Portfolio Hedging and Cross-Greeks

When a trading book contains options on multiple underlyings, the risk management problem becomes genuinely multi-dimensional. Cross-Greeks --- sensitivities involving more than one underlying --- capture correlation-driven risks that are invisible to single-asset analysis. This section develops the framework for portfolio-level hedging with emphasis on the cross-gamma matrix and its implications for hedging error.

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Multi-Asset Sensitivity Framework

Portfolio Value and First-Order Sensitivities

Consider a portfolio of options depending on \(d\) underlying assets with prices \(S = (S_1, \ldots, S_d)^T\). The portfolio value \(\Pi(t, S)\) admits a Taylor expansion:

\[ \delta\Pi \approx \sum_{i=1}^d \Delta_i\,\delta S_i + \Theta\,\delta t + \frac{1}{2}\sum_{i=1}^d \sum_{j=1}^d \Gamma_{ij}\,\delta S_i\,\delta S_j + \sum_{i=1}^d \mathcal{V}_i\,\delta\sigma_i \]

where the sensitivities are:

Greek	Definition	Interpretation
\(\Delta_i = \frac{\partial \Pi}{\partial S_i}\)	Delta vector	Directional exposure to asset \(i\)
\(\Gamma_{ij} = \frac{\partial^2 \Pi}{\partial S_i \partial S_j}\)	Gamma matrix	Second-order (cross-)sensitivity
\(\Theta = \frac{\partial \Pi}{\partial t}\)	Theta	Time decay
\(\mathcal{V}_i = \frac{\partial \Pi}{\partial \sigma_i}\)	Vega vector	Volatility exposure to asset \(i\)

Vector and Matrix Notation

The delta vector and gamma matrix in compact form:

\[ \boldsymbol{\Delta} = \nabla_S \Pi \in \mathbb{R}^d, \qquad \boldsymbol{\Gamma} = \nabla_S^2 \Pi \in \mathbb{R}^{d \times d} \]

The second-order Taylor expansion becomes:

\[ \boxed{\delta\Pi \approx \boldsymbol{\Delta}^T \delta\mathbf{S} + \Theta\,\delta t + \frac{1}{2}\,\delta\mathbf{S}^T \boldsymbol{\Gamma}\,\delta\mathbf{S}} \]

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Cross-Gamma: Definition and Structure

Definition

The cross-gamma between assets \(i\) and \(j\) is:

\[ \Gamma_{ij} = \frac{\partial^2 \Pi}{\partial S_i \partial S_j}, \quad i \neq j \]

The diagonal entries \(\Gamma_{ii}\) are the standard single-asset gammas. The full gamma matrix \(\boldsymbol{\Gamma}\) is symmetric: \(\Gamma_{ij} = \Gamma_{ji}\).

When Cross-Gamma Arises

Cross-gamma is nonzero when the portfolio contains instruments that depend on multiple underlyings simultaneously:

• Basket options: Options on \(\sum_i w_i S_i\) have cross-gammas proportional to \(w_i w_j\).
• Spread options: Options on \(S_1 - S_2\) have \(\Gamma_{12} < 0\).
• Quanto options: Currency-adjusted options create cross-gamma between the asset and the exchange rate.
• Correlation-dependent structures: Worst-of, best-of, and rainbow options.

Single-Asset Options

A portfolio containing only single-asset options (each depending on only one \(S_i\)) has \(\Gamma_{ij} = 0\) for \(i \neq j\). The gamma matrix is diagonal, and the multi-asset hedging problem decouples into \(d\) independent single-asset problems.

Cross-Gamma for a Two-Asset Basket

Consider a call on the basket \(B = w_1 S_1 + w_2 S_2\) with strike \(K\). The payoff is \((B - K)^+\). By the chain rule:

\[ \Delta_i = w_i \frac{\partial C}{\partial B}, \qquad \Gamma_{ij} = w_i w_j \frac{\partial^2 C}{\partial B^2} \]

where \(C(B)\) is the call price as a function of the basket level. Since \(\frac{\partial^2 C}{\partial B^2} > 0\) (the call has positive gamma in \(B\)), the cross-gamma \(\Gamma_{12} = w_1 w_2 \frac{\partial^2 C}{\partial B^2} > 0\) for positive weights.

