P&L Explanation Under Heston

Introduction

In the Black--Scholes world, a delta-hedged portfolio's profit and loss decomposes neatly into three terms: theta (time decay), gamma (realized versus implied variance), and financing. Because volatility is constant, this decomposition is exact --- every dollar of P&L is explained. The Heston model breaks this simplicity. Variance \(v_t\) is now a second stochastic factor, so the option price \(V(t, S_t, v_t)\) depends on three variables instead of two. Applying Ito's lemma to this two-factor system introduces additional terms --- a vega P&L from variance moves and a vanna P&L from the correlation between stock and variance --- that have no analogue in Black--Scholes. Understanding these terms is essential for any practitioner who delta-hedges under stochastic volatility.

This section derives the full P&L decomposition for a delta-hedged option in the Heston model, identifies each contributing term, and explains the sources of unexplained P&L that arise in practice.

Prerequisites

• Heston SDE and Parameters (the bivariate SDE system)
• Greeks via Characteristic Function Differentiation (delta, gamma, vega definitions)
• Vega Surface and Vol-of-Vol (variance sensitivities)

Learning Objectives

By the end of this section, you will be able to:

1. Derive the instantaneous P&L of a delta-hedged option under the Heston model using two-dimensional Ito's lemma
2. Identify the theta, gamma, vega, vanna, and volga contributions to P&L
3. Explain why unexplained P&L arises from vol-of-vol, discrete rehedging, and correlation
4. Compare the Heston P&L decomposition with its Black--Scholes counterpart

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The Heston Pricing PDE

Two-Factor Dynamics

Recall the Heston model under the risk-neutral measure \(\mathbb{Q}\):

\[ dS_t = (r - q) S_t \, dt + \sqrt{v_t} \, S_t \, dW_t^{(1)} \]

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

with \(d\langle W^{(1)}, W^{(2)} \rangle_t = \rho \, dt\), where \(r\) is the risk-free rate, \(q\) is the continuous dividend yield, \(\kappa\) is the mean-reversion speed, \(\theta\) is the long-run variance, \(\xi\) is the vol-of-vol, and \(\rho\) is the correlation.

The PDE

The European option price \(V(t, S, v)\) satisfies the Heston PDE:

\[ \frac{\partial V}{\partial t} + \frac{1}{2} v S^2 \frac{\partial^2 V}{\partial S^2} + \rho \xi v S \frac{\partial^2 V}{\partial S \, \partial v} + \frac{1}{2} \xi^2 v \frac{\partial^2 V}{\partial v^2} + (r - q) S \frac{\partial V}{\partial S} + \kappa(\theta - v) \frac{\partial V}{\partial v} - r V = 0 \]

This PDE encodes the no-arbitrage condition under \(\mathbb{Q}\). Each term in the PDE will correspond to a specific component of the P&L decomposition.

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Two-Dimensional Ito Expansion

Applying Ito's Lemma

To understand P&L, we work under the physical measure \(\mathbb{P}\), where the dynamics may differ from \(\mathbb{Q}\) through the market price of risk. The option price \(V(t, S_t, v_t)\) is a function of three variables. By two-dimensional Ito's lemma:

\[ dV = \frac{\partial V}{\partial t} \, dt + \frac{\partial V}{\partial S} \, dS_t + \frac{\partial V}{\partial v} \, dv_t + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} (dS_t)^2 + \frac{\partial^2 V}{\partial S \, \partial v} \, dS_t \, dv_t + \frac{1}{2} \frac{\partial^2 V}{\partial v^2} (dv_t)^2 \]

Substituting the quadratic variation terms:

\[ (dS_t)^2 = v_t S_t^2 \, dt, \qquad dS_t \, dv_t = \rho \xi v_t S_t \, dt, \qquad (dv_t)^2 = \xi^2 v_t \, dt \]

we obtain the full Ito expansion:

\[ dV = \frac{\partial V}{\partial t} \, dt + \frac{\partial V}{\partial S} \, dS_t + \frac{\partial V}{\partial v} \, dv_t + \frac{1}{2} v_t S_t^2 \frac{\partial^2 V}{\partial S^2} \, dt + \rho \xi v_t S_t \frac{\partial^2 V}{\partial S \, \partial v} \, dt + \frac{1}{2} \xi^2 v_t \frac{\partial^2 V}{\partial v^2} \, dt \]

Greek Notation

We introduce standard notation for the partial derivatives (Greeks):

Greek	Symbol	Definition
Delta	\(\Delta\)	\(\partial V / \partial S\)
Gamma	\(\Gamma\)	\(\partial^2 V / \partial S^2\)
Theta	\(\Theta\)	\(\partial V / \partial t\)
Vega (variance)	\(\mathcal{V}\)	\(\partial V / \partial v\)
Vanna (cross)	\(\mathcal{A}\)	\(\partial^2 V / (\partial S \, \partial v)\)
Volga (variance gamma)	\(\mathcal{G}\)	\(\partial^2 V / \partial v^2\)

Vega Convention

In the Heston model, vega is naturally defined as the sensitivity to the variance \(v\), not to the volatility \(\sigma = \sqrt{v}\). The relationship is \(\partial V / \partial \sigma = 2\sqrt{v} \cdot \partial V / \partial v\). Market practitioners often quote vega with respect to implied volatility, so care is needed when comparing.

