigma_{\text{IV}}(K)}{\partial t} \]

3. Stochastic Volatility Perspective¶

In a stochastic volatility model:

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

The implied volatility depends on both $S_t$ and $v_t$:

\[ \sigma_{\text{IV}}(K, T; S_t, v_t) \]

The dynamics are:

\[ d\sigma_{\text{IV}} = \Sigma_S dS + \Sigma_v dv + \Sigma_t dt + \text{higher order} \]

Smile Dynamics and P&L Attribution¶

1. Option P&L Decomposition¶

The total P&L of a delta-hedged option position can be decomposed:

\[ \text{P\&L} = \underbrace{\Theta \cdot dt}_{\text{time decay}} + \underbrace{\frac{1}{2}\Gamma (dS)^2}_{\text{gamma P\&L}} + \underbrace{\mathcal{V} \cdot d\sigma_{\text{ATM}}}_{\text{parallel vol}} + \underbrace{\text{Smile effects}}_{\text{residual}} \]

The "smile effects" capture: - Changes in skew - Changes in curvature - Non-parallel volatility moves

2. Detailed P&L Attribution¶

Expanding the vega term:

\[ \mathcal{V} \cdot d\sigma = \mathcal{V} \cdot \left(\Sigma_S dS + \Sigma_v dv + \Sigma_t dt\right) \]

Cross-gamma (Vanna P&L):

\[ \text{P\&L}_{\text{vanna}} = \mathcal{V} \cdot \Sigma_S \cdot dS \]

This captures the interaction between spot moves and volatility changes.

Pure volatility P&L:

\[ \text{P\&L}_{\text{vol}} = \mathcal{V} \cdot \Sigma_v \cdot dv \]

This captures changes in the underlying volatility state.

3. Second-Order Attribution¶

Including second-order terms:

\[ \text{P\&L} = \Delta \cdot dS + \Theta \cdot dt + \frac{1}{2}\Gamma (dS)^2 + \mathcal{V} \cdot d\sigma + \text{Vanna} \cdot dS \cdot d\sigma + \frac{1}{2}\text{Volga} \cdot (d\sigma)^2 \]

Term	Greek	Source
$\Delta \cdot dS$	Delta	Spot move
$\Theta \cdot dt$	Theta	Time decay
$\frac{1}{2}\Gamma (dS)^2$	Gamma	Spot convexity
$\mathcal{V} \cdot d\sigma$	Vega	Vol level change
$\text{Vanna} \cdot dS \cdot d\sigma$	Vanna	Spot-vol cross
$\frac{1}{2}\text{Volga} \cdot (d\sigma)^2$	Volga	Vol convexity

Model-Based Smile Dynamics¶

1. Local Volatility Dynamics¶

In local volatility models, the smile dynamics are fully determined by the local volatility surface:

\[ \sigma_{\text{loc}}(S, t) = \sigma_{\text{loc}}(S, t) \quad \text{(fixed)} \]

Key property: The local vol model produces sticky strike behavior: - IV at each strike is fixed - When spot moves, the option "visits" different parts of the local vol surface - No smile-related P&L at fixed strike

Forward smile: The local vol model implies a specific forward smile that typically flattens over time:

\[ \sigma_{\text{fwd}}(K, T_1, T_2) \to \text{flat as } T_1 \to \infty \]

This is inconsistent with persistent empirical skew, a major shortcoming of local vol.

2. Stochastic Volatility Dynamics (Heston)¶

The Heston model produces richer smile dynamics:

ATM volatility:

\[ \sigma_{\text{ATM}}^2 \approx v_t + \text{corrections} \]

So ATM volatility moves with the variance process $v_t$.

Skew:

\[ \text{Skew} \propto \rho \cdot \xi \]

The spot-vol correlation $\rho$ generates skew, and vol-of-vol $\xi$ affects its magnitude.

