# Numerical Sensitivities

Initially, the forward value is zero because we lend and borrow exactly the same amounts valued in a single currency. After initiating the contract, all market parameters move, but the final values are exchanged at the forward exchange rate: $10,000,000 and €7619,048.

The initial value of the forward contract is zero because the present values of the long and short legs of the forward are identical in a common currency: $10,000,000/(1 + /$) = $952,380.95, with *i$ —* 5%. At the initial spot rate this is identical to the value of the long leg in€, or €7619,048/(1 + 4%) (Table 35.8).

For calculating the sensitivities numerically, we start from initial values of risk factors, and we change the values of each one of them, one at a time, by a small amount. With the new set of values, there is a new value of the forward contract derived with the analytical formula. Instead of deriving the changes of values using the constant sensitivity assumption, the variation of value is obtained by revaluation of the contract with the new values of the risk factors. The difference with the initial value zero is the sensitivity with respect to the modified risk factor. Revaluation and differences are calculated numerically starting from the initial value as of the initial date of the contract.

Sensitivities of the forward contract are variations of its value in € when each one of the market parameters changes by a small amount; they are summarized in Table 35.9, where both initial and final values of the contract are shown.

The numerical sensitivities are derived from revaluation of the forward contract when each factor changes by an arbitrary and small amount. The variation of each factor appears in bold in Table 35.10.

The numerical values are identical to the mathematical values derived above. Note that the calculation of sensitivities is entirely different depending if we use the formulas for sensitivities directly or when revaluing the contract, as done above. In the first case, sensitivities are calculated analytically directly, while on the second case, the variations of the value of the contract for a given change of market parameters derives from the initial value of zero and a final value using as inputs their initial values plus a small change. The sensitivities become inputs for further calculations.

TABLE 35.9 **Numerical sensitivities in Euro value**

TABLE 35.10 **Detailed calculations of the forward sensitivities**

The usage of full revaluation of the contract is the unique method used in the simulations methodology, developed in the next chapter, when we cannot consider sensitivities as constant, which is the case for options. The difference with the above full revaluation used for calculating the sensitivities only once is that we need to revalue the contract for all hypothetical scenarios in simulations.

# Decomposition of the Forward as a Linear Function of Elementary Positions

The variation of the forward value as of date zero is a linear function of the small changes of each risk factor:

The forward is equivalent to three simple positions of which variations of values are proportional to the random changes of each of the three risk factors, expressed in units consistent with the numerical values of the sensitivities.