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Portfolio-Level Delta Hedging

The Multi-Asset Delta Hedge

To delta-hedge the portfolio, hold \(-\Delta_i\) shares of asset \(i\) for each \(i = 1, \ldots, d\). The hedged portfolio is:

\[ \Pi_{\text{hedged}} = \Pi - \sum_{i=1}^d \Delta_i S_i \]

The residual P&L after delta hedging is:

\[ \delta\Pi_{\text{hedged}} \approx \Theta\,\delta t + \frac{1}{2}\,\delta\mathbf{S}^T \boldsymbol{\Gamma}\,\delta\mathbf{S} \]

Hedged P&L in Terms of Correlations

Under geometric Brownian motion for each asset (\(dS_i = \mu_i S_i\,dt + \sigma_i S_i\,dW_i\) with \(\operatorname{Corr}(dW_i, dW_j) = \rho_{ij}\)), the expected delta-hedged P&L over \(\delta t\) is:

\[ \mathbb{E}[\delta\Pi_{\text{hedged}}] = \left(\Theta + \frac{1}{2}\sum_{i,j} \Gamma_{ij} \rho_{ij} \sigma_i \sigma_j S_i S_j\right)\delta t \]

The cross-gamma terms \(\Gamma_{ij}\rho_{ij}\sigma_i\sigma_j S_i S_j\) contribute to the expected P&L through the correlation structure. If correlations change, these terms produce unexpected gains or losses.

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Gamma Hedging in Multiple Dimensions

The Hedging System

To neutralize the entire gamma matrix, we need hedging instruments with their own gamma matrices. Let \(H^{(l)}\) be the \(l\)-th hedging instrument with gamma matrix \(\boldsymbol{\Gamma}^{(l)}\). To achieve \(\boldsymbol{\Gamma}_{\text{portfolio}} = \mathbf{0}\), solve:

\[ \sum_{l=1}^L n_l\,\boldsymbol{\Gamma}^{(l)} = -\boldsymbol{\Gamma}_{\text{existing}} \]

Since \(\boldsymbol{\Gamma}\) is a \(d \times d\) symmetric matrix with \(d(d+1)/2\) independent entries, we need at least \(L = d(d+1)/2\) hedging instruments to fully neutralize the gamma matrix.

Dimensionality Curse

For \(d = 2\) assets: need at least 3 instruments (\(\Gamma_{11}, \Gamma_{12}, \Gamma_{22}\)).

For \(d = 5\) assets: need at least 15 instruments.

For \(d = 10\) assets: need at least 55 instruments.

Full gamma neutralization becomes impractical for large \(d\). In practice, traders focus on the most significant gamma exposures.

Partial Gamma Hedging

When full neutralization is infeasible, prioritize:

1. Diagonal gammas \(\Gamma_{ii}\): Hedge single-asset gammas first using single-asset options.
2. Largest cross-gammas: Hedge the \(\Gamma_{ij}\) entries with the greatest dollar exposure \(|\Gamma_{ij} \rho_{ij} \sigma_i \sigma_j S_i S_j|\).
3. Accept residual: Leave smaller cross-gamma exposures unhedged and monitor.

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Cross-Gamma Effects on Hedging Error

Variance of Delta-Hedged P&L

The variance of the delta-hedged P&L over one step \(\delta t\) involves the fourth moments of the joint returns:

\[ \operatorname{Var}(\delta\Pi_{\text{hedged}}) \approx \frac{1}{2}\sum_{i,j,k,l} \Gamma_{ij}\Gamma_{kl}\,C_{ijkl}\,(\delta t)^2 \]

where \(C_{ijkl} = \operatorname{Cov}(\delta S_i\,\delta S_j,\; \delta S_k\,\delta S_l)\) is the fourth-order covariance tensor. For jointly normal returns:

\[ C_{ijkl} = (\rho_{ik}\rho_{jl} + \rho_{il}\rho_{jk})\sigma_i\sigma_j\sigma_k\sigma_l S_i S_j S_k S_l\,(\delta t)^2 \]

Two-Asset Special Case

For \(d = 2\), the hedged P&L variance simplifies to:

\[ \operatorname{Var}(\delta\Pi_{\text{hedged}}) \approx \left[\frac{1}{2}(\Gamma_{11} S_1^2 \sigma_1^2)^2 + 2\rho_{12}^2 \Gamma_{11}\Gamma_{22}S_1^2 S_2^2 \sigma_1^2\sigma_2^2 + \frac{1}{2}(\Gamma_{22}S_2^2\sigma_2^2)^2 + 2(\Gamma_{12}S_1 S_2 \sigma_1\sigma_2)^2(1 + \rho_{12}^2)\right](\delta t)^2 \]

The cross-gamma \(\Gamma_{12}\) contributes an additional variance term that depends on \((1 + \rho_{12}^2)\), which is always positive and amplified by high correlation.