Smile dynamics: - When $v_t$ increases, the entire smile shifts up - When $S_t$ decreases (with $\rho < 0$), $v_t$ tends to increase, steepening the skew - This creates leverage-like dynamics

3. SABR Dynamics¶

The SABR model:

\[ \begin{align} dF_t &= \alpha_t F_t^\beta dW_t^F \\ d\alpha_t &= \nu \alpha_t dW_t^\alpha \\ d\langle W^F, W^\alpha \rangle &= \rho dt \end{align} \]

Backbone parameter $\beta$: - $\beta = 1$: Lognormal dynamics, sticky strike behavior - $\beta = 0$: Normal dynamics, sticky delta behavior - $0 < \beta < 1$: Intermediate

SABR smile dynamics: - The parameter $\alpha_t$ (ATM vol) evolves stochastically - Skew is controlled by $\rho$ and $\nu$ - The model can match both smile shape and basic dynamics

4. Bergomi's Framework¶

Bergomi models the forward variance curve directly:

\[ \xi_t^T = \mathbb{E}_t[\sigma_T^2] \]

The dynamics:

\[ d\xi_t^T = \xi_t^T \cdot \omega(T-t) \cdot dZ_t \]

Key insight: By specifying how forward variances evolve, Bergomi's model can match: - Realistic smile dynamics - Forward smile behavior - Term structure of skew

This framework provides a unified approach to smile dynamics and is widely used in equity derivatives.

Hedging Implications¶

1. Delta Hedging Under Smile Dynamics¶

The "true" delta depends on smile dynamics:

\[ \Delta_{\text{true}} = \Delta_{\text{BS}} + \mathcal{V} \cdot \Sigma_S \]

where $\Sigma_S = \frac{\partial \sigma_{\text{IV}}(K)}{\partial S}$ is the smile sensitivity to spot.

Sticky strike: $\Sigma_S = 0$, so $\Delta_{\text{true}} = \Delta_{\text{BS}}$

Sticky delta: $\Sigma_S \neq 0$, requiring adjustment

Model-implied: Use the specific model's prediction for $\Sigma_S$

2. Vega Hedging Across Strikes¶

A single ATM option hedge provides parallel vega protection but not skew protection.

Skew risk: If the portfolio has different vega exposures at different strikes, a change in skew creates P&L:

\[ \text{P\&L}_{\text{skew}} = \sum_i \mathcal{V}_i \cdot \Delta\sigma(K_i) - \mathcal{V}_{\text{hedge}} \cdot \Delta\sigma_{\text{ATM}} \]

Skew hedge: Use risk reversals (long OTM put, short OTM call) to hedge skew exposure.

3. Vega Hedging Across Maturities¶

Different maturities respond differently to volatility shocks:

Term structure risk: The term structure may steepen or flatten:

\[ \Delta\sigma(T_{\text{long}}) \neq \Delta\sigma(T_{\text{short}}) \]

Calendar vega: Use calendar spreads to hedge term structure exposure.

Vega bucketing: Decompose vega by maturity bucket and hedge each bucket separately.

4. Vanna Hedging¶

Vanna exposure creates P&L when spot and volatility move together:

\[ \text{P\&L}_{\text{vanna}} \approx \text{Vanna} \cdot \Delta S \cdot \Delta\sigma \]

Empirical correlation: For equities, spot and volatility are negatively correlated:

\[ \mathbb{E}[\Delta S \cdot \Delta\sigma] < 0 \]

Vanna hedge: Position in options or variance swaps to neutralize vanna.

5. Volga Hedging¶

Volga exposure creates P&L from volatility convexity:

\[ \text{P\&L}_{\text{volga}} \approx \frac{1}{2} \text{Volga} \cdot (\Delta\sigma)^2 \]

Large OTM options: Have significant positive volga, benefiting from vol-of-vol.

Volga hedge: Trade options at different strikes to neutralize volga exposure.

Multi-Factor Hedging¶

1. The Hedging Problem¶

A portfolio of exotic options has exposures to: - Spot ($\Delta$) - Gamma ($\Gamma$) - ATM volatility ($\mathcal{V}_{\text{ATM}}$) - Skew ($\mathcal{V}_{\text{skew}}$) - Term structure ($\mathcal{V}_{T_1}, \mathcal{V}_{T_2}, \ldots$) - Higher-order Greeks (Vanna, Volga)

Objective: Find hedge ratios $h_1, h_2, \ldots$ for available instruments such that total exposure is minimized.

