# Modelling Multi-Curve Interest Rates

**Title:** Modelling Multi-Curve Interest Rates

**Responsible: **Matthias Bollwein

**Supervisor: **Christian Fries

**Dates: **SS 2015

**Abstract:**

The credit crunch starting in 2007 has had a profound impact on the interest rate market, making it necessary to re-think and extend commonly used market models and practices.

Interest rate curves used to closely follow each other, as liquidity and credit risk had no practical impact on their modeling. The LIBOR market model constitutes the de-facto standard for the modeling of interest rate curves and evaluation of so-called linear interest rate products. However, liquidity and credit risk that became obvious during the financial crisis led to the divergence of curves. In particular, we now distinguish between discount and forward curves. That is, we observe a non-negligible spread between the forward curve that is derived from the curve we use for discounting cash flows, and forward curves underlying interest rate products like caplets or swaptions (in particular, LIBOR and EURIBOR).

There is a range of possible approaches to modeling multiple curves simultaneously, either by modeling different curves directly, or modeling one curve together with the spread. In this thesis we focus on the latter approaches.

We derive and implement an additive model, as well as a multiplicative model for the spread. We then perform numerical experiments in order to compare their properties regarding the calibration to linear interest rate products. In particular, we derive analytic formulas for the pricing of swaptions in each of the models. The implementation is done in Java and the calibration is done using both the analytic pricing formulas and Monte Carlo simulation.