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Correlation Risk

Sensitivity to Correlation

The portfolio vega with respect to correlation \(\rho_{ij}\) is:

\[ \frac{\partial \Pi}{\partial \rho_{ij}} = \text{``correlation vega'' or ``cega''} \]

This sensitivity is particularly important for:

• Basket options (highly sensitive to pairwise correlations)
• Worst-of options (increase in correlation increases value)
• Spread options (increase in correlation decreases value)

Correlation and Cross-Gamma

The delta-hedged P&L depends on correlations through the cross-gamma terms. If the true correlation differs from the model correlation:

\[ \delta\Pi_{\text{hedged}} \approx \frac{1}{2}\sum_{i \neq j} \Gamma_{ij} S_i S_j \sigma_i \sigma_j (\rho_{ij}^{\text{realized}} - \rho_{ij}^{\text{model}})\,\delta t + \cdots \]

This is the correlation P&L --- a source of model risk that is difficult to hedge because correlation derivatives are illiquid.

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Worked Example: Two-Asset Portfolio

A trader holds the following positions:

• Long 100 calls on asset 1 (\(S_1 = 100\), \(K_1 = 100\), \(\sigma_1 = 25\%\))
• Short 50 puts on asset 2 (\(S_2 = 80\), \(K_2 = 85\), \(\sigma_2 = 30\%\))
• Long 20 basket calls on \(0.6 S_1 + 0.4 S_2\) (\(K = 95\))

Per-option Greeks (illustrative values):

	Call on \(S_1\)	Put on \(S_2\)	Basket call
\(\Delta_1\)	0.55	0	0.36
\(\Delta_2\)	0	\(-0.45\)	0.24
\(\Gamma_{11}\)	0.032	0	0.012
\(\Gamma_{22}\)	0	0.028	0.005
\(\Gamma_{12}\)	0	0	0.008

Portfolio-level Greeks:

\[ \boldsymbol{\Delta} = \begin{pmatrix} 100(0.55) + 20(0.36) \\ -50(-0.45) + 20(0.24) \end{pmatrix} = \begin{pmatrix} 62.2 \\ 27.3 \end{pmatrix} \]

\[ \boldsymbol{\Gamma} = \begin{pmatrix} 100(0.032) + 20(0.012) & 20(0.008) \\ 20(0.008) & -50(0.028) + 20(0.005) \end{pmatrix} = \begin{pmatrix} 3.44 & 0.16 \\ 0.16 & -1.30 \end{pmatrix} \]

Delta hedge: Short 62.2 shares of asset 1 and 27.3 shares of asset 2.

Gamma analysis: The portfolio is long \(\Gamma_{11}\) (benefits from moves in \(S_1\)) and short \(\Gamma_{22}\) (exposed to moves in \(S_2\)). The cross-gamma \(\Gamma_{12} = 0.16\) means correlated moves in both assets create additional P&L.

To fully gamma-hedge, the trader would need at least 3 additional instruments with independent gamma structures to neutralize \(\Gamma_{11}\), \(\Gamma_{22}\), and \(\Gamma_{12}\) simultaneously.

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Practical Portfolio Risk Management

Bucketed Sensitivities

Large trading books aggregate Greeks into buckets:

• By underlying: Total delta, gamma per asset
• By maturity: Short-dated vs. long-dated gamma
• By strike: ATM vs. wing gamma
• By correlation pair: Cross-gamma for each asset pair

Risk Limits

Portfolio risk limits are typically expressed as:

Metric	Definition	Typical limit
Net delta per asset	$	\Delta_i
Dollar gamma per asset	\(\frac{1}{2}\Gamma_{ii}S_i^2\)	$X per 1% move
Cross-gamma	\(\Gamma_{ij}S_iS_j\)	$Y per 1% correlated move
Vega per asset	\(\mathcal{V}_i\)	$Z per 1 vol point

Hedging Priority

In practice, the hedging priority for a multi-asset book is:

1. Delta: Neutralize directional risk using the underlyings.
2. Largest gammas: Hedge dominant diagonal gammas using ATM options.
3. Vega: Manage volatility exposure using options at various strikes and maturities.
4. Cross-gamma: Address if material, using multi-asset derivatives or proxy hedges.
5. Higher-order: Speed, charm, vanna --- monitor but rarely hedge directly.

