A Master’s Thesis to obtain the degree in Financial Econometrics
M O D E L I N G B O T H W O R L D S
A Joint-Measure Model for The Risk-Neutral and Real-WorldCarried out at Sprenkels & Verschuren
Author: Mark Verschuren
Student nr: 5957540
Email: mark.verschuren@sprenkelsenverschuren.nl
Date: April 12, 2015
Supervisor: Prof. Dr. Ir. M.H.Vellekoop In-company supervisors: Ir. B.J.W. Kobus
A B S T R A C T
In this thesis joint-measure models are constructed and estimated, that on one hand are consistent with the observed prices of at-the-money swaptions and bonds and on the other hand provide plausible real-world scenarios for risk management. Our analysis shows that the calibrated risk-neutral G2++ model (Brigo and Mercurio,2006) outperforms the Hull-White one-factor model (Hull and White, 1990) based on both the in-sample fit and the out-of-sample pricing performance. However, historical cross-section estimation shows that its calibrated parameters are less stable than in the Hull-White case due to the larger number of parameters. We have used the estimation technique ofHull et al.(2014) and have extended it for the G2++ model to estimate our real-world parameters. Our results show that the two models under the real-world measure produce similar future short-rate distributions. These distribu-tions are especially different in the short term from their risk-neutral versions, confirming that for adequate risk management real-world dis-tributions should be used. Our sensitivity analysis shows that the real-world short-rate distributions are highly dependent on the functional form that is chosen for the market price(s) of risk. Therefore both the functional form of the market price(s) of risk and the historical estima-tion period are important aspects in the estimaestima-tion of joint-measure models.
I would like to take this opportunity to show my gratitude to a number of people who have made it possible for me to write my master’s thesis successfully.
First, I want to thank by university supervisor Prof. Dr. Ir. M.H. Vellekoop for his great guidance during the last couple of months. In between the technical details of mathematical finance and the program-ming in Matlab he made sure that I never lost sight of the practical goal of this master’s thesis for Financial Econometrics. During our sessions he was always able to provide fundamental insight and clarifications to any problems. Therefore I am very grateful.
Furthermore, I want to thank Sprenkels&Verschuren for giving me the opportunity to write my master’s thesis at Sprenkels&Verschuren. My two in-company supervisors Bertjan Kobus and Bart ter Veer have sup-ported me during the entire process. Bertjan challenged me from the start to pinpoint the subject of my thesis, while Bart as a mentor made sure I had enough time to write my thesis and encouraged me to keep going.
At last I would like to thank my family and friends for their support throughout the entire process. They always supported me and encour-aged me to keep going.
Hereby I proudly present my master’s thesis to obtain the Master’s in Financial Econometrics at the University of Amsterdam.
Mark Verschuren
C O N T E N T S
1 i n t ro d u c t i o n 1
2 m o d e l c h o i c e a n d l i t e r at u r e r e v i e w 3 2.1 Model Choice 3
2.2 Previous studies 4
2.3 Criteria Model Performance 6 3 s h o rt - r at e m o d e l s 9
3.1 Hull-White one-factor model 9 3.1.1 Real-World Dynamics 9
3.1.2 Price of a Zero-Coupon Bond 10 3.1.3 Options on bonds 11
3.1.4 Swaption Price 11 3.2 G2++ model 12
3.2.1 Risk-Neutral Dynamics 12 3.2.2 Real-World Dynamics 13
3.2.3 Price of a Zero-Coupon Bond 13 3.2.4 Price of a European Swaption 14
4 c a l i b r at i o n o f t h e s h o rt r at e m o d e l s 17 4.1 Data 18
4.2 Optimization Problem 20
4.2.1 Global & Local Optimization 20
4.3 Calibrating the Hull-White One-Factor model 21 4.3.1 Calibration Method 21
4.3.2 Results for Calibration of the Hull-White One-Factor model 22
4.4 Calibrating the G2++ model 24
4.4.1 Results for Calibration of G2++ 24 4.5 Historical Estimates 27
4.6 Out-of-Sample Pricing Performance 31 4.6.1 Concluding remarks 33
5 e s t i m at i n g t h e m a r k e t p r i c e s o f r i s k 35 5.1 Literature & Estimation Principle 35
5.2 Data & Optimization 37
5.3 Hull-White One-factor Model 39
5.3.1 Estimated Market Price of Risk 40
5.4 G2++ 43
5.4.1 Estimated Market Price of Risk 44 5.5 Comparison 46 5.5.1 Concluding Remarks 48 6 c o n c l u s i o n 49 i a p p e n d i x 51 a b a s i c t h e o ry 53 v
a.1 Interest Rates and Bonds 53 a.1.1 Bond 55
a.2 Interest Rate Swap 55
a.2.1 Valuation of Interest Rate Swaps 57 a.3 Swaption 58
a.3.1 Black’s Model 59 a.4 Risk-Neutral Pricing 60
b d e r i vat i o n z e ro - c o u p o n p r i c e s h u l l - w h i t e 63 c data s e p t e m b e r 1 0 t h , 2 0 1 4 . 67
d h i s t o r i c a l data 69 b i b l i o g r a p h y 73
L I S T O F F I G U R E S
Figure 1 The relationship between the funding ratio and interest rates and stock prices. Ultimo 2008 has been chosen as reference point (index 100). Source: http://www.dnb.nl/nieuws/nieuwsoverzicht-en-archief/dnbulletin-2014/dnb307060.jsp. 1 Figure 2 Calibration process. 17
Figure 3 Data from Bloomberg of September 10th, 2014. 18 Figure 4 Term structure of zero rates on September 10th,
2014 after cubic spline interpolation. 19 Figure 5 Basins of attraction. 21
Figure 6 Calibration results Hull-White One factor model: percentage deviation from market prices Septem-ber 10th, 2014. 23
Figure 7 Calibration results G2++ model. 26
Figure 8 Historical zero rates as from 2006 up and till end 2014. 27
Figure 9 Calibration results of the Hull-White one-factor model since 2006 at the end of each quarter. 29 Figure 10 Calibration results G2++ model since 2006 at
the end of each quarter. From top to bottom and left to right: a, b, σ, η and ρ. 30
Figure 11 Root mean square error time-series. 31 Figure 12 Mean of the historical short rates. 38 Figure 13 Short-rate distributions (Hull-White) 1 and 20
years in the future under both the risk-neutral and the real-world measure. 42
Figure 14 Short-rate distributions of the G2++ model 1 and 20 years in the future under both the risk-neutral and the real-world measure. 46 Figure 15 Short-rate distributions of the Hull-White
one-factor- and the G2++ model one year from now. 47 Figure 16 Short-rate distributions of the Hull-White
one-factor- and the G2++ model twenty years from now. 48
Figure 17 Timeline showing different interest rates. 53 Figure 18 Graphical display of coupon bond cash flows. 55 Figure 19 Interest rate swap mechanics. 56
Figure 20 Graphical display of a swaption. 58
Figure 21 The implied at-the-money swaption volatilities of September 10th, 2014. 60
Figure 22 Implied market volatilities of swaptions with ma-turity one year and different tenors. The data is from 2006 up and till the end of 2014. 69 Figure 23 Implied market volatilities of swaptions with
ma-turity two years and different tenors. The data is from 2006 up and till the end of 2014. 69 Figure 24 Implied market volatilities of swaptions with
ma-turity three years and different tenors. The data is from 2006 up and till the end of 2014. 70 Figure 25 Implied market volatilities of swaptions with
ma-turity four years and different tenors. The data is from 2006 up and till the end of 2014. 70 Figure 26 Implied market volatilities of swaptions with
ma-turity five years and different tenors. The data is from 2006 up and till the end of 2014. 71 Figure 27 Implied market volatilities of swaptions with
ma-turity ten years and different tenors. The data is from 2006 up and till the end of 2014. 71
L I S T O F TA B L E S
Table 1 The properties of different short-rate models. 4 Table 2 Calibration results of the Hull-White one-factor
model with two different solvers and two differ-ent weights on data from September 10th, 2014. In addition results from previous studies are dis-played. 22
Table 3 Implied volatility by the Hull-White-one-factor model prices. 24
Table 4 Calibration results of the G2++ model with two different solvers and two different sets of weights. 25 Table 5 Implied volatility by the G2++ model prices. 26 Table 6 The mean and standard deviation of the
esti-mated parameters of the Hull-White one-factor model for quarterly data from the first quarter of 2006 till the end of 2014. 28
Table 7 The mean and standard deviation of the cali-brated parameters of the G2++ model for quar-terly data from the first quarter of 2006 till the end of 2014. 29
Table 8 Four statistics for the measurement of the out-of-sample pricing errors: the average pricing er-ror, the average absolute pricing erer-ror, the av-erage percentage pricing error and the avav-erage absolute pricing error. 32
Table 9 Estimated market price of risk. 41
Table 10 Estimated market prices of risk for the G2++ model. 45
Table 11 Implied volatilities of at-the-money swaptions on September 10th, 2014. Source: Bloomberg. 67 Table 12 Zero curve on September 10th, 2014. Source:
Bloomberg. 67
1
I N T R O D U C T I O N
Since December 2008 the average funding ratio of Dutch pension funds has increased by roughly 20% (figure 1). While stock prices have in-creased by 80%, interest rates have dropped by 20%. An average pen-sion fund invests 40% into equity and hedges 50% of its interest rate exposure. The net result1 is on average a 20% increase in funding ratio. These results confirm that interest rates and stock prices indeed are key determinants of the funding ratio of pension funds.
