Choice of Interest Rate Term Structure Models for Pricing and Hedging Bermudan Swaptions -An ALM Perspective

Choice of Interest Rate Term Structure Models for Pricing and Hedging Bermudan Swaptions -An ALM Perspective Zhenke Guan Bing Gan Aisha Khan Ser-Huang Poon January 15, 2008 Zhenke Guan (zhenke.guan@postgrad.manchester.ac.uk) and Ser-Huang Poon (serhuang.poon@mbs.ac.uk) are at Manchester Business School, Crawford House, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Tel: +44 161 275 0431, Fax: +44 161 275 4023. We would like to thank Peter van der Wal, Vincent van Bergen and Jan Remmerswaal from ABN AMRO for manu useful suggestions and data, Michael Croucher for NAG software support, and Dick Stapleton and Marti Subrahmanyam for many helpful advice. 1

Abstract Asset Liability Management (ALM) departments in big financial institutions have the problem of choosing the right interest rate model for managing their books, which typically consists of assets and liabilities denominated in different currencies. Should a bank adopt a single term structure (TS) model or is it appropriate to allow different branches use different models for ALM? Each TS model is based on different measure of interest rates (e.g. spot, forward or swap rates) with a different assumed dynamics for the key rate. Is it appropriate to use a single TS model for ALM when the bank is subject to interest rate exposure in very different types of economy? This paper aims to address these questions by comparing, empirically, four one-factor TS models, viz. Hull-White, Black- Karasinski, Swap Market Model and Libor Market Model, for pricing and hedging long term Bermudan swaptions which resemble mortgage loans in banks book. In contrast to pricing need in the front office, a single-factor model is preferred for ALM purpose because of its focus in long term risk management. Managing long term interest rate risk using a multi-factor models is cumbersome and the multi-factors could potentially introduce more input errors over long horizon. The test involved calibrating the four TS models to European swaption prices for EURO and USD over the period February 2005 to September 2007. The calibrated models are then used to price and hedge 11-year Bermudan swaption with a 1-year holding period. The empirical results show that, the calibrated parameters of all four models are very stable and their pricing error is small. The pricing performance of four models is indistinguishable in both currencies. The hedging P&L from the four models is similar for the Euro market. For the USD market, the short rate models preforms marginally better than SMM and LMM. The performances of HW and BK are indistinguishable.

1 Introduction In this paper, we study the choice of interest rate term structure models for assetliability management in a global bank. In particular, we compare the performance of Hull-White, Black-Karasinski, Swap market model and Libor Market Model for hedging a 11-year Bermudan swaption on an annual basis from February 2005 to September 2006. The 11-year Bermudan swaption is chosen because it resembles a loan portfolio with early redemption feature, an important product for most banks. Unlike the short-term pricing problem, the one-factor model is often preferred for the longer term ALM purpose because of its simplicity. For long horizon hedging, the multi-factor model could produce more noise as it requires more parameter input. Also, for this ALM study, we do not consider the smile effect for the same reason. Pricing performance measures a model s capability of capturing the current term structure and market prices of interest rate sensitive instruments. Pricing performance can always be improved, in an almost sure sense, by adding more explanatory variables and complexities to the dynamics. However, pricing performance alone cannot reflect the model s ability in capturing the true term structure dynamics. To assess the appropriateness of model dynamics, one has to study model forecasting and hedging performance. The last few decades have seen the development of a great variety of interest rate models for estimating prices and risk sensitivities of interest rate derivatives. These models can be broadly divided into short rate, forward rate and market models. The class of short-rate models, among others, includes Vasicek (1977), Hull and White (1990), and Black and Karasinski (1991). A generalized framework for arbitrage free forward-rate modelling originates from the work of Heath, Jarrow and Morton (HJM, 1992). Market models are a class of models within the HJM framework that model the evolution of rates that are directly observable in the market. Brace, Gatarek and Musiela (1997) firstly introduced the arbitrage free process for forward libor rates which lead to the Libor Market Model (LMM, also known as BGM model) under the HJM framework. Miltersen, Sandmann and Sondermann (1997) derive a unified interest rates term structure which gives closed form solution for caplet under the assumption of log-normally distributed interest rates. Different methods of parameterization and calibration of LMM are examined in Brigo, Mercurio and Mrini (2003). The term structure of swap rates is firstly developed in Jamshidian (1997) which is known as Swap Market Model (SMM). Jamshidian (1997) also proposed the concept of co-terminal market model at the earliest. Base on the research of the above papers, Galluccio, Huang, Ly and Scaillet (2007) use graph theory to classify the admissible market models into three subclasses named co-initial, co-sliding and co-terminal. Among other things, they show that the LMM is the only admissible model for swaps of a co-sliding type. All these models have their own strengths and weaknesses. Short-rate models are tractable, easy to understand and implement but do not provide complete 2

