Delta-Gamma-Vega Hedging With Two Options

Options Pricing · Medium · Free problem

You hold a portfolio whose current Greeks are $(0, 0, V_0)$ for delta, gamma, and vega respectively. You can trade three instruments:

  • The underlying stock, with Greek vector $(1, 0, 0)$.
  • Option 1, with Greek vector $(\Delta_1, \Gamma_1, \mathcal{V}_1)$.
  • Option 2, with Greek vector $(\Delta_2, \Gamma_2, \mathcal{V}_2)$.

Let $x_S, x_1, x_2$ be the number of units you trade in the stock, Option 1, and Option 2 respectively (positive = buy).

  1. Write the $3 \times 3$ linear system for $(x_S, x_1, x_2)$ that makes the combined portfolio delta-neutral, gamma-neutral, and vega-neutral.
  1. Give necessary and sufficient conditions on $(\Gamma_1, \mathcal{V}_1)$ and $(\Gamma_2, \mathcal{V}_2)$ for a unique solution to exist.
  1. If no exact solution exists (i.e., the gamma-vega subsystem is singular), formulate a least-squares or constrained optimization objective that minimizes residual gamma and vega exposure subject to maintaining delta neutrality.

Hints

  1. The stock has zero gamma and zero vega, so it can only help with delta. The real hedging problem is a
\times 2$ system in the option positions.
  • Write out the \times 2$ gamma-vega subsystem and think about when its determinant $\Gamma_1 \mathcal{V}_2 - \Gamma_2 \mathcal{V}_1$ is nonzero.
  • When the subsystem is singular, introduce a quadratic objective in the residual Greeks and enforce delta neutrality as a hard constraint.

    • Worked Solution

    How to Think About It: This is a bread-and-butter desk problem. You have unwanted gamma and vega in your book and need to flatten them out. The stock gives you a free delta dial -- it has no gamma or vega -- so the real question is whether your two options can independently zero out gamma and vega. That is a \times 2$ linear system, and it has a unique solution exactly when the two options' gamma-vega profiles are not proportional to each other. Once you solve for the option positions, you mop up the leftover delta with stock. If the \times 2$ system is singular (both options have the same gamma-to-vega ratio), you cannot perfectly hedge both, and you fall back to optimization.

    Quick Sanity Checks: - The stock should only appear in the delta equation. Its gamma and vega are zero, so it cannot help with those. - If both options are identical (same Greeks), you effectively have one instrument for two constraints -- you are stuck. - If $\Gamma_1 \mathcal{V}_2 \neq \Gamma_2 \mathcal{V}_1$, the options span the gamma-vega plane, and a unique solution exists.

    Derivation:

    *Part (i): The linear system*

    After trading, the portfolio Greeks become:

    $\text{Delta: } 0 + x_S \cdot 1 + x_1 \Delta_1 + x_2 \Delta_2 = 0$ $\text{Gamma: } 0 + x_S \cdot 0 + x_1 \Gamma_1 + x_2 \Gamma_2 = 0$ $\text{Vega: } V_0 + x_S \cdot 0 + x_1 \mathcal{V}_1 + x_2 \mathcal{V}_2 = 0$

    Note the initial delta and gamma are zero, and the initial vega is $V_0$. In matrix form:

    $\begin{pmatrix} 1 & \Delta_1 & \Delta_2 \\ 0 & \Gamma_1 & \Gamma_2 \\ 0 & \mathcal{V}_1 & \mathcal{V}_2 \end{pmatrix} \begin{pmatrix} x_S \\ x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ -V_0 \end{pmatrix}$

    The system decouples. Rows 2 and 3 determine $x_1$ and $x_2$ independently of $x_S$. Then row 1 gives $x_S$.

