Topic 19 – Sensitivities (Buckets, Cross-Greeks, Risk Factor Mapping)

1. What a sensitivity actually is

Idea:

“If this risk factor moves a tiny bit, how much does my PV change?”

Mathematically (single factor):

$$\Delta \text{PV} \approx s \cdot \Delta f$$

• $f$: risk factor (forward price, rate, vol, FX, etc.).
• $s = \partial \text{PV} / \partial f$: sensitivity to that factor.

In vector form:

$$\Delta \text{PV} \approx \sum_i s_i \Delta f_i = s^\top \Delta f$$

On a desk:

• Sensitivities are used for:
  • Risk reports (bucket deltas, PV01, vega),
  • Limits,
  • Scenario P&L (sensitivities × shocks),
  • P&L explain (predicted P&L).

2. Buckets: turning curves into factor sets

We don’t hedge “the curve” as a continuum.

We define buckets:

• Tenor buckets:
  • e.g. Cal-26, Cal-27, Winter-28, Summer-28,
  • or 1Y, 2Y, 5Y, 10Y for rates.
• Product buckets:
  • Base vs peak,
  • Different locations (hub vs zone).

Each bucket is a risk factor. Sensitivities are:

• “Delta to Winter-27 base”,
• “PV01 to 5Y swap rate”,
• “Vega to 1Y ATM vol”.

Bucket choice is crucial:

• Too fine → noisy, unstable, reports unreadable.
• Too coarse → P&L not captured by risk factors.

3. Mapping trades to risk factors

Each trade has:

• A payoff that depends on some underlying primitives:
  • Forwards at different maturities,
  • Volatilities at different strikes/tenors,
  • FX, spreads, etc.

Library logic:

1. Define risk factor universe:
   • e.g. POWER_FWD[hub, tenor bucket], GAS_FWD[...], VOL[tenor, delta].
2. Each instrument maps to a subset of those factors.
3. Pricer computes sensitivities to those factors:
   • Analytic (closed-form Greeks) where possible,
   • Numeric (finite differences, AAD) elsewhere.

Portfolio sensitivities = sum over trades (netting across books).

4. Cross-Greeks and non-linearities

First-order sensitivities (delta, bucket PV01, vega) assume:

$$\Delta \text{PV} \approx \sum_i s_i \Delta f_i$$

But for non-linear books:

• Gamma: second-order sensitivity wrt one factor:
  • curvature in spot, forward, rate, etc.
• Cross-gamma: second derivative wrt two factors:
  • e.g. joint curvature in power and gas prices (spark spread risk),
  • joint curvature in two tenors (calendar spread).

You rarely publish a full gamma tensor, but you should know:

• Large gamma/cross-gamma → linear hedges leak for large moves.
• Structured power/gas (storage, swing, spark) often carry spread gamma:
  • They like volatility in differences between factors.

5. How sensitivities are actually used

1. Risk reports:
   • Show bucket deltas/PV01/vega for each book and desk.
   • Aggregate to legal entity and firm.
2. Scenario P&L (fast):
   • For small shocks, approximate scenario P&L as: $$\Delta \text{PV}_{\text{scenario}} \approx s^\top \Delta f_{\text{scenario}}$$
   • For big stresses, revalue fully.
3. P&L explain:
   • Take yesterday’s sensitivities and today’s factor moves:
     • Predicted P&L ≈ sensitivities × moves.
   • Compare vs realised P&L.

6. Interview soundbite

You can summarise like this:

“Sensitivities live in factor space. We define a consistent set of risk factors — forward curve buckets, vol buckets, spreads, basis — and map every trade to those. The pricers then produce deltas, PV01s, vegas, etc., and we net them across trades to get book-level risk. For non-linear power and gas products, cross-Greeks matter: books can have big gamma to curve shape and spreads, so linear hedging only works locally.”