freedom in choosing the volatility structure. The HJM framework is popular due to its flexibility in terms of the number of factors that can be used and it permits different volatility structures for forward rates with different maturities. Despite these attractions the key problem associated with the HJM is that instantaneous forward rates are not directly observable in the market and hence models under this framework are difficult to calibrate. The market models overcome these limitations but are complex and computationally expensive when compared with the short rate models. The question is whether one should use the simple model like Gaussian HW model or the lognormal BK model, or should one use complicated model like SMM and LMM. In this paper, we look at the difference between the four interest rate models for longer term asset liability management in two interest rate regimes (i.e. Euro and USD). The choice of HW and BK is simple: at the time of writing, they are the most important and popular short rate models used by the industry; the choice of swap market model and Libor market model is also simple because they are new or more recent and are widely used among practitioners. More recently, a number of previous studies have examined various term structure models for pricing and hedging interest rate derivatives. A large part of the term structure literature has focused on the performance of term structure models for bond pricing (see, for example, Dai and Singleton 2000, Pearson and Sun 1994). Andersen and Andreasen (2001) use mean-reverting Gaussian model and lognormal Libor Market Model for pricing Bermudan swaption. They find that for both models, Bermudan swaption prices change only moderately when the number of factors in the underlying interest rate model is increased from one to two. Pietersz and Pelsser (2005) compare single factor Markov-functional and multi-factor market models for hedging Bermudan swaptions. They find that on most trade days the Bermudan swaption prices estimated from these two models are similar and co-move together. Their results also show that delta and delta-vega hedging performances of both models are comparable. Gupta and Subrahmanyam (2005) compare various one- and two-factor models based on the out-of-sample pricing performance, and the models ability to delta-hedge caps and floors. They find that one-factor short rate models with time varying parameters and the two-factor model produce similar size pricing errors. But in terms of hedging caps and floors, the two-factor models are more effective. For both pricing and hedging of caps and floors, the BK model is better than HW model. With regard to both pricing and hedging, their results are in line with those obtained by Fan, Gupta and Ritchken (2006). Driessen, Klaassen and Melenberg (2003) use a range of term structure models to price and hedge caps and (European) swaptions. Regarding hedging, their results show that if the number of hedge instruments is equal to the number of factors then multi-factor models outperform one-factor models in hedging caps and swaptions. However, when a large set of hedge instruments is used then both one-factor and multi-factor models perform well in terms of delta hedging of European swaptions. Fan, 3

Gupta and Ritchken (2006) show that one- and two-factor models are as capable of accurately price swaptions as higher order multi-factor models. However, regarding hedging, their results show that multi-factor models are significantly better than one-factor models. In addition, their results for swaptions show that using multiple instruments within a lower order model does not improve hedging performance. These results differ from Driessen et al (2003) and Andersen and Andreasen (2001), who lend support for the one-factor models. This paper contributes to the literature and the existing empirical test in: i) By comparing a wide choice of models including the latest SMM, ii) by hedging the more complicated Bermudan swaption (instead of caplet or floorlet) and iii) by focusing on ALM perspective. Specifically, the objective is to select a interest rate term structure model which best price and delta hedge a 11-year Bermudan swaption in different currencies-eur, USD. On the contract starting date, we constructed a hedged portfolio for the 11-year Bermudan swaption; one year later, we get the new value of the hedged portfolio. The hedging profit and loss is calculated as the difference between the value of the options and portfolios. The data are from Datastream and ABN AMRO. This remaining of the paper is organized as follows. Next section briefly describes the models we use and the implementation design. Data is described in Section 3. Section 4 deals with calibration procedures of all models. Calibration instruments and calibration algorithm are discussed. In Section 5, Bermudan swaption prices are compared within four models. Hedging performance and Profit and Loss are reported in Section 6 and finally, we conclude in Section 7. 2 Term Structure Models We are interested in comparing the pricing and hedging performance of four widely used models-hw Model, BK model, Swap market model and Libor Market model. In this section, we briefly describe the four models and there characteristics. There are numerous models for pricing interest rate derivatives,which, broadly speaking, can be divided into two categories: sport rate models and forward rate models. Both HW and BK model are short rate models which specify the behavior of short-term interest rate, r. Short rate model, as it evolved in the literature, can be classified into equilibrium and no-arbitrage models. Equilibrium models are also referred to as endogenous term structure models because the term structure of interest rates is an output of, rather than an input to, these models. If we have the initial zero-coupon bond curve from the market, the parameters of the equilibrium models are chosen such that the models produce a zero-coupon bond curve as close as possible to the one observed in the market. Vasicek (1977) is the earliest and most famous general equilibrium short rate model. Since the 4

equilibrium models cannot reproduce exactly the initial yield curve, most traders have very little confidence in using these models to price complex interest rate derivatives. Hence, no-arbitrage models designed to exactly match the current term structure of interest rates are more popular. It is not possible to arbitrage using simple interest rate instruments in this type of no-arbitrage models. Two of the most important no-arbitrage short-rate models are the Hull-White model (1990) and the Black-Karasinski (1991) model. 2.1 The Hull-White and BK Model Hull and White (1990) propose an extension of the Vasicek model so that it can be consistent with both the current term structure of spot interest rates and the current term structure of interest-rate volatilities. According to the Hull- White model, also referred to as the extended-vasicek model, the instantaneous short-rate process evolves under the risk-neutral measure as follows: dr t = [θ t a t r t ]dt + σ t dz, (1) where θ, a and σ are deterministic functions of time. The function θ t is chosen so that the model fits the initial term structure of interest rates. The other two timevarying parameters, a t and σ t, enable the model to be fitted to the initial volatility of all zero coupon rates and to the volatility of short rate at all future times. 1 Hull and White (1994) note, however, that while a t and σ t allow the model to be fitted to the volatility structure at time zero, the resulting volatility term structure could be non-stationary in the sense that the future volatility structure implied by the model can be quite different from the volatility structure today. On the contrary, when these two parameters are kept constant, the volatility structure stays stationary but model s consistency with market prices of e.g. caps or swaptions can suffer considerably. Thus there is a trade-off between tighter fit and model stationarity. In HW model the distribution of short rate is Gaussian. Gaussian distribution leads to a theoretical possibility of short rate going below zero. Like the Vasicek model the possibility of a negative interest rate is a major drawback of this model. A model that addresses the negative interest rate issue of the Hull-White model is the Black and Karasinski (1991) model. In this model, the risk neutral process for logarithm of the instantaneous spot rate, ln r t is d ln r t = [θ t a t ln r t ] dt + σ t dz, (2) where r 0 (at t = 0) is a positive constant, θ t, a t and σ t are deterministic functions of time. Equation (2) shows that the instantaneous short rate evolves as the 1 The initial volatility of all rates depends on σ(0) and a(t). The volatility of short rate at future times is determined by σ(t) (Hull and White 1996, p.9). 5

