## OIS Curve

### FinPricing offers:

#### Four user interfaces:

**Data API**.**Excel Add-ins**.**Model Analytic API**.**GUI APP.**

**Data API**.**Excel Add-ins**.**Model Analytic API**.**GUI APP.**

FinPricing offers the following curves for various currencies via API. All the interest rate curves have data points up to 50 years.

- OIS curves
- RFR (risk free rate) curves
- SOFR, €STR (ESTR, ESTER), SONIA, TONA, CORRA, AONIA, SARON curves
- IBORs (LIBOR, EURIBOR, TIBOR, CDOR, EONIA, etc.) fallback rate curves
- Swap rate curves
- Basis curves
- Spot rate or zero rate curves
- Forward rate curves
- Discount curves
- Inflation Swap rate (CPI, RPI, HICP) curves
- Nordic electricity futures curve
- VIX futures curve
- S&P 500 futures curve

1. OIS Discounting Introduction |

Overnight index swaps OIS curves became the market standard for discounting collateralized cashflows. The reason
often given for using the OIS rate as the discount rate is that it is derived from the fed funds rate and the fed
funds rate is the interest rate usually paid on collateral. As such the fed funds rate and OIS rate are the relevant
funding rates for collateralized transactions.

In the past, a classic yield curve, such as 3 month LIBOR was used for both discounting
cashflows and projecting forward rates. However, this classic viewpoint is too simplistic. It does not take into
account the relative credit risk between lending money forward over a short period of time at
interbank rates, versus the risk involved in short-term funding of those loans.

Prior to the 2007 financial crisis, market practitioners considered the
swap curve as a proxy for the risk-free curve and
used it for discounting cashflows. LIBOR is the short-term borrowing rate of AA-rated financial institutions, but still
is not risk-free. Specifically, in stressed market conditions as the 2007 financial crisis, TED-which is the spread
between three month US LIBOR and three month US treasury rate-increased dramatically; in October 2008, it reached over
450 basis points. Besides that, the 2007 financial crisis triggered high basis spreads
for swaps characterized by different underlying rate tenors. As a result, the
swap curve data can not be regarded as risk free anymore.

Market practitioners started to use a new valuation methodology referred to as dual curve discounting, OIS
discounting or CSA discounting. OIS curves became the market standard for discounting collateralized cashflows.
This curve represents the market expectations of the Federal Reserve daily target for the overnight lending rate.

The reason often given for using the OIS rate as the discount rate is that it is derived from the fed funds rate
and the fed funds rate is the interest rate usually paid on collateral. As such the fed funds rate and OIS rate are
the relevant funding rates for collateralized transactions. Many banks now consider that overnight indexed swap OIS
rates should be used for discounting when collateralized portfolios are valued and that LIBOR should be used for
discounting when portfolios are not collateralized.

2. OIS Curve Construction and Bootstrapping Overview |

The most liquid instruments that can be used to build OIS curve are Fed Fund Futures and
OIS swaps that pay at the daily compounded Fed Fund rate. However,
Fed Fund Futures are currently only liquid up to two years and OIS
swaps up to ten years. Beyond ten years, the most liquid instruments are Fed Fund versus
3M LIBOR basis swaps, which are liquid up to thirty years.

The problem is that to price these basis swaps one needs both the OIS curve, to
project the Fed Fund rate, and the LIBOR curve, to project the LIBOR rate.
In the past one could have generated the LIBOR curve data separately, by using the
single curve for both forward projection and discounting.

However, in the modern market LIBOR swaps are quoted using OIS discounting. This means that in order to generate a forward LIBOR curve from LIBOR swap quotes one must first have the OIS curve already constructed so that one knows how to discount the cashflows. So neither the OIS curve nor the Libor curve can be built without the other. The two curves must be generated simultaneously.

The central tenet of curve construction under OIS discounting is to bootstrap multiple curves simultaneously.
One needs two term structure inputs for curve construction under OIS discounting: a term structure of OIS
instruments and a term structure of swaps.

This method proceeds as follows:

- From the underlying instruments, determine which define a point on the OIS curve and which define a point on the swap curve.
- All missing values have to be interpolated using linear interpolation on rates.
- Create these two sets of unknown curve points and make an initial guess for their values.
- Price all of the given instruments using the initial guess of the two curves.
- Compare the prices with the market quotes and adjust the initial guess accordingly. This would be performed by the same numerical optimization routine as is currently used in FinPricing, the Levenberg-Marquardt algorithm. The only difference is the greater number of points that will be perturbed in the algorithm.
- Repeat the pricing and adjustment until the error reaches acceptable levels.

The other important elements in curve construction are interpolation and optimization for root finding.

3. Interpolation |

Most popular interpolation algorithms in yield curve bootstrapping are linear, log-linear and
cubic spline. They can be applied to either zero rates or
discount factors.

Some critics argue that some of those simple interpolations cannot generate smooth forward rates and the others may
be able to produce smooth forward rates but fail to match the market quotes. Also they cannot guarantee the continuity
and positivity of forward rates.

The monotone convex interpolation is more rigorous. It meets the following essential criteria:

- Replicate the quotes of all input underlying instruments.
- Guarantee the positivity of the implied forward rates
- Produce smooth forward curves.

Although the monotone convex interpolation rule sounds almost perfectly, it is not very popular with market
practitioners.

4. Optimization |

As described above, the bootstrapping process needs to solve a yield using a root finding algorithm. In other words,
it needs an optimization solution to match the prices of curve-generated instruments to their market quotes.

FinPricing employs the Levenberg-Marquardt algorithm for root finding, which is very common in curve construction. Another popular algorithm is the Excel Solver, especially in Excel application.

Related Topics |