Valuing derivatives is a two-step process.
Step 1 forecasts forward rates. Step 2 discounts them to PV using a curve of risk-free rates. Prior to the GFC of 2008, a single Libor curve was used to perform both steps for IR derivatives. After the Lehman collapse, markets realised that interbank 3M and 6M lending rates embedded in Libor curves were not risk-free. Instead, two sets of curves were required. The first was risk-free curves based on O/N lending rates known as OIS curves. These are used for Step 2. The second is forward projection curves calculated as spreads over the OIS curves. These are used for Step 1. This post-GFC approach for building yield curves is known as the multi curve approach.
The pre-crisis quantitative method for creating yield curves from single Libor curves is known as bootstrapping. It converts the rates of liquid deposits, futures and swaps into DFs. It does this for the shortest dated instrument first and then moves to the next tenor. It solves for the DF of the latest tenor using an optimization process that iterates until the latest DF leads to a model price for latest instrument that equals its market price. That is, model price minus market price should = 0. The optimization process that finds this DF subject to the [ModelPrice – MarketPrice=0] constraint is Newton’s method. It uses the derivative of f[ModelPrice – MarketPrice] wrt the DF to find the DF that makes [ModelPrice – MarketPrice] = 0.
In a multi curve setting, while the objective is still the same, i.e. set the pricing error, [ModelPrice – MarketPrice], =0, the problem is extended from a single variable optimization problem that uses the Newton derivative of f(DF) to a multi variable optimization problem that needs to find the values of the parameters of a vector that allow a series of PV formulas to calculate rates that match the market rates of instruments on input curves.
This multi-curve global solver problem uses the Newton vector method to achieve [ModelPrices – MarketPrices =0]. The parameters of the vector are the tenors of the OIS curve and the 3M and 6M forward projection curves. The tenors of basis swap spreads are also sometimes parameters. As are cross currency basis swap spreads if the global solver is solving for multi-curves for more than one currency.
The global solver uses multi-variable calculus to minimise the pricing error. A Jacobian matrix uses the derivative of the pricing errors wrt to the parameters of the Newton vector to iterate towards its pricing error = 0 solution.
