A simple and fundamental question in derivatives pricing is how (contingent) cash-flows should be discounted. As cash can generally not be invested at Libor, the Libor curve is probably not the right discounting curve, even for Libor derivatives. The impact on derivative pricing of changing the...

Even if the name futures indicates a simple instrument, bond futures are complex. Several special features are embedded in the instrument. In particular the future is not written on one specific bond but on a basket of bonds, from which the short side can deliver the cheapest. This paper focuses...

Bond futures are liquid but complex instruments. Here they are analysed in a one-factor Gaussian HJM model. The in-the-model delta and out-of-the-model delta and gamma are studied. An explicit formula is provided for in-the-model delta. The out-of-the-model delta and gamma are equivalent to...

The twin brothers Libor Market and Gaussian HJM models are investigated. A simple exotic option, floor on composition, is studied. The same explicit approach is used for both models. Using an approximation the LLM price is obtained without Monte Carlo simulation. The results of the approximation...

Even if the name futures indicate a simple instrument, bond futures are complex. Several special features are embedded in the instrument. In particular the future is not written on one specific bond but on a basket of bonds, from which the short side can choose the cheapest-to-deliver. This...

In this note we give pricing formulas for different instruments linked to rate futures (euro-dollar futures). We provide the future price including the convexity adjustment and the exact dates. Based on that result we price options on futures, including the mid-curve options

Leveraging the explicit formula for European swaptions and coupon-bond options in HJM one-factor model, we develop a semi-explicit formula for 2-Bermudan options (also called Canary options). We first extend the European swaption formula to future times. So equipped, we are able to reduce the...

A popular way to value (Bermudan) swaption in a Hull-White or extended Vasicek model is to use a tree or PDE approach. A more direct approach through iterated numerical integration is developed. A brute force numerical integration would lead to a complexity exponential in the number of exercise...

The original Libor Market Model (LMM) has been extended to several dynamics (or local volatilities) for the underlying Libor rates. The main result presented here is a generic approximation that provides an explicit swaptions price for local volatilities LMM. The approximation is base on an...