Brownian Bridge™

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:

Netting: which positions offset each other.
Bucketing: grouping by the risk factor that drives each position.
Ladder construction: expressing sensitivities as a vector indexed by market pillars.
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.

Portfolio Greeks Aggregation