2. Linear Hedging¶

For first-order exposures, solve:

\[ \begin{pmatrix} \Delta_{\text{port}} \\ \mathcal{V}_{\text{ATM,port}} \\ \mathcal{V}_{\text{skew,port}} \end{pmatrix} + \begin{pmatrix} \Delta_1 & \Delta_2 & \Delta_3 \\ \mathcal{V}_{\text{ATM},1} & \mathcal{V}_{\text{ATM},2} & \mathcal{V}_{\text{ATM},3} \\ \mathcal{V}_{\text{skew},1} & \mathcal{V}_{\text{skew},2} & \mathcal{V}_{\text{skew},3} \end{pmatrix} \begin{pmatrix} h_1 \\ h_2 \\ h_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \]

This requires at least as many hedging instruments as exposures.

3. Practical Instrument Selection¶

Liquid instruments for hedging:

Exposure	Typical Hedge
Delta	Underlying, futures
ATM vega	ATM options, variance swaps
Skew	Risk reversals (25D)
Term structure	Calendar spreads
Gamma	ATM options
Vanna	Risk reversals, OTM options
Volga	Strangles, butterflies

4. Hedge Ratio Estimation¶

Model-based: Use a calibrated model to compute Greeks and solve for hedge ratios.

Empirical: Estimate sensitivities from historical data:

\[ \Delta\sigma_{\text{port}} = \beta_1 \Delta\sigma_{\text{ATM}} + \beta_2 \Delta\text{skew} + \epsilon \]

Hybrid: Combine model guidance with empirical adjustments.

Numerical Example: Smile Hedging¶

1. Setup¶

Portfolio: Long 1-year ATM call, short 6-month 25-delta put

Market data: - $S_0 = 100$, $r = 5\%$, $q = 0$ - ATM vol (1Y): 22% - ATM vol (6M): 20% - 25D put vol (6M): 25%

Greeks:

Position	Delta	Vega (ATM)	Vega (25D)
Long 1Y ATM call	0.57	+25.5	0
Short 6M 25D put	+0.15	0	-12.3
Total	0.72	+25.5	-12.3

2. Scenario Analysis¶

Scenario A: Parallel vol up 2%

\[ \text{P\&L} \approx 25.5 \times 0.02 - 12.3 \times 0.02 = 0.51 - 0.25 = +\$0.26 \]

Scenario B: Skew steepens (25D put vol up 3%, ATM unchanged)

\[ \text{P\&L} \approx 0 - 12.3 \times 0.03 = -\$0.37 \]

Scenario C: Spot down 5%, vol up 2%

\[ \text{P\&L}_{\Delta} = 0.72 \times (-5) = -\$3.60 \]

\[ \text{P\&L}_{\text{vol}} \approx +\$0.26 \text{ (as in A)} \]

\[ \text{Total} \approx -\$3.34 \]

3. Hedge Construction¶

Objective: Neutralize delta, ATM vega, and skew exposure.

Instruments: - Underlying (hedge delta) - 6M ATM straddle (hedge ATM vega) - 6M 25D risk reversal (hedge skew)

Solution: (simplified) - Short 72 shares of underlying - Short some ATM straddles to reduce ATM vega - Long 25D risk reversal to offset short 25D put exposure

Dynamic Consistency Considerations¶

1. The Dynamic Consistency Problem¶

A model is dynamically consistent if: - Calibrated to today's surface - Evolved forward under the model - The resulting surface matches future recalibration

Violation: Most models exhibit dynamic inconsistency: - Local vol: Forward smile flattens unrealistically - Stochastic vol: Better but not perfect - Market-implied: Not a model, just interpolation

2. Consequences for Hedging¶

If the model is dynamically inconsistent: - Today's hedge ratios may be wrong for tomorrow's market - Recalibration introduces P&L noise - Hedging effectiveness degrades over time

Mitigation: - Use models with better dynamic properties - Hedge with robust instruments (variance swaps) - Frequent recalibration and hedge adjustment

3. Forward Smile as a Diagnostic¶

The forward smile reveals model dynamics:

\[ \sigma_{\text{fwd}}(K, T_1, T_2) = \text{IV of forward-start option} \]

Local vol: Forward smile flattens quickly Stochastic vol: Forward smile persists but may steepen/flatten Empirical: Forward smile should reflect expected future dynamics

Comparing model-implied forward smiles to historical smile behavior is a key model validation tool.