Using this notation, the Ito expansion becomes:

\[ dV = \Theta \, dt + \Delta \, dS_t + \mathcal{V} \, dv_t + \frac{1}{2} \Gamma \, v_t S_t^2 \, dt + \mathcal{A} \, \rho \xi v_t S_t \, dt + \frac{1}{2} \mathcal{G} \, \xi^2 v_t \, dt \]

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Delta-Hedged P&L Decomposition

The Hedged Portfolio

A trader holds the option \(V\) and delta-hedges by shorting \(\Delta\) shares of the underlying. The portfolio value is:

\[ \Pi_t = V(t, S_t, v_t) - \Delta \, S_t \]

The instantaneous P&L of the hedged portfolio is:

\[ d\Pi_t = dV - \Delta \, dS_t - r \Pi_t \, dt \]

where the last term accounts for the financing cost of the portfolio. Substituting the Ito expansion and cancelling the \(\Delta \, dS_t\) terms:

\[ d\Pi_t = \Theta \, dt + \mathcal{V} \, dv_t + \frac{1}{2} \Gamma \, v_t S_t^2 \, dt + \mathcal{A} \, \rho \xi v_t S_t \, dt + \frac{1}{2} \mathcal{G} \, \xi^2 v_t \, dt - r(V - \Delta S_t) \, dt \]

Decomposition Into Named Components

Theorem (Heston P&L Decomposition)

The instantaneous P&L of a continuously delta-hedged option under the Heston model decomposes as:

\[ d\Pi_t = \underbrace{\Theta \, dt}_{\text{theta}} + \underbrace{\frac{1}{2} \Gamma \, v_t S_t^2 \, dt}_{\text{gamma}} + \underbrace{\mathcal{V} \, dv_t}_{\text{vega}} + \underbrace{\mathcal{A} \, \rho \xi v_t S_t \, dt}_{\text{vanna}} + \underbrace{\frac{1}{2} \mathcal{G} \, \xi^2 v_t \, dt}_{\text{volga}} - \underbrace{r(V - \Delta S_t) \, dt}_{\text{financing}} \]

Each term captures a distinct source of risk:

1. Theta P&L (\(\Theta \, dt\)): Time decay. For a long option position, \(\Theta < 0\) --- the option loses value as time passes, all else equal.

2. Gamma P&L (\(\frac{1}{2} \Gamma v_t S_t^2 \, dt\)): Convexity. The delta-hedged portfolio benefits from realized variance exceeding the level priced into the option. For a long gamma position (\(\Gamma > 0\)), this term is always non-negative.

3. Vega P&L (\(\mathcal{V} \, dv_t\)): Sensitivity to variance moves. This is the key term absent from Black--Scholes. When variance increases (\(dv_t > 0\)) and the trader is long vega (\(\mathcal{V} > 0\)), this contributes positively.

4. Vanna P&L (\(\mathcal{A} \rho \xi v_t S_t \, dt\)): Cross-sensitivity between spot and variance. This term captures the interaction between the two risk factors through their correlation and the cross-partial derivative.

5. Volga P&L (\(\frac{1}{2} \mathcal{G} \xi^2 v_t \, dt\)): The "gamma of vega." This term captures the convexity of the option price with respect to variance, scaled by the vol-of-vol squared.

6. Financing (\(r(V - \Delta S_t) \, dt\)): Cost of carrying the hedged position.

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Theta-Gamma Relationship Under Heston

Using the PDE

In the Black--Scholes model, the theta-gamma relationship \(\Theta + \frac{1}{2} \sigma^2 S^2 \Gamma + r S \Delta - r V = 0\) links theta, gamma, delta, and the option value. Under Heston, the analogous relationship comes directly from the pricing PDE:

\[ \Theta = -\frac{1}{2} v S^2 \Gamma - \rho \xi v S \, \mathcal{A} - \frac{1}{2} \xi^2 v \, \mathcal{G} - (r - q) S \Delta - \kappa(\theta - v) \mathcal{V} + r V \]

Substituting into the P&L decomposition, the deterministic terms simplify.

Theorem (Theta-Gamma-Vega Identity)

Under continuous delta hedging and risk-neutral pricing, the deterministic component of the delta-hedged P&L satisfies:

\[ \Theta + \frac{1}{2} v S^2 \Gamma + \rho \xi v S \, \mathcal{A} + \frac{1}{2} \xi^2 v \, \mathcal{G} + (r - q) S \Delta + \kappa(\theta - v) \mathcal{V} - r V = 0 \]

This is simply a restatement of the Heston PDE with Greek notation. The deterministic terms in the P&L cancel exactly if the option is priced consistently with the Heston model.