Figure 1: The relationship between the funding ratio and interest rates and stock prices. Ultimo 2008 has been chosen as reference point (in-dex 100). Source: http://www.dnb.nl/nieuws/nieuwsoverzicht-en-archief/dnbulletin-2014/dnb307060.jsp.
Since the introduction of the Financial Assessment Framework (Finan-cieel Toetsingskader) in 2007 pension funds are obligated to value both their assets and liabilities at fair value. The liabilities are thus deter-mined by discounting the future cash flows of the pension obligations with the current market interest rate term structure. For risk manage-ment it is therefore important to produce possible scenarios of this term structure of interest rates. This scenario analysis should be based on the real-world distribution of interest rates.
A decrease in the above mentioned term structure of interest rates is a serious risk for pension funds. Due to a duration-mismatch between the assets and liabilities a decrease in interest rates leads to a decrease in funding ratio with potentially a reduction of acquired benefits. To prevent this pension funds (partly) hedge their interest rate risk with for example interest rate derivatives such as swaptions. These derivatives are priced risk-neutrally, where the price of a derivative equals the risk-neutral expectation of the discounted payoff of the derivative. To assess portfolios that incorporate derivatives to hedge short-term risks
1 There are also other factors that effect the development of the funding ratio.
a model is needed that can calculate both the risk-neutral and the real-world distribution: a joint-measure model. In this area, Hull et al. (2014) have taken the lead by constructing an one-factor term structure model which provides term structure movements under both measures. The performance of different portfolios that include derivatives can be assessed by using such a joint-measure model. This thesis therefore will address the following central research question:
"How can we construct and estimate a joint-measure in-terest rate model that on one hand is consistent with the observed prices of interest rate derivatives under the risk-neutral measure and on the other hand provides plausible real-world scenarios for risk management purposes?"
In general, there is a difference between the estimation of the param-eters of a risk-neutral and a real-world model. A risk-neutral interest rate model is derived by choosing a functional form for the model. The parameters of this model are then fitted in such a way that the model option prices are as close as possible to the observed prices in the mar-ket on a particular day. This process is called the calibration of the model. Real-world interest rate models on the other hand, use histori-cal data (time-series) to estimate the parameters of the model. In this thesis, calibration and historical estimation are combined to estimate the complete joint-measure model. The market prices of swaptions show us the expectation under the risk-neutral measure. Under this measure all securities earn the risk-free rate, which is not realistic for risk man-agement. The market therefore expects a compensation/risk premium for the associated risk when moving from the risk-neutral to the real-world measure. This risk premium in our case is based on a market price of interest rate risk. We use the approach ofHull et al. (2014) for the estimation of this market price of interest rate risk. They assume that the interest rate process is stationary and that observations in the past can be seen as random samples of the short-rate distribution under real-world measure. The market price of risk is then estimated through an iterative process. This is done in such a way that the expected short rate at future time T is equal to the average of the observed short-rates corresponding to maturity T in the past.
The remainder of this thesis is organized as follows. First, we choose two interest rate models based on certain criteria and give an overview of the literature on the calibration of these models and results of previous studies in chapter 2. Next, we discuss the Hull-White one-factor model (Hull and White, 1990) and the two-factor G2++-model (Brigo and Mercurio,2006) and introduce model pricing formulas in chapter 3. In chapter 4, we describe our data and the calibration set up, and discuss our calibration results for both models. In chapter 5 we estimate the market price(s)of interest rate risk. We conclude this thesis in chapter 6.
2
M O D E L C H O I C E A N D L I T E R AT U R E R E V I E W
The goal of this thesis is to estimate a joint-interest rate model, which is able to capture the distributions under the real-world (P ) measure and the risk-neutral measure (Q). Such a model can be used for two purposes, namely the pricing of contingent claims and risk management. We want to find a model that is able to fit to market prices of bonds and European swaptions and is able to produce realistic scenarios of the term structure of interest rates, that are used in risk management. In appendix A a detailed explanation of several interest rates and in-terest rate derivatives like swaptions is given.
This chapter consists of three sections. In the first section we discuss which properties are desirable in interest-rate models and which models are capable of capturing these properties. From several models we have chosen two. In section2.2we discuss several empirical studies that have performed a calibration for our models to market data. Finally, we end the chapter by introducing three criteria to compare the performance of our calibrated models.
2.1 m o d e l c h o i c e
We are looking for a model that is able to price interest rate derivatives and produce and simulate the entire term structure of interest rates. There are several types of models that are able to do this: for example market models and short-rate models. Market models are expressed in terms of forward/swap rates, while short-rate models model the instan-taneous short rate. Since market models have many parameters, the calibration and simulation is time consuming. Therefore, we have cho-sen to limit our analysis to short-rate models. Ideally the short-rate model should possess the following properties:
1. Mean reversion: historical data show that interest rates have a higher probability to go down when high and go up when low. They revert to a mean level.
2. Market consistency: the model should give an exact fit to the market term structure.
3. Negative interest rates: some short-term interest rates are cur-rently negative. A realistic model should be able to capture the possibility of negative rates.
4. Closed form/analytical pricing formulas: a model is more tractable whenever there are closed form pricing formulas. The parameters of the models are calibrated to the market prices of swaptions. Such a procedure involves the calculation of thousands of prices. This can be done more efficiently whenever there are analytical pricing formulas.
5. Arbitrage free: the model should assign values to prices of deriva-tives in such a way that it is impossible to construct arbitrages. Nowadays many different short-rate models exist. We have compared several models in table1. On one hand, we have chosen one-factor mod-els that have one source of uncertainty: the Vasicek model (Vasicek, 1977), The Hull-White one-factor model (HW1F) (Hull and White, 1990), the Cox, Ingersoll and Ross (CIR) model (Cox et al., 1985) and the Black Karinski (BK) model (Black and Karasinski,1991). On the other hand, two two-factor models are chosen: the G2++ model (Brigo and Mercurio, 2006) and the CIR2++ model (Brigo and
Mer-curio, 2006). These models have been compared on the properties of mean reversion, market consistency, the possibility of negative inter-est rates, on the availability of analytical pricing formulas and if the model is arbitrage free. Table 1 shows that only two models satisfy all five properties: the Hull-White one-factor model and the G2++ model/Hull-White two-factor model.
m o d e l va s i c h w 1 f c i r b k g 2++ c i r 2++
Mean Reversion + + + + + +
Market consistency - + - + + +
Negative interest rates + + - - +
-Analytical formulas + + + - +/-
+/-Arbitrage free - + - + + +
Table 1: The properties of different short-rate models.
