Swaption Pricing Approximations for LIBOR Market Models

Yueci Li

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam Faculty of Economics and Business Amsterdam School of Economics

Author: Yueci Li Student nr: 11850655 Email: [email protected] Date: July 15, 2018 Supervisor: prof. dr. ir. M.H. (Michel) Vellekoop Second reader: dr. S.U. (Umut) Can Supervisor: MSc R. (Remco) Stam J.M. (Jan) Nooren Statement of Originality

This document is written by Student Yueci Li who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents. Swaption Pricing Approximations for LMMs — Yueci Li iii

Abstract

The recent negative interest rate environment has created great challenges for financial institutions, since many traditional interest rate models fail in such an environment, and among those who do not, many are not designed for the pricing of complicated financial instruments, such as caps, floors and swaptions. In this thesis, I will study the LIBOR market model which can be calibrated for such purposes. In practice, the LIBOR market model is first calibrated using some basic financial instruments to obtain the parameter values of the model. The model is then used to price exotic financial instruments whose prices are not on the market. In this thesis, emphasis is put on the approximation of European swaption pricing. The purpose is to analyze computational cost and accuracy from the obviously time-consuming Monte Carlo simulation processes when certain approximations are used.

Keywords Swaption pricing, LIBOR market model, Interest rate model, Monte Carlo, Black’s formula Contents

Preface vi

1 Introduction1

2 Backgrounds3 2.1 Zero-coupon Bonds...... 3 2.2 Forward Rate Agreements and Swaps...... 3 2.2.1 Forward Rate Agreements...... 4 2.2.2 Swaps...... 4 2.3 Options, Caps and Floors...... 4 2.3.1 Options...... 5 2.3.2 Caps and Floors...... 5 2.4 European Swaptions...... 6 2.5 Pricing of Linear and Non-Linear Financial Instruments...... 6

3 LIBOR Market Model7 3.1 Heath-Jarrow-Morton (HJM) Framework...... 7 3.2 Standard LIBOR Market Model (LMM)...... 7 3.3 Drawbacks of the LIBOR Market Model...... 9

4 Swaption Pricing and Approximations 10 4.1 Swaption Pricing based on Monte Carlo Method...... 10 4.2 Approximation Methods...... 11 4.2.1 Swaption Dynamics and Volatilities...... 12 4.2.2 Rebonato’s Method...... 12 4.2.3 Hull-White Method...... 14

5 Numerical Implementation and Results 15 5.1 Numerical Implementation...... 15 5.1.1 Generation of LIBOR Forward Model (LFM) Paths...... 15 5.1.2 Calculating Swaption Prices Based on Simulations and Black’s Formula...... 19 5.1.3 Test on Approximation Methods...... 20 5.2 Numerical Results and Indications...... 21 5.2.1 Forward Rate Processes...... 21 5.2.2 Swaption Pricing...... 21 5.2.3 Simulation and Time Step Choices...... 28

6 Conclusions and Further Study 34

Appendix A: Parameter Values and Initial Forward Rates 35

Appendix B: Correlation Matrix 36

Appendix C: Instantaneous Forward Rate Volatility σi,j 37

iv Swaption Pricing Approximations for LMMs — Yueci Li v

Appendix D: Volatility vα,β for Black’s Formula 38

References 39 Preface

I would like to take this opportunity to express my great gratitude to my supervisors Michel Vellekoop, Remco Stam, Jan Nooren and Umut Can for making this thesis possible. Many incidents have happened during the process and I am happy that this arduous journey is ﬁnally nearing its end.

I would also like to thank my classmates at University of Amsterdam and my col- leagues at PwC, you provided me with so many brilliant ideas that this thesis would never be complete without you.

And I would like to dedicate this thesis to my family and friends, thank you all for always being there for me and supporting all my crazy ideas.

vi Chapter 1

Introduction

When reliable stochastic interest rate models first emerged in the 1980’s, the most basic form was the Ho-Lee model, where interest rates follow a Gaussian process. Later on, the interest rates were constructed in a way such that they follow a mean-reversion process as can be seen in the well-known Vasicek model. However, both Ho-Lee model and Vasicek model generate negative interest rates. It is not hard to see why this used to be seen as a severe drawback, since the interest rates were as high as over 10% then. Soon enough, a new adjusted model emerged, proposed by the professionals of Gold- man Sachs, which is known as the Black-Derman-Toy model. It assumes log-normal and mean-reversion interest rate processes. The interest rates generated from this model are no longer possible to become negative. Nevertheless, ”Every dog has his day.” No one in the 1980’s would have believed that the interest rates could become so low that they became officially negative in the European Union. On 11, June, 2014, European Central Bank(2016) introduced negative interest rate into the Euro system for the first time ever in history. And it is no coincidence that shortly afterwards, on the other side of the globe, Japan also declared negative interest rate in January, 2016 Yoshino(2017). There is no doubt that negative interest rates have become contagious among major economy entities throughout the world, and the majority believe that such trends will continue in the foreseeable future, as the inflation rate has also remained low since the tremendous financial crisis in 2008. What seemed once to be a serious disadvantage of the Ho-Lee model and the Vasicek model has become their great advantage. But we must not neglect another drawback of all the previously mentioned models: due to the small number of factors included in the models, they are mostly only applicable to linear financial instruments, by which I mean that they can only have closed-form formulas to price bonds, futures, forwards, and products alike in practice, but rarely options, or more synthesised products built on the basis of options. Fortunately, in the late 1980’s, an interest rate model for the pricing of non-linear financial instruments has already emerged. The ancestral framework was formed from the work of Heath et al.(1992). The model could hardly be applied into practice as the required input is not readily available on the market Hull(2012). Nevertheless, the model inspired many other practically useful models among which the LIBOR market model is one of the most promising. Although we usually assume log-normal forward rates for the LIBOR market model, by adding displaced diffusion terms to negative forward rates, the LIBOR market model can be used without further problem. The LIBOR market model is mostly used to price exotic financial instruments, whose prices are not available on the market, for instance, the Bermuda swaptions. The common practice when pricing such financial instruments is as follows: firstly, we generate forward rate processes based on the LIBOR market model. The starting value of the parameters are usually taken from literature or set according to the professional’s cali-

1 2 Yueci Li — Swaption Pricing Approximations for LMMs bration experience. Then we use the generated forward rate processes to price common type swaptions whose prices can be found on market, for example, European swaptions, as will be used in this thesis. Implementing the Monte Carlo method, we generate enough forward rate processes for the pricing purpose, and take means of the resulting prices of the swaptions. By comparing the resulting swaption prices with the market prices, we are able to find the correct values of the LIBOR market model parameters. Next, with the calibrated LIBOR market model, we again run many Monte Carlo simulations to generate forward rate processes, and price the exotic financial instruments. Taking the means of all the prices resulted from the simulations, we can get a reliable price of the value of the financial instrument. As can be seen from above, the current practice of exotic financial product pricing using LIBOR market model may involve many Monte Carlo simulations. And the LIBOR market model itself, as will be shown in the following chapters, is a very complicated model and is itself fairly high in computational cost. In this case, any kind of simplification in the process is appreciated, either in the parameter calibration part, or in the financial instrument pricing part. The research question of this thesis is thus:

• How accurate are simpliﬁcation methods for LIBOR European swaption pricing?

• What can we do to increase the eﬃciency of swaption pricing while maintaining reasonable levels of accuracy?

In this thesis, I will not perform my own calibration. Fortunately, a good guideline of the model is provided in Brigo and Mercurio(2007), from where, I will take initial parameter values and other data. My major focus is the pricing of European swaptions. As swaptions are based on swaps, a great number of summations and hence loops are involved in programming, which add up to a very high computation time. Nevertheless, it can be observed that the changes of certain parameters over time are very small, so we can try to fix some parameter values at their very first moments so that a large amount of computation time can be saved, especially for Monte Carlo simulations of more than 10000 times. I will investigate two types of approximation methods in this thesis. The methods are the Rebonato’s method, and the slightly more complicated method, the Hull-White method. I will test the accuracy of these two methods by comparing their results to that of the simulations, and see if they are accurate enough to be implemented in practice. The rest of this thesis is structured as follows: In Chapter 2, I will provide some backgrounds in the pricing of financial instruments, both linear and non-linear; In Chapter 3, I will give the derivation of the standard LIBOR market model. Some of the problems embedded in the model will also be discussed. Next, in Chapter 4, I will introduce the calibration methods used in generating the simulations related to the log-normal forward LIBOR model (LFM), and also, the two approximation methods for the pricing of European swaptions. And in Chapter 5, the detailed numerical implementation methods will be explained and the results will be given and discussed. Finally, in Chapter 6, conclusions will be drawn and indications for future studies will be provided. Chapter 2

Backgrounds

I will start this chapter by giving the definitions of some fixed income instruments: the zero-coupon bonds, caps and floors, as well as the European swaptions that we are going to price using the LIBOR market model in the following chapters. I will also define some other intermediate fixed income instruments in order to make the evolution process more consistent and complete. Next, I will give a brief explanation on the reason why we cannot use models such as Ho-Lee model, Vasicek model and especially the Black-Derman-Toy model for the pricing of non-linear financial instruments. This explanation should provide good motivation for the following chapter, where I will give a formal introduction to the Heath-Jarrow- Morton framework and the LIBOR market model.

2.1 Zero-coupon Bonds

We start by giving the definition of the most basic but essential fixed income instruments on the market, the zero-coupon bonds. Definition 1. A T -maturity zero-coupon bond is a bond that pays off a unit of a given currency at maturity T and has no other intermediate interest payment during its life time. In other words, the buyer buys the bond and gets the face value back when the bond matures. Following conventions and to simplify the derivations later on, the principal of the zero coupon bond is set to 1. Naturally, we have, P (T,T ) = 1. (2.1) The T −year zero-coupon interest rate is sometimes referred to as T -year spot rate or T -year zero rate. The price at an earlier time t ≤ T is defined as P (t, T ) Hull(2012). If the compounding convention is simple compounding with a fixed interest rate, 1 h R (t, T )in(T −t) = 1 + n , (2.2) P (t, T ) n where Rn(t, T ) represents the simply-compounded annual zero-coupon rate for the time period (t, T ] with compounding frequency n. Not only are zero-coupon bonds very commonly used financial instruments in practice, they also play an essential and indispensable role in the pricing of other more complicated financial instruments.

2.2 Forward Rate Agreements and Swaps

The zero-coupon bond market seems to be quite stable in most times, I will now discuss forwards based on zero-coupon bonds.