Residual P&L

After using the PDE to eliminate the deterministic terms, the remaining P&L is purely stochastic:

\[ d\Pi_t = \mathcal{V}\left[dv_t - \kappa(\theta - v_t) \, dt\right] = \mathcal{V} \, \xi \sqrt{v_t} \, dW_t^{(2)} \]

Under the risk-neutral measure with continuous hedging and correct model, the expected P&L is zero. Under the physical measure, or with model misspecification, this residual drives the unexplained P&L.

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Comparison with Black--Scholes P&L

The Black--Scholes Decomposition

In the Black--Scholes model (\(v_t = \sigma^2\) constant), the delta-hedged P&L is:

\[ d\Pi_t^{\text{BS}} = \Theta^{\text{BS}} \, dt + \frac{1}{2} \Gamma^{\text{BS}} S_t^2 \, (dS_t / S_t)^2 - r(V^{\text{BS}} - \Delta^{\text{BS}} S_t) \, dt \]

Using the Black--Scholes PDE (\(\Theta^{\text{BS}} + \frac{1}{2} \sigma^2 S^2 \Gamma^{\text{BS}} + rS\Delta^{\text{BS}} - rV^{\text{BS}} = 0\)), this reduces to the well-known gamma P&L:

\[ d\Pi_t^{\text{BS}} = \frac{1}{2} \Gamma^{\text{BS}} S_t^2 \left[\left(\frac{dS_t}{S_t}\right)^2 - \sigma^2 \, dt\right] \]

The P&L is driven solely by the difference between realized and implied variance.

Additional Terms in Heston

The Heston model introduces three extra terms relative to Black--Scholes:

Term	Source	Hedgeable?
\(\mathcal{V} \, dv_t\)	Stochastic variance	Only with variance swaps or VIX options
\(\mathcal{A} \, \rho \xi v_t S_t \, dt\)	Spot-variance correlation	Partially, through delta adjustment
\(\frac{1}{2} \mathcal{G} \, \xi^2 v_t \, dt\)	Vol-of-vol convexity	Only with second-order variance instruments

Practical Implication

A trader who delta-hedges an option using the Heston model but does not hedge variance risk will have P&L driven by \(\mathcal{V} \, dv_t\). The magnitude of this unhedged term depends on the option's vega and the realized variance path. For long-dated options with large \(\mathcal{V}\), variance moves can dominate the gamma P&L.

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Sources of Unexplained P&L

Defining Unexplained P&L

In practice, the theoretical P&L decomposition never perfectly matches the actual P&L of a trading book. The difference is called unexplained P&L (sometimes "P&L leakage" or "P&L noise"):

\[ \text{Unexplained P\&L} = \text{Actual P\&L} - \sum_{\text{Greeks}} \text{Greek} \times \text{Risk factor move} \]

Sources Under Heston

Several factors contribute to unexplained P&L when hedging under the Heston model:

1. Discrete Rehedging. The decomposition assumes continuous delta hedging. In practice, rehedging occurs at discrete intervals \(\Delta t\). The discretization error scales as \(\mathcal{O}(\Delta t)\) for each Greek term:

\[ \text{Discrete error} \approx \frac{1}{2} \frac{\partial^2 V}{\partial S^2} S^2 v \left[(\Delta S / S)^2 - v \, \Delta t\right] \]

This "gamma slippage" depends on the realized path and is non-diversifiable across time steps.

2. Higher-Order Terms. The Ito expansion truncates at second order. Third-order terms in \(\Delta S\) and \(\Delta v\) contribute \(\mathcal{O}((\Delta t)^{3/2})\) errors that become visible in volatile markets or for highly convex instruments.

3. Model Misspecification. If the true data-generating process differs from Heston --- for example, if jumps are present or if the vol-of-vol is itself stochastic --- the P&L decomposition based on Heston Greeks will leave a residual. The Heston model cannot capture:

• Jump-induced P&L (requires Bates or Merton extensions)
• Rough volatility effects (\(H < 0.5\) Hurst parameter)
• Regime changes in mean-reversion speed \(\kappa\)

4. Correlation Instability. The parameter \(\rho\) is assumed constant, but empirical correlation between spot returns and variance changes fluctuates. When \(\rho\) varies, the vanna term \(\mathcal{A} \, \rho \xi v S \, dt\) generates unexplained P&L proportional to \(\Delta\rho\).

5. Vol-of-Vol Contribution. The volga term \(\frac{1}{2} \mathcal{G} \xi^2 v \, dt\) is often small for vanilla options but becomes significant for:

• Variance options (where \(\mathcal{G}\) is large)
• Deep out-of-the-money options (where convexity in \(v\) is high)
• Long-dated options (where accumulated vol-of-vol exposure is large)

Vol-of-Vol and Unexplained P&L

The vol-of-vol parameter \(\xi\) controls the curvature of the implied volatility smile. When a trader hedges using a model with incorrect \(\xi\), the volga P&L is systematically misestimated. This is a common source of persistent unexplained P&L in equity derivatives desks.