exponential of an Ornstein-Uhlenbeck process with time-dependent coefficients. The function θ t is chosen so that the model fits the initial term structure of interest rates. Functions a t and σ t are chosen so that the model can be fitted to the chosen market volatility. Both HW and BK model are implemented by constructing a recombining trinomial lattice for the short-term interest rate. 2.2 Libor Market Model In the next two subsections, we are going to describe the two most advanced model in interest rates-swap Market Model and Libor Market Model. They have attracted a lot of interests from both academic and practitioners mostly because of their characteristics - they are consistent with market standard - using Black formula to price (European) swaption and caplet. Let L i (t), i = 1, 2,...N as the forward Libor rates. δ as the time interval between two Libor rates. The dynamics of forward Libor rate under the T n+1 terminal measure is by dl i (t) L i (t) = N j=i+1 δ j L j (t)σ ij (t) 1 + δ j L j (t) dt + σ i(t) dw QT n+1(t) (3) for 0 t min (T i, T n+1 ), i = 1,, n,, N. Note that L n (t) is driftless under the T n+1 terminal measure when B n+1 (t) is used as the numeraire. 2.2.1 Model Implementation Unlike caplet valuation, to implement LMM for pricing swaptions, it is necessary to simulate all forward rates under one measure. This means we need the dynamics (the drift term) for all L n (t) under the terminal measure. Here, we use a Euler discretization scheme for the forward rate simulation. While Glasserman and Zhao (1999) note that forward prices of bonds are not arbitrage free under the Euler discretization scheme, we adopt the Euler scheme nevertheless due to its simplicity. The approximation error is not severe as long as the step size is small. Following Glasserman and Zhao (1999), the discretized version of the LMM dynamics under the T N+1 terminal measure in (3) is, for a time step size h, L i ((k + 1)h) = L i (kh) exp [(µ i (kh) 12 σ i(kh) 2 )h + σ i (kh) ] hɛ j+1, where µ i (kh) = N j=i+1 δl j (kh)σ ij (kh). 1 + δl j (kh) 6

Here, we have chosen n = N and T N+1 as the terminal measure. Note that under the T N+1 terminal measure, when i = N, the drift of L i = L N is zero and L N is log-normally distributed. Predictor-corrector scheme is used here for more accuracy(see Joshi and Stancy 2005 for more details). 2.3 Swap Market Model Similar as Libor Market Model, swap market model has the same good property. However, it is not as intensively studied as LMM. Recently, people begin to realize the usefulness of co-terminal swap market model. Given the tenor structure, a co-terminal Swap Market Model refers to a model which assigns the arbitrage free dynamic to a set of forward swap rates that have different swap starting date T n (n = 1,, M 1) but conclude on the same maturity date T M. One important feature about the co-terminal swaption is that it is internally consistent with a Bermudan swaption that gives the holder the right to enter into a swap at each reset date during period T 1 to T M 1 with T M being the terminal maturity of the underlying swap. When considering at each tenor date whether or not to exercise the option to enter into a swap contract, the holder need to consider the forward swap rate dynamics from that tenor date till final maturity, which is actually driven by the volatility prevailing at that time. The advantage of co-terminal SMM over other market models in pricing Bermudan swaptions has already been noted and discussed in Jamshidian (1997) and Galluccio et al (2007). In Galluccio et al (2007) the co-terminal SMM is defined by introducing a collection of mutually equivalent probability measures and a family of Brownian motions such that for any the forward swap rate satisfies a SDE for all. Here we follow their approach. 2.3.1 Drift of Co-terminal SMM under the Terminal Measure Let C n,m (t) = m i=n+1 τb i (t) represent the value of the annuity from T n+1 to T m. The co-terminal forward swap rates satisfy the following general SDE under the terminal measure: ds n,m (t) = S n,m (t) σ n,m (t) dw M 1,M t + drift for n = 1,, M 1 (4) where W M 1,M t is the Wiener process under the terminal measure Q M 1,M. We follow Joshi and Liesch (2006), who recommend using the cross-variation in assessing impact of changing numeraire on a drift. The drift of a forward swap rate S n,m (t), n = 1,, M 1 under the terminal measure Q M 1,M is given by: E M 1,M [ds n,m (t)] = µ n,m = C M 1,M (t) Bn (t) B m (t), C n,m (t) C n,m (t) = C M 1,M (t) C n,m (t) S n,m (t), C n,m (t) C M 1,M (t) 7 C n,m (t) C M 1,M (t) (5)