Empirical Smile Dynamics¶

1. Stylized Facts¶

Equity indices (SPX, EURO STOXX): - Negative spot-vol correlation: $\rho \approx -0.7$ - Skew steepens after down moves - Vol spikes decay within days to weeks - Term structure inverts during stress

FX markets: - Spot-vol correlation varies by pair - EUR/USD: near zero correlation - EM pairs: often positive correlation

Individual stocks: - More idiosyncratic behavior - Earnings-driven dynamics - Jump risk dominates

2. Quantitative Measures¶

Skew-spot beta:

\[ \frac{d(\text{skew})}{d(\text{log spot})} \approx -0.3 \text{ to } -0.5 \text{ for SPX} \]

Vol-spot beta:

\[ \frac{d(\sigma_{\text{ATM}})}{d(\text{log spot})} \approx -1.5 \text{ to } -2.5 \text{ for SPX} \]

Vol-of-vol:

\[ \text{Realized vol of ATM IV} \approx 3\% \text{ to } 5\% \text{ annualized} \]

3. Time Scales¶

Effect	Time Scale
Spot-vol correlation	Intraday to daily
Vol mean reversion	Days to weeks
Skew normalization	Weeks to months
Term structure normalization	Months

Summary¶

Smile dynamics are central to volatility hedging:

1. Static vs. Dynamic¶

Static smile: Fixed in time, delta = BS delta
Dynamic smile: Evolves with spot, vol, time
Reality: Smiles are highly dynamic

2. Model Perspectives¶

Model	Smile Dynamics	Forward Smile
Local vol	Sticky strike	Flattens
Stochastic vol	Leverage-driven	Persists
SABR	Adjustable via $\beta$	Model-dependent
Bergomi	Realistic	Calibrated

3. Hedging Requirements¶

Delta: Adjust for smile dynamics ($\Sigma_S$)
Vega: Bucket by strike and maturity
Vanna: Hedge spot-vol cross-effects
Volga: Hedge vol convexity

4. P&L Attribution¶

\[ \text{P\&L} = \Delta \cdot dS + \Theta \cdot dt + \frac{1}{2}\Gamma (dS)^2 + \mathcal{V} \cdot d\sigma + \text{Vanna} \cdot dS \cdot d\sigma + \frac{1}{2}\text{Volga} \cdot (d\sigma)^2 \]

5. Dynamic Consistency¶

Most models are dynamically inconsistent
Causes hedging degradation over time
Forward smile diagnostics help identify issues

6. Empirical Regularities¶

Negative spot-vol correlation in equities
Skew steepens after down moves
Vol mean-reverts over days to weeks
Term structure inverts during stress

Effective volatility hedging requires understanding and managing all these dimensions of smile dynamics.

Exercises¶

Exercise 1. A delta-hedged option portfolio has $\Gamma = 0.05$, $\mathcal{V} = 15.0$, $\text{Vanna} = -0.12$, $\text{Volga} = 2.5$, and $\Theta = -0.08$. The spot moves $\Delta S = -3$ and implied volatility moves $\Delta\sigma = +0.015$ over one day ($\Delta t = 1/252$). Compute the full second-order P&L decomposition and identify which term contributes the most.

Solution to Exercise 1

We compute each term in the P&L decomposition using the given Greeks and market moves.

Given: $\Gamma = 0.05$, $\mathcal{V} = 15.0$, $\text{Vanna} = -0.12$, $\text{Volga} = 2.5$, $\Theta = -0.08$, $\Delta S = -3$, $\Delta\sigma = +0.015$, $\Delta t = 1/252$.