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Worked Example

Setup

Consider a European call option under the Heston model with the following parameters:

Parameter	Symbol	Value
Spot price	\(S_0\)	$100
Strike	\(K\)	$100
Risk-free rate	\(r\)	3%
Dividend yield	\(q\)	0%
Initial variance	\(v_0\)	0.04
Mean reversion	\(\kappa\)	2.0
Long-run variance	\(\theta\)	0.04
Vol-of-vol	\(\xi\)	0.5
Correlation	\(\rho\)	\(-0.7\)
Time to maturity	\(T\)	0.5 years

Greeks at Inception

Using Fourier inversion (e.g., the COS method), the option price and Greeks at \(t = 0\) are approximately:

Greek	Value	Units
\(V\)	$6.42	dollars
\(\Delta\)	0.567	per dollar of \(S\)
\(\Gamma\)	0.0298	per dollar\(^2\) of \(S\)
\(\Theta\)	\(-0.0215\)	per day (calendar)
\(\mathcal{V}\)	17.8	per unit of variance
\(\mathcal{A}\)	\(-0.48\)	cross-sensitivity
\(\mathcal{G}\)	62.5	per unit of variance\(^2\)

One-Day P&L Attribution

Suppose over one trading day (\(\Delta t = 1/252\)):

• The stock moves from $100.00 to $101.20 (\(\Delta S = +1.20\))
• Variance moves from 0.0400 to 0.0385 (\(\Delta v = -0.0015\))

The P&L components are:

\[ \text{Theta P\&L} = \Theta \times \Delta t = -0.0215 \times (1/252) \approx -\$0.0001 \]

\[ \text{Delta P\&L} = \Delta \times \Delta S = 0.567 \times 1.20 = +\$0.680 \]

\[ \text{Gamma P\&L} = \frac{1}{2} \Gamma (\Delta S)^2 = \frac{1}{2} \times 0.0298 \times 1.44 = +\$0.021 \]

\[ \text{Vega P\&L} = \mathcal{V} \times \Delta v = 17.8 \times (-0.0015) = -\$0.027 \]

\[ \text{Vanna P\&L} = \mathcal{A} \times \rho \xi v S \times \Delta t \approx -0.48 \times (-0.7)(0.5)(0.04)(100) \times (1/252) \approx +\$0.003 \]

\[ \text{Volga P\&L} = \frac{1}{2} \mathcal{G} \xi^2 v \times \Delta t = \frac{1}{2} \times 62.5 \times 0.25 \times 0.04 \times (1/252) \approx +\$0.001 \]

P&L Attribution Summary

Component	Value	Fraction
Delta	+$0.680	100.4%
Gamma	+$0.021	3.1%
Vega	\(-\)$0.027	\(-\)4.0%
Theta	\(-\)$0.0001	\(\approx 0\)%
Vanna	+$0.003	0.4%
Volga	+$0.001	0.1%
Total explained	+$0.677

The delta-hedged P&L (removing the delta component) is approximately \(+\$0.021 - \$0.027 + \$0.003 + \$0.001 \approx -\$0.002\). The vega term (from variance declining) nearly offsets the gamma gain, illustrating how stochastic volatility fundamentally changes the P&L profile relative to Black--Scholes.

Observation

In this example, the stock rose 1.2% while variance fell by 3.75%. The negative \(\rho = -0.7\) makes these moves consistent: rising stock prices are associated with falling variance (the leverage effect). The vega P&L from the variance decline partially offsets the gamma P&L from the stock move --- a signature feature of stochastic volatility that is invisible under Black--Scholes.

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Summary

Concept	Formula / Description
Heston Ito expansion	\(dV = \Theta \, dt + \Delta \, dS + \mathcal{V} \, dv + \frac{1}{2}\Gamma v S^2 \, dt + \mathcal{A} \rho \xi v S \, dt + \frac{1}{2}\mathcal{G} \xi^2 v \, dt\)
Delta-hedged residual	\(d\Pi = \mathcal{V} \xi \sqrt{v} \, dW^{(2)}\) (under continuous hedging)
Black--Scholes gamma P&L	\(\frac{1}{2}\Gamma S^2[(\Delta S/S)^2 - \sigma^2 \Delta t]\)
Unexplained P&L sources	Discrete rehedging, higher-order terms, model misspecification, \(\rho\) instability, vol-of-vol

Key Takeaways

1. Two-factor Ito expansion: Under Heston, the P&L decomposes into six terms --- theta, delta, gamma, vega, vanna, and volga --- compared to three under Black--Scholes.

2. Vega dominates for long-dated options: The \(\mathcal{V} \, dv_t\) term can be the largest contributor to delta-hedged P&L when variance moves are large relative to gamma gains.

3. Theta-gamma-vega identity: The Heston PDE links all deterministic terms; under continuous hedging with the correct model, the residual P&L is purely driven by variance Brownian motion.

4. Unexplained P&L: Discrete rehedging, higher-order Greeks, model misspecification, and parameter instability all generate unexplained P&L that practitioners must monitor and attribute.

