Key Tolerance in refers to (roughly) the threshold that the difference xᵢ₊₁(t) - xᵢ(t) between two successive guesses xᵢ₋₁(t) and xᵢ(t) of the variable x(t) needs to reach in order for the second guess xᵢ(t) to be considered a solution in the Brent solver method for the optimal value of the variable x(t) at time t
The variable x(t) is chosen through the
For example, let = Discount, which results in the variable x(t) being the discount factor as a function of maturity time t
Then at each pillar date t (typically the maturity of a market instrument) the Brent method produces successive guesses of the discount factor t
Each guess implies quotes for the affected market instruments that deviate somewhat from the supplied market quotes.
The sum of the absolute values of all these deviations is the "error" implied by the current guess.
In the Brent method, this error becomes smaller with each successive guess.
The process of taking guesses stops when (roughly) a guess xᵢ(t) differs from the previous guess by an amount less than the entry defined here.