2.2 p r e v i o u s s t u d i e s
Several empirical studies have been performed on the calibration of the Hull-White one-factor model and/or the G2++ model. Gurrieri et al. (2009) for example provide a documentation and numerical calculation
of several calibration methods for the Hull-White one-factor model. They consider a two year period from November 2007 till Septem-ber 2009 where the calibration is performed weekly on the basis of 120 swaptions. In their analysis they compare different strategies such as constant or time-dependent mean reversion and volatility, local or
2.2 previous studies 5
global optimization, an SMM approximation or exact analytical prices and the choice for one two-dimensional optimization or two subsequent one-dimensional optimizations. The authors argue that several of these methods are acceptable on the basis of parameter stability and the per-centage fitting errors. According to the authors there is not one single best method and the choice of method depends on the user preferences regarding fitting quality, run-time and ease of implementation.
Another study is performed by Driessen et al. (2003), who provide an empirical analysis and comparison of several one-factor models, in-cluding the Hull-White one-factor model, and multifactor term struc-ture models based on their pricing and hedging performance. They use weekly quotes on US caps and swaptions from January 2, 1995 until June 11, 1999. They find that the average mean-reversion param-eter over these 232 weekly observations is slightly positive on average, but not significantly different from 0. The pricing performance of the models is measured by the out-of-sample fit to derivative prices. This procedure is as follows. At day 1 they estimate the parameters of the model given the information up to that day. Next, caps and swaptions are valued at day 1+τ (τ > 0) using these calibrated parameters and the actual term structure of that day. The model prices are then com-pared with the observed prices, giving an out-of-sample error. They find that the Hull-White one-factor model has the lowest absolute pre-diction errors, which on average are equal to 8.5%. Furthermore they find no empirical evidence to back the theoretical claim of Rebonato (1999) that multiple-factor models have a better swaption pricing per-formance since they can incorporate non-perfect correlation between interest rates of different maturities.
Several empirical studies have compared the Hull-White one-factor to the two-factor Hull-White model/ G2++ model.Koopman (2013) has investigated the impact of varying modeling assumptions on the holistic balance sheet. One part of this research focused on the calibration of the Hull-White one-factor model and the two-factor variant. However, he does not analyze the different results and only gives the estimated parameter values. Nonetheless these results are useful since data is used from 2011 and 2012 on at-the-money swaptions and different calibration methods are used with respect to the handling of the term structure of interest rates, the observed premiums or implied volatilities, a SMM approximation or exact formulas, local or global optimization and rela-tive or absolute weights. Bloch Mikkelsen (2011) investigated the pric-ing and hedgpric-ing performance of the Libor, Hull-White one-factor and G2++ model in his master-thesis. He followed the work of Gupta and Subrahmanyam (2005) and compared the models based on in-sample estimation and out-of-sample pricing. Based on Danish data on inter-est rate caps from April 29, 2002 till December 1, 2009 he found that
the G2++ model outperforms the Hull-White one-factor model in both in-sample estimation and out-of-sample pricing performance.
The former studies show that the calibration of the short-rate models involves at least the making of the following choices:
• What calibration instruments should we use?
• How do we calculate the entire term structure of interest rates: do we use a Nelson-Siegel function or do we interpolate the missing maturities for example with a cubic spline?
• Do we use exact pricing formulas or a SMM approximation (Hull-White one-factor model)?
• Should we use relative or absolute weights? These weights are used in the minimization criterion to either minimize the sum of squared percentage pricing errors (relative) or the sum of squared pricing errors (absolute).
• Do we use a local or a global optimizer?
2.3 c r i t e r i a m o d e l p e r f o r m a n c e
In order to draw conclusions about the pricing performance of mod-els we need criteria to compare the modmod-els. In this thesis we follow Driessen et al. (2003) and Gupta and Subrahmanyam (2005) and use the following three criteria to compare the two models:
1. Goodness-of-fit measure, 2. Stability of the parameters,
3. Out-of-sample pricing performance or forecasting performance. Almost all of the empirical studies above have used the following func-tion to determine the goodness-of-fit:
min x N X i M X j wi,j
Pi,jmarket− Pi,jmodel(x)2, (1)
where the market price of a swaption with a certain maturity (i) and tenor (j) is given by Pi,jmarket. The price of the model is given by Pi,jmodel(x), which is a function of the vector of parameters x of the short-rate model. There are in total N different swaption maturities and M different tenors. The function is minimized with respect to x. The weights wi,j can be used to give different weights to different swap-tions:
2.3 criteria model performance 7
2. Absolute weights: wi,j=1.
Often relative weights are used, which means that the goodness-of-fit measure is equal to the sum of squared percentage pricing errors. An-other measure would be the sum of squared pricing errors (absolute weights). However, this measure places more weight on expensive swap-tions creating a better fit for swapswap-tions with longer maturities, which are more expensive, and a worse fit for the shorter maturities.
The stability of the parameters is straightforward. If the calibrated pa-rameters of a model are volatile this can produce changing future term structure movements. For risk-management purposes this is undesir-able. This can be measured through the standard deviation of the cal-ibrated parameters to cross-section swaption data. The out-of-sample performance or forecasting performance is the last criterion and it in-volves the following procedure. Suppose we have cross-sectional swap-tion data on 10 days. Then we calibrate the parameters of the model to the swaption data of the first day. Next we "forecast" the prices of the swaptions in the other 9 days by using the calibrated parameters and the actual term structures of those dates. These "forecast" prices are then compared with the actual market prices. The out-of-sample errors are then used to assess the performance of the models. To follow previous studies by Gupta and Subrahmanyam (2005) and Amin and Morton (1994) these out-of-sample pricing errors are measured accord-ing to four different statistics:
1. Average pricing error,
2. Average absolute pricing error, 3. Average percentage pricing error,
3
S H O RT - R AT E M O D E L S
In the previous chapter we chose two short-rate models based on sev-eral criteria. In this chapter we discuss both models. In section 3.1the first short-rate model is described: the Hull-White one-factor model and closed form pricing formulas are given for zero-coupon bonds, op-tions on bonds and swapop-tions. We end this chapter by doing the same analysis for the G2++ model.
3.1 h u l l - w h i t e o n e - f ac t o r m o d e l
In 1990Hull and Whiteintroduced a model that describes the behavior of the short rate and can provide an exact fit to the initial term struc-ture. The dynamics of the short rate rtunder the risk-neutral measure Q are described by:
dr(t) =λ(θ(t)− r(t))dt+ηdWQ(t). (2) The parameters of this model have the following interpretation:
λ mean-reversion rate, where λ > 0,
θ(t) mean-reversion level, as a function of time t, η volatility of the short rate, with η > 0.
The short rate r(t)is mean reverting, which means that the short rate is pulled back towards a long-term level θ(t)when it diverges from this level. The function θ(t)is used to exactly fit the initial term structure of interest rates. The parameter λ determines how fast the short-rate process is pulled back when it diverges from θ(t). Finally, the parameter η represents the volatility of the short rate. The uncertainty is modeled through the Brownian motion WQ(t).