5. Hedging implications: To fully hedge under Heston, a trader needs instruments sensitive to variance (variance swaps, VIX options) in addition to delta hedging with the underlying.

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What's Next

Section	Topic
Greeks via Characteristic Function Differentiation	Analytic computation of Heston Greeks
Greeks via Finite Differences	Numerical Greek computation by bumping
Vega Surface and Vol-of-Vol	The term structure of variance sensitivity
Variance Swaps (Closed-Form)	Hedging the vega P&L component

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Exercises

Exercise 1. In Black-Scholes, the delta-hedged P&L over an interval \([t, t+dt]\) is \(dP\&L = \frac{1}{2}\Gamma S^2(\sigma_{\text{real}}^2 - \sigma_{\text{imp}}^2)dt + \Theta\,dt\). Under Heston, an additional vega term appears: \(\mathcal{V}\,dv_t\). Explain the financial meaning of this term. If \(\mathcal{V} > 0\) (long vega) and variance increases (\(dv > 0\)), is the P&L contribution positive or negative?

Solution to Exercise 1

The vega P&L term \(\mathcal{V}\,dv_t\) captures the change in the option's value due to changes in the instantaneous variance \(v_t\), holding all other variables fixed. This term arises because variance is a second stochastic factor in the Heston model: unlike Black-Scholes where \(\sigma\) is constant, the Heston variance \(v_t\) fluctuates randomly according to the CIR process \(dv_t = \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t^{(2)}\).

Financial meaning. If \(\mathcal{V} > 0\) (the trader is long vega, i.e., the option price increases when variance rises), then:

• When variance increases (\(dv > 0\)): the vega P&L contribution is \(\mathcal{V} \times dv > 0\), which is positive. The option becomes more valuable because higher variance implies larger expected future moves in \(S\), increasing the option's time value.
• When variance decreases (\(dv < 0\)): the contribution is negative. The option loses value because the expected future volatility has declined.

For a long call position, \(\mathcal{V} > 0\) always (call prices are increasing in variance). So a variance increase (\(dv > 0\)) produces a positive P&L contribution.

This term is absent from Black-Scholes because \(v\) is constant, so \(dv = 0\) identically. Under Heston, the vega P&L can be the dominant component of delta-hedged P&L for long-dated options, since variance moves \(dv_t\) can be large (especially when \(\xi\) is high) while the gamma P&L depends on realized stock moves which are of order \(\sqrt{v\,dt}\).

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Exercise 2. The Heston P&L decomposition includes a vanna term \(\partial^2 V / \partial S \partial v \cdot S\sqrt{v}\rho\xi v\,dt\). Explain why this term arises from the correlation between \(dS\) and \(dv\). For \(\rho < 0\), does a simultaneous drop in \(S\) and rise in \(v\) produce a positive or negative vanna P&L for a long call position?

Solution to Exercise 2

The vanna term in the P&L decomposition is:

\[ \text{Vanna P\&L} = \mathcal{A}\,\rho\,\xi\,v_t\,S_t\,dt \]

where \(\mathcal{A} = \partial^2 V / \partial S \,\partial v\).

Why it arises from correlation. In the two-dimensional Ito expansion of \(dV(t, S_t, v_t)\), the cross-term is:

\[ \frac{\partial^2 V}{\partial S\,\partial v}\,dS_t\,dv_t \]

Computing the quadratic covariation:

\[ dS_t\,dv_t = \bigl(\sqrt{v_t}\,S_t\,dW_t^{(1)}\bigr)\bigl(\xi\sqrt{v_t}\,dW_t^{(2)}\bigr) = \xi\,v_t\,S_t\,dW_t^{(1)}\,dW_t^{(2)} = \rho\,\xi\,v_t\,S_t\,dt \]

This is nonzero precisely because \(\rho \neq 0\): the stock and variance Brownian motions are correlated. If \(\rho = 0\), the vanna P&L vanishes entirely.

Sign analysis for \(\rho < 0\) and a long call. Consider a simultaneous drop in \(S\) and rise in \(v\) (the leverage effect, which is the typical scenario when \(\rho < 0\)):

• \(\Delta S < 0\) and \(\Delta v > 0\)

For a long call position, the vanna \(\mathcal{A} = \partial^2 V / \partial S\,\partial v\) is typically negative for ATM and ITM options. Intuitively: when \(S\) increases, the option moves deeper ITM and its sensitivity to variance (vega) decreases (because deep ITM options are less sensitive to volatility). So \(\partial\mathcal{V}/\partial S < 0\), i.e., \(\mathcal{A} < 0\).