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Summary

Concept	Key point
Delta vector	\(\boldsymbol{\Delta} \in \mathbb{R}^d\) captures directional exposure to each asset
Gamma matrix	\(\boldsymbol{\Gamma} \in \mathbb{R}^{d \times d}\) (symmetric); diagonal = single-asset, off-diagonal = cross
Cross-gamma	Arises from multi-asset instruments; drives correlation-dependent P&L
Full gamma hedge	Requires \(d(d+1)/2\) instruments --- impractical for large \(d\)
Correlation risk	Cross-gamma terms couple the P&L to realized correlations
Hedged P&L	\(\delta\Pi_{\text{hedged}} \approx \frac{1}{2}\delta\mathbf{S}^T\boldsymbol{\Gamma}\,\delta\mathbf{S} + \Theta\,\delta t\)
Practical approach	Prioritize diagonal gamma, hedge largest cross-gammas selectively

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Exercises

Exercise 1. For a portfolio with delta vector \(\boldsymbol{\Delta} = (30, -20)^T\) and gamma matrix \(\boldsymbol{\Gamma} = \begin{pmatrix} 2.0 & 0.5 \\ 0.5 & 1.5 \end{pmatrix}\), compute the delta-hedged P&L from a simultaneous move \(\delta S_1 = +3\), \(\delta S_2 = -2\). Decompose the P&L into contributions from \(\Gamma_{11}\), \(\Gamma_{22}\), and \(\Gamma_{12}\).

Solution to Exercise 1

The delta-hedged P&L from second-order terms is:

\[ \delta\Pi_{\text{hedged}} \approx \frac{1}{2}\,\delta\mathbf{S}^T \boldsymbol{\Gamma}\,\delta\mathbf{S} = \frac{1}{2}\begin{pmatrix} 3 & -2 \end{pmatrix}\begin{pmatrix} 2.0 & 0.5 \\ 0.5 & 1.5 \end{pmatrix}\begin{pmatrix} 3 \\ -2 \end{pmatrix} \]

First compute \(\boldsymbol{\Gamma}\,\delta\mathbf{S}\):

\[ \begin{pmatrix} 2.0 & 0.5 \\ 0.5 & 1.5 \end{pmatrix}\begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 2.0(3) + 0.5(-2) \\ 0.5(3) + 1.5(-2) \end{pmatrix} = \begin{pmatrix} 5.0 \\ -1.5 \end{pmatrix} \]

Then:

\[ \delta\Pi_{\text{hedged}} = \frac{1}{2}\begin{pmatrix} 3 & -2 \end{pmatrix}\begin{pmatrix} 5.0 \\ -1.5 \end{pmatrix} = \frac{1}{2}(15.0 + 3.0) = \frac{1}{2}(18.0) = 9.0 \]

Decomposition by component:

• From \(\Gamma_{11}\): \(\frac{1}{2}\Gamma_{11}(\delta S_1)^2 = \frac{1}{2}(2.0)(3)^2 = 9.0\)
• From \(\Gamma_{22}\): \(\frac{1}{2}\Gamma_{22}(\delta S_2)^2 = \frac{1}{2}(1.5)(-2)^2 = 3.0\)
• From \(\Gamma_{12}\): \(\Gamma_{12}\,\delta S_1\,\delta S_2 = 0.5(3)(-2) = -3.0\)

(The cross-gamma term appears twice in the sum due to symmetry: \(\Gamma_{12}\delta S_1\delta S_2 + \Gamma_{21}\delta S_2\delta S_1 = 2 \times 0.5 \times 3 \times (-2) = -6.0\), but in the quadratic form this is already captured as \(\Gamma_{12}\delta S_1\delta S_2 = -3.0\) with the factor of 2 from symmetry included.)

Total: \(9.0 + 3.0 + (-3.0) = 9.0\), confirming the matrix computation.

