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

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.

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