Three Essays on Caps Market and Unspanned Volatility

Abstract

In this thesis we study the caps market. Caps are a contract where the interest rates are capped at some fixed value r. Purchasers of caps pay the prevailing interest rate if it is below r, but pay the interest rate r if the prevailing rate is above r. In the latter case the sellers of caps pay the difference. Therefore, the purchasers are able to prevent risks associated with the future change in interest rates. Caps consist of caplets which are European options on the forward rates called LIBOR (the London Inter-Bank Offer Rates). Caps are the derivatives of LIBOR. However, their relation is not so simple as the relation between a stock and its derivatives. One important difference is that the volatility in one market does not affect the volatility of the other market as much as in the stock and its derivative markets. This phenomenon is termed unspanned stochastic volatility (USV) and various research has been done. There are arguments supporting and against USV. This motivates further study of USV.In Chapter 2, we study the modeling and calibration of LIBOR caps. We use an unspanned stochastic volatility model to discuss the pricing of caps. We calibrate the cap prices using the model and compare them with the market caps. We choose the parameters of the model so that the difference between prices of the theoretical (the model based) caps and those of the market caps will be minimum. A goal of this chapter is to examine the effects of jumps in interest rates and in stochastic volatility. We compare the calibrations of USV models without jumps, with jumps in both interest rates and volatility, with jumps in interest rates only, and with jumps in volatility only. The calibration of Vasicek model is also performed for the sake of reference. Another goal is to obtain the calibration results based on a few days of data. So far most calibration results are based on one-day data. We find out that the calibrations with jumps give better results than without jumps and between the jumps in interest rates and volatility the jumps in interest rates give better calibration results. The results indicate that the more detailed study of effects of jumps is important.In Chapter 3, we examine the factors affecting the price of caps, so that accurate and efficient pricing and hedging of caps are possible. Since they are bond derivatives and the bonds are affected by the economic activities, it is interesting to examine how the economic activities influence the cap prices. We study empirically the effects of macroeconomic announcements and fed announcements on the implied volatility of difference caps. As mentioned earlier it has been observed that the prices of caps are driven by risk factors not spanned by the factors explaining LIBOR rates, even though caps are derivatives of LIBOR. We perform the regression analysis on the implied volatility of caps for all maturities and strike rates to see how the economic activities affect the cap prices. Using 21 series of macroeconomic announcements, first we do the event study to see which…