Exercise: Vega-Neutral Portfolio Construction

Prerequisites: Vega, Delta

Problem

Setup: $S = 100, r = 0.05, q = 0$. Available instruments with $\sigma_{\text{imp}} = 0.20$:

• $C_1$: 3-month ATM call ($K = 100, T = 0.25$)
• $C_2$: 1-year ATM call ($K = 100, T = 1.0$)
• underlying stock at $100

1. Compute delta, gamma, vega of $C_1$ and $C_2$.

2. Construct a portfolio $P = \alpha C_1 + \beta C_2 + \gamma S$ that is delta-neutral AND vega-neutral. Express the constraints and solve for $(\alpha, \beta, \gamma)$, normalising $\alpha = 1$.

3. Compute the portfolio's gamma. Is it positive or negative? What does its sign tell you?

4. Scenario test. If implied vol rises from 20% to 21% uniformly across all maturities, verify that the portfolio's P&L is approximately zero (vega neutrality held).
5. Term-structure risk. If 3-month IV rises to 21% but 1-year IV stays at 20% (term-structure twist), compute the approximate P&L. Interpret.

Hint

Vega neutrality: $\alpha\mathcal{V}_1 + \beta\mathcal{V}_2 = 0$ (stock has zero vega). Delta neutrality: $\alpha\Delta_1 + \beta\Delta_2 + \gamma = 0$. Two equations, three unknowns, so fix $\alpha = 1$ and solve for $\beta, \gamma$.