The full second-order P&L is:

\[ \text{P\&L} = \Theta\,\Delta t + \tfrac{1}{2}\Gamma(\Delta S)^2 + \mathcal{V}\,\Delta\sigma + \text{Vanna}\,\Delta S\,\Delta\sigma + \tfrac{1}{2}\text{Volga}\,(\Delta\sigma)^2 \]

Computing each term:

Theta: $\Theta\,\Delta t = -0.08 \times (1/252) = -0.000317$
Gamma: $\frac{1}{2}\Gamma(\Delta S)^2 = \frac{1}{2}\times 0.05 \times 9 = +0.225$
Vega: $\mathcal{V}\,\Delta\sigma = 15.0 \times 0.015 = +0.225$
Vanna: $\text{Vanna}\,\Delta S\,\Delta\sigma = -0.12 \times (-3)\times 0.015 = +0.0054$
Volga: $\frac{1}{2}\text{Volga}\,(\Delta\sigma)^2 = \frac{1}{2}\times 2.5 \times 0.000225 = +0.000281$

Total P&L:

\[ \text{P\&L} \approx -0.000317 + 0.225 + 0.225 + 0.0054 + 0.000281 \approx +0.4554 \]

The two dominant contributors are the gamma P&L ($+0.225$) and the vega P&L ($+0.225$), each contributing roughly equally and together accounting for nearly all of the total. Theta, vanna, and volga are comparatively negligible. In this scenario, the portfolio benefits from both the realized large spot move (gamma) and the concurrent rise in implied volatility (vega).

Exercise 2. Explain the difference between a static smile and a dynamic smile. In the context of the Heston model, describe how each of the parameters $\rho$, $\xi$, and $\kappa$ affects the dynamics of the smile when the spot price decreases by 5%.

Solution to Exercise 2

A static smile assumes the implied volatility surface is fixed except for deterministic time decay. It does not change in response to spot movements or volatility regime shifts. A dynamic smile evolves as a function of spot price, variance, and other state variables, capturing real-market behavior.

In the Heston model, the dynamics are governed by:

\[ dv_t = \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t^v, \quad d\langle W^S, W^v\rangle = \rho\,dt \]

When the spot decreases by 5%:

$\rho$ (spot-vol correlation): With $\rho < 0$ (typical for equities), a spot decrease is associated with an increase in $v_t$. This shifts the entire smile upward and steepens the skew, because the negative correlation amplifies the leverage effect: falling prices coincide with rising volatility.
$\xi$ (vol-of-vol): A larger $\xi$ means $v_t$ responds more violently to the Brownian shock $dW_t^v$. When spot drops and vol rises (via $\rho < 0$), a high $\xi$ amplifies the variance increase, making the smile shift more dramatic and increasing the curvature (wings) of the smile.
$\kappa$ (mean-reversion speed): A larger $\kappa$ dampens the effect of spot-driven volatility changes because variance reverts more quickly to $\theta$. With high $\kappa$, the initial vol spike after a 5% spot drop is partially absorbed, and the smile shift is more transient. With low $\kappa$, the volatility increase persists longer and the dynamic smile effects are more pronounced.

Exercise 3. The "true" delta under smile dynamics is $\Delta_{\text{true}} = \Delta_{\text{BS}} + \mathcal{V} \cdot \Sigma_S$, where $\Sigma_S = \partial \sigma_{\text{IV}}(K)/\partial S$. For an ATM call with $\Delta_{\text{BS}} = 0.55$, $\mathcal{V} = 20$, and $\Sigma_S = -0.003$ (implied by leverage effect), compute $\Delta_{\text{true}}$. How many additional shares should the hedger hold compared to the Black-Scholes delta?