Note that when n = M 1, µ M 1 = 0 because S n,m (t), 1 = 0. In Joshi and Liesch (2006), the S n,m (t) term inside the square brackets is simplified into independent Wiener process W k, which explains why their drift term does not contain the correlation terms ρ n,m. Although for one factor model, the correlation terms ρ n,m = 1 can be ignored, it cannot be omitted when the dimension of W k is two or more. Expanding the cross-variation term in equation (5), the drift becomes a complicated function of numeraires and cross-variation of two forward swap rates. By induction, the general form of drift under terminal measure can also be written as: µ n,m = M 2 i=n ( ) C i+1,m i τ i+1 ρ n,i+1 S n,m σ n,m S i+1,m σ i+1,m C j=n+1 (1 + τ j S j,m ) M 1,M Details of the derivation are given in the working paper by Gan et al. However, the above equation is not an explicit function of S n,m (t) for n = 1,, M 1, because the C n,m term is derived from a set of S i,m (t) for n i M 1. This drift term is complicated in appearance but not time-consuming in computation. Similar as Libor market model, SMM is also implemented using Monte Carlo simulation, in the interest of computational efficiency. Predictor-corrector drift approximation is employed for accuracy. (6) 3 Research Design and Data Let t denote a particular month in the period from February 2005 to September 2007. The procedure for calibrating, hedging and unwinding a 10 1 Bermudan swaption are as follows: (i) At month t, the interest rate model is calibrated to 10 ATM co-terminal European swaptions underlying the 10 1 ATM Bermudan swaption by minimising the root mean square of the pricing errors. (ii) The calibrated model from (i) is then used to price the 10 1 ATM Bermudan swaption at t, and to calculate the hedge ratios using, as hedge instruments, 1-year, 5-year and 11-year swaps, all with zero initial swap value at time t. (iii) A delta hedged portfolio is formed by minimising the amount of delta mismatch. (iv) At time t + 1 (i.e. one year later), the short rate model is calibrated to 9 co-terminal ATM European swaptions (9 1, 8 2,...) underlying the Bermudan swaption from (ii) which is now 9 1. 8

(v) The calibrated model from (iv) is used to price the 9 1 Bermudan swaption, and the time t + 1 yield curve is used to price the three swaps in (ii), which are now 0 year, 4 years and 10 years to maturity. (vi) The profit and loss is calculated for the delta hedged portfolio formed at t and unwound at t + 1. (vii) Steps (i) to (vi) are repeated every month for t + 1, t + 2, till T 1 where T is the last month of the sample period. To perform the valuation and hedging analyses described above, the following data sets are collected from Datastream: (a) Monthly prices (quoted in Black implied volatility) of ATM European swaptions in Euro and USD. Two sets of implied volatilities were collected: from February 2005 to September 2006, prices of co-terminal ATM European swaptions underlying the 10 1 Bermudan swaption, and from February 2006 to September 2007, prices of co-terminal ATM European swaptions underlying the 9 1 Bermudan swaption. These prices are quoted in Black implied volatility. The implied volatility matrix downloaded has a number of missing entries especially in the earlier part of the sample period. The missing entries were filled in using log-linear interpolation following Brigo and Morini (2005, p 9, 24 and 25). (b) Annual yields, R 0,t, for maturities up to 11 years are downloaded from Datastream. 2 All other yields needed for producing the trinomial tree are calculated using linear interpolation. These annual yields are converted to continuously compounded yields, r 0,t = ln(1 + R 0,t ). (c) Monthly data of the annual yield curve for the period January 1999 to July 2007, i.e. total 103 observations are downloaded for Euro and USD. This data was transformed into discrete forward rates (as in LMM) for use in the principal component analysis. Also, Swap rate can be derived from the forward libor rates. (iv) Monthly data of 1-month yield for the period January 2000 to July 2007 was downloaded for estimating the mean-reversion parameter. In the implementation, a time step ( t) of 0.1 year is used for constructing the trinomial tress, which means that the rates on nodes of the tree are continuously compounded t-period rates. Here we have used the one-month yield as a proxy for the first 0.1-year short rate. 2 The time step ( t) for the trinomial tress in the C++ program is 0.1 year. The C++ program linearly interpolates all the required yields. 9

4 Model Calibration Before we price Bermudan swaptions, we need to find the parameters that give the correct European swaptions to avoid any arbitrage opportunities. This section discusses different calibration method for each model. 4.1 HW/BK Model Calibration In this section, we discuss HW/BK model calibration. As we know, for both models, we have the mean reversion rate, time-dependent volatility σ t and the constant rate of mean reversion a. The mean reversion rate parameter for the two models has been extracted from historical interest rate data. Practitioners and econometricians often use historical data for inferring the rate of mean reversion (Bertrand Candelon and Luis A. Gil-Alana (2006)). In this study we calibrate the two models with time-dependent short rate volatility, σ(t), and constant rate of mean reversion, a. 4.1.1 Parameterizations of σ(t) There can be different ways to parameterise the time-dependent parameter: it can be piecewise linear, piecewise constant or some other parametric functional form can be chosen. In this study, the volatility parameter has been parameterised as follows: The last payoff for the all the instruments that need to be priced in this study would be at 11 years point of their life. We explicitly decided these three points, because values of σ(t), t = 0, 3, 11 on these three points can be interpreted as instantaneous, short term and long term volatility. These three volatility parameters are estimated through the calibration process. The volatilities for the time periods in between these points are linearly interpolated. 4.1.2 Choosing calibration instruments A common financial practice is to calibrate the interest rate model using the instruments that are as similar as possible to the instrument being valued and hedged, rather than attempting to fit the models to all available market data. In this study the problem at hand is to price and hedge 10 1 Bermudan. For this 10 1 Bermudan swaption the most relevant calibrating instruments are the 1 10, 2 9, 3 8,., 10 1 co-terminal European swaptions (A n m swaption is an n-year European option to enter into a swap lasting for m years after option maturity.). The intuition behind this strategy is that the model when used with the parameters that minimize the pricing error of these individual instruments would price any related instrument correctly. Therefore these 10 European swaptions are used for calibrating the two models for pricing 10 1 10