The cross-gamma reduces the P&L by \(\$3.00\) because the assets moved in opposite directions (\(\delta S_1 > 0\), \(\delta S_2 < 0\)) while \(\Gamma_{12} > 0\).

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Exercise 2. A basket call on \(B = 0.6S_1 + 0.4S_2\) has gamma \(\frac{\partial^2 C}{\partial B^2} = 0.03\). Compute the cross-gamma \(\Gamma_{12} = w_1 w_2 \frac{\partial^2 C}{\partial B^2}\) and the diagonal gammas \(\Gamma_{11}\) and \(\Gamma_{22}\). Verify that the gamma matrix is positive semidefinite.

Solution to Exercise 2

For a basket call on \(B = w_1 S_1 + w_2 S_2\) with \(w_1 = 0.6\), \(w_2 = 0.4\), and \(\frac{\partial^2 C}{\partial B^2} = 0.03\):

Cross-gamma:

\[ \Gamma_{12} = w_1 w_2 \frac{\partial^2 C}{\partial B^2} = (0.6)(0.4)(0.03) = 0.0072 \]

Diagonal gammas:

\[ \Gamma_{11} = w_1^2 \frac{\partial^2 C}{\partial B^2} = (0.6)^2(0.03) = 0.0108 \]

\[ \Gamma_{22} = w_2^2 \frac{\partial^2 C}{\partial B^2} = (0.4)^2(0.03) = 0.0048 \]

Positive semidefiniteness. The gamma matrix is:

\[ \boldsymbol{\Gamma} = \frac{\partial^2 C}{\partial B^2}\begin{pmatrix} w_1^2 & w_1 w_2 \\ w_1 w_2 & w_2^2 \end{pmatrix} = 0.03\begin{pmatrix} 0.36 & 0.24 \\ 0.24 & 0.16 \end{pmatrix} \]

This is a rank-1 matrix: \(\boldsymbol{\Gamma} = 0.03\,\mathbf{w}\mathbf{w}^T\) where \(\mathbf{w} = (0.6, 0.4)^T\). A rank-1 outer product \(\mathbf{w}\mathbf{w}^T\) is always positive semidefinite (its eigenvalues are \(\|\mathbf{w}\|^2 > 0\) and \(0\)). Multiplying by the positive scalar \(0.03\) preserves positive semidefiniteness.

Alternatively, verify: \(\det(\boldsymbol{\Gamma}) = 0.03^2(0.36 \times 0.16 - 0.24^2) = 0.0009(0.0576 - 0.0576) = 0\), and \(\operatorname{tr}(\boldsymbol{\Gamma}) = 0.03(0.36 + 0.16) = 0.0156 > 0\). Both eigenvalues are non-negative, confirming positive semidefiniteness.

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Exercise 3. Full gamma neutralization for \(d = 3\) assets requires \(d(d+1)/2 = 6\) hedging instruments. If only 3 single-asset options are available (one per underlying), which gamma matrix entries can be neutralized and which remain unhedged? Describe the residual risk.

Solution to Exercise 3

For \(d = 3\) assets, the gamma matrix has \(3(3+1)/2 = 6\) independent entries: \(\Gamma_{11}\), \(\Gamma_{22}\), \(\Gamma_{33}\), \(\Gamma_{12}\), \(\Gamma_{13}\), \(\Gamma_{23}\).

With 3 single-asset options (one per underlying), each option \(l\) has a gamma matrix with only one nonzero entry: \(\Gamma_{ll}^{(l)} \neq 0\) and all other entries zero (since single-asset options have zero cross-gamma).

What can be neutralized: The three diagonal entries \(\Gamma_{11}\), \(\Gamma_{22}\), \(\Gamma_{33}\) can each be independently neutralized by choosing appropriate positions in the corresponding single-asset options.

What remains unhedged: The three cross-gamma entries \(\Gamma_{12}\), \(\Gamma_{13}\), \(\Gamma_{23}\) cannot be affected by single-asset options at all. These entries remain at their original values.

Residual risk. The unhedged cross-gammas create P&L exposure to correlated moves:

\[ \delta\Pi_{\text{residual}} \approx \Gamma_{12}\,\delta S_1\,\delta S_2 + \Gamma_{13}\,\delta S_1\,\delta S_3 + \Gamma_{23}\,\delta S_2\,\delta S_3 \]

This residual risk is driven by the realized correlations between the assets. To hedge it, the trader would need multi-asset derivatives (e.g., basket options, spread options, or correlation swaps) that generate nonzero off-diagonal gamma entries.

