Deriving Sensitivities
Sensitivities can be derived analytically, as first derivatives of the contract with respect to the risk factors, or numerically by changing slightly the value of each risk factor and recalculating the contract value. It is feasible in this example to calculate analytically the first derivative with respect to each risk factor using the function relating the value of the contract to risk factors.
TABLE 35.6 Initial values of risk factors
Risk factor |
Value |
Change |
Spot rate |
0.7692 €/$ |
0.01 |
"/ $" (annual rate) |
5.00% |
0.010% |
"/ €" (annual rate) |
4.00% |
0.010% |
The alternate numerical methodology is applicable to any such function and when values are calculated numerically when there is no closed form formula. With a residual maturity of one year, the analytical formula is:
The initial values and changes of risk factors are shown in Table 35.6.
The formula can be used for deriving analytical sensitivities or for determining them by applying the small changes of values considered for the risk factors and revaluing the contract with the new value. Value and value changes are calculated at inception of the contract, when its initial value is zero.
Mathematical Sensitivities
Sensitivities S(m) with respect to each market parameter (risk factor) "m" are the first derivatives for the forward with respect to each factor. The sensitivity values are the product of these derivatives with the change considered for the risk factor.
The forward is equivalent to three positions, each one of them depending on one risk driver only, the other risk drivers being held constant. Numerical values are derived from closed form formulae above, where LP is the long position (in €) and SP is the short position (in $). Plugging in the initial values of the parameters, we find the numerical values of the derivatives (Table 35.7).
The sensitivities apply to a unit change of the risk factor. For small changes, they are multiplied by the changes considered above.
TABLE 35.7 Sensitivities in € value
TABLE 35.8 Calculation of the initial value of the contract