3.1.1 Real-World Dynamics
Interest rate derivatives are priced under the risk-neutral measure. The model in equation (2) provides a way to do this. However, for risk management the distribution of the short rate under the real-world measure P is needed. The real-world and the risk-neutral world are connected by the market price of risk, denoted by µ(t). A change of measure can be made such that:
WQ(t) =WP(t) +
Z t 0
µ(s)ds, (3)
where WP(t)is a Brownian Motion under P . This in turn leads to the dynamics under the real-world measure:
dr(t) =λ θ(t) + µ(t)η λ − r(t) dt+ηdWP(t). (4) In the equation above, there is a new parameter µ(t), indicating the market price of interest-rate risk. The market price of risk can been seen as the excess return with respect to a risk-free investment per unit of risk. The model is not fully specified until an assumption is made about the functional form of the market price of risk process. This process could be constant, time-dependent or even a functional form of the short rate. The real-world measure and the risk-neutral measure are thus connected through µ(t). In order to construct a joint-measure model, the market price of risk is estimated together with the other parameters of the model in equation (2).
3.1.2 Price of a Zero-Coupon Bond
Since interest rate derivatives can be easily written in terms of coupon bonds, it is convenient to use an analytical expression for a zero-coupon bond.Hull and White(1990) derived such an expression under the assumption that the short rate follows their Hull-White model. In this section, the price of a zero-coupon bond is given in terms of the parameters of the Hull-White model. A complete derivation is given in appendix B. One of the reasons for choosing the Hull-White model is that it can reproduce the market discount curve T 7→ PM(0, T) exactly. This can be done through the market instantaneous forward rates fM(0, T)in the equation for θ(t):
θ(t) =fM(0, t) + 1 λ ∂ ∂tf M(0, t) + η2 2λ2(1 − e −2λt). (5) The price of a zero-coupon bond with maturity T at time t, P(t, T), can be expressed as:
P(t, T) =A(t, T)e−B(t,T)r(t), A(t, T) = P M(0, T) PM(0, t) exp ( B(t, T)fM(0, t)−η 2 4λ(1 − e −2λt)B(t, T)2 ) , B(t, T) = 1 λ h 1 − e−λ(T −t)i. (6)
3.1 hull-white one-factor model 11
3.1.3 Options on bonds
Jamshidian (1989) showed that it is convenient to write a European option on a coupon-bearing bond as the sum of European options on zero-coupon bonds. Therefore, a formula to price a European option on a zero-coupon bond is given in this subsection. The short rate has a normal distribution in the Hull-White model. The price of an option on a zero-coupon bond can therefore be given by a Black (1976)-like formula. The price of a call option with strike K and maturity T on a zero-coupon bond with maturity S, is at time t equal to:
ZBc(t, T , S, K) =P(t, S)Φ(d1)− KP(t, T)Φ(d2), (7) where: d1= σ1 pln P(t,S) P(t,T)K +σp 2 , d2=d1 − σp, σp =η s 1 − e−2λ(T −t) 2λ B(T , S).
In this formula the cumulative normal distribution is given by Φ and B(T , S)is as given in (6) .
3.1.4 Swaption Price
The parameters of the model are calibrated to the market prices of bonds and European swaptions. An explicit formula for the price of a European swaption is therefore useful under the Hull-White model. Suppose we have a swaption with a maturity of T0 and strike rate
K. The underlying swap exchanges cash flows at times T1, .., Tn and the notional value is given by N . The times between the payments are equally spaced and the time is given by δ. In the Hull-White one-factor model the price of a zero-coupon bond is a monotone decreasing function in the short rate. Jamshidian (1989) developed a smart trick by using this observation, which makes it possible to write European options on a coupon bearing bond as the sum of European options on zero-coupon bonds. Jamshidian’s decomposition consists of three steps:
1. In the first step we find the rate r∗ for which the following holds: P(T0, Tn; r∗) +KδPin=1P(T0, Ti; r∗) =1;
2. In the second step, the prices of options on zero-coupon bonds, that replicate the payoff of the coupon bearing bond, are calcu-lated. The strike price of each option equals the corresponding zero-coupon bond at time T0 with r=r∗: Ki =P(T0, Ti; r∗);
3. The price of the coupon bearing bond or swaption is then equal to the sum of the options on the zero-coupon bonds in step 2. The price of such a receiver swaption at time t ≤ T0 is given by:
P(t, T0, T1, ..., Tn, N , K) =N(ZBc(t, T0, Tn, Kn) +Kδ n X i=1 ZBc(t, T0, Ti, Ki)). (8) The price of the swaption is the sum of the options on zero-coupon bonds ZB, where the formula for these options is given by equation (7).
3.2 g 2++ m o d e l
In the previous section we discussed the Hull-White one-factor model. One-factor models have the disadvantage that at every time instant, rates for all maturities are perfectly correlated. However, it seems rather unrealistic to assume that the 3-month yield is perfectly correlated with the 30-year yield. Also, the payoffs of swaptions depend on interest rates of different maturities, which makes the correlation between the inter-est rates important. Two-factor models have two sources of uncertainty, described by Brownian motions which are possibly correlated. There-fore these models are capable of giving non-perfect correlation between rates of different maturities, which makes them especially useful for the calibration to swaptions. BesidesJamshidian and Zhu(1997) show that two-factor models are able to explain upto 90% of the variation in the yield curve.
3.2.1 Risk-Neutral Dynamics
The G2++ model is a two-factor short-rate model, where the short rate is modeled as the sum of two factors x and y and a deterministic function φ. The dynamics under the risk-neutral measure Q are:
r(t) =x(t) +y(t) +φ(t), r(0) =φ(0), (9) where the two factors satisfy:
dx(t) =−ax(t)dt+σdW1Q(t), x(0) =0, dy(t) =−by(t)dt+ηρdW1Q(t) +η q 1 − ρ2dWQ 2 (t), y(0) =0. (10)
This representation is obtained through a Cholesky decomposition of the original dynamics ofBrigo and Mercurio(2006). We have used this representation since it makes the process under the real-world mea-sure more clear. In the above representation there are two independent sources of uncertainty given by the Brownian motions W1Q and W2Q.
3.2 g2++ m o d e l 13
Through the deterministic φ the current term structure is exactly fit-ted. The parameters a, σ, b and η are strictly positive constants. For the parameter ρ, the instantaneous correlation between the Brownian motions in the original process, we have the constraint −1 ≤ ρ ≤ 1. As mentioned before, there is a deterministic function φ through which the initial term structure can be exactly fitted. This fit is only exact if and only if for each T the following holds for φ:
φ(T) =fM(0, T) + σ 2 2a2(1 − e −aT)2+ η2 2b2(1 − e −bT)2 +ρση ab(1 − e −aT)(1 − e−bT). (11)
Finally, we would like to point out that the G2++ model is equal to a Hull-White two-factor model. See Brigo and Mercurio (2006) for equivalent parameter definitions of the Hull-White two-factor model.
3.2.2 Real-World Dynamics
In risk management the distribution of interest rates under the real-world measure is relevant. The change from the risk-neutral measure to the real-world measure can be made by setting:
W1Q(t) =W1P(t) + Z t 0 µ1(s)ds, (12) W2Q(t) =W2P(t) + Z t 0 µ2(s)ds. (13)
This gives the dynamics of the model under the real-world measure:
r(t) =x(t) +y(t) +φ(t), r(0) =φ(0). (14) Also, dx(t) = (−ax(t) +σµ1(t))dt+σdW1P(t), x(0) =0, dy(t) = −by(t) +ηρµ1(t) +η q 1 − ρ2µ 2(t) dt+ηρdW1P(t) +η q 1 − ρ2dWP 2 (t), y(0) =0. (15)
The real-world process of the G2++ model is now known.
