Portfolio Greeks Aggregation

Setup

From Single Option to Book-Level Risk

A single option's Greeks are straightforward to compute. A trading book is a collection of hundreds to thousands of positions across different underlyings, strikes, maturities, and option types. Aggregating these into meaningful, actionable risk metrics requires:

1. Netting: which positions offset each other.
2. Bucketing: grouping by the risk factor that drives each position.
3. Ladder construction: expressing sensitivities as a vector indexed by market pillars.
4. Cross-Greeks: second-order sensitivities that matter for books with large vega.

None of these is trivial. A common error is to aggregate Greeks naively and miss the distinction between a flat ladder (no net risk) and a butterfly (zero net vega but large volga). The two have identical total vega but completely different risk profiles.

Notation and Sign Conventions

Throughout:

• $V = \sum_{k} n_k C_k$: portfolio value, where $n_k$ is the position (positive = long, negative = short) and $C_k$ the per-unit option price.
• Portfolio Greeks are additive for positions in the same underlying: $\Delta_{\mathrm{portfolio}} = \sum_k n_k \Delta_k$. This is exact under the Black-Scholes flat-smile model; under smile models, the aggregation is approximate (because delta depends on the smile model, which is a portfolio-level calibration).
• All Greeks computed at market mid-prices under a consistent model (e.g., Black-Scholes with the implied vol for each option).

Netting and Bucketing

Delta Netting

Delta is aggregated across all positions in the same underlying:

$\Delta_{\mathrm{net}} = \sum_{k} n_k \Delta_k.$

A delta-neutral book requires $\Delta_{\mathrm{net}} = 0$, achieved by holding $-\Delta_{\mathrm{net}}$ shares of the underlying. Note: delta netting applies within a single underlying. Across underlyings, deltas cannot be netted (a long delta in EURUSD and a short delta in GBPUSD are distinct risks).

Gamma Bucketing

Gamma is additive across positions in the same underlying:

$\Gamma_{\mathrm{net}} = \sum_{k} n_k \Gamma_k.$

However, the total net gamma is less informative than the gamma profile — the gamma as a function of spot. A long ATM straddle and a short OTM strangle may have similar total gammas but completely different profiles: the straddle has concentrated gamma at current spot, while the strangle has distributed gamma at the wings. The profile determines the P&L distribution under different spot moves.

Gamma bucketing by moneyness: divide positions into buckets by $\ln(K/F)$ (log-moneyness) and report $\Gamma_{\mathrm{bucket}}$ for each bucket. This reveals concentrations.

Theta Aggregation

$\Theta_{\mathrm{net}} = \sum_{k} n_k \Theta_k.$

For a delta-and-gamma-neutral book, the net theta is:

$\Theta_{\mathrm{net}} \approx -\frac{1}{2}\sigma^2 S^2 \Gamma_{\mathrm{net}},$

from the BS PDE. A flat gamma book has flat theta. A long gamma book pays theta.

Vega Ladder Construction

The Implied Vol Surface as a Risk Factor

The implied vol surface $\hat{\sigma}(K, T)$ is a function of two variables. The option book's P&L from a move in the surface is:

$\delta V_{\mathrm{vega}} = \sum_{i,j} \nu_{ij} \cdot \delta\hat{\sigma}(K_i, T_j),$

where $\nu_{ij} = \partial V / \partial\hat{\sigma}(K_i, T_j)$ is the vega ladder entry at pillar $(K_i, T_j)$. The vega ladder has dimensions (number of strike pillars) $\times$ (number of maturity pillars).

Constructing the Vega Ladder

For each surface pillar $(K_i, T_j)$:

$\nu_{ij} = \frac{V(\hat{\sigma} + \delta_{ij}\,\mathbf{e}_{ij}) - V(\hat{\sigma} - \delta_{ij}\,\mathbf{e}_{ij})}{2\,\delta_{ij}},$

where $\mathbf{e}_{ij}$ is the unit bump at pillar $(K_i, T_j)$ and $\delta_{ij} = 0.01$ (1 vol point). Each entry requires two full portfolio repricings (or one complex-step repricing). For a surface with $5 \times 7 = 35$ pillars, this is 70 repricings (or 35 complex steps) per risk run.

Term structure of vega. Summing across strikes for each maturity bucket:

$\nu_j = \sum_i \nu_{ij}, \qquad j = 1, \ldots, N.$

The vega term structure shows how the book's vol sensitivity is distributed across maturities — short-dated vega is more volatile (short-dated vol moves more) but also hedged more cheaply. Long-dated vega is more stable but harder to hedge (fewer liquid instruments at long maturities).

Vega-weighted by maturity: some desks report $\nu_j^* = \nu_j / \sqrt{T_j}$ (vega normalised by the vol scaling $\sqrt{T}$), which gives a "volatility-normalised" sensitivity comparable across maturities.