3 4 Yueci Li — Swaption Pricing Approximations for LMMs

2.2.1 Forward Rate Agreements

From zero-coupon rates, we can deﬁne the forward rates, Fn(t, T1,T2) Veronesi(2011).

Deﬁnition 2. A forward rate Fn(t, T1,T2) is the future yield on a bond for the period (T1,T2] as observed at the current moment t with t ≤ T1 ≤ T2. n is the compounding frequency during the period.

We have,

Rn(t, T ) = Fn(t, t, T ) (2.3)

From the current term structure, we can derive the forward rate as follows,

P (t, T2) = P (t, T1)P (T1,T2) (2.4)

Therefore, " 1 # P (t, T1) n(T2−T1) Fn(t, T1,T2) = n × − 1 (2.5) P (t, T2)

2.2.2 Swaps

With Fn(t, T1,T2), we can easily set up forward rate agreements, with which we agree to purchase (sell) at a certain agreed forward interest rate. And it is also natural for us to set up a series of such contracts, so that we can exchange ﬂoating rates and ﬁxed rates at agreed reset dates Veronesi(2011).

Definition 3. A vanilla fixed-for-floating swap contract is an agreement that one party pays n times per year at a pre-agreed annualised fixed swap rate S until TM ; the other party makes payments at the same periods according to a floating rate.

A vanilla swap contract can either be a fixed-for-floating contract, a floating-for-fixed contract Saunders and Marcia(2008). We can derive the prices of swaps directly from that of forward rate agreements, for a payer’s swap with a principal of one unit,

M S X 1 − × P (T ,T ) + 1 × P (T ,T ) (2.6) n i j i M j=i+1 where S is the swap rate and M the maturity. The value of the swap contract at initial is always 0. Thus we have, for instance, the swap rate S of a payer swap,

1 − P (t, T ) S = n × M (2.7) PM j=1 P (t, Tj)

2.3 Options, Caps and Floors

Although forward rate agreements make it easier for people to secure a price they desire, people went another step further, they invented the options family, with which, they can reap the diﬀerence of market price and expected price (which is called the strike price or the execution price), meanwhile, remain worry-free of the potential downside risk. Swaption Pricing Approximations for LMMs — Yueci Li 5

2.3.1 Options Deﬁnition 4. An option is a contract that entitles the owner the right, but not the obligation to purchase or sell the underlying security at a future time for an agreed price. We refer to the agreed price as strike price.

Many diﬀerent forms of options exist in the world: European options, American options, to name but a few. A European option can only be exercised at its expiry date, while an American option can be exercised throughout its whole life-time before the expiry date. I will only focus on the European-style options in this thesis, as not only are they the simplest form, but also the most fundamental and important ones. And they are the ones that are always used for the calibration of LIBOR market model in practice. We follow Black’s formula in option pricing. Naturally, we assume our market to be a Black-Scholes complete market. Since European calls, puts have symmetric forms, we only cite the example of a standard European caplet option, the payoﬀ of which is + (R(T,TM ) − K) , where R(T,TM ) is the interest rate at expiry date T for the period (T,TM ] and K is the strike price Black(1976). In continuous form, the value of a caplet at the current moment is, Capleti = F (t, Ti−1,Ti)N(d1) − KN(d2) P (t, Ti−1) (2.8) F (t,Ti−1,Ti) v2 log K + 2 Ti−1 d1 = √ (2.9) v Ti−1 p d2 = d1 − v Ti−1 (2.10) where Capleti is the price of the European caplet option at origin; F (t, Ti−1,Ti) is the forward rate for the period (Ti−1,Ti] as observed at present t; K is the strike price agreed; v is the implied volatility of the forward rates; N(·) is the cumulative distribution function, in this case, as assumed by Black, the distribution is the standard normal distribution.

2.3.2 Caps and Floors Deﬁnition 5. A cap is a series of caplets which have the same strike price K and with reset dates T1,T2,T3, ..., TM . Each caplet is a European call option.

Similarly, for ﬂoors,

Definition 6. A floor is a series of floorlets which have the same strike price K and with reset dates T1,T2,T3, ..., TM . Each floorlet is a European put option.

Note that the payoﬀs depend on the spot rates observed at T0,T1,T1, ..., TM−1. We also mention a vanilla cap, the payoﬀ of a unit of which at Ti is (Ti − Ti−1) × + R(Ti−1,Ti) − K, 0 , the interest rates are determined at time Ti−1 and paid out at Ti:

M X Cap = Capleti (2.11) i=1 Capleti = F (t, Ti−1,Ti)N(d1) − KN(d2) P (t, Ti−1) (2.12) F (t,Ti−1,Ti) v2 log K + 2 Ti−1 d1 = √ (2.13) v Ti−1 p d2 = d1 − v Ti−1 (2.14)

The notations are the same as of the European calls. 6 Yueci Li — Swaption Pricing Approximations for LMMs

2.4 European Swaptions

After setting up both the vanilla swaps and the caps, it leads us directly to the combination of the two, the European swaptions, or European swap options. Similar to swaps, there are two types of swaptions, a payer swaption and a receiver swaption. We give the example of a payer swaption.

Deﬁnition 7. A payer European swaption is an agreement that gives the purchaser of the swaption the right but not the obligation to enter a swap at a ﬁxed rate at a pre-agreed time in the future.

In the following chapters, we will particularly focus on the pricing of European swaptions. The detailed derivation will be given in chapter 4.

2.5 Pricing of Linear and Non-Linear Financial Instru- ments

Before the emergence of options, most financial products were linear products, and no intermediate decisions had to be made. For options and financial instruments constructed on the basis of options, decision(s) had to be made during the life of the financial instruments, and they are not linear any more. The intermediate interest rates now play an essential role in deciding the current price of the instrument, and for that, the so called ”equilibrium interest rate models”, such as the pre-mentioned Ho-Lee model, Vasicek model and the Black-Derman-Toy model can no longer be used. Although they try to create as close a fit as possible, they generally create very large fitting errors, and such errors can be magnified when pricing non-linear financial instruments. Chapter 3

LIBOR Market Model

We start by introducing the Heath-Jarrow-Morton Framework, the standard model which the LIBOR market model can be considered as a special case of. I will introduce the Monte Carlo simulation of the forward rate processes, which we will be using in the following chapters.

3.1 Heath-Jarrow-Morton (HJM) Framework

The risk-neutral models used before the discovery of the Heath-Jarrow-Morton model Heath et al.(1992) were mostly one-factor models and this has strong limitations on choosing volatility structures. The Heath-Jarrow-Morton Model shed light on the development of interest rate models in that it provides a modelling framework under which the full dynamics of the entire forward rate curve are captured: Take T1 = T and T2 = T + ∆t, and let ∆t → 0, so that F (t, T1,T2) can be denoted as F (t, T ) Hull(2012).

Z T dF (t, T ) = σ(t, T ) σ(t, u)dudt + σ(t, T )dWt (3.1) t where σ(t, T ) is the standard deviation of F (t, T ). The HJM model was a big leap in the development of the interest rate models. How- ever, it has some severe drawbacks, which prevent it from being easily used in practice. The major drawback is that the HJM model is expressed in terms of instantaneous forward rates, which are not directly observable on the market. And the model itself cannot be calibrated to use tradable ﬁnancial instruments in any way.

3.2 Standard LIBOR Market Model (LMM)

The LIBOR market model has advantages over the HJM model in that the sets of forward rates are readily observable on the market. We follow the conventions and derivations of Hull and White(2000) and Hull(2012), which are no big difference from Brace et al.(1997) in constructing the LIBOR market model. First, we make an assumption that the model is implemented in a rolling forward risk-neutral world. We define the present moment to be t = 0 and the following reset dates to be T0,T1,...,Tβ−1. The compounding period between each reset date is thus defined as τi = Ti+1 − Ti, ∀ 0 ≤ i ≤ β − 1 (3.2) By defining the model in a ”rolling forward risk-neutral world”, we assume that the zero rate we observe at time Ti persists until Ti+1. In other words, we can discount back from Ti+1 to Ti without worrying about a change of interest rate in the process. The

7 8 Yueci Li — Swaption Pricing Approximations for LMMs

last reset date is set to be at Tβ−1, and the amount determined will only be paid off a period later at Tβ. For the convenience of model construction, we also define here m(Ti), which is the smallest integer that satisfies the condition Ti ≤ Tm(Ti) for the immediate next reset date Ti. The interest rates used here are in fact discrete time forward rates observed directly from the market. This is one of the major improvements of LIBOR market model compared to the HJM model as mentioned above. We denote the forward rate Fn(t, Ti,Ti+1) with Fi(t) where n is assumed to be 1. Next, we define a Z numeraire:

Z(t, Ti) = 1 + τiFi(t) (3.3) Z(t, Ti+1) where Z(t, Ti) represents the discount rate at time t that matures at time Ti. The compounding convention used here, as shown above, is discrete compounding, the same as is used in the market. By taking logarithms on both sides of equation (3.3), we can get, equivalently,

logZ(t, Ti) − logZ(t, Ti+1) = log[1 + τiFi(t)] (3.4)

It is not hard to get from the above linear equation that

dlogZ(t, Ti) − dlogZ(t, Ti+1) = d log[1 + τiFi(t)] (3.5)

The stochastic process of the forward rates can be deﬁned as

p 1 X dFi(t) = (··· ) dt + σi,q(t)Fi(t)dZq (3.6) q=1 where σi,q(t) represents the qth component of the volatility term σi(t) and p represents the number of factors that aﬀect the process. We assume that all the components of the volatility processes are independent of each other. σi(t) is the implied instantaneous volatility of Fi(t) at time t. Same as the forward rate Fi(t), σi(t) is also constant throughout the period (Ti,Ti+1]. Similarly, we deﬁne the stochastic process of the numeraire Z(t, Ti) to be,

p 2 X dZ(t, Ti) = (··· ) dt + si,q(t)Z(t, Ti)dZq (3.7) q=1 where si,q(t) represents the qth component of the volatility term si(t). We change the pricing numeraire from Z(t, Ti+1) to Z(t, Tm(t)) of (3.6) by increasing Pp the expected growth rate of the market price of risk by q=1 σi,q sm(t),q(t) − si+1,q(t) Hull(2012) 3 and using Ito’s Lemma,

p p dFi(t) X X = σ [s (t) − s (t)]dt + σ (t)dZ (3.8) F (t) i,q m(t),q i+1,q i,q q i q=1 q=1