The vanna P&L over a discrete interval is approximately:

\[ \mathcal{A}\,\Delta S\,\Delta v \approx \mathcal{A} \times (\text{negative}) \times (\text{positive}) < 0 \quad \text{if } \mathcal{A} < 0 \]

Wait -- we must be more careful. The continuous-time vanna P&L from the Ito expansion is the deterministic cross-variation term \(\mathcal{A}\,\rho\,\xi\,v\,S\,dt\), not \(\mathcal{A}\,\Delta S\,\Delta v\). For \(\rho < 0\), \(\mathcal{A} < 0\), \(\xi > 0\), \(v > 0\), \(S > 0\):

\[ \mathcal{A}\,\rho\,\xi\,v\,S\,dt = (\text{negative})(\text{negative})(\text{positive})(\text{positive})(\text{positive})(\text{positive}) > 0 \]

The vanna P&L is positive for a long call with \(\rho < 0\). This makes financial sense: the leverage effect (negative correlation) means that when stocks drop, volatility rises, providing a partial natural hedge. The vanna term captures this beneficial correlation effect --- the option "benefits" from the systematic relationship between spot and variance.

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Exercise 3. A trader sells an ATM call and delta-hedges daily. Over one month, the unexplained P&L (residual after delta, gamma, theta, and vega terms) has a standard deviation of $0.50 per option. Identify three sources of this unexplained P&L: (a) discrete hedging error, (b) unhedged volga (\(\partial^2 V/\partial v^2\)) exposure, (c) model mis-specification. Which is likely the dominant source?

Solution to Exercise 3

The three sources of unexplained P&L are:

(a) Discrete hedging error. The P&L decomposition assumes continuous delta hedging (\(dt \to 0\)), but in practice the trader rebalances at discrete intervals \(\Delta t\) (e.g., daily). The discretization error over each rebalancing interval is approximately:

\[ \text{Discrete error} \approx \frac{1}{2}\Gamma\,S^2\left[\left(\frac{\Delta S}{S}\right)^2 - v\,\Delta t\right] + \text{higher-order terms} \]

This is the gamma slippage: the realized squared return \((\Delta S/S)^2\) deviates from its expected value \(v\,\Delta t\) over finite intervals. The standard deviation of this term scales as \(\mathcal{O}(\sqrt{\Delta t})\) per step and \(\mathcal{O}((\Delta t)^{0})\) cumulatively over \(T/\Delta t\) steps (since errors are partially diversified but not independent due to serial correlation in \(v_t\)). For daily hedging (\(\Delta t = 1/252\)), this can produce P&L noise of order $0.10--$0.50 per option over a month.

(b) Unhedged volga exposure. The volga term \(\frac{1}{2}\mathcal{G}\,\xi^2\,v\,dt\) is a deterministic contribution to the P&L decomposition, but it is often omitted from simplified attribution models. More importantly, the trader does not hedge the stochastic part of the volga exposure: changes in \(\mathcal{G}\) itself as \((S, v)\) evolve, and the fact that \(\xi\) may not be constant. For an ATM call, \(\mathcal{G}\) is moderate, but for OTM options or variance-sensitive instruments, this can be significant. Over one month, the cumulative unhedged volga P&L is approximately:

\[ \sum_{i=1}^{N} \frac{1}{2}\mathcal{G}_i\,\xi^2\,v_i\,\Delta t \approx \frac{1}{2}\bar{\mathcal{G}}\,\xi^2\,\bar{v}\,T_{\text{month}} \]

For \(\bar{\mathcal{G}} \approx 60\), \(\xi = 0.5\), \(\bar{v} = 0.04\), \(T_{\text{month}} = 1/12\): approximately \(\frac{1}{2} \times 60 \times 0.25 \times 0.04 / 12 \approx \$0.025\).

(c) Model misspecification. The Heston model assumes specific functional forms for the dynamics (CIR variance, constant parameters, no jumps). If the true process has:

• Jumps in \(S\) or \(v\): the Ito expansion misses the jump terms entirely
• Time-varying \(\kappa\), \(\theta\), \(\xi\), or \(\rho\): the Greeks computed with static parameters are systematically wrong
• Rougher volatility dynamics (\(H < 0.5\)): the CIR process is too smooth, and the Heston Greeks understate the true sensitivities at short horizons

Dominant source. For a standard ATM call hedged daily over one month, the discrete hedging error is likely the dominant source, producing the largest contribution to the $0.50 standard deviation. The gamma slippage from daily rebalancing is inherently noisy and scales with \(\Gamma\,S^2\,v\) (which is large for ATM options). The volga and model misspecification terms are smaller for vanilla ATM options but become more important for exotic or long-dated positions.

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Exercise 4. The gamma P&L is \(\frac{1}{2}\Gamma S^2 (dS/S)^2 \approx \frac{1}{2}\Gamma S^2 v\,dt\). The vega P&L is \(\mathcal{V}\,dv\). For typical equity Heston parameters (\(v_0 = 0.04\), \(\xi = 0.5\)), estimate the order of magnitude of each term for an ATM call with \(T = 0.5\), \(S_0 = 100\). Which contributes more to daily P&L variance?

