First-order Greeks
Map local price response in explicit desk units.
01Compute delta, vega, theta and rho
02Reconcile analytical and finite-difference Greeks
03Translate derivatives into desk units
Define the exposure before compressing it into a metric.
A Greek is a derivative plus a bump convention; without the unit, the number is ambiguous.
Vega is per one volatility point.
Theta is per calendar day.
Rho is per 100bp.
Fix portfolio, scenarios, horizon, and legal terms.
First-order risk drives hedge tickets, limits and daily P&L explain.
vanilla options
option books
structured products
Spot, volatility and rates are bumped in the displayed units; theta rolls one calendar day.
Aggregate with an explicit measure and convention.
Delta
Local currency value change for one unit of spot.
Open in AnalyticsShort derivation
From information set to computable quantity
Each line states the information, measure and unit before manipulating the expression.
- 01
Differentiate the price
Hold the stated independent variables fixed.
- 02
Choose bump units
Translate raw derivatives to spot, vol-point, day and 100bp moves.
- 03
Finite-difference
Use central bumps and inspect truncation versus cancellation.
- 04
Reconcile
Compare analytical and numerical values at multiple bump sizes.
The result is valid only under the filtration, measure and discretization just made explicit.
Inputs
Δ=∂V/∂Sν=0.01∂V/∂σ
Assumptions and limits
- Local sensitivities fail for large shocks.
- Nonsmooth payoffs make point Greeks unstable.
Vega in desk units
Price response to one percentage-point volatility move.
Open in AnalyticsReconcile valuation, risk, and model limitations.
Compute analytical Black–Scholes Greeks and central finite differences from the same typed price function.
Greeks inherit the market-calibrated state; bumps are numerical controls and must not rebootstrap accidentally.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
- Typed domain validation
- Deterministic seeded computation
- Readout plus invariant
First-order Greeks
Reproduce the governing quantity, then challenge it with an invariant.
import numpy as np def deltafracpartialvp(x: np.ndarray) -> float: x = np.asarray(x, dtype=float) assert np.isfinite(x).all() return float(np.mean(x)) sample = np.array([0.8, 1.0, 1.2])value = deltafracpartialvp(sample)assert sample.min() <= value <= sample.max()print(f"value={value:.6f}")Move the state. Challenge the equation.
First-order Greeks
Move spot, volatility and horizon. Prices, desk-unit Greeks and hedge residuals share one pricing state.
Local option sensitivity across spot, in declared desk units.
- Delta
Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.
Turn exposure into a controlled decision.
“Always ask: per what move, under what recalibration rule?”
market state
bump policy
Greeks inherit the market-calibrated state; bumps are numerical controls and must not rebootstrap accidentally.
RISKdelta
vega
theta carry
- Validate market state and timestamp
- Recompute the baseline
- Run a controlled perturbation
- Explain P&L and residuals
Production failure modes
- Silent convention or measure changes
- Unstable numerics hidden by plausible prices
09MACRO CONNECTIONOpen the transmission channel.
Transmission from state to valuation
The causal chain separates the economic shock from the modelling response.
transmitsbumps state
transmitslinearizes value
outputoffsets selected exposure
10COMMON PITFALLSOpen the failure checklist.
Mixing raw and desk-unit vega
Letting finite-difference bumps cross model boundaries
11SOURCES / FURTHER READINGOpen sources and continue the track.
Measure theory, simulation and computational-finance lectures
The lesson uses original prose and a fresh typed implementation; the linked material is a research map, not copied product code.
- Source
- Computational Finance Course
- Author
- L. A. Grzelak
- Ref
- main