3.2.3 Price of a Zero-Coupon Bond
The G2++ model allows for closed form solutions for the price of zero-coupon bonds. The derivation of this formula can be found in (Brigo
and Mercurio, 2006). The price of a zero-coupon bond with maturity T at time t, P(t, T), can be written as:
P(t, T) =A(t, T)e−B(a,t,T)x(t)−B(b,t,T)y(t), (16) where A(t, T) = P M(0, T) PM(0, t)e 1 2[V(t,T)−V(0,T)+V(0,t)], B(z, t, T) = 1 − e −z(T −t) z , V(t, T) = σ 2 a2[T − t+ 2 ae −a(T −t)− 1 2ae −2a(T −t)− 3 2a] +η 2 b2[T − t+ 2 be −b(T −t)− 1 2be −2b(T −t)− 3 2b] +2ρση ab[T − t+ e−a(T −t)− 1 a + e−b(T −t)− 1 b −e −(a+b)(T −t)− 1 a+b ]. (17)
3.2.4 Price of a European Swaption
Suppose we have a European swaption with the following characteris-tics:
X the strike rate,
T maturity of the swaption, N nominal value of the swap,
T ={t1, .., tn} payment dates of the swap, τi year fraction between dates ti−1 and ti, ci net cashflow at time i.
The cash flows equal ci = Xτi for i = 1, .., n − 1 and 1+Xτn at maturity. The price at time 0 of a European swaption with the above characteristics can be obtained by numerically computing the following integral: P(0, T , T , N , X, ω) =N ωP(0, T) Z ∞ −∞ e−12( x−µx σx ) 2 σx √ 2π [Φ(−ωh1(x)) − n X i=1 λi(x)eκi(x)Φ(−ωh2(x))]dx. (18)
In this formula ω = 1 (ω = −1) corresponds to a payer (receiver) swaption. The components of this formula are expressed in terms of the parameters of the G2++ model:
3.2 g2++ m o d e l 15 h1(x) = ¯y − µy σy q 1 − ρ2 xy −ρxy(x − µx) σx q 1 − ρ2 xy , h2(x) =h1(x) +B(b, T , ti)σy q 1 − ρ2 xy, λi(x) =ciA(T , ti)e−B(a,T ,ti)x, κi(x) =−B(b, T , ti) µy− 1 2(1 − ρ 2 xy)σy2B(b, T , ti) +ρxyσy x − µx σx . The variable ¯y= ¯y(x) in h1(x) is the unique solution of the following equation: n X i=1 ciA(T , ti)e−B(a,T ,ti)x−B(b,T ,ti)y¯ =1. (19) Where: µx =−MxT(0, T), µy =−MyT(0, T), σx =σ s 1 − e−2aT 2a , σy =η s 1 − e−2bT 2b , ρxy = ρση (a+b)σxσy h 1 − e−(a+b)Ti. (20) And as last: MxT(s, t) = σ 2 a2 +ρ ση ab ! h 1 − e−a(t−s)i − σ 2 2a2 h e−a(T −t)− e−a(T+t−2s)i − ρση b(a+b) h e−b(T −t)− e−bT −at+(a+b)si, MyT(s, t) = η 2 b2 +ρ ση ab ! h 1 − e−b(t−s)i − η 2 2b2 h e−b(T −t)− e−b(T+t−2s)i − ρση a(a+b) h e−a(T −t)− e−aT −bt+(a+b)si. (21)
Now that we have (semi-)analytical formulas for the price of a European swaption the parameters of our joint-measure models can be estimated. This is done in two steps. First, we calibrate the parameters of the
risk-neutral models to the market prices of at-the-money swaptions in chapter 4. Finally, we estimate the market price(s) of risk by using historical data in chapter 5, so that the model can be used to create real-world scenarios.
4
C A L I B R AT I O N O F T H E S H O RT R AT E M O D E L S
In the previous chapter we gave (semi-) analytical formulas for the price of a European swaption. In this chapter we use these formulas to calibrate two different models to the observed market prices of at-the-money European swaptions. The set up of the calibration process is shown in figure 2.
The first step is gathering market data and using this data to find the market prices of swaptions. The data and the calculation of market prices are explained in section 4.1. In the second step (section4.2), we fit the models to the market prices of swaptions by using the swaption model formulas in sections 3.1.4 and 3.2.4. An optimization criterion or goodness-of-fit measure is chosen, after which a routine is carried out in matlab. The parameters of the models are estimated in such a way that the optimization criterion is minimized. In section 4.3, the exact optimization in matlab for the Hull-White one-factor model is described. The results are then presented and discussed. In section4.4, the exact optimization for the G2++ model is described and the results are discussed. In section 4.5 we test the stability of the estimated pa-rameters by performing the same calibration on a quarterly basis since 2006. Finally, the out-of-sample pricing prediction errors are calculated for both models in section 4.6. The two models are thus assessed on the basis of the three assessment criteria in section 2.3.
Figure 2: Calibration process.
4.1 data
Both models are calibrated to at-the-money (ATM) Euro swaptions. There are up to 200 different ATM Euro swaptions traded in the market. We have chosen swaptions that have appropriate maturities and tenors for pension fund purposes:
a. Maturities: the swaptions have maturities of 1, 2, 3, 4, 5 and 10 years.
b. Tenors: the underlying swap has a tenor of 5, 10, 15, 20, 25 and 30 years.
The calibration is performed on data of swaptions and the term struc-ture of interest rates of September 10th, 2014. ATM swaptions traded in EUR are quoted in terms of Black volatilities and can be extracted from Bloomberg, see figure 3(a). The underlying swap has a notional of 1, the fixed leg pays once a year and the floating leg resets semi-annually.
(a) ATM swaption implied volatility surface.
(b) Calculated market prices of ATM swaptions of September 10th, 2014.
4.1 data 19
The implied volatility surface in figure 3(a) shows the implied Black volatility of 36 ATM swaptions. In this figure, the implied Black volatil-ity has been plotted against the maturvolatil-ity of the swaption and the tenor of the underlying swap. The implied volatility surface shows a clear pat-tern. First, the implied volatility of ATM swaptions increases when the maturity of the option decreases, often referred to as the volatility smile. Second, a decrease in the tenor of the underlying swap increases the im-plied volatility of ATM swaptions. This leads to a high imim-plied volatility for a ATM swaption with a small tenor and small maturity. We would like both models to fit to the term structure of interest rates and the market prices of ATM swaptions. The market however, quotes the im-plied Black volatilities. We therefore use Black’s formula in equation (55) to derive the market prices of ATM swaptions. Since the swaptions are at-the-money, the corresponding strike rates are obtained by using the zero curve and equation (53). The calculated market prices are de-picted in figure 3(b).
Figure 4: Term structure of zero rates on September 10th, 2014 after cubic spline interpolation.
The term structure of interest rates of September 10th, 2014 is used. The zero curve that is provided by Bloomberg however is incomplete. First, since we plan on using the models to simulate the short rate in the next chapter we need to choose the starting value r(0). The Euro OverNight Index Average (Eonia) seems reasonable, but this rate is very volatile. Therefore we follow Enev (2011) and use the 3 month EURIBOR. This rate is converted into a continuously compounded rate that will be used as the starting value r(0)in the simulation of the short rate. Second, the corresponding zero rates obtained from Bloomberg do not include all maturities. This problem could for example be solved by constructing a term structure of interest rates through a Nelson-Siegel function or obtaining the missing maturities by interpolation.
We follow Koopman (2013) and use a cubic spline to interpolate the data points. Splines are sufficiently smooth polynomial functions, where these functions are defined between two subsequent data points. This procedure is performed in matlab by using interp1 function with the option spline. In this way all zero rates for the maturities 1 up to and including 50 are derived, see figure 4. The market prices of swaptions are obtained and we can now proceed with the actual calibration. In the calculations, day-count conventions are ignored.
