Erasmus University Rotterdam Erasmus School of Economics Master Thesis Quantitative Finance Empirical Pricing Analysis of Caps and Swaptions using Multi-Factor Models D. Balt 412346 Supervisor: prof. dr. M. van der Wel Second assessor: dr. X. Xiao July 22, 2019 The content of this thesis is the sole responsibility of the author and does not reflect the view of either Erasmus School of Economics or Erasmus University. Abstract The aim of this paper is to price caps and swaptions using multi-factor term structure models in a low interest rate environment. We price these products within the Heath-Jarrow-Morton framework, condition on the term structure, and investigate the pricing errors to detect possible modeling biases. The volatility functions are modeled using principal component analysis of up to three factors using interest-rate or option data with either constant or time-varying parameters. We use weekly data of US swaps, caps and swaptions from 2013-2019. The results show that option-based estimation outperforms interest-rate based estimation, and that the multi-factor models generally outperform one-factor models in pricing. We find that pricing errors vary over time, and are related to the term structure shape and level, and moneyness, but not to time-to-maturity, option expiry, and swap tenor. We further find that the low interest rate environment has a negative influence on pricing errors. Table of Contents 1 Introduction 1 2 Literature 3 3 Methodology 6 3.1 Heath-Jarrow-Morton Framework . .6 3.1.1 Theoretical Framework . .6 3.1.2 Volatility Functions . .7 3.1.3 Pricing Formulas . .8 3.2 Empirical Implementation . .9 3.2.1 Yield Curve Smoothing . .9 3.2.2 Interest-Rate-Based Parameter Estimation of Volatility Functions . 10 3.2.3 Option-Based Parameter Estimation of Volatility Functions . 11 3.3 Pricing and Evaluation . 12 3.3.1 Conditional Pricing Setup . 12 3.3.2 Model Forecast Comparison Test . 13 3.3.3 Pricing Prediction Error Evaluation . 14 3.3.4 In-Sample Pricing . 16 4 Data 16 5 Results 21 5.1 Parameter Estimates . 21 5.2 Pricing Prediction . 23 5.3 Model Forecast Comparison Test . 27 5.4 Error Analysis . 30 5.5 In-Sample Pricing . 34 6 Conclusion 35 Appendix 41 A Bloomberg Tickers . 41 B Libor Rate . 42 C Black's Formula Implementation . 43 D Supplementary Results . 44 E Cap Prediction Errors . 45 F Swaption Prediction Errors . 47 G Model Forecast Comparison Test (Standard Normal) . 49 1 Introduction The market for interest rate derivatives is very large and has rapidly expanded over the last decades. Among these interest rate derivatives are caps and swaptions. According to the International Swaps and Derivatives Association (ISDA), the notional outstanding for swaptions and interest rate options such as caps reached $30 and $12 trillion US Dollar (USD) respectively in 2014.1 These derivatives can be used for hedging or speculating on interest rates. Swaptions are often used by investors that have a large portion of liabilities, such as insurers and pension funds, because they provide a hedge against rising rates whilst still being able to profit from declining rates. Interest rate caps are, among others, useful for borrowers paying a floating LIBOR that wish to limit their risk exposure to increasing rates. It is important for investors to properly price and hedge these products in order to successfully manage their risk exposure. In the existing literature, research has been done on the modeling of interest rate derivatives. This followed after the extensive literature on term structure models. A good term structure model should be able to price and hedge interest rate derivatives. The majority of the interest rate derivatives research has been done around the 1990s, this was mainly theoretical due to the difficulty in obtaining data.2 In the early 2000s more empirical research came to light.3 The empirical papers distinguish themselves from each other in that they use a different modeling framework, estimation approach, underlying instrument, and/or evaluation criteria. The direction of this paper is similar to Driessen et al. (2003), who analyze the pricing and hedging performance of caps and swaptions in the Heath et al. (1992) (HJM) framework. They use both an interest-rate-based and option-based estimation method and assume that the interest rates follow either a Gaussian or log-normal distribution. The parameters of the volatil- ity functions are estimated with either Principal Component Analysis (PCA) or Generalized Method of Moments, using a time-series data set from 1995 to 1999. The pricing and hedging performance are both analyzed in an out-of-sample setting. Our approach for pricing purposes is similar, except we only use the models that assume a Gaussian distribution on interest rates, and only use PCA to estimate volatility functions. In this paper we investigate which combination of volatility function model and estimation method has a good pricing and hedging performance for caps and swaptions within the HJM framework. We investigate up to three factor models for the volatility functions, estimated 1Source: https://www.isda.org/a/qJEDE/isda-final-2014.pdf. 2See, among others, Amin and Morton (1994), Brace and Musiela (1994a) and Heath et al. (1992). 