Cross-Greeks: Vanna and Volga in Portfolio Context

Why Cross-Greeks Matter

For a portfolio with significant vega, the second-order P&L from simultaneous moves in spot and vol is:

$\delta V \approx \Delta \cdot \delta S + \frac{1}{2}\Gamma \cdot (\delta S)^2 + \nu \cdot \delta\sigma + \mathrm{Vanna} \cdot \delta S \cdot \delta\sigma + \frac{1}{2}\mathrm{Volga} \cdot (\delta\sigma)^2.$

For equity options, $\delta S$ and $\delta\sigma$ are negatively correlated (leverage effect: when spot falls, vol rises). This means the vanna term $\mathrm{Vanna} \cdot \delta S \cdot \delta\sigma$ is typically negative for a long call book (positive vanna, negative correlation → negative cross-term P&L on a vol spike combined with a spot drop).

Vanna at Portfolio Level

Portfolio vanna is additive:

$\mathrm{Vanna}_{\mathrm{net}} = \sum_k n_k \mathrm{Vanna}_k = -\sum_k n_k \frac{n(d_1^k)\,d_2^k}{\sigma_k}.$

Sign structure:

• OTM calls ($d_2 < 0$): positive vanna. As vol rises, delta increases (options become more likely to expire ITM).
• ITM calls ($d_2 > 0$): negative vanna. As vol rises, delta decreases (delta moves toward 1 more slowly).
• At-the-money ($d_2 \approx 0$): vanna $\approx 0$.

A risk-reversal (long OTM call, short OTM put) is long vanna: its delta increases when vol rises and decreases when vol falls — consistent with negative spot-vol correlation in equity markets, which means the risk-reversal profits from the correlation.

Volga at Portfolio Level

$\mathrm{Volga}_{\mathrm{net}} = \sum_k n_k \frac{\nu_k \cdot d_1^k d_2^k}{\sigma_k}.$

Sign structure:

• OTM options (both $|d_1|$ and $|d_2|$ large, same sign): positive volga. These options have convex prices in vol — they benefit from large vol moves in either direction.
• ATM options ($d_1 \approx \sigma\sqrt{\tau}/2$, $d_2 \approx -\sigma\sqrt{\tau}/2$): volga proportional to $-(\sigma\sqrt{\tau}/2)^2 < 0$... wait, for ATM: $d_1 d_2 \approx -\sigma^2\tau/4 < 0$, so volga $< 0$ at-the-money.

Correction: recall $d_2 = d_1 - \sigma\sqrt{\tau}$, so for ATM ($S = K$, $r = 0$): $d_1 = \sigma\sqrt{\tau}/2 > 0$, $d_2 = -\sigma\sqrt{\tau}/2 < 0$. Product $d_1 d_2 = -(\sigma\sqrt{\tau}/2)^2 < 0$. So ATM options have negative volga under BS — surprising, and correct: the ATM vega is maximised, so its second derivative in vol is negative (it's a maximum). OTM options have positive volga (their vega is increasing in vol at OTM strikes).

A strangle (long OTM call + long OTM put) is long volga: it profits from large vol moves in either direction, making it a pure vol convexity position.

Vanna-Volga Pricing of Exotics

The vanna-volga method is a practitioner approach for pricing FX exotic options. Given the prices of three vanilla options (ATM, 25-delta call, 25-delta put), one can exactly replicate the vanna and volga of any exotic by a linear combination of these three. The cost of this replication in the market is used as a correction to the BS price:

$V_{\mathrm{exotic}} = V_{\mathrm{BS}} + \text{vanna cost} + \text{volga cost}.$

The method is approximate (it assumes the third-order terms are negligible) but widely used in FX options for first-cut pricing of barrier options when a full stochastic vol calibration is unavailable.

DV01 and Fixed Income Aggregation

For books with both equity and rates exposure (e.g., convertible bonds, equity-linked notes), the interest rate sensitivity is reported via DV01:

$\mathrm{DV01} = -\frac{\partial V}{\partial r} \times 0.0001,$

expressed as the P&L from a 1bp decrease in rates (hence the negative sign: lower rates → higher bond values → positive DV01).

DV01 ladder: sensitivity to a 1bp shift at each tenor point of the yield curve (1m, 3m, 6m, 1y, 2y, 5y, 10y, 20y, 30y). A bond with 10-year maturity has DV01 concentrated in the 10y bucket; a swap portfolio has DV01 distributed across all its fixed and floating legs.

Key vs. parallel: a DV01 ladder shows the key-rate sensitivities (shift only one tenor). The total parallel DV01 (sum of all entries) measures sensitivity to a parallel shift of the entire curve.