Solution to Exercise 3

The true delta under smile dynamics is:

\[ \Delta_{\text{true}} = \Delta_{\text{BS}} + \mathcal{V}\cdot\Sigma_S \]

Substituting $\Delta_{\text{BS}} = 0.55$, $\mathcal{V} = 20$, and $\Sigma_S = -0.003$:

\[ \Delta_{\text{true}} = 0.55 + 20 \times (-0.003) = 0.55 - 0.06 = 0.49 \]

The true delta is 0.49, compared to the Black-Scholes delta of 0.55. The hedger should hold $0.49 - 0.55 = -0.06$ additional shares per option compared to the Black-Scholes delta, meaning 6 fewer shares per 100 options.

The negative $\Sigma_S$ reflects the leverage effect: when spot rises, IV tends to fall, which partially offsets the price increase. The correct hedge therefore requires a smaller position in the underlying.

Exercise 4. Compare the forward smile behavior of local volatility models versus stochastic volatility models. (a) Why does the local volatility model produce a forward smile that flattens over time? (b) Why is this inconsistent with empirical observations? (c) How do stochastic volatility models improve upon this?

Solution to Exercise 4

(a) In the local volatility model, the local volatility surface $\sigma_{\text{loc}}(S,t)$ is calibrated to match today's implied volatility surface exactly. The forward smile (the smile of forward-starting options) is determined by the future local volatility values along paths. Since $\sigma_{\text{loc}}(S,t)$ is a fixed deterministic function, the dispersion of future paths narrows as time progresses. For a forward-start option beginning at $T_1$, the effective volatility variation across strikes is driven only by the local vol surface in $[T_1, T_2]$. As $T_1$ increases, the conditional distribution of $S_{T_1}$ concentrates (by the law of large numbers for the diffusion), so the relevant portion of the local vol surface is traversed in a narrower range, producing a flatter forward smile.

(b) Empirically, the volatility smile is persistent: the skew observed in short-dated options reappears in forward-start options and in the realized smiles of future option chains. The local volatility prediction of a flattening forward smile contradicts this stylized fact. Market participants consistently observe that forward skew remains steep, especially in equity indices.

(c) Stochastic volatility models (e.g., Heston) introduce an additional random factor $v_t$ that evolves independently of the diffusion. Even conditioned on future time $T_1$, the variance $v_{T_1}$ remains random, so the forward smile retains curvature and skew. The correlation $\rho$ between spot and variance ensures that skew persists in the forward smile. This matches empirical observations far better than local volatility.

Exercise 5. A portfolio is long a 1-year ATM call (vega = $25.5 at ATM, zero at 25-delta) and short a 6-month 25-delta put (vega = $0 at ATM, $12.3 at 25-delta). Construct a hedge using a 6-month ATM straddle and a 6-month 25-delta risk reversal. Set up the linear system to solve for the hedge ratios that neutralize both ATM vega and 25-delta vega exposures.

Solution to Exercise 5

The portfolio exposures are:

	ATM vega	25-delta vega
Long 1Y ATM call	$+25.5$	$0$
Short 6M 25D put	$0$	$-12.3$
Portfolio	$+25.5$	$-12.3$

Let the hedge instruments have the following vega profiles. Denote the 6M ATM straddle vega at ATM as $\mathcal{V}_{\text{str}}^{\text{ATM}}$ and at 25-delta as $\mathcal{V}_{\text{str}}^{25}$, and the 6M 25D risk reversal vega at ATM as $\mathcal{V}_{\text{RR}}^{\text{ATM}}$ and at 25-delta as $\mathcal{V}_{\text{RR}}^{25}$.

Let $h_1$ = number of straddles and $h_2$ = number of risk reversals. The linear system to neutralize both exposures is:

\[ \begin{pmatrix} \mathcal{V}_{\text{str}}^{\text{ATM}} & \mathcal{V}_{\text{RR}}^{\text{ATM}} \\ \mathcal{V}_{\text{str}}^{25} & \mathcal{V}_{\text{RR}}^{25} \end{pmatrix} \begin{pmatrix} h_1 \\ h_2 \end{pmatrix} = \begin{pmatrix} -25.5 \\ +12.3 \end{pmatrix} \]

The right-hand side is the negative of the portfolio exposures. In a typical setup, the ATM straddle has large ATM vega and small 25-delta vega, while the risk reversal has small ATM vega but significant 25-delta vega. This makes the matrix well-conditioned, and the system has a unique solution $h_1, h_2$ that simultaneously neutralizes both vega buckets.