Bermudan swaption. 3 When the models are used for pricing 9 1 Bermudan swaption we use nine European swaptions; 1 9, 2 8,., 9 1 for calibrating the models. (σ 0, σ 3 and σ 11 for the 10x1 Bermudan swaption and σ 0, σ 2 and σ 10 for the 9 1 Bermudan swaption) 4.1.3 Goodness-of-fit measure for the calibration The models are calibrated by minimizing the sum of squared percentage pricing errors between the model and the market prices of the co-terminal European swaptions, i.e. the goodness-of-fit measure is n ( ) 2 Pi,n,model min 1 P i,n,market i=1 where P i,n,market is the market price and P i,n,model is the model generated price of the i (n i) European swaptions, with n = 11 when models are calibrated to price 10 1 year Bermudan swaption and n = 10 when models are calibrated to price 9 1 year Bermudan swaption. Instead of minimising the sum of squared percentage price errors alternatively we could have minimised the sum of squared errors in prices. However, such a minimization strategy would place more weight on the expensive instruments. Minimization of squared percentage pricing error is typically used as a goodness-of-fit measure for similar calibrations in literature and by practitioners. 4.1.4 NAG routine used for calibration We need to use some optimization technique to solve the minimization problem mentioned in the last section. Various off-the shelf implementations are available for the commonly uses optimization algorithms. We have used an optimisation routine provided by the Numerical Algorithm Group (NAG) C library. The NAG routine (e04unc) solves the non-linear least-squares problems using the sequential quadratic programming (SQP) method. The problem is assumed to be stated in the following form: min x R nf (x) = 1 2 n {y i f i (x)} 2 i=1 where F (x) (the objective function) is a nonlinear function which can be represented as the sum of squares of m sub-functions (y 1 f 1 (x)), (y 2 f 2 (x)),, (y n f n (x)). The ys are constant. The user supplies an initial estimate of the solution, together with functions that define f(x) = (f 1 (x), f 2 (x),, f n (x)) and as many first partial derivatives as possible; unspecified derivatives are approximated by finite differences. 3 Pietersz and Pelsser (2005) followed the same approach. 11

In order to use this routine, for our calibration purpose we did following: (i) Set s to 1. (ii) n = 10/9, depending on the number of calibration instruments (iii) f i (x) = P i,n,model P i,n,market (iv) Set initial estimates for all the three parameters as 0.01 i.e. 1%. (v) Set a lower bound of 0.0001 and upper bound of 1 for the three parameters to be estimated. Because of the complexity of our objective function partial derivatives are not specified and therefore the routine itself approximates partial derivatives by finite differences. 4.1.5 Estimating mean reversion parameter (a) As stated in the start of this section, the mean-reversion parameter has been estimated from historical data of interest rates. To measure the presence of mean reversion in interest rates we need large data set. This study spans over a period of one year, and within this one-year period models are calibrated every month. Hence it does not make sense to estimate this parameter for any economy on a monthly basis. Therefore, using historical data, once we estimate the value of this parameter and then use this value in all the tests. For the rate of mean reversion parameter a first order autocorrelation of the 1 month interest rate series has been used. Here we present the basic idea behind the estimation procedure used. Under the HW model, the continuous time representation of the short rate process is dr t = [θ t a r t ]dt + σ t dz, The discrete-time version of this process would be r t+1 r t = [θ t ar t ] + ε t+1, r t+1 = θ t + (1 a)r t + ε t+1 46 (7) where ε t+1 is a drawing from a normal distribution. Equation (46) represents an AR(1) process. An autoregressive (AR) process is one, where the current values of a variable depends only upon the values that variable took in previous periods plus an error term. A process y t is autoregressive of order p if y t = φ 0 + φ 1 y t 1 + φ 2 y t 2 +... + ε t, ε t N(0, σ 2 ). 12