Solution to Exercise 4

Gamma P&L (per day). The expected magnitude of the gamma P&L per day is:

\[ \left|\frac{1}{2}\Gamma\,S_0^2\,v_0\,\Delta t\right| = \frac{1}{2} \times \Gamma \times 100^2 \times 0.04 \times \frac{1}{252} \]

For an ATM call with \(T = 0.5\), using the worked example values, \(\Gamma \approx 0.0298\):

\[ \frac{1}{2} \times 0.0298 \times 10{,}000 \times 0.04 \times \frac{1}{252} \approx \frac{1}{2} \times 0.0298 \times 400 \times 0.00397 \approx \$0.024 \]

The variance of the daily gamma P&L (driven by the randomness of \((\Delta S/S)^2\)) scales as \(\Gamma^2 S^4 v^2 \Delta t / 2\), giving a daily standard deviation of approximately:

\[ \sigma_{\text{gamma}} \approx \Gamma\,S_0^2\,v_0\,\sqrt{\Delta t / 2} \approx 0.0298 \times 10{,}000 \times 0.04 \times \sqrt{1/504} \approx 11.92 \times 0.0445 \approx \$0.53 \]

Vega P&L (per day). The vega P&L is \(\mathcal{V}\,\Delta v\). The daily change in variance under the Heston model has standard deviation:

\[ \sigma_{\Delta v} = \xi\sqrt{v_0\,\Delta t} = 0.5 \times \sqrt{0.04 / 252} = 0.5 \times 0.0126 \approx 0.0063 \]

With \(\mathcal{V} \approx 17.8\):

\[ \sigma_{\text{vega}} = |\mathcal{V}| \times \sigma_{\Delta v} \approx 17.8 \times 0.0063 \approx \$0.112 \]

Comparison. The daily expected gamma P&L (\(\approx \$0.024\)) is larger than the daily expected vega P&L (which is approximately \(\mathcal{V}\,\kappa(\theta - v_0)\,\Delta t = 0\) since \(v_0 = \theta\)). However, in terms of P&L variance (which drives hedging risk):

\[ \sigma_{\text{gamma}} \approx \$0.53, \qquad \sigma_{\text{vega}} \approx \$0.11 \]

The gamma P&L contributes more to daily P&L variance in this example, roughly 5 times more in standard deviation. However, the vega contribution is not negligible and would dominate for longer-dated options (where \(\mathcal{V}\) is larger and \(\Gamma\) is smaller) or during periods of high vol-of-vol (\(\xi > 0.5\)).

Over longer horizons, the vega P&L can accumulate because variance changes are autocorrelated (mean-reverting but persistent), while gamma P&L noise is more diversified across independent daily stock moves. This means the vega contribution to cumulative P&L variance grows faster than \(\sqrt{N}\) (where \(N\) is the number of days), potentially becoming dominant over monthly or quarterly horizons.

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Exercise 5. To hedge the vega P&L, a trader can add a variance swap to the portfolio. Explain how: if the delta-hedged call has vega \(\mathcal{V}\) and the variance swap has vega \(\mathcal{V}_{\text{VS}} = \partial(\text{VS price})/\partial v_0\), what notional of the variance swap neutralizes the vega exposure?

Solution to Exercise 5

A variance swap pays the difference between realized and fixed (strike) variance. Its price under Heston is a function of \(v_0\), and its vega is:

\[ \mathcal{V}_{\text{VS}} = \frac{\partial V_{\text{VS}}}{\partial v_0} \]

For a variance swap with maturity \(T_{\text{VS}}\), the fair strike is:

\[ K_{\text{var}} = \mathbb{E}^{\mathbb{Q}}\!\left[\frac{1}{T}\int_0^T v_t\,dt\right] = \theta + (v_0 - \theta)\frac{1 - e^{-\kappa T}}{\kappa T} \]

The sensitivity to \(v_0\) is:

\[ \mathcal{V}_{\text{VS}} = \frac{\partial K_{\text{var}}}{\partial v_0} \times \text{notional} = \frac{1 - e^{-\kappa T}}{\kappa T} \times \text{notional} \]

Hedging the vega exposure. The delta-hedged call has unhedged vega P&L \(\mathcal{V}_{\text{call}}\,dv_t\). To neutralize this, the trader adds \(N\) units of variance swap notional such that the total vega is zero:

\[ \mathcal{V}_{\text{call}} + N \times \mathcal{V}_{\text{VS}} = 0 \]

Solving:

\[ N = -\frac{\mathcal{V}_{\text{call}}}{\mathcal{V}_{\text{VS}}} \]

For example, with \(\mathcal{V}_{\text{call}} = 17.8\) and a variance swap with \(T = 0.5\), \(\kappa = 2.0\):

\[ \frac{1 - e^{-\kappa T}}{\kappa T} = \frac{1 - e^{-1}}{1} = 1 - 0.368 = 0.632 \]

If the variance swap notional is quoted per unit of variance, \(\mathcal{V}_{\text{VS}} = 0.632\) per unit notional. The required notional is:

\[ N = -\frac{17.8}{0.632} \approx -28.2 \]

The trader should sell approximately 28.2 units of variance swap notional (since \(\mathcal{V}_{\text{call}} > 0\), the long call is long vega, and selling variance swaps provides short vega exposure).