Then, we use Ito’s Lemma on the right-hand-side of equation (3.3) and we get,

p τi X d log[1 + τ F (t)] = (··· )dt + σ (t)dZ (3.9) i i 1 + τ F (t) i,q q i i q=1

1We follow the convention of Hull and White(2000) and Hull(2012). This part will not aﬀect the following derivations thus is not given. 2Same as above 3The full derivation of change of numeraire can be found in section 27.8 of Hull(2012). Swaption Pricing Approximations for LMMs — Yueci Li 9

Similarly, we use Ito’s Lemma on the left-hand-side of equation (3.3) and we get,

Z(t, ti) d log = (··· )dt + [sm(t),q(t) − si+1,q(t)]dZq (3.10) Z(t, ti+1) Combining (3.5), (3.9) and (3.10), we have

i X τjFj(t)σj,q(t) sm(t),q(t) − si+1,q(t) = (3.11) 1 + τjFj(t) j=m(t) After substitution, i Pp p dFi(t) X τjFj(t) q=1 σj,q(t)σi,q(t) X = dt + σi,q(t)dZq (3.12) Fi(t) 1 + τjFj(t) j=m(t) q=1

3.3 Drawbacks of the LIBOR Market Model

It is true that the LIBOR market model brought new solutions to the long-lingering problem of exotic financial instruments pricing, especially under the negative interest rate environment. However, it is also believed by a range of people that the LIBOR market model has some embedded problems since it might not be a suitable modelling choice when the interest rate is very close to zero. This problem was addressed in Hull and White(2000) when they set up the model. From their experiments, when the interest rate goes as low as 1%, the stationary assumption of volatilities would fall apart. In that case, it might not be appropriate to implement the LIBOR market model in simulating interest rates. That said, they also pointed out the fact that if the interest rate goes up again and breaks out of the low range, the assumption would become valid again. One of the major reasons that we favour the LIBOR market model, in the recent five years especially, is that the interest rates are so low that they have already reached or have great risk reaching the low range zone. Thus on the one hand, if the model does not hold in that zone, then the significance and attractiveness of the model in practice would be largely reduced. On the other hand, if we use the model anyway, there is a great possibility that the volatility structure cannot be fully covered in the model, which will also lead to some errors in the results. The calibration of the volatility structure has been particularly focused on in recent literature: a few complicated volatility and correlation models have been created in order to cope with the changes in interest rate volatilities in hope to keep the model. However, significantly long computation time has also been spent on such calculations. In addition to the deviation of the stationary assumption of the volatility structure, as put in Hagan and Lesniewski(2008), the standard LIBOR market model has the problem of matching a volatility smile which is very often observed in the market. The Stochastic Alpha, Beta, Rho (SABR) model provides a good solution to this problem. It constructs also a stochastic process for the volatility σi(t) so that the stochastic risks in the market are captured more accurately. Also, the good news is that the SABR model can also be calibrated to suit a negative interest rate market. A shifted SABR model can be used where a displaced diffusion term is added to the forward rate. The concept of the adjusted model is simple, however, the determination of the displaced diffusion term still remains to be a problem. Although the term is normally a fixed number, it is proved to be difficult to calibrate in practice: mostly an arbitrary value is taken according to the professional’s own judgement. In this thesis we do not explore the possibility of including displaced diffusion terms, we only discuss the fundamental standard LIBOR market model. However, since displaced diffusion terms are only arbitrary values added directly to the forward rates in practice, including them when necessary is not a big jump from the standard model. Chapter 4

Swaption Pricing and Approximations

In this chapter, we introduce ﬁrst a relatively accurate approach of European swaption pricing employing the Monte Carlo method. Next, we will cover two similar approximation methods for the swaption pricing, the Rebonato’s method and the Hull-White method respectively. We hope that the proposed methods will shed light on the calibration of derivative pricing based on methods identical to LIBOR market model.

4.1 Swaption Pricing based on Monte Carlo Method

Following the convention of previous chapters, we use the example of a payer’s swaption. In order to calculate the value of the swaption, we will need to calculate first its payoff and then discount it back. As a combination of swaps and options, the payoff is not hard to construct, following previous notations, the payoff will be

+ (Sα,β(Tα) − K) (4.1) where Tα and Tβ represent the expiry date of the swaption and the last payment date of the swap underlying the swaption respectively. Sα,β(t) is the price of the swap right at time t. To get the actual price of the swaption, we need to discount back to the moment t. This leads us to β X + Et P (t, Ti)τi Sα,β(Tα) − K (4.2) i=α+1 where τi = Ti+1 − Ti and P (t, Ti) is the pricing numeraire. The pricing numeraire P (t, Ti) is simple in this case, as we have already explained how to simulate the necessary forward rate paths. They should be the cumulative products of the numeraires based on forward rates of spot rate form. 1 P (Ti−1,Ti) = (4.3) 1 + τiFi(t) Note that the forward rates are determined one step ahead of the actual payment time, thus for pricing numeraire P (Ti−1,Ti), the ﬁrst forward rate is determined at Ti−1. In order to derive the price of the swaption, we start from the underlying swap, the value of which at present time t is:

β h X i Et P (t, Ti)τi Fi(t) − K (4.4) i=α+1

10 Swaption Pricing Approximations for LMMs — Yueci Li 11

We consider ”at-the-money” swaptions with strike thus,

K = Sα,β(t) (4.5)

Combining (4.3) and (4.5), by setting (4.4) to 0 according to the Law of One Price, we can solve for the strike price K,

β h X i Et P (t, Ti)τi Fi(t) − K = 0 i=α+1 (4.6) β β h X X i Et P (t, Ti)τiFi(t) − K P (t, Ti)τi = 0 i=α+1 i=α+1

Since, h i h i Et P (t, Ti)τiFi(t) = Et P (t, Ti−1) − P (t, Ti) (4.7) Thus,

β β β h X i h X X i Et P (t, Ti)τiFi(t) = Et P (t, Ti−1) − P (t, Ti) i=α+1 i=α+1 i=α+1 (4.8) h i = Et P (t, Tα) − P (t, Tβ)

We can replace the corresponding terms in (4.6) and get, " # P (t, T ) − P (t, T ) S (t) = K = α β (4.9) α,β Et Pβ i=α+1 P (t, Ti)τi

To facilitate the derivations following, we set up a new concept, the forward discount factor D(t; Tα,Ti) that, P (t, Ti) D(t; Tα,Ti) = (4.10) P (t, Tα) By dividing both nominator and denominator of (4.9), and plug in (4.3), we get,

Qβ 1 1 − D(t; T ,T ) 1 − j=α+1 1+τ F (t) S (t) = α β = j j . (4.11) α,β Pβ Pβ Qi 1 τiD(t; Tα,Ti) i=α+1 i=α+1 j=α+1 1+τj Fj (t)

The above equation (4.11) holds for all t ≤ Tα. Thus we can easily get the strike price K for which t = 0 and Sα,β(Tα) for which t = Tα. As can be seen from the derived equation (4.11), although the computational cost for K is negligible, that for Sα,β(Tα) is too big to be overlooked, especially when putting into the Monte Carlo simulations. The computation time for generating diﬀerent forward rate paths and calibration of parameters of the LIBOR market model by marking the swaption prices to the market is considerable. In this case, methods of any type that can reduce the computational cost are appreciated.

4.2 Approximation Methods

We hereby introduce two approximation methods that can noticeably save computational cost. We start from the dynamics of Sα,β(t) and the derivation of swaption’s Black-style volatility based on that. Next, we will introduce the Rebonato’s method, and wrap up with the more complicated Hull-White method. 12 Yueci Li — Swaption Pricing Approximations for LMMs

4.2.1 Swaption Dynamics and Volatilities As introduced in Brigo and Mercurio(2007) 1, a log-normal swaption pricing dynamics can be assumed: α,β dSα,β(t) = σα,β(t)Sα,β(t)dWt (4.12) where σα,β(t) is the instantaneous swaption volatility at t. Applying Ito’s Lemma,

1 d log S (t) = σ (t)dW α,β − σ2 dt (4.13) α,β α,β t 2 α,β

The Black-style swaption volatility2 is,

Z Tα 2 2 vα,β · (Tα) = σα,β(t)dt (4.14) 0 Notice that by (4.13),

2 d log Sα,β(t) d log Sα,β(t) = σα,β(t) dt (4.15)

Combining (4.14) and (4.15), we ﬁnally arrive at the general formula of Black-style 2 swaption volatility vα,β · (Tα) ,

Z Tα 2 vα,β · (Tα) = d log Sα,β(t) d log Sα,β(t) (4.16) 0

Thus in order to simplify the process for obtaining vα,β, the major task rests in simplifying the process getting Sα,β(t)’s. This is the focus of the rest of this chapter.

4.2.2 Rebonato’s Method The Rebonato’s method Brigo and Mercurio(2007) is based directly on the derivations introduced above. We start with equation (4.6),

β β X X P (t, Ti)τiFi(t) − P (t, Ti)τiK = 0 i=α+1 i=α+1 Pβ P (t, Ti)τiFi(t) K = i=α+1 Pβ P (t, Tj)τj j=α+1 (4.17) Pβ D(t; TαTi)τiFi(t) = i=α+1 Pβ j=α+1 D(t; TαTj)τj β X D(t; TαTi)τi = F (t) Pβ i i=α+1 j=α+1 D(t; TαTj)τi where D(t; TαTi) is deﬁned in equation (4.10). A certain structure can be observed from (4.17), we deﬁne a new parameter ωi(t), which serves as the ith weight in the formula, Sα,β(t) = K is now a weighted summation of forward rates Fi(t)’s, with the weight ωi(t):

D(t; T T )τ ω (t) = α i i i Pβ (4.18) j=α+1 D(t; TαTj)τj

1Chapter 6, page 240 of Brigo and Mercurio(2007) 2 Note that this swaption volatility is squared and is multiplied by Tα Swaption Pricing Approximations for LMMs — Yueci Li 13

Now the formula for Sα,β(t) can be written as,

β X Sα,β(t) = ωi(t)Fi(t) (4.19) i=α+1

We assume the current moment to be t = 0.