An ordinary least square (OLS) estimate of coefficient (1 a) in equation (46) ˆβ would be 4 1 a = ˆβ = ρσ(r t+1)σ(r t ) = ρ σ 2 (r t ) a = 1 ρ where ρ is the correlation coefficient between r t+1 and r t and can be easily calculated using Excel. For the BK model we perform this regression using time series of ln(r), where as before r, is 1M interest rate. 4.2 Libor Market Model Calibration This section deals with the issue of Libor Market Model calibration. We observed in the previous section that, in LMM, the joint evolution of forward Libor rates under the pricing measure is fully determined by their instantaneous volatilities and correlations. Therefore, a well specified parameters is essential for obtaining correct prices and hedge ratios for exotic interest rate derivatives. Unfortunately, in general, they are not directly observable. Therefore, model parameters must be calibrated, i.e., they must be inferred either from time series of forward rates or from market prices of vanilla derivative contracts, or from a combination of both. Here we follow Lvov(2005) the diagonal recursive calibration procedure to the co-terminal European swaptions. The diagonal recursive calibration need to make assumptions of volatility of forward rates. Assumption: The instantaneous volatility term structure has the following form σ k (t) = φ k ν(t k t, α) (8) where T k is the expiry time of rate k, ν is a function imposing a qualitative shape on the volatility term structure, α is a vector of volatility parameters and φ k is a rate-specific multiplier. Brigo and Mercurio (2001) demonstrated that this form has a potential to produce an economically meaningful volatility structure while allowing for a satisfactory calibration to market data. 4.2.1 Tested Parameterizations We could have different parameterizations of the volatility term structure. This volatility parametric form was suggested by Rebonato (1999) which is tested extensively in the literature and is proved to possess very good properties. 4 For the regression y i = α + βx i + ε i,, the OLS estimate of β is ˆβ xy = ρ xyσ x σ y σ 2 xy 13

The parametric form as in Rebonato(1999) as σ i (t) = φ(i)[(a + b(t i t))exp( c(t i t)) + d] (9) In our test, we make φ(i) = 1. This parametric specification of the instantaneous volatility, in contrast with a piecewise-const form, preserves the qualitative shape of the implied volatility observed in the market, i.e., a hump at around two years. 4.3 Swap Market Model Calibration Brigo, Mercurio and Morini(2003), Rebonato(2003), and other literatures attribute a lot to the parameterisation and calibration of LMM. Based on their research, Galluccio et al (2007) test calibration of co-terminal SMM s using similar methodology as the one advocated by Rebonato(2003) in the case of LMM. By calibrating to swaption and caplet ATM volatilities, they finally get different set of parameters {φ j, a j, b j, c j, d j } where j = 1,, M 1 for M 1 number of co-terminal swaptions under different forward measure. Since the model built in this paper is under one measure and the volatility of different co-terminal swap rate will affect the drift terms in a complicated way, it is unrealistic to use different parameter set for different co-terminal swaption. Therefore, a similar parameter formula is followed here, but four common parameters will be allocated to volatilities of all co-terminal swaptions. In order to capture the term structure of swaption volatilities which are periodically deterministic, the linear-exponential formulation proposed in Brigo, Mercurio and Morini (2003) is adopted here as swaption volatility function. Further more the swaption volatility is formulated as σ n,m (t) = φ n,m ψ (T n t, a, b, c, d) ψ (T n t, a, b, c, d) = [a + b (T n t)] e c(tn t) + d14 (10) The term ψ can preserve the well-known humped shape of market quoted Black implied volatility. The method adopted in this paper is to calibrate parameter a, b, c, and d to the implied volatilities of a set of co-terminal swaptions with different maturities but associated with same length of swaps. The parametric form is as below v (t, M) = ψ (t, a, b, c, d) = [a + bt] e ct + d for T 0 < t < T M where v (t, M) is the market implied volatility, and t denotes the maturity of swaption. Because of its simple log-linear character, calibration algorithm using this parametric formula is quite fast and robust. When performing calibration with DRC method, we use Sequential Quadratic Programming algorithm from NAG C library to find the optimal parameters. This routine is based on the algorithm suggested in Gill et al (1986). Due to the 14

complexity of our target function, we approximate all partial derivatives using finite differences. This however does not appear to preclude the convergence of the optimization routine. 4.4 Calibration Results Calibration results are summarized in Table(1). The calibration is performed on the last business date of each month from February 2005 to September 2006. Table 1, Figure 1 and Figure 2 describe the errors by the calibration from different point of view. Generally, the root mean square error is quite small and we say that the model is very well calibrated. Insert Table 1,Figure 1,Figure 2 Here We also want to have the calibrated parameters to be stationary. In Table 2 and Figure 3, we find that given our parametric form of volatility, the parameters are indeed stable. Insert Table 2(a) and 2(b) and Figure 3 here Several conclusions could be drawn at this stage. First, it has been shown that all four models could calibrate to co-terminal European swaption implied volatilities matrix pretty well. The calibration methods are quite stable for all the testing dates in our data sets. This gives us firm ground to proceed with pricing and hedging Bermudan swaptions. 5 Pricing Bermudan Swaption We have recombining trinomial tree for pricing Bermudans swaptions with short rate models. For Swap Market Model and Libor Market Model, we perform Monte Carlo simulation with Longstaff-Schwartz Least Square Method.As Bermudan swaption are exotic interest rate derivative product, there is no market quoted price for it. We compare the price calculated from HW, BK, SMM and LMM. In theory, if calibrated appropriately, all models could give similar prices for the same Bermudan swaption. The first part, we are pricing a 11-year Bermudan swaption whereas in the second part we are pricing a 10-year Bermudan swaption (with the same strike as 1-year before). We price the same Bermudan swaption on the date of holding the option. Insert Table 3(a) and 3(b) and Figure 4 here The results are summarized in Table (3) and Figure 4. As we could see from the graph and tables, on most dates, the prices given by four models are indistinguishable. 15