4.2 o p t i m i z at i o n p ro b l e m
The short-rate models are calibrated to the market prices of swaptions. This is done by minimizing the following function:
min x X i X j wi,j
Pi,jmarket− Pi,jmodel(x)2. (22)
The market price of a swaption with a certain maturity and tenor is given by Pi,jmarket. The price of the model is given by Pi,jmodel(x), which is a function of the vector of parameters x of the short-rate model. The function is minimized with respect to x. The weights wi,j can be used to give different weights to different swaptions. Two different sets of weights are used:
1. Relative weights: wi,j = (Pi,jmarket)−2, 2. Absolute weights: wi,j=1.
Now that the optimization function has been defined we can proceed to the methods of optimization in the next subsection.
4.2.1 Global & Local Optimization
The models are calibrated by finding a solution to the optimization problem in equation (22). The minimization is performed in matlab and since this is a non-linear least squares problem, a minimum can be found by the local matlab solver lsqnonlin. However, an optimiza-tion problem could potentially have several local optima and we would like to find the global optimum.
Suppose the optimization function has multiple local minima such as in the left graph of figure 5, that are depicted by the blue spikes. At a certain starting point, the local solver finds the solution in a "basin of attraction", which is indicated by the arrows into the blue contour plot in the right graph. Once the local solver has found this point it does not search outside the basin anymore. Choosing the right starting
4.3 calibrating the hull-white one-factor model 21
Figure 5: Basins of attraction.
point therefore seems fundamental for finding the global optimum. matlab also offers solvers that are less reliable upon the chosen start-ing point. One of these solvers is multistart. multistart generates n random starting points and then runs the local solver lsqnonlin. It saves all the local optima and at the end compares the values to find the best optimum. It therefore searches different basins. Although multistart cannot guarantee that our solution is global it does make our initial "guess" for the starting point less important, and makes the procedure more robust.
4.3 c a l i b r at i n g t h e h u l l - w h i t e o n e - f ac t o r m o d e l In this section we discuss the calibration of the risk-neutral version of the Hull-White one-factor model to the data mentioned before.
4.3.1 Calibration Method
Four different calibrations are performed for the Hull-White one-factor model by using two different solvers and two different weights, namely absolute and relative weights.
l s q n o n l i n : The standard optimization options are used. The start-ing point is [λ, η] = [0.05, 0.01]. The solution is restricted to [0, 1]×[0, 1]. In the first step of Jamshidian (1989), the r∗ is found by solving an equation. This is done in matlab through fzero with starting point 2.5%.
m u lt i s ta rt : The multistart procedure involves the creation of n random starting points. From these random starting points the local solver lsqnonlin is run. The local solver has the same settings as in the above description. In total 50 starting points have been created at random. Before starting the procedure, the random number generator is set at rng(14).
4.3.2 Results for Calibration of the Hull-White One-Factor model The in-sample results of all four calibrations are shown in table2. The parameter estimates for λ, η and the Root Mean Square Error (RMSE) are given for each of the four calibrations. As a comparison the results of Plomp(2013) and Koopman (2013) are also displayed. The RMSE is calculated through the following formula:
RM SE(x) = v u u t P6 i=1 P6 j=1
Pi,jmarket− Pmodel i,j (x) 2 36 . (23) s o lv e r w e i g h t s λ η r m s e Lsqnonlin Relative 1.0908x10−10 0.0086 0.0160 Absolute 0.0108 0.0102 0.0136 MultiStart Relative 9.0880x10−12 0.0086 0.0160 Absolute 0.0108 0.0102 0.0136 p r e v i o u s s t u dy w e i g h t s λ η r m s e Plomp (2013) Relative 0.0234 0.0082 -Koopman (2013) Relative 0.0000 0.0081 0.3561 Koopman (2013) Absolute 0.0400 0.0107 0.4482
Table 2: Calibration results of the Hull-White one-factor model with two dif-ferent solvers and two difdif-ferent weights on data from September 10th, 2014. In addition results from previous studies are displayed.
The estimated value for λ is almost equal to 0 for both solvers in case of relative weights. This means that there is almost no mean-reversion. In section 4.5 we will test on the basis of a larger dataset if λ is signif-icantly different from zero. Our study is not the only empirical study that reports such a parameter estimate, Koopman (2013) for example also report a mean-reversion rate of 0.000 based on swaption data from ultimo 2012 and a similar calibration set up by using relative weights and cubic splines to interpolate the term structure. Plomp(2013) esti-mated a value of 0.0234. The estiesti-mated value of σ for both solvers in case of relative weights is close to 0.008. Again both Koopman (2013) and Plomp (2013) show a similar estimate of 0.008. In general, the parameter values increase when we change from relative weights to ab-solute weights. The Root Mean Square error (RMSE) decreases due to this change. Although, the global solver multistart gives lower val-ues for the parameter λ, the RMSE does not change compared to the local solver lsqnonlin . The use of a global solver therefore does not change the results significantly in this case. These parameter estimates are comparable to other research,Koopman(2013) estimated λ = 0.040
4.3 calibrating the hull-white one-factor model 23
and σ =0.0107 and a decrease in RMSE when changing from relative to absolute weights. It is difficult to compare the RMSE of our results toKoopman (2013) since they do not report the principal value of the underlying swap and use more swaptions.
(a) Lsqnonlin and relative weights. (b) Lsqnonlin and absolute weights.
(c) MultiStart and relative weights. (d) MultiStart and absolute weights.
Figure 6: Calibration results Hull-White One-factor model: percentage devia-tion from market prices September 10th, 2014.
It is not evident from table 2 if the model provides a good fit to the data. Therefore the percentage deviation of model prices from market prices is plotted in figure 6. The percentage deviation is between -50% to +150%. The figure shows the graph for the different combinations of weights and solvers. Each graph has the highest positive deviation of around 100% at maturity 1 and tenor 5. Swaptions with a low maturity and low tenor are overvalued in our model compared to the market. The highest negative deviation is shown at maturity 1 and tenor 30. These swaptions are undervalued by our model. Since the use of rela-tive weights involves the minimization of the sum of relarela-tive deviations, figures 6(a) and 6(c) show less percentage deviation than when using absolute weights. This is what we expect to find.
It is market convention to quote swaptions through their implied volatil-ity. Table 3 gives the implied Hull-White one-factor volatility of the
lsqnonlin calibration with relative weights. The market implied (Black) volatility table is given in AppendixC. Again the largest deviations are shown at short maturities, indicating that the model has trouble fitting the short end of the volatility surface. In the next section we investigate whether a two-factor model is able to give a better fit to the market data. m at / t e n o r 5 1 0 1 5 2 0 2 5 3 0 1 0.9734 0.4584 0.3640 0.3386 0.3331 0.3344 2 0.6597 0.3795 0.3253 0.3142 0.3165 0.3225 3 0.4461 0.2976 0.2705 0.2695 0.2771 0.2860 4 0.3187 0.2356 0.2239 0.2290 0.2398 0.2505 5 0.2398 0.1897 0.1866 0.1951 0.2075 0.2192 10 0.1455 0.1342 0.1444 0.1603 0.1762 0.1914 Table 3: Implied volatility by the Hull-White-one-factor model prices.
4.4 c a l i b r at i n g t h e g 2 + + m o d e l
The second model that we calibrate, is the G2++ model. Again we perform four different calibrations by using the two different solvers and two different sets of weights.
l s q n o n l i n : The standard optimization options of the solver are used. The starting point is [a, b, σ, η, ρ] = [0.1, 0.5, 0.01, 0.006, −0.7]. The solution is restricted to [10−7, 1] for all parameters except ρ, which is restricted to [−1, 1]. The model price of a swaption involves the calculation of the integral in equation (18). This is done in matlab by using linspace with ±5 × σ as boundaries and 500 evaluation points. The function is evaluated by the func-tion trapz, which uses the trapezoid method. The value of ¯y (equation (19)) is calculated using fsolve.
m u lt i s ta rt : We chose a total of 30 starting points and set the ran-dom number generator at rng(14).