The ﬁrst approximation made in the Rebonato’s method is to set all ωi(t)’s equals to ωi(0), this is because the ωi(t)’s are considered fairly stable when compared to Fi(t)’s. With that, and diﬀerentiate on both sides of (4.19), and take the one-factor version of (3.6), we have, β X dSα,β(t) = ωi(0)dFi(t) i=α+1 (4.20) β X = ωi(0) (··· )dt + σi(t)Fi(t)dZi(t) i=α+1

Since the following relation between the Brownian motions holds,

dZi(t)dZj(t) = ρi,jdt (4.21)

We have,

β X dSα,β(t)dSα,β(t) ≈ ωi(0)ωj(0)Fi(t)Fj(t)ρi,jσi(t)σj(t)dt (4.22) i,j=α+1

Hence, dSα,β(t) dSα,β(t) d log Sα,β(t) d log Sα,β(t) = Sα,β(t) Sα,β(t) Pβ (4.23) i,j=α+1 ωi(0)ωj(0)Fi(t)Fj(t)ρi,jσi(t)σj(t)dt ≈ 2 Sα,β(t)

Another seemingly bold approximation is made in Rebonato’s method, all the Fi(t)’s are frozen at the very ﬁrst moment as well, such that Fi(t) = Fi(0), ∀t. In the end, for Rebonato’s method, we have,

β X Sα,β(t) = ωi(0)Fi(0) = Sα,β(0) (4.24) i=α+1

To ﬁnd the swaption volatilities, we can transform part of the integral into summations and arrive at the ﬁnal result,

β Z Tα 2 X ωi(0)ωj(0)Fi(0)Fj(0)ρi,j v · (T ) = σ (t)σ (t)dt (4.25) α,β α S (0)2 i j i,j=α+1 α,β 0

Contrary to what might be thought of of such a bold method, the approximation result of the Rebonato’s method, as will be shown in the next chapter, is indeed reasonably accurate. Another similar approximation method will be introduced in the next section. 14 Yueci Li — Swaption Pricing Approximations for LMMs

4.2.3 Hull-White Method If we modify the ﬁrst approximation made in the Rebonato’s method by taking the ωi(t)’s not equal to their initial values we ﬁnd Brigo and Mercurio(2007),

β X dSα,β(t) = ωi(t)dFi(t) + Fi(t)dωi(t) + (··· )dt i=α+1 (4.26) β X ∂ωi(t) = ω (t)I + F (t) dF (t) + (··· )dt h h,i i ∂F h i=α+1 h where Ih,i is an indicator function and is true only when i = h. We can reach the same structure of weighted summation of Fi(t)’s as in (4.19) by deriving the following formula from (4.26),

β X ∂ωi(t) ω¯ (t) = ω (t) + F (t) (4.27) h h i ∂F i=α+1 h where the partial derivative term can be derived directly by diﬀerentiating ωi(t)’s as in (4.18) with regard to Fh,

∂ω (t) τ 1 2 i = − h 2 β k ∂Fh 1 + τ F (t) P τ Q 1 h h k=α+1 k j=α+1 1+τj Fj (t) (4.28) β k i 0 X Y 1 Y 1 × Ii,h − τk τi 1 + Fh(t) 1 + τjFj(t) 1 + τjFj(t) k=α+1 j=α+1 j=α+1

0 where Ii,h is the indicator function that is true only when h ≤ i. We can arrive at the ﬁnal formula for ∂ωi(t) , ∂Fh

Pβ Qk 1 ∂ω (t) τ ω (t) k=h τk j=α+1 1+τ F (t) i = h i j j − I0 . (4.29) β k 1 i,h ∂Fh 1 + τhFh(t) P τ Q k=α+1 k j=α+1 1+τj Fj (t)

Now dSα,β(t) has the same structure as the Rebonato’s method. And by freezing theω ¯i(t)’s and Fi(t)’s at their very ﬁrst moments, we can ﬁnd a similar formula for the Black-style volatility as in equation (4.25),

β Z Tα 2 X ω¯i(0)¯ωj(0)Fi(0)Fj(0)ρi,j v · (T ) ≈ σ (t)σ (t)dt (4.30) α,β α S (0)2 i j i,j=α+1 α,β 0

As can be seen from the above derivations, the Hull-White method is slightly more complicated than the Rebonato’s method, with less approximations made in the process. The actual outcome of these two approximation methods will be compared in the next chapter. Chapter 5

Numerical Implementation and Results

In this chapter, we will give the numerical results for the European swaption prices based on both the Monte Carlo simulations as well as the other approximation methods. We will ﬁrst introduce the numerical implementations, and then, compare and interpret the numerical results we get.

5.1 Numerical Implementation

As explained in the previous chapters, the most accurate method for calculating the swaption price based on the LIBOR market model is using the Monte Carlo method, without making approximations. However, it is obvious that such a method would induce tremendous computational cost, which is deﬁnitely not favourable in daily use.

5.1.1 Generation of LIBOR Forward Model (LFM) Paths In chapter 3, we derived the full-set of LIBOR market model, but in practice we barely use the most sophisticated one. ”Brevity is the soul of wit”: sometimes, a simpliﬁed model works just as good as the complex ones, or even better, as they save the precious computation time. We only use the one-factor version of LIBOR market model:

i Pp p dFi(t) X τjFj(t) q=1 σj,q(t)σi,q(t) X = dt + σi,q(t)dZi,q(t) (5.1) Fi(t) 1 + τjFj(t) j=m(t) q=1

The reasons doing this are as follows: If we employ the full model, with as many Brownian motion terms as forward rate processes, that is, with p = N, where N represents the number of diﬀerent forward rate processes involved in the whole forward portfolio. In this thesis, we retain the simplest form of the LIBOR market model, with only one σi(t)Zi(t) for each Fi(t) at time t. However, I do not assume the correlation ρ between the forward rate processes to be simply 1, as in the Hull and White(2000). The correlations are calculated while calibrating the LIBOR market model, which I take, as the rest of the parameter values, from Brigo and Mercurio(2007) 1. Furthermore, following actual practice, I assume the forward rates to follow a log- normal distribution and a minor adjustment is made to the standard LIBOR market model for this. Note that in order to be implemented in a negative interest rate environment, a displaced diﬀusion term should be added to the forward rate Fi(t). This is

1See Appendix A

15 16 Yueci Li — Swaption Pricing Approximations for LMMs out of the scope of this thesis, but a brief discussion of the possibilities doing this will be provided in chapter 6. The LIBOR market model I use for the numerical implementation is,

i X ρk,jτjσj(t)Fj(t) dFk(t) = σk(t)Fk(t) dt + σk(t)Fk(t)dZk(t) (5.2) 1 + τjFj(t) j=m(t) with k = 1, 2, . . . β − 1 and dZk(t) the Brownian motion term with correlation matrix {ρi,j}, i, j = 1, 2, . . . , β − 1. Applying Ito’s Lemma,

k 1 X ρk,jτjσj(t)Fj(t) 1 1 2 2 d log Fk(t) = 0 + σk(t)Fk(t) − 2 σk(t)Fk (t) dt Fk(t) 1 + τjFj(t) 2 F (t) j=m(t) k 1 + σk(t)Fk(t)dZk(t) Fk(t) k X ρk,jτjσj(t)Fj(t) 1 2 = σk(t) dt − σk(t)dt + σk(t)dZk(t) 1 + τjFj(t) 2 j=m(t) (5.3) In order to implement in practice, I transform the continuous formula (5.3) to its discrete version Brigo and Mercurio(2007):

k 2 X ρk,jτjσj(t)Fj(t) σk(t) log F (t+∆t) = log F (t)+σ (t) ∆t− ∆t+σ (t)Z (t+∆t)−Z (t) k k k 1 + τ F (t) 2 k k k j=α+1 j j (5.4)

Context Set-up I assume that the current moment is t = 0, and each year in the future is denoted as Ti. As the main target of the LIBOR market model is swaptions, we also define the reset dates, payment dates and the length of time period between them. The reset dates are T0,T1,T2,...,Tβ−1 and the payment dates are therefore T1,T2,T3 ...,Tβ. The time period in between is defined as τi = Ti+1 − Ti. However, the time step for the forward rate processes cannot be as large as 1 year, thus the time steps between each reset dates are also divided into small time steps, each time step is denoted as ∆t. At the forward rate paths generation stage, 19 reset dates are involved: T0,T1,...,T18 1 and the payment dates are thus T1,T2,...,T19. And the time step ∆t is set to be 6 . This number can be smaller, so that more time steps are taken between (Ti,Ti+1], and correspondingly, the simulation results are presumed to be more accurate but with potentially a very heavy cost in computation. We aim to price 70 swaptions that expire in 1, 2, 3,..., 10 years, and the lengths of the underlying swaps of which are 1, 2, 3, 4, 5, 7, 10 years. For convenience and because the market observation of forward rate volatilities are relatively stable, we assume that the instantaneous forward rate volatilities are piecewise-constant, which in other words means that the instantaneous forward rate volatilities are constant during the entire time period (Ti,Ti+1]. Since in our swaption set, the first swaption is paid in 2 years and the last in 20 years 2, we need in total 19 forward rate processes:

2The last swaption payment is for the swaption that expires in 10 years and has an underlying swap of 10-year length. The last payment is determined at year 19 and paid out at year 20. Swaption Pricing Approximations for LMMs — Yueci Li 17

t = 0 (0,T0] (T0,T1] ··· (T17,T18] F1(0) F1(∆t), ··· ,F1(T0) Dead ··· Dead F2(0) F2(∆t), ··· ,F2(T0) F2(T0 + ∆t), ··· ,F2(T1) ··· Dead ··· ··· ··· ··· Dead F19(0) F19(∆t), ··· ,F19(T0) F19(T0 + ∆t), ··· ,F19(T1) ··· F19(Tα−2 + ∆t), ··· ,F19(T18)

Table 5.1: Discrete Forward Rate Processes

Parameter Calibration – Instantaneous Volatility

As can be seen from equation (5.4), the indispensable parameters in the model are σk(t)’s and ρk,j’s. The σk(t)’s represent the volatility of the process and the ρk,j’s the correlation between diﬀerent forward rate processes. First of all, as pre-assumed earlier, the instantaneous forward rate volatilities are constant over each year. Also, they expire at the same time as the forward rate agreements. The following table indicates how instantaneous forward rate volatilities proceed during the process.

t = 0 (0,T0] (T0,T1] ··· (T17,T18] σ1(0) Φ1ψ1 Dead ··· Dead σ2(0) Φ2ψ2 Φ2ψ1 ··· Dead ··· ··· ··· ··· Dead σ19(0) Φ19ψ19 Φ19ψ18 ··· Φ19ψ1

Table 5.2: Instantaneous Forward Rate Volatilities

The instantaneous volatility of a forward rate obviously depends on its expiry date Φk, which is constant for the whole life time of each forward rate process, k = 1, 2, 3,..., 19. Also, the time left until expiry date also plays an important role in a forward rate process, so another parameter ψk−(β(t)−1) is included. The parameter β(t) in the subscript here is the ﬁnal payment date of the forward rate process. As shown in table 5.2, the instantaneous volatilities are computed as:

σk(t) = Φkψk−(β(t)−1) (5.5)

In practice, the instantaneous volatility calibrated using (5.5) is considered complex enough.