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Exercise 4. The correlation P&L is \(\frac{1}{2}\sum_{i \neq j}\Gamma_{ij}S_iS_j\sigma_i\sigma_j(\rho_{ij}^{\text{realized}} - \rho_{ij}^{\text{model}})\,\delta t\). For a two-asset portfolio with \(\Gamma_{12} = 0.5\), \(S_1 = S_2 = 100\), \(\sigma_1 = \sigma_2 = 0.20\), compute the daily P&L impact if the realized correlation is \(\rho^{\text{real}} = 0.8\) while the model assumes \(\rho^{\text{model}} = 0.6\).

Solution to Exercise 4

The correlation P&L formula for two assets (the sum over \(i \neq j\) gives two terms \(\Gamma_{12}\) and \(\Gamma_{21}\), which are equal):

\[ \text{P\&L}_{\text{corr}} = \Gamma_{12} S_1 S_2 \sigma_1 \sigma_2 (\rho^{\text{real}} - \rho^{\text{model}})\,\delta t \]

(The factor of \(\frac{1}{2}\) in the sum and the two symmetric terms combine to give a single factor of 1.)

Substituting \(\Gamma_{12} = 0.5\), \(S_1 = S_2 = 100\), \(\sigma_1 = \sigma_2 = 0.20\), \(\rho^{\text{real}} = 0.8\), \(\rho^{\text{model}} = 0.6\), and \(\delta t = 1/252\):

\[ \text{P\&L}_{\text{corr}} = 0.5 \times 100 \times 100 \times 0.20 \times 0.20 \times (0.8 - 0.6) \times \frac{1}{252} \]

\[ = 0.5 \times 10000 \times 0.04 \times 0.2 \times \frac{1}{252} \]

\[ = 0.5 \times 80 \times \frac{1}{252} = \frac{40}{252} \approx \$0.159 \text{ per day} \]

The positive cross-gamma combined with realized correlation exceeding the model correlation produces a small daily gain. Over a month (21 trading days), this would accumulate to approximately \(\$3.33\). While small in absolute terms, this systematic bias compounds and represents a genuine model risk if the correlation mismatch persists.

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Exercise 5. Using the worked example portfolio (\(\boldsymbol{\Gamma} = \begin{pmatrix} 3.44 & 0.16 \\ 0.16 & -1.30 \end{pmatrix}\)), compute the variance of the delta-hedged P&L over one day assuming \(\sigma_1 = 0.25\), \(\sigma_2 = 0.30\), \(\rho_{12} = 0.5\), \(S_1 = 100\), \(S_2 = 80\). Which term dominates: the diagonal gammas or the cross-gamma?

Solution to Exercise 5

From the worked example: \(\boldsymbol{\Gamma} = \begin{pmatrix} 3.44 & 0.16 \\ 0.16 & -1.30 \end{pmatrix}\), \(\sigma_1 = 0.25\), \(\sigma_2 = 0.30\), \(\rho_{12} = 0.5\), \(S_1 = 100\), \(S_2 = 80\), \(\delta t = 1/252\).

Using the two-asset variance formula, define the dollar-gamma quantities:

• \(a_1 = \Gamma_{11}S_1^2\sigma_1^2 = 3.44 \times 100^2 \times 0.25^2 = 3.44 \times 10000 \times 0.0625 = 2150\)
• \(a_2 = \Gamma_{22}S_2^2\sigma_2^2 = (-1.30) \times 80^2 \times 0.30^2 = -1.30 \times 6400 \times 0.09 = -748.8\)
• \(a_{12} = \Gamma_{12}S_1 S_2 \sigma_1\sigma_2 = 0.16 \times 100 \times 80 \times 0.25 \times 0.30 = 0.16 \times 600 = 96\)

The variance of the hedged P&L (dropping the \((\delta t)^2\) factor and restoring it at the end):

\[ \operatorname{Var}(\delta\Pi_{\text{hedged}}) \approx \left[\frac{1}{2}a_1^2 + 2\rho_{12}^2 a_1 a_2 \cdot \frac{1}{2} + \frac{1}{2}a_2^2 + 2a_{12}^2(1+\rho_{12}^2)\right]\frac{(\delta t)^2}{4} \]