Exercise 6. Define dynamic consistency for a volatility model. Explain why local volatility models fail the dynamic consistency test in practice. What diagnostic tool (involving the forward smile) can be used to detect dynamic inconsistency?

Solution to Exercise 6

A volatility model is dynamically consistent if, after calibrating to today's implied volatility surface and evolving the model forward in time, the resulting implied volatility surface at a future date matches what the market would produce upon recalibration. Formally, the model's predicted future smile should be consistent with the smile that would be calibrated from future option prices.

Local volatility models fail dynamic consistency because:

The local vol surface $\sigma_{\text{loc}}(S,t)$ is fixed at calibration. When the market evolves and the model is recalibrated, a different $\sigma_{\text{loc}}$ surface is obtained.
The model predicts that forward smiles flatten over time (as discussed in Exercise 4), but empirical recalibration produces persistent skew.
Daily recalibration produces a sequence of different local vol surfaces, each inconsistent with the previous day's model evolution.

Diagnostic tool: The forward smile serves as a key diagnostic. One computes the model-implied forward smile (the implied volatility surface of forward-starting options) and compares it to:

Historically realized future smiles, and
The market-implied forward smile extracted from calendar spread prices.

If the model's forward smile flattens significantly faster than observed in the market, the model is dynamically inconsistent. Stochastic volatility models generally produce forward smiles that are closer to market observations, though they are not perfectly consistent either.

Exercise 7. Using the empirical data for SPX (vol-spot beta $\approx -2.0$ and skew-spot beta $\approx -0.4$), estimate the change in ATM implied volatility and skew when the SPX drops by 3%. If a trader holds a portfolio that is vega-neutral but has positive skew exposure, will the portfolio gain or lose money? Explain.

Solution to Exercise 7

The vol-spot beta relates ATM IV changes to log-spot changes:

\[ \Delta\sigma_{\text{ATM}} \approx \text{(vol-spot beta)} \times \frac{\Delta S}{S} \]

For a 3% drop ($\Delta S/S = -0.03$):

\[ \Delta\sigma_{\text{ATM}} \approx (-2.0)\times(-0.03) = +0.06 = +6\% \]

ATM implied volatility rises by approximately 6 percentage points.

The skew-spot beta relates skew changes to log-spot changes:

\[ \Delta(\text{skew}) \approx (-0.4)\times(-0.03) = +0.012 \]

The skew steepens by approximately 1.2 percentage points (becomes more negative in the conventional sense that OTM put IVs rise more than ATM).

For the trader who is vega-neutral but has positive skew exposure:

Vega-neutral: The parallel rise of 6% in ATM vol does not directly affect the portfolio (net vega is zero).
Positive skew exposure: The portfolio benefits when skew steepens. Since the 3% spot drop causes the skew to steepen (OTM put IVs rise more than ATM IVs), the positive skew position gains money.

The P&L from the skew move is approximately proportional to the skew exposure times $\Delta(\text{skew}) = +0.012$. The portfolio profits because the skew steepening is in the direction that benefits a long skew position.

	ATM vega	25-delta vega
Long 1Y ATM call	\(+25.5\)	\(0\)
Short 6M 25D put	\(0\)	\(-12.3\)
Portfolio	\(+25.5\)	\(-12.3\)

Term	Greek	Source
\(\Delta \cdot dS\)	Delta	Spot move
\(\Theta \cdot dt\)	Theta	Time decay
\(\frac{1}{2}\Gamma (dS)^2\)	Gamma	Spot convexity
\(\mathcal{V} \cdot d\sigma\)	Vega	Vol level change
\(\text{Vanna} \cdot dS \cdot d\sigma\)	Vanna	Spot-vol cross
\(\frac{1}{2}\text{Volga} \cdot (d\sigma)^2\)	Volga	Vol convexity