Parameter Calibration – Instantaneous Correlation

According to section 6.9 of Brigo and Mercurio(2007), a two-factor correlation matrix is used. A correlation matrix {ρk,j} can always be transformed and written as

0 {ρk,j} = PDP (5.6) where matrix D is a diagonal matrix with the positive eigenvalues at the diagonal; And corresponding to D, we have matrix P which is orthogonal, in other words, PP0 = 0 P P = I19. The columns of matrix P are composed of the eigenvalues with the same order as in the diagonal of matrix D.

  e1 0 ··· 0  0 e2 ··· 0  D =   (5.7)  . . ..   . . . 0  0 0 ··· (e19) 18 Yueci Li — Swaption Pricing Approximations for LMMs

Since all the elements in D are non-negative, we can take square root of D and deﬁne a new diagonal matrix T: √  e 0 ··· 0  1 √  0 e2 ··· 0  T =  . . .  (5.8)  . . .. 0   √  0 0 ··· ( e19) And by setting A = PT, we see that the following equation holds:

0 0 0 0 AA = P(TT )P = PDP = {ρk,j} (5.9)

The decomposition of the correlation matrix {ρk,j} provides a good idea for lowering the rank of the correlation matrix. It is possible to ﬁnd an α × N matrix B such that 0 BB ≈ {ρk,j} according to Rebonato(1999). By lowering the rank of the correlation matrix from α to N in this way, we can rewrite the Brownian motion term. Now,

0 0 0 0 (B dW)(B dW) = B (WW )B = BB dt = {ρˆk,j}dt (5.10) where {ρˆk,j} is the parameterised correlation matrix that we try to make as close to {ρk,j} as possible. We successfully constructed a new instantaneous volatility matrix B such that the rank of the correlation matrix is lowered to N. Now the question left is how do we ﬁnd the elements of B. Rebonato(2005) came up with a very smart solution based on a transformation from the Cartesian coordinate system to the polar coordinate system. The method is based on the feature of a correlation matrix, the elements on the diagonal of such matrices are always 1. Hence we shall always have, for the elements on the ith row of matrix B,

N X 2 bi,j = 1 (5.11) j=1 This coincides with a polar coordinate system. Take N = 2 for example, for i = 1, 2, 3,..., 19, take

bi,1 = cos θi (5.12) bi,2 = sin θi Since obviously 2 2 cos θi + sin θi = 1 (5.13) PN 2 which is equivalent to j=1 bi,j = 1, this transformation holds. In addition, it is an inspiring approach since instead of putting a constraint when calibrating for B, it directly includes the constraint when setting up matrix B. By doing so, the calibration procedure is simplified and a large amount of computation time is saved. The same concept can be extended to larger N’s. In general, as given by Rebonato (2005), instead of fitting the bi,j’s in a circle, we now fit them in a unit-radius sphere, for N = 3, and a hypersphere for N > 3. According to Rebonato(2005), the formula for a higher N should be

j−1 Y bi,j = cos θi,j sin θi,k j = 1, 2, ··· , β − 1 k=1 (5.14) j−1 Y bi,j = sin θi,k j = α k=1 Swaption Pricing Approximations for LMMs — Yueci Li 19

As for this thesis, following Brigo and Mercurio(2007) experiment, I only use the result for N = 2. It is also noteworthy that according to the test conducted by Rebonato (2005), and taking into consideration the industry practice, N = 2 or 3 is usually more than enough. The parameterised correlation matrix is constructed in the following way:   1 cos(θ1 − θ2) ··· cos(θ1 − θ19)  cos(θ2 − θ1) 1 ··· cos(θ2 − θ19)  {ρˆ } =   (5.15) k,j  . . ..   . . . cos(θ18 − θ19) cos(θ19 − θ1) cos(θ19 − θ2) ··· 1

The parameter value θi, i = 1, 2,..., 19 can be found by minimising the squared distance between the historic correlation matrix {ρk,j} and the parameterised matrix {ρˆk,j}. Rebonato(2005) compared a few market models of correlation coeﬃcients, and the most practical one is, ρi,j = LongCorr + (1 − LongCorr) exp γ | Ti − Tj | (5.16)

where γ = d1 − d2 max(Ti − Tj) (5.17) The parameter values are set to be

LongCorr = 0.3 (5.18)

d1 = −0.12 (5.19)

d2 = 0.005 (5.20)

And the squared distance function χ2(t) is deﬁned as,

19 2 X 2 χk,j = ρk,j − ρˆik,j (5.21) k,j=1

With all the necessary parameters successfully calibrated, we can easily generate as many forward rate paths as needed using (5.4). In this thesis, however, I did not calibrate these parameters myself, instead I take all the parameter values as provided in Brigo and Mercurio(2007) 3.

5.1.2 Calculating Swaption Prices Based on Simulations and Black’s Formula With the parameters found by the ”marking to market” procedure, we can now build forward rate processes that are consistent with the curves on the market. Taken from (4.2), with some minor adjustments, we have the current price of the swaption as,

β h + X i Et Sα,β(Tα) − K τiP (t, Tα) P (Tα,Ti) (5.22) i=α+1

Since P (t, Tα) is observed at present, it is actually deterministic and can be calculated directly using the initial forward rate values. Pβ The rest of the pricing numeraire term, i=α+1 P (Tα,Ti), which is a summation of pricing numeraires observed at expiry date Tα and are paid at diﬀerent payment dates starting Ti, i = α + 1, can also be easily calculated using the forward rate paths generated.

3See Appendix A 20 Yueci Li — Swaption Pricing Approximations for LMMs

+ The tricky part left is Sα,β(Tα) − K , which is in fact an option and for a payer’s swaption, the form is a call option. The volatilities quoted by Brigo and Mercurio(2007) are based on Black’s formula for swaptions which gives the price of the call option: Sα,β(0)N(d1) − KN(d2) P (t, Tα)

Sα,β (0) 1 2 log K + 2 vα,βTα d1 = √ (5.23) vα,β Tα p d2 = d1 − vα,β Tα where vα,β is the implied volatility calculated for swaption pricing using Black’s formula. In practice as we are trying to ﬁnd the price of the swaption at the current moment (t = 0), when the strike price of the swaption equals the current value of Sα,β(t), formula (5.23) can be transformed into, KP (t, Tα) 2N(d1) − 1 1 p (5.24) d = v T 1 2 α,β α

5.1.3 Test on Approximation Methods As explained previously, what really makes the long and arduous process of forward rate paths generation worthwhile is applying those paths to financial instrument pricing. And according to experience, some approximations during this process can really make a difference. The main structures of the Rebonato’s method and Hull-White method are very similar, what actually distinguishes the two is the construction of the weight terms. For Rebonato’s method, the weight term is ωi(t) and for the Hull-White method,ω ¯h(t) is constructed specifically. Except for this difference, all the other structures as well as approximations made remain exactly the same. The ωi(t) andω ¯h(t) can be constructed easily according to formula (4.18) and (4.27), (4.28) respectively. The tricky part is the construction of volatilities,4, the general formula for the volatilities is, β Z Tα 2 X ωi(0)ωj(0)Fi(0)Fj(0)ρi,j v · (T ) ≈ σ (t)σ (t)dt (5.25) α,β α S (0)2 i j i,j=α+1 α,β 0 The nominator of the fraction can be constructed by creating a vector for a new term ωi(t)Fi(t) and then make a matrix of ωi(0)ωj(0)Fi(0)Fj(0)ρi,j using matrix multi- plication. And as for Sα,β(0), since it equals K and is independent of i and j, this term is deterministic and can be taken out of the summation. The question left is how to calculate the integral term. In fact, although the term R Tα 0 σi(t)σj(t)dt has the form of an integral, since the σi(t)’s are constant for each small time periods (Ti−1,Ti], the integral is in fact a summation,

Z Tα σi(t)σj(t)dt = σi(T0)σj(T0) + σi(T1)σj(T1) + ··· + σi(Tα)σj(Tα) 0 α−1 (5.26) X = σi(Tk)σj(Tk) k=0

With all the terms properly set, we can get the Black-style volatilities vα,β’s without a problem. It is noteworthy here that the result we get is multiplied by Tα, thus we need to divide it with Tα to get the vα,β we need. Now we can plug in the resulted vα,β’s into the Black’s formula as indicated in (5.24) and get the approximated results.

4 Since the (4.25) and (4.30) have exactly same structure, for simpliﬁcation reasons, we use ωi(t) for both ωi(t) andω ¯h(t) in the following formula. Swaption Pricing Approximations for LMMs — Yueci Li 21

5.2 Numerical Results and Indications

5.2.1 Forward Rate Processes Following the standard Monte Carlo method, we generate forward rate paths sets5:

Figure 5.1: Example of Simulated Forward Rate Paths (Expire in 10 years)

Note that for some forward rates with farther expiry dates, the forward rate paths might look strange, for instance, the paths for forward rates that would expire in 19 years as shown in Figure 5.2. The reason is that the σi(t) values used contain some 0’s, thus the stochastic terms become 0 as well, which makes the forward rate paths constant at some periods. Con- sidering the great amount of parameters included in the LIBOR market model, it is not uncommon that zero σi(t)’s appear. However, for a forward rate path, especially for a forward rate path with small ∆t, this phenomenon is hard to explain and will have a slight inﬂuence on the accuracy of the ﬁnal swaption prices calculated in the end.

5.2.2 Swaption Pricing With the generated forward rate paths, we can get the simulated swaption prices according to equation (5.22) directly by calculating Sα,β(Tα) using (4.11). In the following sections, if not speciﬁcally mentioned, the number of simulations is taken to be 20000 1 6 and ∆t is taken to be 6 . The simulated swaption prices are shown in Table 5.3 and Figure 5.3. Small humps can be observed from the curves, which can be eliminated presumably by increasing the number of simulations or the number of time steps during each year.

