Three Essays on Caps Market and Unspanned Volatility

In this thesis we study the caps market. Caps are a contract where the interest rates are capped at some fixed value r¯. 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 macroeconomic announcements affect the implied volatility. We also regress them with the principal components of macroeconomic announcements. Thirdly, we examine the effect of the fed announcements. For the regressions with the principal components and fed announcements, we compare the two ways to do the regressions. One way is to select the days when at least one new macroeconomic announcement becomes available to public and construct the time series with the data on the selected days. Another way is to construct the time series of a macroeconomic announcement by filling the days without that announcement the most recent data. We see vivid difference between the two regressions. We observe that the latter gives better results. This may show that the effects of macroeconomic announcements are not instantaneous but rather transient or persistent.;In Chapter 4 we study the caps term premiums. Cochrane and Piazzesi [16, 17] did the regression analysis of the bond term premium with the forward rates and observed that the term premium is regressed by a hump shaped linear combination of forward rates very well. The term premiums are the difference in the cost between holding one unit of T-year bonds and holding one unit of one-year bonds and then one unit of (T -- 1)-year bonds after one year. This is also termed excess return and measures the risk of holding longer maturity bonds. We examine whether the similar results hold for caps. If there is no uncertainty, the caps term premium is zero, because holding longer term caps and holding two shorter term caps would not make any difference. Uncertainty causes the deviation from being equal. We regress the term premium of caps with the difference caps and observe persistent patterns for longer maturity caps term premiums. This would aid the predictability of cap prices. (Abstract shortened by UMI.)