6 Hedging Bermudan Swaption Changes in the term structure can adversely affect the value of any interest rate based asset or liability. Therefore, protecting fixed income securities from unfavorable term structure movements or hedging is one of the most demanding tasks for any financial institutions and for the ALM group in particular. Inefficient hedging strategies can cost big prices to these institutes. In order to protect a liability from possible future interest rate changes first one need to generate realistic scenarios and then need to know how the impacts of these scenarios can be neutralized. Thus, two important issues to be addressed by any interest rate risk management strategy are: (i) how to perturb the term structure to imitate possible term structure movements (ii) how to immunise the portfolio against these movements. Next sections explain the methodologies applied in this study for (i) estimating the perturbations by which the input term structure had been bumped to simulate the possible future changes in the input term structure (ii) selecting hedge instruments (iii) delta-hedging the underlying Bermudan swaption using the selected instruments. (iv) calculating possible profits and losses (P&L). In each section we have examples from literature have been referred to justify the choices made. 6.1 Perturbing the term structure Over the years researchers and practitioners have been using duration analysis for interest rate risk management, i.e. they shift the entire yield curve upward and downward in a parallel manner and then estimate how the value of their portfolio is affected as a result of these parallel perturbations. They then hedge themselves against these risks. Parallel shifts are unambiguously the most important kind of yield curve shift but alone cannot explain completely explain the variations of yield curve observed in market. Three most commonly observed term structure shifts are: Parallel Shift where the entire curve goes up or down by same amount; Tilt, also known as slope shift, in which short yields fall and long yields rise (or vice versa); Curvature shift in which short and long yields rise while mid-range yields fall (or vice versa). These three shifts together can explain almost all the variance present in any term-structure and thus and one should not completely rely on duration and convexity measures for estimating the risk sensitivity of a fixed income security. There are numerous examples in literature to support this 16

argument. Here, we mention a few studies that lead to this conclusion. Litterman and Scheikman (1991) performed principal components analysis (PCA) and found that on average three factors, referred to as level (roughly parallel shift), slope, and curvature, can explain 98.4% of the variation on Treasury bond returns. They suggested that by considering the effects of each of these three factors on a portfolio, one can achieve a better hedged position than by holding a only a zeroduration portfolio. 5 Knez, Litterman, and Scheikman (1994) investigated the common factors in money markets and again they found that on average threefactors can explain 86% of the total variation in most money market returns whereas on average four factors can explain 90% of this variation. Chen and Fu (2002) performed a PCA on yield curve and found that the first four factors capture over 99.99% of the yield curve variation. They too claimed that hedging against these factors would lead to a more stable portfolio and thus superior hedging performance. Insert Figure4, Figure 5 and Figure 6 Here Based on the findings of these studies, in this study we performed PCA on historical data for estimating more realistic term structure shifts. In this study we performed PCA on annual changes of the forward rates and used scores of the first three principal components for estimating the shifts by which we bumped the forward rate curves. There are not many examples of estimating price sensitivities w.r.t. multiple factors (like 3 principal components here) with a one-factor term structure model. Generally risk sensitivities are calculated by perturbing only the model intrinsic factors i.e. for the one-factor model, only one-factor is perturbed and so on. PCA has been performed on annual changes of forward Libor rates. Annual changes have been used because each hedge is maintained for one year. The reason for using forward rates rather than yield curve for doing PCA is two folds: first forward Libor rates are directly observable in market. Second using forward curves easily we can construct zero-curve and swap curve (needed for estimating interest rate sensitivities of swaps that are used for hedging). Without going into the mathematical details of this procedure, here, we briefly describe how PCA has been done for estimating the term structure shifts/bumps in the study: 6 Monthly observations of 11 forward Libor rates (f 0,0,1, f 0,1,2, f 0,2,3,..., f 0,10,11 ) for the period Jan 1999 to Dec 2006 have been used for PCA. Explicitly these rates have been used because we give annual zero curve of maturity till 11 years as input to our short rat models and we want to estimate bumps for all these maturities. 7 5 Litterman and Scheikman (1991, 54) 6 For details on PCA refer http://csnet.otago.ac.nz/cosc453/student tutorials/principal components.pdf 7 As we know that this dissertation is a part of a big project. The models to be implemented are LMM and SMM. These maturity forward rates are also needed by these two models. 17

Using these monthly observations of the 11 forward rates we calculate annual changes for each of the 11 forward rates as follows. Suppose we have monthly observations of the 11 forward rates for a period of n years i.e. in total we have 12 n monthly observations of forward Libor rates f 0,τj,τ j +1(t i ), where i = 1, 2,, 12 n and τ j = 0, 1, 2,.., 10. When the observations are arranged in ascending order (by date) then annual change for the forward Libor rates can be calculated as cl i,j = f 0,τj,τ j +1(t i + 12) f 0,τj,τ j +1(t i ) now i = 1, 2,..., 12 (n 1). These values would form a 12 (n-1) x 11 matrix i.e. we have 12 less entries. To clear this, say t i = Jan 1999, and τ j = 0. Then f 0,0,1 (t i )is value of f 0,0,1 on Jan 1999; f 0,0,1 (t i + 12)is the value of f 0,0,1 on Jan 2000 and cl is the one-year change in the market observed value of f 0,0,1. The matrix cl i,j is given as input to the NAG routine (g03aac) that performs principal component analysis on the input data matrix and returns principal component loadings and the principal component scores. The other important statistics of the principal component analysis reported by the routine are : the eigen values associated with each of the principal components included in analysis and the proportion of variation explained by each principal component. We used the scores of first three factors to calculate the three types of shifts for the 11 forward Libor rates using following regression: f 0,τj,τ j +1 = α + β τj,1p 1 + β τj,2p 2 + β τj,2p 3 + ε Where τ j = 0, 1, 2,.., 10 and P k is the vector of scores for the k th factor k = 1, 2, 3. Nag routine (g02dac) has been employed to perform this regression. The routine computes parameter estimates, the standard errors of the parameter estimates, the variance covariance matrix of the parameter estimates and the residual sum of squares. After performing the regressions specified above, the 11 forward rates are bumped by shocks corresponding to the first three factors as: f ± 0,τ j,τ j +1 = f 0,τj,τ j +1 ± β τj,k P k where τ j = 0, 1, 2,.., 10, and k = 1, 2, 3. 6.2 Choosing the hedge instruments Selecting appropriate hedge instruments is a critical part of a successful hedging strategy. In literature there are evidences of two hedging strategies, factor 18