5To make the ﬁgures clearer, Figure 5.1 and Figure 5.2 only include 100 generated forward rate paths 1 with ∆t = 6 . 6 1 In Brigo and Mercurio(2007), the number of simulations is 200000 and ∆ t = 4 . It is very surpris- 1 ing that ∆t = 4 would satisfy the accuracy requirements of the calculations. Considering the heavy computational cost performing more simulations, we take the above initial values for now. 22 Yueci Li — Swaption Pricing Approximations for LMMs

Figure 5.2: Example of Simulated Forward Rate Paths (Expire in 19 years)

1 2 3 4 5 6 7 8 9 10 1 0.003292 0.005229 0.006266 0.006649 0.006880 0.007077 0.006794 0.006913 0.006619 0.006431 2 0.005858 0.008587 0.010584 0.010563 0.010976 0.011589 0.011352 0.011466 0.011920 0.010896 3 0.008098 0.011559 0.013602 0.014450 0.014414 0.016237 0.015094 0.015840 0.015074 0.014903 4 0.010272 0.013804 0.016409 0.017140 0.018090 0.019192 0.017833 0.019554 0.018370 0.017880 5 0.012056 0.016515 0.019326 0.021037 0.020680 0.021526 0.021713 0.022429 0.021161 0.021252 7 0.015065 0.021617 0.024872 0.026250 0.027014 0.028128 0.028410 0.027873 0.026991 0.026120 10 0.019400 0.026552 0.030481 0.033303 0.033359 0.035895 0.033675 0.032567 0.031383 0.031560

Table 5.3: Swaption Prices (Simulations)

+ We have similar curves when implementing Black’s formula for Sα,β(Tα) − K part according to (5.24)7 while keeping the discounting part the same as in the above simulations. The results are shown in Table 5.4 and Figure 5.4:

1 2 3 4 5 6 7 8 9 10 1 0.002979 0.004289 0.004780 0.005057 0.005241 0.005359 0.005315 0.005228 0.005039 0.004928 2 0.006616 0.008408 0.009092 0.009421 0.009827 0.010002 0.010001 0.009769 0.009586 0.009342 3 0.009910 0.012380 0.013211 0.013448 0.013899 0.014210 0.014153 0.014011 0.013775 0.013522 4 0.012618 0.015309 0.016066 0.016417 0.017029 0.017298 0.017434 0.017258 0.017133 0.016791 5 0.014585 0.017922 0.018881 0.019323 0.019787 0.020120 0.020263 0.020268 0.019919 0.019683 7 0.018189 0.022304 0.023511 0.024072 0.024575 0.025235 0.025168 0.024904 0.024605 0.023940 10 0.022687 0.026405 0.027976 0.028916 0.029424 0.030028 0.029884 0.029023 0.028144 0.027143

Table 5.4: Swaption Prices (Black’s Formula)

We can see that the results using Black’s formula are smoother than that of pure simulations. This is not unexpected as Fi(t)’s are used for the calculation of Sα,β(Tα)− + K in pure simulation method, while only Sα,β(0) = K and the implied volatilities vα,β are used for the the previous, both of which are deterministic. The pure simulation method is more aﬀected by the ﬂuctuation of the forward rate processes and thus the results are not as smooth as the Black’s formula results.

7See Appendix D for the Black-style volatilities’ used Swaption Pricing Approximations for LMMs — Yueci Li 23

Figure 5.3: Swaption Prices (Simulations)

Figure 5.4: Swaption Prices (Black’s Formula)

Despite the slightly rough curves observed from the simulation results, we also see an unexpected large diﬀerence in the two results. The deviation of the two methods is calculated according to the following formula:

Simulation Results − Black’s Formula Results Deviation = (5.27) Simulation Results We choose to calculate the deviation with regard to simulation results instead of Black’s formula results because the Black-style volatilities vα,β’s have to be calculated 24 Yueci Li — Swaption Pricing Approximations for LMMs when implementing Black’s formula. And due to the rounding issues of the results we take from Brigo and Mercurio(2007) 8, the simulation results might be more accurate in this case. The deviations are shown in Table 5.5:

1 2 3 4 5 6 7 8 9 10 1 0.095160 0.179700 0.237149 0.239392 0.238267 0.242754 0.217661 0.243699 0.238705 0.233737 2 -0.129331 0.020840 0.140965 0.108063 0.104644 0.136916 0.119017 0.148021 0.195803 0.142630 3 -0.223695 -0.071011 0.028721 0.069316 0.035718 0.124839 0.062327 0.115472 0.086198 0.092628 4 -0.228408 -0.109007 0.020872 0.042201 0.058680 0.098692 0.022421 0.117434 0.067314 0.060890 5 -0.209764 -0.085187 0.023026 0.081489 0.043182 0.065331 0.066794 0.096345 0.058671 0.073847 7 -0.207335 -0.031794 0.054755 0.082946 0.090277 0.102841 0.114124 0.106500 0.088396 0.083464 10 -0.169432 0.005515 0.082203 0.131732 0.117953 0.163447 0.112561 0.108829 0.103214 0.139947

Table 5.5: Deviation of Simulation Results and Black’s Formula Results

According to (5.24), the potential deviation in vα,β is magnified through the implementation of Black’s formula, especially with the increase of underlying swap length or the increase of expiry date Tα. This coincides with what is observed from Table 5.5. It is noteworthy that for swaptions that expire too soon or with too short underlying swap lengths, the results can be more volatile as they are more easily affected by the fluctuation of the forward rates. It is expected that with more simulations and smaller time step ∆t’s involved, and with more accurate Black-style volatility vα,β’s, we can have more identical results of simulations’ and Black’s formula’s. In the following sections, we use the simulation results as model results when calculating the accuracy of the approximation methods.

Approximation Results As shown previously, great similarity is observed in the two approximation methods. They have exactly same structures, the only diﬀerence is in the construction of the weight term. According to (4.27) and (4.29), the weight term of the Hull-White method is:

β X ∂ωi(t) ω¯ (t) = ω (t) + F (t) (5.28) h h i ∂F i=α+1 h while term ωh(t) is exactly the weight term for Rebonato’s method. We freeze both ωi(t) andω ¯i(t) at t = 0. ωi(0) as calculated directly from (4.18) is always larger than 0:

D(0; T T )τ ω (0) = α i i > 0 i Pβ (5.29) j=α+1 D(0; TαTj)τj

The partial derivative term ∂ωi(t) according to (4.29): ∂Fh

Pβ Qk 1 ∂ω (t) τ ω (t) k=h τk j=α+1 1+τ F (t) i = h i j j − I0 (5.30) β k 1 i,h ∂Fh 1 + τhFh(t) P τ Q k=α+1 k j=α+1 1+τj Fj (t)

∂ωi(t) Forω ¯i(t), its component can be negative occasionally, for example when α+1 = ∂Fh 0 1, β = 2, and i = h = 2: the Indicator function Ii,h equals 1 while the fraction of Pβ Qk 1 τk k=h j=α+1 1+τj Fj (t) ∂ωi(t) summations Pβ Qk 1 < 1. The term ∂F in this case is negative. τk h k=α+1 j=α+1 1+τj Fj (t)

8The values taken from Brigo and Mercurio(2007) are only rounded to 3 digits, which is not considered accurate enough. Swaption Pricing Approximations for LMMs — Yueci Li 25

It should be noted that in most cases, even with some negative ∂ωi(t) terms involved, ∂Fh theω ¯i(0) does not necessarily go negative. In fact, onlyω ¯2(0) is observed to be negative in our implementation. The otherω ¯i(0)’s remain positive because we make summation Pβ ∂ωi(t) Fi(t) , with longer underlying swap lengths, more positive components are i=α+1 ∂Fh added up, the summations remain positive, and thus the Hull-White weightsω ¯i(0)’s are larger than Rebonato’s weights ωi(0)’s in most cases. The deviation of the two approximation methods is calculated according the the following formula: Hull-White Results − Rebonato’s Results Deviation = (5.31) Hull-White Results where Rebonato’s method is compared to the Hull-White method as the latter makes slightly less approximations and is assumed to be more accurate. In implementation, the diﬀerence between the swaptions’ Black-style volatilities vα,β’s computed based on the two approximation methods according to formula (5.31) are observed to be very small. The approximated volatilities vα,β are shown in Table 5.6 and Table 5.7 respectively:

1 2 3 4 5 6 7 8 9 10 1 0.180319 0.191464 0.186202 0.177329 0.167915 0.158090 0.152664 0.148744 0.144730 0.141306 2 0.159155 0.159803 0.160988 0.143298 0.136494 0.137494 0.129427 0.130095 0.132227 0.115059 3 0.144713 0.145915 0.137246 0.132260 0.118852 0.127316 0.113374 0.122253 0.115381 0.103391 4 0.135726 0.133035 0.130115 0.122152 0.116617 0.116342 0.106423 0.114469 0.105011 0.097577 5 0.126471 0.127891 0.123865 0.119878 0.113103 0.110060 0.105025 0.108056 0.098497 0.094308 7 0.121921 0.122000 0.118761 0.113512 0.108867 0.107806 0.101665 0.099652 0.092551 0.091029 10 0.116889 0.113449 0.113027 0.108417 0.102677 0.101712 0.095827 0.091186 0.084039 0.084253

Table 5.6: Volatility vα,β of Rebonato’s Method

1 0.180319 0.191464 0.186202 0.177329 0.167915 0.158090 0.152664 0.148744 0.144730 0.141306 2 0.159131 0.159925 0.161152 0.143459 0.136654 0.137710 0.129626 0.130320 0.132421 0.115231 3 0.144820 0.146267 0.137635 0.132671 0.119303 0.127847 0.113889 0.122799 0.115827 0.103835 4 0.136083 0.133677 0.130833 0.122929 0.117493 0.117277 0.107356 0.115376 0.105788 0.098396 5 0.127155 0.128940 0.125041 0.121184 0.114504 0.111488 0.106435 0.109392 0.099709 0.095614 7 0.123689 0.124312 0.121326 0.116195 0.111618 0.110524 0.104280 0.102138 0.095017 0.093736 10 0.121296 0.118433 0.118321 0.113823 0.108095 0.107066 0.101072 0.096255 0.089102 0.089583

Table 5.7: Volatility vα,β of Hull-White Method

The deviation of swaptions’ Black-style volatilities vα,β’s calculated using the two approximation methods are shown in Table 5.8:

1 2 3 4 5 6 7 8 9 10 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 -0.000153 0.000765 0.001020 0.001124 0.001167 0.001570 0.001531 0.001722 0.001471 0.001486 3 0.000738 0.002409 0.002820 0.003097 0.003776 0.004153 0.004527 0.004444 0.003850 0.004279 4 0.002623 0.004803 0.005487 0.006321 0.007451 0.007971 0.008691 0.007862 0.007347 0.008318 5 0.005381 0.008140 0.009400 0.010779 0.012231 0.012810 0.013248 0.012214 0.012158 0.013654 7 0.014294 0.018596 0.021149 0.023086 0.024648 0.024587 0.025075 0.024342 0.025952 0.028878 10 0.036338 0.042085 0.044740 0.047489 0.050118 0.050003 0.051894 0.052665 0.056817 0.059500

