Sensitivity ladders

The purpose of the sensitivity ladder is to increase the PNL calculation accuracy by computing the sensitivities at different market configurations.

Input data

The system is fed with a table of sensitivities shifted from the current market value.

For instance:

Shift Delta Gamma
-50% 9,944,227.53 -0.00019762
-25% 9,924,531.37 -492,746.23
-15% 9,501,104.56 -6,432,540.09
-10% 8,668,816.62 -18,453,313.35
-7% 7,569,003.25 -33,894,475.23
-5% 6,499,116.86 -42,468,327.80
-3% 5,306,289.78 -44,427,553.97
-1% 4,157,778.67 -41,418,870.50
0% 3,633,050.69 -38,673,906.17
1% 3,153,644.65 -35,250,054.12
3% 2,353,529.88 -27,317,737.12
5% 1,765,164.79 -19,914,252.52
7% 1,348,405.85 -14,333,377.94
10% 934,996.82 -9,143,072.19
15% 539,326.70 -5,111,151.27
25% 171,999.71 -2,095,135.72
50% 2,613.78 -59,994.40

Calculation theory

The purpose is to calculate the price based on the delta or gamma.

As we have:

We will have 

and also

Discrete calculation

Delta effect

We will sum the surface under the delta curve. The delta curve is formed by taking the closest delta value to 0.

In our sample:

  • Between -3% and -1% we take delta of -1%
  • Between -1% and 1% we take delta of 0%
  • Between 1% and 3% we take delta of 1%
  • Between 3% and 5% we take delta of 3%

So for PNL(4.3%) we will have the sum of:

  • Between 0% and 1% : 1% x delta of 0%
  • Between 1% and 3% : 2% x delta of 1%
  • Between 3% and 4.3% : 1.3% x delta of 3%

Gamma effect

The gamma expresses the slope of the delta curve.

It is sampled in the same way as delta.

So for the gamma part of the PNL(4.3%) we will have the sum of:

  • Between 0% and 1% : square of 1% x gamma of 0% divided by 2
  • Between 1% and 3% : square of 2% x gamma of 1% divided by 2
  • Between 3% and 4.3% : square of 1.3% x gamma of 3% divided by 2

note

The gamma can be implied by taking the slope between two deltas.

Implementation

Input files

Ladder pillar definition

Column name Content Sample
RiskClass The name of the risk class the ladder is intended to be used for Equity
ShifType The type of shift used to construct the ladder:

* S = Spot => Si = Li
* A = Absolute =>Si = Li + S
* R = Relative => Si = (Li + 1) * S
R
Scale The ladder scale to use. -0.5;-0.25;-0.15-;-0.1;-0.07;-0.05;-0.03;-0.01;0;0.01;0.03;0.05;0.07;0.1;0.15;0.25;0.5

Ladder file

Column name Content Sample
AsOfDate The current working date. 2018-09_28
TradeId The trade identifier. EQ_FUT_Merck & Co. b661100e
RiskClass Equity
RiskFactor The name of the risk axis Equity_Merck & Co._Repo rate
TenorLabels The list of the tenors 1Y;10Y
TenorDates The associated dates
MaturityLabels The list of maturities
MaturityDates The associated dates
Moneyness The list of moneyness
Order1 The matrix of the delta ladders The 1Y ladders; The 10Y ladders
Order2 The matrix of the gamma ladders, can be null

Datastore

The ladder definition is stored as is in a standalone store.

The ladder files are stored in the Ladder datastore.

For a scalar model, each line is split by tenor, maturity, and moneyness according to the scalar model of data.

In the standard model, the lines are stored with the bucket expansion.

There is a relation between the TradeSensitivities store and the Ladder store. So for delta plus gamma the store will be joined twice.

Cube

The Order1 and Order2 columns are populated by two native measures in the sensitivity cube. The cube will also have an analysis dimension on the ladder definition. So with an expand post-processor you can view the content of the ladder.

If there is no vanilla sensitivity, the ladder will not be displayed in the cube.

warning

We also need two extra dimensions that display the actual presence of the Order1 / Order2 ladder for the calculation.

Calculation logic

Vanilla sensitivity Ladder order 1 Ladder order 2 calculation
Delta / Vega No No Standard calculation
Delta / Vega Yes No calculation of the delta effect of the Ladder by using the order 1 values
Delta / Vega No Yes Not handled => Standard calculation
Delta / Vega Yes Yes calculation of the delta effect of the Ladder by using the order 1 values
Gamma / Volga No No Standard calculation
Gamma / Volga Yes No calculation of the gamma effect of the Ladder by using the order 1 values
Gamma / Volga No Yes calculation of the gamma effect of the Ladder by using the order 2 values
Gamma / Volga Yes Yes calculation of the gamma effect of the Ladder by using the order 2 values