Dollar Delta and Dollar Gamma - RiskMetrics Group Online User ...

Product Technical NoteDollar Delta and Dollar GammaJorge Minajorge.mina@riskmetrics.comSummary: This note describes the calculation of the dollar delta and dollar gamma statisticsKeywords: Greeks, sensitivities, hedge ratiosThis note specifies the calculations for the dollar delta and dollar gamma statistics. Dollar deltacan be roughly described as the delta hedge of the position expressed in dollars. In other words, itindicates how much of the underlying I have to buy/sell in order to delta hedge a position or groupof positions. Similarly, dollar gamma can be described as the change in the delta hedge (or dollardelta) when the underlying moves by a certain amount.We have not identified an industry standard for calculating these statistics. We have gotten requestsfrom different clients to incorporate variations in the calculations across two dimensions: the shift tobe applied to the underlying, and whether the statistics should be calculated as one-sided or two-sided(average of up and down moves) differences. Our implementation allows for any combination ofthese two parameters. In addition, we incorporate a third parameter that allows the user to stressthe underlying before calculating delta and gamma. In other words, the user can obtain the valuesthat delta and gamma would take if the underlying (i.e., equity prices, commodity prices, currencies,interest rates, spreads, or implied volatilities) moved by a certain amount. We divide this note in twosections: the first discusses the calculation of delta and gamma for equities, currencies, commodities,and volatilities; the second discusses the calculations for interest rates and spreads.1 Equities, currencies, commodities, and volatilitiesIn this section we will use to refer to , , , and . 11 has not been implemented in the system.

2. determines the shift to be applied to the underlying before calculating delta.In other words, if = 10 we calculate delta after applying a 10% shock tothe underlying. This is an optional parameter and its default value is 0.3. is a boolean that determines whether delta is calculated as a one-sided ortwo-sided difference ( = 1 means two-sided). This parameter is optionaland its default value is 1.Dollar delta is then defined aswhere = δ(h δ ,s,d δ ), (6)h δ = ,100(7)s = 100(8)d δ = . (9)1.2 Dollar gammaDollar gamma () has five parameters (four of which are optional):1. determines the shift to be applied in the calculation of dollar gammaexpressed as a percentage (=10 means that the shift is 10%.) Thisparameter is required and can take any value between −100 and ∞.2. determines the shift to be applied in the calculation of dollar deltaexpressed as a percentage (=10 means that the shift is 10%.) Thisparameter is optional and its default is . This parameter can take anyvalue between −100 and ∞.3. determines the shift to be applied to the underlying before calculatinggamma. In other words, if = 10 we calculate gamma after applying a10% shock to the underlying. This is an optional parameter and its default value is 0.3

4. is a boolean that determines whether gamma is calculated as a onesidedor two-sided difference ( = 1 means two-sided). This parameteris optional and its default value is 1.5. is a boolean that determines whether delta is calculated as a one-sided ortwo-sided difference ( = 1 means two-sided). This parameter is optionaland its default value is 1.Dollar gamma is then defined as = Ɣ(h δ ,h Ɣ ,s,d δ ,d Ɣ ), (10)whereh δ = ,100(11)h Ɣ = ,100(12)s = 100(13)d δ = , (14)d Ɣ = . (15)2 Interest rates and spreadsIn this section we will use to refer to and .Let us first define three auxiliary functions:• Value functionV(x)= for = x. (16)In other words, V(x) is the present value of the position when we apply a parallel shift tointerest rates of x basis points.4

• First-order difference functionδ(h δ ,h Ɣ , 1) = V(h Ɣ + h δ ) − V(h Ɣ − h δ ),2h δ(17)δ(h δ ,h Ɣ , 0) = V(h Ɣ + h δ ) − V(h Ɣ ).h δ(18)• Second-order difference functionƔ(h δ ,h Ɣ ,s,d δ , 1) = δ(h δ,h Ɣ + s, d δ ) − δ(h δ , −h Ɣ + s, d δ ), (19)2Ɣ(h δ ,h Ɣ ,s,d δ , 0) = δ(h δ ,h Ɣ + s, d δ ) − δ(h δ ,s,d δ ). (20)Where h δ is the basis point move for delta, d δ is the boolean that indicates whether a one-sided ortwo-sided difference will be applied for delta (d δ = 1 indicates two-sided), h Ɣ is the basis pointmove for gamma, s is a stress amount in basis points that will be used in the definition of stresseddelta and gamma, d Ɣ is the boolean that indicates whether a one-sided or two-sided difference willbe applied for gamma (d Ɣ = 1 indicates two-sided).2.1 Dollar deltaDollar delta () has three parameters (two of which are optional):1. determines the parallel shift in basis points. This parameter isrequired and can take any value between −∞ and ∞.2. determines the shift to be applied to the rates/spreads before calculatingdelta. In other words, if = 10 we calculate delta after applying a 10basis point parallel shock to the rates/spreads. This is an optional parameter and its defaultvalue is 0.3. is a boolean that determines whether delta is calculated as a one-sided ortwo-sided difference ( = 1 means two-sided). This parameter is optionaland its default value is 1.5

Dollar delta is then defined as = δ(h δ ,s,d δ ), (21)whereh δ = , (22)s = , (23)d δ = . (24)2.2 Dollar gammaDollar gamma () has five parameters (four of which are optional):1. determines the parallel shift to be applied in the calculation ofdollar gamma expressed in basis points. This parameter is required and can take any valuebetween −∞ and ∞.2. determines the parallel shift to be applied in the calculation of dollardelta expressed in basis points. This parameter is optional and its default is .This parameter can take any value between −∞ and ∞.3. determines the shift to be applied to the rates/spreads before calculatinggamma. In other words, if = 10 we calculate gamma after applying a10 basis point parallel shock to the rates/spreads. This is an optional parameter and its defaultvalue is 0.4. is a boolean that determines whether gamma is calculated as a onesidedor two-sided difference ( = 1 means two-sided). This parameteris optional and its default value is 1.5. is a boolean that determines whether delta is calculated as a one-sided ortwo-sided difference ( = 1 means two-sided). This parameter is optionaland its default value is 1.Dollar gamma is then defined as = Ɣ(h δ ,h Ɣ ,s,d δ ,d Ɣ ), (25)6

whereh δ = , (26)h Ɣ = , (27)s = , (28)d δ = , (29)d Ɣ = . (30)Version 1.0 September 19, 20037