Important caveat. This hedge neutralizes only the first-order vega exposure \(\mathcal{V}\,dv\). The residual P&L still contains the volga term \(\frac{1}{2}\mathcal{G}\,\xi^2\,v\,dt\) and any mismatch in the vega term structure (the call's vega decays differently with \(T\) than the variance swap's vega). A more precise hedge would match the vega at multiple maturities or use vega-weighted positions in variance swaps of different tenors.

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Exercise 6. Simulate a delta-hedging experiment under Heston: sell a 6-month ATM call, delta-hedge daily using the Heston delta, and record the total P&L across 10,000 paths. Decompose the P&L into gamma, theta, and vega components. Verify that the mean P&L is approximately zero (the hedge is unbiased) and that the standard deviation reflects the unhedged variance risk.

Solution to Exercise 6

Simulation setup.

1. Model parameters: \(S_0 = 100\), \(K = 100\), \(T = 0.5\), \(r = 0.03\), \(q = 0\), \(v_0 = 0.04\), \(\kappa = 2.0\), \(\theta = 0.04\), \(\xi = 0.5\), \(\rho = -0.7\).
2. Discretization: Daily steps, \(N = 126\) steps (\(T = 0.5\) years \(\times\) 252 days/year).
3. Paths: \(M = 10{,}000\) independent paths.
4. Simulation scheme: Use the QE (quadratic exponential) scheme for the variance process to ensure \(v_t \geq 0\), and log-Euler for the stock price.

Delta-hedging algorithm. On each path \(m\) and at each time step \(t_i\):

1. Compute the Heston delta \(\Delta_i^m = \Delta(t_i, S_i^m, v_i^m)\) using Fourier inversion (or a fast approximation).

2. The hedged portfolio P&L over \([t_i, t_{i+1}]\) is:

   \[ \Delta\Pi_i^m = V(t_{i+1}, S_{i+1}^m, v_{i+1}^m) - V(t_i, S_i^m, v_i^m) - \Delta_i^m(S_{i+1}^m - S_i^m) - r\bigl(V(t_i, S_i^m, v_i^m) - \Delta_i^m S_i^m\bigr)\Delta t \]

3. The total hedged P&L is \(\Pi^m = \sum_{i=0}^{N-1} \Delta\Pi_i^m\).

P&L decomposition. At each step, decompose \(\Delta\Pi_i^m\) into:

• Gamma component: \(\frac{1}{2}\Gamma_i^m (S_i^m)^2 \left[(\Delta S_i^m / S_i^m)^2 - v_i^m \Delta t\right]\)
• Theta component: \(\Theta_i^m \Delta t + \frac{1}{2}\Gamma_i^m (S_i^m)^2 v_i^m \Delta t - r(V_i^m - \Delta_i^m S_i^m)\Delta t\) (the deterministic part)
• Vega component: \(\mathcal{V}_i^m \Delta v_i^m\)
• Residual: everything not captured by the above (vanna, volga, higher-order, discretization)

Expected results.

• Mean P&L \(\approx 0\). Under the risk-neutral measure with correct model pricing, the expected delta-hedged P&L is zero: \(\mathbb{E}[\Pi] = 0\). Across 10,000 paths, the sample mean should be close to zero (within \(\pm 2\sigma/\sqrt{M}\)).

• Standard deviation. The residual P&L after delta hedging is driven by the unhedged variance risk. From the continuous-time result \(d\Pi = \mathcal{V}\,\xi\sqrt{v}\,dW^{(2)}\), the variance of the cumulative P&L is:

  \[ \operatorname{Var}(\Pi) = \mathbb{E}\!\left[\int_0^T \mathcal{V}^2(t)\,\xi^2\,v_t\,dt\right] \]

  With \(\mathcal{V} \approx 17.8\) (at inception, decreasing over time), \(\xi^2 = 0.25\), \(\bar{v} \approx 0.04\):

  \[ \operatorname{Var}(\Pi) \approx \bar{\mathcal{V}}^2\,\xi^2\,\bar{v}\,T \approx (12)^2 \times 0.25 \times 0.04 \times 0.5 \approx 0.72 \]

  So \(\operatorname{Std}(\Pi) \approx \$0.85\). (Here \(\bar{\mathcal{V}} \approx 12\) is the time-averaged vega.)

• Decomposition verification. Summing the gamma, theta, and vega components across all time steps should approximately equal the total P&L for each path. The unexplained residual (from discretization, vanna, volga, higher-order terms) should have a standard deviation of approximately $0.10--$0.20 per option, much smaller than the $0.85 total P&L standard deviation.

• Gamma P&L statistics. The cumulative gamma P&L should have mean \(\approx 0\) (since realized variance equals implied on average under the risk-neutral measure) and standard deviation of approximately $0.30--$0.50.

• Vega P&L statistics. The cumulative vega P&L should also have mean \(\approx 0\) (variance innovations have zero mean) but with a standard deviation of approximately $0.60--$0.80, confirming that variance risk is the dominant source of hedging uncertainty.