Table 5.8: Deviation of Volatility vα,β of Hull-White and Rebonato’s Method

v2,1 is negative, and the reason has been discussed above. The volatilities of swaptions with only 1 year underlying swap length are exactly the same according the formulas of the two methods, as ∂ωi(t) = 0, ∀i for these swaptions according to equation (5.30). All ∂Fh the other Hull-White volatilities are larger than that of the Rebonato’s method. Next, we use Black’s formula as indicated in (5.24) to price the swaptions with vα,β calculated using the approximation methods and compare the pricing results to that of simulations. The swaption prices of Rebonato’s method are shown in Table 5.9 and Figure 5.5 and that of the Hull-White Method are shown in Table 5.10 and Figure 5.6. 26 Yueci Li — Swaption Pricing Approximations for LMMs

1 2 3 4 5 6 7 8 9 10 1 0.003275 0.005193 0.006086 0.006485 0.006602 0.006554 0.006426 0.006308 0.006063 0.005936 2 0.005950 0.008612 0.010373 0.010300 0.010556 0.011081 0.010604 0.010670 0.010818 0.009428 3 0.008152 0.011657 0.013045 0.014002 0.013433 0.014946 0.013489 0.014634 0.013820 0.012383 4 0.010138 0.013954 0.016205 0.016850 0.017119 0.017651 0.016426 0.017793 0.016363 0.015184 5 0.011679 0.016494 0.018861 0.020138 0.020160 0.020314 0.019708 0.020467 0.018695 0.017863 7 0.015298 0.021097 0.024068 0.025296 0.025720 0.026406 0.025332 0.025067 0.023246 0.022709 10 0.019647 0.026050 0.030397 0.031978 0.032129 0.032827 0.031461 0.030068 0.027506 0.027225

Table 5.9: Swaption Prices (Rebonato’s Method)

Figure 5.5: Swaption Prices (Rebonato’s Method)

1 2 3 4 5 6 7 8 9 10 1 0.003275 0.005193 0.006086 0.006485 0.006602 0.006554 0.006426 0.006308 0.006063 0.005936 2 0.005949 0.008619 0.010383 0.010311 0.010568 0.011098 0.010620 0.010688 0.010834 0.009442 3 0.008158 0.011685 0.013082 0.014045 0.013484 0.015007 0.013550 0.014699 0.013873 0.012436 4 0.010165 0.014021 0.016294 0.016956 0.017247 0.017792 0.016569 0.017933 0.016483 0.015310 5 0.011742 0.016629 0.019039 0.020357 0.020408 0.020576 0.019971 0.020718 0.018923 0.018108 7 0.015519 0.021496 0.024586 0.025891 0.026366 0.027067 0.025980 0.025688 0.023862 0.023379 10 0.020387 0.027192 0.031816 0.033566 0.033816 0.034545 0.033173 0.031729 0.029153 0.028936

Table 5.10: Swaption Prices (Hull-White Method)

The two methods produce very similar results, and the curves appear to be identical to that of the simulations as well. To determine which approximation method produces more accurate results, we calculate the deviation according to the following formula:

Approximation Results − Simulation Results Deviation = (5.32) Simulation Results The deviations of the two methods are shown in Table 5.11 and 5.12 respectively: It can be seen that the levels of deviation of the approximation methods are ac- ceptable when the expiry date of the swaption is below 59 years, where the deviation is within 5%. As for swaptions that will expire more than 5 years later, since in both approximation methods, the weights ωi(t)’s and the forward rates Fi(t)’s are frozen at

9It is assumed that if we do more simulations with smaller ∆t’s, there is a possibility that this number can be extended to 5 years, or even further. But as emphasised a lot in this thesis, the trade-oﬀ of computational cost and simulation accuracy is an essential topic to consider. Swaption Pricing Approximations for LMMs — Yueci Li 27

Figure 5.6: Swaption Prices (Hull-White Method)

1 2 3 4 5 6 7 8 9 10 1 -0.005359 -0.006931 -0.028712 -0.024636 -0.040393 -0.073917 -0.054146 -0.087522 -0.084052 -0.076953 2 0.015727 0.002930 -0.019927 -0.024879 -0.038211 -0.043838 -0.065890 -0.069440 -0.092429 -0.134751 3 0.006585 0.008464 -0.040916 -0.030989 -0.068042 -0.079518 -0.106321 -0.076144 -0.083202 -0.169070 4 -0.013033 0.010833 -0.012443 -0.016954 -0.053700 -0.080297 -0.078930 -0.090087 -0.109258 -0.150764 5 -0.031284 -0.001297 -0.024084 -0.042711 -0.025146 -0.056297 -0.092336 -0.087498 -0.116533 -0.159483 7 0.015428 -0.024052 -0.032339 -0.036335 -0.047911 -0.061216 -0.108326 -0.100657 -0.138751 -0.130589 10 0.012737 -0.018897 -0.002779 -0.039779 -0.036872 -0.085468 -0.065737 -0.076741 -0.123552 -0.137372

Table 5.11: Deviation of Rebonato’s Method in Swaption Pricing

1 2 3 4 5 6 7 8 9 10 1 -0.005359 -0.006931 -0.028712 -0.024636 -0.040393 -0.073917 -0.054146 -0.087522 -0.084052 -0.076953 2 0.015573 0.003695 -0.018933 -0.023790 -0.037095 -0.042348 -0.064472 -0.067852 -0.091109 -0.133477 3 0.007328 0.010891 -0.038217 -0.027996 -0.064530 -0.075711 -0.102288 -0.072061 -0.079694 -0.165531 4 -0.010441 0.015697 -0.007017 -0.010732 -0.046637 -0.072958 -0.070909 -0.082940 -0.102720 -0.143698 5 -0.026050 0.006876 -0.014859 -0.032331 -0.013140 -0.044127 -0.080230 -0.076304 -0.105741 -0.147936 7 0.030135 -0.005607 -0.011507 -0.013663 -0.023974 -0.037694 -0.085535 -0.078372 -0.115957 -0.104922 10 0.050879 0.024109 0.043767 0.007893 0.013705 -0.037600 -0.014897 -0.025722 -0.071059 -0.083152

Table 5.12: Deviation of Hull-White Method in Swaption Pricing time t = 0, the longer it takes for the swaption to mature, the more inaccurate the approximations become. In order to determine the accuracy of the two approximation methods, we calculate the Mean Squared Errors (MSE). We denote Xi,j as the model (simulated) swaption prices and Xˆi,j the approximated swaption prices where i = 1, 2,..., 7 and j = 1, 2,..., 10. There are in total 70 Xi,j and Xˆi,j respectively. The corresponding swaption prices are given in Table 5.3, 5.9 and 5.10.

7 10 1 X X 2 MSE = X − Xˆ (5.33) 70 i,j i,j i=1 j=1

The MSEs of Rebonato’s method and Hull-White method are shown in Table 5.13, since the MSEs calculated are very small, for the convenience of comparison, we use log(MSE) instead. 28 Yueci Li — Swaption Pricing Approximations for LMMs

Rebonato Hull-White log(MSE) -12.8856 -13.3804

Table 5.13: log(MSE) of Approximation Methods

We also calculate MSEs in relative terms. The formula we use is,

7 10 ˆ 1 X X Xi,j − Xi,j 2 MSE = (5.34) Relative 70 X i=1 j=1 i,j

Same as in absolute terms, we also compare log(MSERelative)’s. The results are shown in Table 5.14: Rebonato Hull-White log(MSERelative) -5.208254 -5.471122

Table 5.14: log(MSERelative) of Approximation Methods

As can be seen, in both absolute terms and relative terms, Hull-White method is shown to be slightly more accurate than the Rebonato’s method. And using Hull-White method does not put any additional pressure on the simulation process as it does not use any simulated forward rates, and is itself not too much more complicated than the Rebonato’s method. In the following section, we base our discussion of computational cost and simulation and time step choices on Hull-White method.

5.2.3 Simulation and Time Step Choices Proving that the Hull-White method is more accurate than the Rebonato’s method and can be used when the swaption we price is less than 5 years to expire is not the end of the story. To give some idea about the efficiency of this method, we also test different numbers of simulations with different ∆t’s. 1 In addition to 20000 simulations and ∆t = 6 , we also test 10000, 5000 and 1000 1 1 simulations, and ∆t = 4 and ∆t = 2 . We calculate the log(MSE) and log(MSERelative) of each arrangement following (5.33). The results are shown in Table 5.15 and 5.16,

1 1 1 ∆t = 2 ∆t = 4 ∆t = 6 1000 -12.64138 -12.68478 -12.88030 5000 -13.15012 -13.17324 -13.18654 10000 -13.28045 -13.25785 -13.33987 20000 -13.33145 -13.34669 -13.38040

Table 5.15: log(MSE) of Diﬀerent Simulation and ∆t Choices (Hull-White Method)

We follow the conventions and compare log(MSE) and log(MSERelative) of different ∆t’s, the results are shown in Figure 5.7 and 5.8. From bottom to top, the curves in Figure 5.7 are calculated with 20000, 10000, 5000 and 1000 simulations respectively. And we use same colour for same combinations in Figure 5.8 as in 5.7. Although some unexpected increases exist when reducing ∆t’s, the general trend shows that reducing ∆t’s has little effect on improving the level of accuracy, especially when the number of simulations is larger. The effect of increasing the number of simulations is on the other hand significant though the effect becomes much smaller when the number of simulations reaches a high level. In order to see whether it is worthwhile to reduce ∆t, we do a further test: Swaption Pricing Approximations for LMMs — Yueci Li 29

1 1 1 ∆t = 2 ∆t = 4 ∆t = 6 1000 -4.994444 -4.922840 -4.985984 5000 -5.271046 -5.314305 -5.229721 10000 -5.437702 -5.389646 -5.432375 20000 -5.397976 -5.429024 -5.471122

Table 5.16: log(MSERelative) of Diﬀerent Simulation and ∆t Choices (Hull-White Method)