hedging and bucket hedging. For factor hedging, in a K-factor model, K different instruments (together with the money market account) are used to hedge any derivative. The choice of hedge instruments is independent of the derivative to be hedged i.e., the same K hedging instruments can be used for hedging any derivative in a K-factor model, and depend only on the number of factors in the model. For bucket hedging the choice of hedge instruments depend on the instrument to be hedged and not on the factors in the model. In this hedging strategy number of hedge instruments is equal to the number of total payoffs provided by the instrument. The hedge instruments are chosen so their maturities correspond to different payment or decision dates of the underlying derivative. On using any other criteria for selecting the hedge instruments, the number of hedge instruments will lie between the numbers of hedge instruments for these two hedging strategies. Before discussing the instruments that are used to delta-hedge the Bermudan swaption in this study, in this section first we briefly discuss a few examples from literature. Driessen, Klaassen and Melenberg (2002) used (delta-) hedging of caps and swaptions as criteria for comparing the hedge performance of HJM class models and Libor market models. They used zero coupon bonds as hedge instruments. For each model, they considered factor and bucket hedging strategies. DKM show that when bucket strategies are used for hedging, the performance of the one-factor models improves significantly Fan, Gupta and Ritchken (2006) also used effectiveness of delta neutral hedges (for swaptions) as criteria for comparing the hedging performance of single factor and multi-factor factor term structure models. They used discount bonds to deltahedge swaptions. In this study they first applied factor hedging for choosing the hedge instruments and found that in context of hedging performance, multifactor models outperform single factor models. Next in light of the DKM results, they repeated their experiments using additional hedging instruments. They found that for the one-factor and two factor models adding more instruments did not result into better hedge results. Pietersz and Pelsser (2005) compared joint delta-vega hedging performance (for 10x1 Bermudan swaption) of single factor Markov-functional and multi-factor market models. They used the bucket hedging strategy and set up hedge portfolios using 11 discount bonds, one discount bond for each tenor time associated with the deal. They found that joint delta-vega hedging performance of both models is comparable The general implications of these examples are: (i) Effectiveness of delta neutral hedges is often used to evaluate the hedging performance of term structure models. (ii) Using multiple instruments can improve the hedge performance of one factor models. Practitioners also favour this practice Therefore, in this study, we decide to use two different swaps of maturities 5 and 11years as hedge instruments. Maturity of 11-year swap coincides with the 19

maturity of the co-terminal Bermudan swaption to be hedged and the length of other swap is almost half way the life of this Bermudan swaption. We could have used discount bonds to hedge this Bermudan swaption 8 but use of swap is more in line with the general practitioners practice. Studies suggest that large banks tend to use interest rate swaps more intensively for hedging. 6.3 Constructing delta hedged portfolio Delta hedging is the process of keeping the delta of a portfolio equal to or as close as possible to zero. Since delta measures the exposure of a derivative to changes in the value of the underlying, the overall value of a portfolio remains unchanged for small changes in the price of its underlying instrument. A delta hedged portfolio is established by buying or selling an amount of the underlier that corresponds to the delta of the portfolio. For interest rate derivatives,if the entire initial term structure is perturbed by same amount say ε (parallel shift), then the risk sensitivity of a fixed income security w.r.t. this perturbation can be estimated as V (ε) V ε where V is the value of the derivative calculated using initial term structure and V (ε) is the value of the derivative after the initial term structure is perturbed by ε. If we first increase the entire initial term structure by ε and then next decrease it by ε, then the risk sensitivity can estimated as V (ε + ) V (ε ) 2ε (11) where V (ε + ) is the value of the derivative calculated after initial term structure has been shifted up by ε and V (ε ) is the value of the derivative after the initial term structure has been shifted down by ε. In our case we have bumped the initial forward rate curve by three factors. Also for each factor, we have bumped the forward rate curve both up and down. Using the idea presented in equation (11), we estimate the sensitivity (delta) of the Bermudan swaption and the two swaps w.r.t the three factors as follows B k = B = B+ k B k P k P k P + k = B+ k B k 2 P k S 5 k = S 5 = (S 5) + k B k P k P k S 11 k = S 11 P k P + k = (S 11) + k B k P + k P k = (S 5) + k B k 2 P k = (S 11) + k B k 2 P k 8 Pietersz and Pelsser (2005) used discount bonds to hedge Bermudan swaption 20