Traditionally practitioners have used LIBOR and LIBOR-swap rates as proxies for risk-free rates when valuing derivatives. This practice has been called into question by the credit crisis that started in 2007. Many banks now consider that overnight indexed swap (OIS) rates should be used as the...

In this paper we illustrate and explain the relationship between interest rates (or forward rates) and discount factors. We present the forward-discount factor relationship, which is popular and widely used in financial markets for yield curve construction, and derive the exact formulae using a...

SABR stochastic volatility model is appealing for modeling smile and skew of option prices. Hagan, who first proposed this model, derived a closed form approximation for european options and showed that it provides consistent and stable hedges. Here I prove a new exact closed formula for the...