# Prisfastsættelse af first-to-default swaps

Anna Østergaard Andersen & Stine Corlin Christensen

Student thesis: Master thesis

## Abstract

In this master’s thesis we price First-to-Default swaps (FTDs) using intensity models and exponentially affine models. We derive a pricing model from a known Credit Default Swap (CDS) pricing model. Despite being the first traded credit derivative, FTDs have not gained the same popularity as CDSs. Therefore, the literature on FTD pricing is limited compared to the literature concerning CDS pricing. To a wide extent one can describe an FTD contract by describing a CDS contract. For this reason we present a thorough introduction to CDS contracts before we describe the FTD contract in detail. For the same reason we base our FTD pricing model on a CDS pricing model. CDSs and FTDs are priced by calculating the expected payments on the premium leg and the protection leg. We use an intensity-based model where the survival probability is described by an exponentially affine model. In an affine setup the survival probability can be written on closed form which makes semi-analytical solutions possible. In addition to this, the affine setup allows us to divide the survival probability in two, and by letting it consist of a systematic and idiosyncratic factor. This is useful when finding a common survival probability for the entities in the FTD. The systematic factor, described by the Markit iTraxx Europe Index, holds information on correlation between the reference entities. All theory relevant to the pricing is described in the first part of the thesis. In the second part we implement the theory on pricing FTDs in an empirical analysis. In this analysis we price a 5 year FTD with Carlsberg, DONG, TDC and Volvo on February 10 2012. As a part of the FTD pricing we calculate CDS prices on the four reference entities. Since FTD quotes cannot be observed in the market, we rely on the CDS prices being accurate. Firstly, we calibrate survival probabilities from CDS quotes observed in the market using a functional form. From these survival probabilities we calibrate parameters for the affine models. The calibration is performed by minimising relative squared errors between observed and calculated survival probabilities. This calibration guarantees that the model finds survival probabilities in accordance with the markets perception. Affine models furthermore give us the possibility of forecasting survival probabilities and prices. Secondly, we calculate CDS and FTD prices using the survival probabilities we found through calibration. Prices are calculated for five maturities on six different observation dates and by simulating future probabilities we calculate 5 year prices at two future dates. We find that deviations on CDS prices are quite low, especially for longer maturities. FTD prices are found to be too high, especially for shorter maturities. Finally, we offer critique of the model and try to foresee the future of FTDs

Educations MSc in Mathematics , (Graduate Programme) Final Thesis Danish 2012 155