Using the simplified version for the dominant contributions:

• \(\frac{1}{2}a_1^2 = \frac{1}{2}(2150)^2 = 2{,}311{,}250\)
• \(\frac{1}{2}a_2^2 = \frac{1}{2}(748.8)^2 = 280{,}351\)
• \(2a_{12}^2(1+\rho_{12}^2) = 2(96)^2(1+0.25) = 2(9216)(1.25) = 23{,}040\)
• Cross term: \(2\rho_{12}^2 \cdot \frac{1}{2} \cdot a_1 \cdot a_2 = 0.25 \times 2150 \times (-748.8) = -402{,}480\) (this represents the \(\Gamma_{11}\Gamma_{22}\) interaction)

The \(\Gamma_{11}\) diagonal term (\(\sim 2.3M\)) clearly dominates, followed by the \(\Gamma_{22}\) term (\(\sim 280K\)). The cross-gamma contribution (\(\sim 23K\)) is relatively small, roughly \(1\%\) of the total variance. This is expected because \(|\Gamma_{12}| = 0.16\) is much smaller than \(|\Gamma_{11}| = 3.44\) and \(|\Gamma_{22}| = 1.30\).

The diagonal gammas dominate the hedging error variance.

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Exercise 6. A risk manager sets limits of \(|\Gamma_{ii}S_i^2/2| \leq \$500\) per asset and \(|\Gamma_{ij}S_iS_j| \leq \$200\) per pair. For the worked example portfolio, check which limits are satisfied and which are breached. Propose a partial hedging plan that brings all exposures within limits using the fewest instruments.

Solution to Exercise 6

From the worked example: \(\boldsymbol{\Gamma} = \begin{pmatrix} 3.44 & 0.16 \\ 0.16 & -1.30 \end{pmatrix}\), \(S_1 = 100\), \(S_2 = 80\).

Check diagonal gamma limits (\(|\Gamma_{ii}S_i^2/2| \leq \$500\)):

• Asset 1: \(|\Gamma_{11}S_1^2/2| = |3.44 \times 100^2 / 2| = |3.44 \times 5000| = \$17{,}200\) --- breached (exceeds \(\$500\) by a factor of \(34\times\))
• Asset 2: \(|\Gamma_{22}S_2^2/2| = |-1.30 \times 80^2 / 2| = 1.30 \times 3200 = \$4{,}160\) --- breached (exceeds \(\$500\) by a factor of \(8.3\times\))

Check cross-gamma limit (\(|\Gamma_{ij}S_iS_j| \leq \$200\)):

• Pair (1,2): \(|\Gamma_{12}S_1 S_2| = |0.16 \times 100 \times 80| = \$1{,}280\) --- breached (exceeds \(\$200\) by a factor of \(6.4\times\))

All three limits are breached.

Partial hedging plan. To bring all exposures within limits using the fewest instruments:

1. Hedge \(\Gamma_{11}\) with a single-asset option on asset 1. Target residual \(|\Gamma_{11}^{\text{res}}| \leq 2 \times 500 / 100^2 = 0.10\). Need to reduce \(\Gamma_{11}\) from \(3.44\) to at most \(0.10\), requiring an option position contributing \(\Gamma \approx -3.34\).

2. Hedge \(\Gamma_{22}\) with a single-asset option on asset 2. Target residual \(|\Gamma_{22}^{\text{res}}| \leq 2 \times 500 / 80^2 = 0.156\). Need to reduce \(|\Gamma_{22}|\) from \(1.30\) to at most \(0.156\), requiring an option position contributing \(\Gamma \approx +1.14\).

3. Hedge \(\Gamma_{12}\) with a multi-asset instrument (e.g., a basket or spread option on assets 1 and 2). Target residual \(|\Gamma_{12}^{\text{res}}| \leq 200 / (100 \times 80) = 0.025\). Need to reduce \(\Gamma_{12}\) from \(0.16\) to at most \(0.025\).

This requires a minimum of 3 instruments: one single-asset option per underlying plus one multi-asset derivative. After adding these instruments, re-delta-hedge both underlyings with shares. If the multi-asset instrument is unavailable, the first two instruments (single-asset options) bring the two diagonal exposures within limits, leaving only the cross-gamma breach, which may be accepted as residual risk given its smaller relative magnitude.