    *Part (ii): Existence of a unique solution*

    The $3 \times 3$ matrix is block upper-triangular with a

    $ in the top-left and the
    \times 2$ block:

    $M = \begin{pmatrix} \Gamma_1 & \Gamma_2 \\ \mathcal{V}_1 & \mathcal{V}_2 \end{pmatrix}$

    The full system has a unique solution if and only if $\det(M) \neq 0$, i.e.,

    $\Gamma_1 \mathcal{V}_2 - \Gamma_2 \mathcal{V}_1 \neq 0$

    This means the two options must have linearly independent $(\Gamma, \mathcal{V})$ vectors. In practical terms, their gamma-to-vega ratios must differ: $\Gamma_1 / \mathcal{V}_1 \neq \Gamma_2 / \mathcal{V}_2$ (assuming nonzero vegas). When this holds, the unique solution is:

    $x_1 = \frac{V_0 \, \Gamma_2}{\Gamma_1 \mathcal{V}_2 - \Gamma_2 \mathcal{V}_1}, \quad x_2 = \frac{-V_0 \, \Gamma_1}{\Gamma_1 \mathcal{V}_2 - \Gamma_2 \mathcal{V}_1}$

    $x_S = -(x_1 \Delta_1 + x_2 \Delta_2)$

    *Part (iii): Optimization when the system is singular*

    When $\Gamma_1 \mathcal{V}_2 = \Gamma_2 \mathcal{V}_1$, the two options are collinear in the gamma-vega plane. You cannot zero out both. The natural fallback is:

    $\min_{x_1, x_2} \; (x_1 \Gamma_1 + x_2 \Gamma_2)^2 + \lambda \, (V_0 + x_1 \mathcal{V}_1 + x_2 \mathcal{V}_2)^2$

    subject to the constraint that delta is neutralized:

    $x_S + x_1 \Delta_1 + x_2 \Delta_2 = 0$

    Here $\lambda > 0$ is a weight reflecting the relative importance of vega risk versus gamma risk. Setting $\lambda = 1$ weights them equally in squared units. In practice, you would scale each term by the inverse variance of the respective Greek's P&L impact, or simply by notional sensitivity.

    Alternatively, you can enforce one constraint exactly and minimize the residual of the other. For example, enforce gamma neutrality exactly ($x_1 \Gamma_1 + x_2 \Gamma_2 = 0$) and minimize the squared residual vega:

    $\min_{x_1, x_2} \; (V_0 + x_1 \mathcal{V}_1 + x_2 \mathcal{V}_2)^2 \quad \text{s.t.} \quad x_1 \Gamma_1 + x_2 \Gamma_2 = 0, \; x_S = -(x_1 \Delta_1 + x_2 \Delta_2)$

    Practical Interpretation: On a trading desk, the key decision is which options to use for hedging. You want options whose gamma-vega profile is as different as possible from each other -- typically a short-dated option (high gamma, low vega) paired with a long-dated option (low gamma, high vega). If you only have options with similar maturities and strikes, their gamma-vega ratios will be close, and the \times 2$ system will be nearly singular -- meaning your hedge positions blow up in size and become unstable. This is why desks monitor the condition number of the hedging matrix, not just whether a solution exists.

    Answer: The system decouples into a \times 2$ gamma-vega block (solved by the two options) and a delta equation (solved by the stock). A unique solution exists iff $\Gamma_1 \mathcal{V}_2 \neq \Gamma_2 \mathcal{V}_1$, i.e., the two options have linearly independent gamma-vega profiles. When the system is singular, minimize a weighted sum of squared residual gamma and vega subject to delta neutrality.

    Intuition

    This problem captures the fundamental tension in options hedging: you need as many independent instruments as you have risk dimensions. Stock gives you one dial (delta), but gamma and vega require options, and not all options are equally useful. Two options with the same gamma-to-vega ratio are essentially redundant for hedging purposes -- they move the same way in the gamma-vega plane. The linear independence condition $\Gamma_1 \mathcal{V}_2 \neq \Gamma_2 \mathcal{V}_1$ is the mathematical expression of "use options with different risk profiles." In practice this means pairing short-dated options (gamma-heavy) with long-dated ones (vega-heavy).

    The fallback to least-squares optimization when the exact system is singular is not just a mathematical curiosity -- it is the everyday reality on a desk. You rarely have the perfect hedging instruments available, so you minimize residual risk subject to hard constraints on the exposures you care most about (usually delta first, then gamma). This connects to broader portfolio optimization: every hedging problem is really a constrained optimization problem, and understanding the rank of your hedging matrix tells you how many independent risks you can eliminate.

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