The Swap-curve framework is based on OIS, FRA, SWAP basis spread, and SWAP market prices. It uses a combination of synthetic deposits, OTC FRA, and SWAP rates to build curves that implement basis spread correctly from beginning to end.
To illustrate this, let's consider the example of building a 6-month Euribor Swap curve. The first instrument we can use is the 6-month Euribor fixing or FRA0x6. Using deposit or shorter Euribor rates would not implement the basis spread correctly.
Synthetic deposits are calculated from OIS forward rates and known OTC FRA rates to create a smooth short-term curve. Backward extrapolation from the first known rate is used to calculate synthetic deposits.
To solve for short-term interest rates, the following equation is used:
(1 + rt) = (1 + OISt) * (1 + ON*t * x)
To calculate synthetic deposits for unknown terms, the ON-x basis is approximated using a polynomial function of degree x, typically 3:
ON-x = α + β * x + γ * x^2
By solving for coefficients α, β, and γ, synthetic deposits can be obtained using the following formula:
Synthetic Deposit = (1 - β * t - γ * t^2) / (1 + r * t)
Finally, the SWAP curve is created by applying synthetic deposits and OTC FRA rates using a discount calculation method.
Overall, this Swap-curve framework allows for accurate valuation and risk assessment of financial instruments, particularly those with basis spread components, by using a combination of synthetic deposits, OTC FRA, and SWAP rates to build curves that reflect market conditions accurately.