Now that the exact settings of the optimization are established, we can proceed to the results.
4.4.1 Results for Calibration of G2++
The calibrated parameters of the G2++ model are shown in table 4 together with the RMSE of each calibration.
4.4 calibrating the g2++ model 25 s o lv e r w e i g h t a b σ η ρ r m s e Lsqnonlin Rel 0.0366 0.6038 0.0108 0.0272 -0.9926 0.0015 Abs 0.0396 0.4902 0.0113 0.0211 -0.9994 0.0014 MultiStart Rel 0.0366 0.6038 0.0108 0.0272 -0.9926 0.0015 Abs 0.0361 0.6030 0.0106 0.0242 -0.9987 0.0013 s t u dy w e i g h t a b σ η ρ r m s e B&M Rel 0.07572 0.5430 0.0058 0.0117 -0.9914 -Koopman Rel 0.08298 0.3103 0.0246 0.0323 -0.9765 0.2208 Koopman Abs 0.08501 0.3111 0.0246 0.0311 -0.9728 0.2008
Table 4: Calibration results of the G2++ model with two different solvers and two different sets of weights.
As a comparison we have added the results fromKoopman(2013) and Brigo and Mercurio(2006) in the table.Brigo and Mercurio(2006) have calibrated their own model to at-the-money Euro cap volatility data of February 13, 2001, with maturities {1, 2, 3, 4, 5, 7, 10, 15, 20}. They minimized the sum of squared percentage differences, similar to our relative weights minimization. Koopman (2013) calibrated the model to data on swaptions with maturities 1 year to 10 years and underlying swap tenors from 1 year to 30 years from ultimo 2012. In general, we can conclude that we find similar parameters estimates as these two previous studies. Finally, in all cases the estimate for ρ is close to -1. When we compare the results from Hull-White calibration to the re-sults of the G2++ model the Root Mean Square Error (RMSE) strikes us. The RMSE is approximately 10 times smaller for all calibrations compared to the Hull-White one-factor model. The exact same solu-tion is found for both the local and global solver in the case of relative weights. In the case of absolute weights the solutions are not the same and the calibrated parameter values differ.
To get a better understanding of the fit of the model, the percentage error of all four calibrations is plotted in figure 7. In comparison to the Hull-White model, the G2++ model shows smaller errors. Where the Hull-White model had errors up to 100%, the G2++ model has smaller errors. In case of relative weights the errors have extremes of ±10% and are on average ±3%. Figure7(b) shows the percentage error for absolute weights and the lsqnonlin solver. The error is large for the swaption with the shortest maturity and shortest tenor, indicated by the white spike. Although this improved by using the global solver multistart there is still a large error of around 25% at the same swap-tion indicating that the short end of the swapswap-tion surface is harder to
(a) Percentage errors with Lsqnonlin and rela-tive weights.
(b) Absolute errors with Lsqnonlin and absolute weights.
(c) Percentage errors with MultiStart and rela-tive weights.
(d) Absolute errors with MultiStart and abso-lute weights.
Figure 7: Calibration results G2++ model.
fit.
Since it is market convention to quote swaptions in terms of their im-plied volatility, the imim-plied volatility of the model prices is given in table 5. The corresponding market implied volatilities are given in Ap-pendix C. m at / t e n o r 5 1 0 1 5 2 0 2 5 3 0 1 0.4728 0.3686 0.3095 0.2804 0.2622 0.2481 2 0.4700 0.3562 0.3068 0.2811 0.2643 0.2504 3 0.4215 0.3369 0.2988 0.2773 0.2624 0.2490 4 0.3745 0.3179 0.2895 0.2718 0.2587 0.2458 5 0.3377 0.3018 0.2808 0.2663 0.2545 0.2421 10 0.2727 0.2698 0.2618 0.2523 0.2402 0.2277
4.5 historical estimates 27
For the data on September 10th, 2014 we conclude that the G2++ model gives a better fit to our 36 at-the-money swaptions than the Hull-White one-factor model. The RMSE is a factor 10 smaller, the percent-age errors are on averpercent-age low and between reasonable bounds and the implied volatilities are closer to the market implied volatilities. Brigo and Mercurio (2006) provide an intuition for this out-performance:
"However, as in this case, it often happens that the ρ value is quite close to minus one, which implies that the G2++ model tends to degenerate into a one-factor (non-Markov) short-rate process. Notice, moreover, that the degenerate process for the short rate is still non-Markovian (if a 6= b), which explains what really makes the G2++ model outper-form its one-factor version."
The two parameters a and b can be seen as influencing different matu-rities. This will come back in our analysis of the market price(s) of risk. Our conclusion that the G2++ model fits better than the Hull-White one-factor model is based on a calibration to market data from a single date. Therefore the same calibration is performed for different dates in the next section.
4.5 h i s t o r i c a l e s t i m at e s
In this section both models are calibrated to end of quarter data, from the first quarter of 2006 up and till the third quarter of 2014 using the local solver lsqnonlin and relative weights, where all the matlab set-tings are as before. Again the data on zero curves is handled as section 4.1. The calculated term structures of interest rates are visualized in figure 8. Appendix D contains additional figures of historical implied volatilities.
The models are calibrated at the end of each quarter to a set of swap-tions and bonds. In table 6 the mean and the standard deviation are given of the calibrated parameters. We have added the results from Driessen et al. (2003) andBloch Mikkelsen (2011) as a comparison.
pa r a m e t e r s tat i s t i c t h i s s t u dy d r i e s s e n b l o c h
λ Mean 6.170% 1.057% 3.217%
S.D. (4.850%) (0.322%) (4.896%)
η Mean 0.022 0.017 0.009
S.D. (0.009) (0.069) (0.002)
Table 6: The mean and standard deviation of the estimated parameters of the Hull-White one-factor model for quarterly data from the first quarter of 2006 till the end of 2014.
Our average of the estimated η, 0.022, is close to the reported average of Driessen et al.(2003). Our standard deviation however is smaller. This is due the fact thatDriessen et al.(2003) use 282 daily estimates based on both cap and swaption data while we have a total of 35 quarterly estimates based on swaption data. Bloch Mikkelsen (2011) reports a smaller average η of 0.009 based on daily cap data from April 29, 2002 till October 31, 2005. The estimates of λ differ more between the stud-ies. We report an average of 6.170%, while both Driessen et al. (2003) andBloch Mikkelsen(2011) report lower values. Again we must add the note that we have used more recent data and only use swaption data. In section4.3.2we estimated a value for λ almost equal to zero for relative weights. Therefore we followDriessen et al.(2003) and test whether the average of the estimates is different from zero. Two t-tests show that both the null hypothesis of mean(λ) = 0 and mean(η) = 0 are re-jected at the 5% significance level. This is different than the results of Driessen et al.(2003) who found that the average of the estimated λ is not significantly different from zero.
Figure 9shows the calibrated parameters of the Hull-White one-factor model from the first quarter of 2006 up and till the third quarter of 2014. The estimates of the volatility η show a very stable pattern through time. The estimates of the mean-reverting rate λ are more volatile, but still quite stable and within the bounds [0, 0.2].
4.5 historical estimates 29
Figure 9: Calibration results of the Hull-White one-factor model since 2006 at the end of each quarter.
In table7the mean and standard deviation are shown for the calibrated parameters of the G2++ model. In addition we have added the reported values byBloch Mikkelsen (2011) who uses cap data.
pa r a m e t e r s tat i s t i c t h i s s t u dy b l o c h a Mean 0.023 0.129 S.D. (0.032) (0.066) b Mean 0.343 0.303 S.D. (0.431) (0.124) σ Mean 0.016 0.189 S.D. (0.020) (0.048) η Mean 0.016 0.292 S.D. (0.022) (0.046) ρ Mean -0.704 -0.867 S.D. (0.556) (0.017)
Table 7: The mean and standard deviation of the calibrated parameters of the G2++ model for quarterly data from the first quarter of 2006 till the end of 2014.