Limitations

Additivity under non-flat smile. Black-Scholes Greeks are additive because the model is linear. Under a stochastic vol model (Heston, SABR), the correct delta is model-dependent and not simply the sum of individual BS deltas. Specifically, the model delta includes a correction for the smile:

$\Delta_{\mathrm{model}} = \Delta_{\mathrm{BS}} + \nu \cdot \frac{\partial \hat{\sigma}}{\partial S},$

where $\partial\hat{\sigma}/\partial S$ captures how the implied vol changes as the spot moves (the smile dynamics). For a portfolio under smile-adjusted deltas, individual option deltas are not additive in a simple way — the whole portfolio must be re-priced under the model.

Cross-position netting of higher-order Greeks. Vanna and volga are additive in principle, but the hedging interpretation of net vanna/volga requires knowing the joint distribution of $\delta S$ and $\delta\sigma$. A book with zero net vanna but large individual vanna positions in different strikes is not truly vanna-neutral — the hedges may cancel on average but not in tail scenarios.

Correlation risk. The interaction term Vanna $\cdot \delta S \cdot \delta\sigma$ assumes a correlation between spot and vol moves. In a crisis, this correlation can change dramatically (e.g., in March 2020, the leverage effect was extreme). Correlation risk is a form of model risk not captured by standard Greeks.

Interview Angle

L1. What is a vega ladder? How many repricings are required to construct a vega ladder over a surface with 5 maturities and 7 strikes using central differences?

A vega ladder is the matrix of sensitivities $\partial V/\partial\hat{\sigma}(K_i, T_j)$ for each surface pillar. For central differences: 2 repricings per pillar $\times$ 35 pillars = 70 repricings. For complex-step: 1 complex repricing per pillar = 35 complex (more expensive per reprice, but fewer). Practical desks use AAD or analytic Greeks where available; bump-reval for the remaining non-analytic exotic positions.

L2. Explain the sign of vanna for OTM calls. Why does a risk-reversal have positive vanna, and why is this relevant for equity options under the leverage effect?

OTM call: $d_2 < 0$ (the risk-neutral probability of expiring ITM is below 0.5). Vanna = $-n(d_1)d_2/\sigma > 0$. As vol rises, $d_1$ and $d_2$ both shift toward zero (the distribution widens), increasing $N(d_1)$ (the delta) for an OTM call. So long OTM calls are long vanna: their delta is positively correlated with vol.

A risk-reversal (long OTM call, short OTM put) combines positive call vanna and negative put vanna. But for an OTM put, $d_2 < 0$ as well (both calls and puts at the same |moneyness| have the same vanna sign for the BS formula applied separately — the difference is in the sign of the position). Net: long OTM call vanna + short OTM put vanna (put vanna is also positive — but the short position flips sign). Actually a risk reversal where you're long OTM call and short OTM put — the call contributes positive vanna, the put contributes negative vanna (short put = short vanna) — so net vanna is long. In equity markets with the leverage effect ($\delta S < 0 \Rightarrow \delta\sigma > 0$): on a spot drop, the risk-reversal loses on vanna: positive vanna × (negative $\delta S$) × (positive $\delta\sigma$) = negative P&L. This is a well-known risk of risk-reversals in equity markets.

L3. Derive the portfolio delta under a sticky-delta smile model. Compare with the BS delta and explain why the two can differ by 0.1 or more for near-the-money options on volatile underlyings.

Under a sticky-delta smile model, implied vol $\hat{\sigma}$ depends on the moneyness ratio $K/S$ — not on absolute $K$. So as $S$ changes by $\delta S$, all strikes' implied vols change to keep $\hat{\sigma}(K/S)$ constant.

The chain rule gives: $\frac{\partial V}{\partial S} = \frac{\partial V_{\mathrm{BS}}}{\partial S}\Big|_{\hat{\sigma}=\text{const}} + \frac{\partial V_{\mathrm{BS}}}{\partial \hat{\sigma}} \cdot \frac{\partial \hat{\sigma}}{\partial S}.$

For a sticky-delta surface with skew slope $\partial\hat{\sigma}/\partial k = s$ (in log-moneyness $k = \ln(K/F)$):

$\frac{\partial \hat{\sigma}}{\partial S} = -\frac{s}{S} \quad \text{(moving spot up flattens the moneyness, changing the effective vol)}.$

So: $\Delta_{\mathrm{sticky-delta}} = N(d_1) - \nu \cdot s / S = N(d_1) - S\sqrt{\tau}\,n(d_1) \cdot s / S = N(d_1) - \sqrt{\tau}\,n(d_1)\,s$.

For ATM with $\sigma = 20\%$, $\tau = 0.25$ (3 months), $s = -0.3$ (a moderate equity skew of 3 vol points per 10% moneyness change): the correction term is $\sqrt{0.25} \cdot n(0) \cdot 0.3 \approx 0.5 \times 0.399 \times 0.3 \approx 0.06$. A 6-point delta correction is substantial — it moves the hedge ratio from 0.5 to 0.56 for an ATM call, changing the number of shares to hold by 12%.