3For example Driessen et al. (2003), Gupta and Subrahmanyam (2005), and Falini (2010). 1 interest-rate-based or option-based, with constant or time-varying parameters. We further investigate which economic factors (e.g. time-to-maturity, option maturity (swaptions), mon- eyness (caps), and interest rate level) influence the pricing and hedging errors of models using an error regression. The main contribution of this paper is the application of an existing pricing framework for caps and swaptions using a recent data set that is characterized by a low interest rate environment. This paper extends on Driessen et al. (2003) in various ways. First, we analyze caps both at-the-money (ATM) and out-of-the-money (OTM), whereas Driessen et al. (2003) only looks at ATM caps. Second, we perform a model forecast comparison test using a novel panel data approach based on Diebold and Mariano (2002) tests using a bootstrapped sampling distribution.4 Third, we further analyze the pricing errors using an approach similar to Gupta and Subrahmanyam (2005), which has not yet been done on the models used by Driessen et al. (2003). Fourth, we investigate the effect of the low interest rate environment on the pricing errors by linear regressions using OLS, LASSO, and ridge estimation methods. Finally, we perform an in-sample test to investigate the model restrictions. The main ingredient for pricing derivatives within the HJM framework is the volatility func- tion. The volatility functions in this paper are estimated using PCA (interest-rate-based esti- mation), with an additional scaling factor for option-based estimation that uses cross-sectional option data. This estimation procedure can use constant parameters, in which we use the first half of the sample as estimation window, or time-varying parameters, in which we use a rolling window of 40 weeks. We subsequently price the caps and swaptions out-of-sample, conditional on the term structure. Caps and swaptions are priced using the derivative pricing formulas of Brace and Musiela (1994a), the implementation of these formulas is not necessarily straightforward. We compare the forecasting performance of the models using a variation on Diebold and Mariano (2002) tests. We perform a Diebold-Mariano test on every product (cap or swaption) for each combination of models and assess the percentage of products in which one model forecast outperforms the other. This allows us to find the model that has the best forecasting performance for the largest amount of products. The Diebold-Mariano statistics are tested against a bootstrapped sampling distribution. The pricing errors in terms of Black implied volatility are regressed on option properties and economic variables in order to find possible sources of modeling bias. We further regress absolute pricing errors on the interest rate level and control variables to assess the effect of the low interest rate environment on pricing 4I have tried to find precedents of a similar approach in the literature, but did not find any. 2 errors; this regression uses OLS, LASSO, and ridge estimation methods. In this research we use a balanced panel data set obtained from Bloomberg that ranges from January 2013 to March 2019 with weekly intervals. We find that the option-based estimation method outperforms the interest-rate estimation method in most situations. The multi-factor models generally outperform the one-factor models in pricing. The conditional pricing forecasts for caps are quite accurate, but not so much for swaptions. The pricing errors of all products are mainly caused by possible model deficiencies with respect to the term structure shape and level. We find that pricing errors vary over time, however, the time-to-maturity, option expiry, and swap tenor do not influence the pricing errors. The low interest rate environment has a negative influence on pricing errors. The in-sample pricing analysis reveals that a near-perfect pricing prediction is unlikely in this sample using PCA based volatility functions. This paper is organized as follows. Section 2 contains an overview of existing literature regarding the pricing of interest rate caps and swaptions. In Section 3, the research methodology is discussed. In Section 4, the data used in this research is described. Section 5 contains the results of this research. In Section 6 the conclusions are discussed. 2 Literature In the current literature a large part of interest-rate-derivative pricing models are modeled with the Heath et al. (1992) framework based on forward rates. This framework has the ability to model the entire yield curve exactly.