We show very different results fromBloch Mikkelsen(2011). Especially the estimates of the volatility parameters η and σ are much lower on average in our case and our estimate of ρ, −0.704, is on average further
away from −1. This is probably due to the fact that Bloch Mikkelsen (2011) uses cap data for his calibration.Brigo and Mercurio(2006) have also calibrated their model to at-the-money swaptions quotes from a single date on February 13, 2001 and found a = 0.774, b = 0.082, σ = 0.022, η = 0.010 and ρ = −0.702. Our results for η, σ and ρ are close to these estimates.
The calibrated parameters of the G2++ model are depicted in the graphs of figure 10. The upper left graph contains the estimates of a, the upper right b. The graph in the middle on the left displays the esti-mates of σ and the graph to its right the estiesti-mates of η. The graph at the bottom shows the estimates of ρ. When looking at the graphs of η and σ we can see that the estimates are quite stable, which corresponds with the standard deviations in table7. The estimates of a are slightly more volatile (s.d. of 0.032) and with a peak at the beginning of 2008. The two most volatile parameters are b and ρ. The graph of b is more volatile at the beginning of 2007 and from the year 2011 until the end of 2014. The graph of ρ shows a striking pattern since its value recipro-cates between -1 and 1. This could be due the fact that we restricted the parameter estimates to a certain region in section 4.4.
Figure 10: Calibration results G2++ model since 2006 at the end of each quarter. From top to bottom and left to right: a, b, σ, η and ρ.
4.6 out-of-sample pricing performance 31
Finally, the two models are compared based on the time-series of their Root Mean Square Error (RMSE), see figure 11. It is evident that the G2++ model outperforms the Hull-White one-factor model based on the RMSE: its RMSE is lower for each calibration date. Now that we have estimated the calibration of a larger historical dataset and have looked at the stability of the parameters we will investigate the pricing performance of both models in the next section.
Figure 11: Root mean square error time-series.
4.6 o u t - o f - s a m p l e p r i c i n g p e r f o r m a n c e
In the last two sections we investigated the in-sample pricing perfor-mance of both models. In this section the out-of-sample pricing per-formance is assessed. This is done by using a part of the dataset to calibrate the models and then "predict" the future swaption prices con-ditional on these calibrated values and the actual term structures of interest rates in the future. The dataset that is used for the calibration contains the quarterly information from March 31, 2006 up and till March 31, 2010. After the calibration the calibrated models in combi-nation with the actual term structures of interest rates from a second dataset are used to predict the swaption prices conditional on the cal-ibrated parameters. By comparing these conditional prices with the actual prices we get the out-of-sample pricing errors. We have chosen 4 quarterly datapoints (1 year) as the second dataset to assess the out-of-sample pricing performance. We have used a time horizon of one year since often pension funds use this horizon in risk-management.
The mean of the calibrated λ for the data from March 31, 2006 up and till March 31, 2010 is equal to 9.82%. The mean of η is equal to 0.0291. These two estimates are used in combination with the ac-tual term structures of interest rates to calculate the conditional prices under both models and the out-of-sample pricing errors. We follow Gupta and Subrahmanyam (2005) and Amin and Morton(1994) and use 4 statistics to measure these pricing errors: the average pricing error (ave), the average absolute pricing error (aape), the average percentage pricing error (appe) and the average absolute percentage pricing er-ror (aappe). These statistics are calculated by taking the mean over all swaptions and all 4 dates. The same is done for the G2++ model, where the calibrated parameter values are equal to a = 0.0261, b = 0.1554, σ = 0.0206, η = 0.0159 and ρ = −0.6652. The results are shown in table 8. m o d e l av e a a p e a p p e a a p p e HW-1F -0.0019 0.0040 -0.52% 6.42% G2++ 0.0096 0.0096 0.16% 0.16% p r e v i o u s s t u dy av e a a p e a p p e a a p p e Driessen - - 3.21% 8.54% Bloch (HW) 0.0001 0.0005 4.65% 6.07% Bloch (G2++) 0.0001 0.0005 0.86% 2.80%
Table 8: Four statistics for the measurement of the out-of-sample pricing er-rors: the average pricing error, the average absolute pricing error, the average percentage pricing error and the average absolute pricing er-ror.
For the Hull-White one-factor model we find an average percentage pric-ing error of −0.52%. This means that on average we under-price the swaptions with this model. On average we slightly overprice by 0.16% with the G2++ model. In general, the G2++ model produces more ac-curate predictions than the Hull-White one-factor model.Driessen et al. (2003) calculated the average percentage pricing error and average ab-solute percentage pricing error for the Hull-White model based on 282 weekly observations and found that on average the model overprices swaptions over a time horizon of 10 days. Their results show a higher pricing error than our results which could be due to the difference in used data. Bloch Mikkelsen (2011) calculated all four error statistics based on a time horizon of a week and daily cap data and also found that the G2++ model displays lower pricing errors compared to the one-factor Hull-White model.
4.6 out-of-sample pricing performance 33
4.6.1 Concluding remarks
In this chapter we found that the G2++ model fits better to the market prices of swaptions than the Hull-White one-factor model. The G2++ model also provides more accurate conditional predictions of swaption prices. These results correspond to our expectation since the G2++ model has more parameters to fit the surface of the swaptions prices. However, due to the higher number of parameters some of these param-eters of the G2++ model are less stable in time. We have calibrated our models to the market prices of swaptions. This is now a Q-measure model, which enables us to price swaptions, but we strive for a joint-measure model which can also be used for risk-management that uses P-scenarios. In the next chapter, the market prices of risk are estimated which completes the joint-measure model.
5
E S T I M AT I N G T H E M A R K E T P R I C E S O F R I S K
In the previous chapter both the Hull-White one-factor- and the G2++ model were calibrated to market data, which enables us to calculate Q-measure distributions. The current calibrated models are able to price swaptions. However, with our joint-measure models we would like to produce realistic P -measure scenarios for risk-management purposes as well. Since there is a change of measure from the Q-measure to the P -measure, a change of drift occurs. This change of drift is given by an interest rate risk premium, which is the product of the short rate volatility and a market price of risk. The market price of risk is defined as the return in excess of the riskfree rate per unit of risk, that the market wants as compensation for taking one unit of extra risk. In this section we estimate the market price(s) of interest rate risk for the two models. First, we discuss some empirical studies that have focused on the estimation of the market prices of risk and discuss the basic prin-ciple of our estimation procedure in section 5.1. The data is described in section 5.2. Next, we choose functional forms for the market prices of risk, introduce simulation schemes and discuss the results in section 5.3 for the Hull-White one-factor model. We conclude the chapter by doing the same analysis for the G2++ model.
5.1 l i t e r at u r e & e s t i m at i o n p r i n c i p l e
The drift of an one-factor short-rate model changes by an interest rate premium π(t)when moving from the risk-neutral(Q) world to the real world (P ). This risk premium is the product of the market price of risk µ(t)and the volatility of the short rate η, which is constant like in the Hull-White one-factor model:
π(t) =µ(t)η. (24)
Since all parameters except for the function µ(t) are known for the P -models, µ(t) needs to be specified and then estimated. Literature however shows that it is very difficult to estimate the market price of interest rate risk and that the results are inconclusive. Stanton (1997) for example does not impose any parametric restrictions on the drift and volatility of the short-rate process and uses non-parametric tech-niques. The drift and volatility are approximated by using 1st, 2nd and 3rd order approximations that are estimated with kernel density estimation. These estimates are based on daily 3-month US Treasury