Figure 5.7: Time Step Choice: log(MSE) of Diﬀerent ∆t’s

1 1 1 1 1 1 We run 100 simulations each for ∆t = 2 , 4 , 6 , 8 , 10 , 12 and calculate and plot all the 100 × 6 log(MSE)’s and log(MSERelative)’s. Next, we calculate the means and standard deviations of the log(MSE)’s and log(MSERelative)’s at each ∆t’s. With which, we can find the 95% confidence intervals of these simulations. The results are plotted in Figure 5.9 and 5.10. The results are surprisingly flat yet this is not entirely unexpected. When we run more simulations in Figure 5.7 and 5.8, we also see an almost constant log(MSE) and log(MSERelative), which indicates that the level of accuracy barely increases when we 1 reduce ∆t’s. We thus can see why ∆t = 4 was chosen in Brigo and Mercurio(2007) while the number of simulations was chosen to be as large as 200000: decreasing ∆t does not have large contribution to the accuracy of swaption pricing using LIBOR market model. On the other hand, significant difference can be observed when we increase the number of simulations. We compare log(MSE) as well as log(MSERelative) of different numbers of simulations in Figure 5.11 and 5.12. Compared to Figure 5.7 and 5.8, much steeper decrease is observed in each curve. It is thus confirmed that increasing the number of simulations has greater effects on the swaption pricing accuracy than reducing ∆t. The question left is what are the effects of changing the number of simulations and ∆t on computational cost. This can be 30 Yueci Li — Swaption Pricing Approximations for LMMs

Figure 5.8: Time Step Choice: log(MSERelative) of Diﬀerent ∆t’s

Figure 5.9: Time Step Choice: log(MSE) against ∆t with Conﬁdence Intervals

evaluated by comparing the computation time of diﬀerent arrangements. Again, we run 1 1 1 1000, 5000, 10000 and 20000 simulations for ∆t = 2 , 4 , 6 . The computation time is shown in Table 5.17: Swaption Pricing Approximations for LMMs — Yueci Li 31

Figure 5.10: Time Step Choice: log(MSERelative) against ∆t with Conﬁdence Intervals

Figure 5.11: Simulation Choice: log(MSE) of Diﬀerent Numbers of Simulation

An almost linear change in computation time can be observed when changing either the number of simulations or ∆t. The computation time when using smaller ∆t’s and when running more simulations are shown in Figure 5.13 and 5.14: 32 Yueci Li — Swaption Pricing Approximations for LMMs

Figure 5.12: Simulation Choice: log(MSERelative) of Diﬀerent Numbers of Simulation

1 1 1 ∆t = 2 ∆t = 4 ∆t = 6 1,000 318.899 638.177 922.153 5,000 1530.939 3138.628 4564.500 10,000 3042.878 6521.944 9329.118 20,000 6067.164 12259.694 18207.437

Table 5.17: Computation Time of Diﬀerent Simulation and ∆t Choices (in Seconds)

Doubling time step per year will result in almost same amount more computation time as doubling the number of simulations, while the resulted improvements in pricing accuracy diﬀer largely. Thus we recommend increasing the number of simulations rather 1 than reducing ∆t. We can take ∆t as large as 4 as in Brigo and Mercurio(2007) without inducing large loss in accuracy, while we still see small margin for improvement of pricing accuracy even when we are already running 20000 simulations. Swaption Pricing Approximations for LMMs — Yueci Li 33

Figure 5.13: Computation Time: Changing ∆t

Figure 5.14: Computation Time: Changing the Number of Simulations Chapter 6

Conclusions and Further Study

Although the LIBOR market model is considered according to both literature and common practice, one of the most promising models in the current economic environment, it is proved in this thesis that the swaption pricing results of this model using simple approximations are in many cases not satisfactory. First of all, there exist some embedded drawbacks in the assumptions of the LIBOR market model. The assumptions of the model only hold when the forward rates are above 1%, however, we aim to use the model also to solve problems when the interest rate is low, if not negative. Both the log-normality assumption and the stationary volatility assumption will not hold in this case. Hence, further study is recommended to include displaced diffusion terms. The displaced diffusion term is normally an arbitrary positive number, which is added to the forward rates to make the new rates satisfy the log- normality assumption again. Including such a term is easy, though only the standard LIBOR market model is discussed in this thesis for more general use. Furthermore, the calibration of the many parameters involved in the LIBOR market model can be difficult. Although such a procedure is not conducted in this thesis, from the parameter values taken from literature and used in this thesis, it can be seen that because of the large amount of parameters involved, the values of many are zeros, which can create strange curves of forward rates. Incorporating such curves in the swaption pricing process will lead to less good results. As for the approximation methods, which are the focus of this thesis, we find both proposed methods reasonably accurate when the swaptions expire in less than 5 years. The deviations of swaption prices when compared to Monte Carlo simulation results are less than 5% in this case. The Hull-White method is shown to provide a slightly more accurate result than the Rebonato’s method; since incorporating the Hull-White method does not induce more computational effort, we recommend using Hull-White method. When doing the calculations, two parameter values have to be determined, the number of simulations and the time step ∆t. Increasing the number of simulations is shown to lead to greater improvement in pricing accuracy while the effect of reducing ∆t is extremely small. Since they induce almost the same computational cost, we recommend running more simulations for more accurate results.

34 Appendix A: Parameter Values and Initial Forward Rates

ψ Φ θ Fi(0) 0 0.046900 1 2.5114 0.0718 1.7864 0.050114 2 1.5530 0.0917 2.0767 0.055973 3 1.2238 0.1009 1.5122 0.058387 4 1.0413 0.1055 1.6088 0.060027 5 0.9597 0.1074 2.3713 0.061315 6 1.1523 0.1052 1.6031 0.062779 7 1.2030 0.1043 1.1241 0.062747 8 0.9516 0.1055 1.8323 0.062926 9 1.3539 0.1031 2.3955 0.062286 10 1.1912 0.1021 2.5439 0.063009 11 0.0000 0.1046 1.6118 0.063554 12 3.3778 0.0844 1.3172 0.064257 13 0.0000 0.0857 1.2225 0.064784 14 1.2223 0.0847 1.0995 0.065312 15 0.0000 0.0869 1.2602 0.063976 16 0.0000 0.0896 1.0905 0.062997 17 0.0000 0.0921 0.8006 0.061840 18 0.1156 0.0946 0.8739 0.060682 19 0.5753 0.0965 1.7096 0.059360

35 Appendix B: Correlation Matrix

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 1.0000 0.9582 0.9626 0.9843 0.8338 0.9832 0.7886 0.9989 0.8202 0.7266 0.9848 0.8919 0.8452 0.7732 0.8647 0.7675 0.5522 0.6118 0.9971 2 0.9582 1.0000 0.8449 0.8925 0.9569 0.8899 0.5796 0.9703 0.9496 0.8928 0.8939 0.7252 0.6568 0.5593 0.6848 0.5519 0.2904 0.3597 0.9334 3 0.9626 0.8449 1.0000 0.9953 0.6531 0.9959 0.9256 0.9492 0.6346 0.5134 0.9950 0.9810 0.9583 0.9160 0.9684 0.9124 0.7573 0.8031 0.9806 4 0.9843 0.8925 0.9953 1.0000 0.7231 1.0000 0.8848 0.9751 0.7062 0.5937 1.0000 0.9578 0.9263 0.8731 0.9399 0.8687 0.6908 0.7419 0.9949 36 5 0.8338 0.9569 0.6531 0.7231 1.0000 0.7192 0.3180 0.8582 0.9997 0.9851 0.7252 0.4940 0.4096 0.2946 0.4437 0.2859 0.0001 0.0733 0.7889 6 0.9832 0.8899 0.9959 1.0000 0.7192 1.0000 0.8875 0.9738 0.7021 0.5891 1.0000 0.9594 0.9284 0.8759 0.9418 0.8715 0.6949 0.7457 0.9943 7 0.7886 0.5796 0.9256 0.8848 0.3180 0.8875 1.0000 0.7595 0.2949 0.1504 0.8834 0.9814 0.9952 0.9997 0.9908 0.9994 0.9481 0.9689 0.8334 8 0.9989 0.9703 0.9492 0.9751 0.8582 0.9738 0.7595 1.0000 0.8456 0.7573 0.9758 0.8702 0.8198 0.7433 0.8408 0.7373 0.5134 0.5748 0.9925 9 0.8202 0.9496 0.6346 0.7062 0.9997 0.7021 0.2949 0.8456 1.0000 0.9890 0.7083 0.4728 0.3874 0.2714 0.4219 0.2627 -0.0241 0.0492 0.7738 10 0.7266 0.8928 0.5134 0.5937 0.9851 0.5891 0.1504 0.7573 0.9890 1.0000 0.5961 0.3373 0.2468 0.1261 0.2832 0.1171 -0.1716 -0.0990 0.6717 11 0.9848 0.8939 0.9950 1.0000 0.7252 1.0000 0.8834 0.9758 0.7083 0.5961 1.0000 0.9569 0.9252 0.8716 0.9388 0.8672 0.6886 0.7399 0.9952 12 0.8919 0.7252 0.9810 0.9578 0.4940 0.9594 0.9814 0.8702 0.4728 0.3373 0.9569 1.0000 0.9955 0.9764 0.9984 0.9744 0.8695 0.9033 0.9240 13 0.8452 0.6568 0.9583 0.9263 0.4096 0.9284 0.9952 0.8198 0.3874 0.2468 0.9252 0.9955 1.0000 0.9924 0.9993 0.9913 0.9123 0.9399 0.8837 14 0.7732 0.5593 0.9160 0.8731 0.2946 0.8759 0.9997 0.7433 0.2714 0.1261 0.8716 0.9764 0.9924 1.0000 0.9871 1.0000 0.9557 0.9747 0.8196 15 0.8647 0.6848 0.9684 0.9399 0.4437 0.9418 0.9908 0.8408 0.4219 0.2832 0.9388 0.9984 0.9993 0.9871 1.0000 0.9856 0.8962 0.9263 0.9007 16 0.7675 0.5519 0.9124 0.8687 0.2859 0.8715 0.9994 0.7373 0.2627 0.1171 0.8672 0.9744 0.9913 1.0000 0.9856 1.0000 0.9583 0.9766 0.8144 17 0.5522 0.2904 0.7573 0.6908 0.0001 0.6949 0.9481 0.5134 -0.0241 -0.1716 0.6886 0.8695 0.9123 0.9557 0.8962 0.9583 1.0000 0.9973 0.6145 18 0.6118 0.3597 0.8031 0.7419 0.0733 0.7457 0.9689 0.5748 0.0492 -0.0990 0.7399 0.9033 0.9399 0.9747 0.9263 0.9766 0.9973 1.0000 0.6707 19 0.9971 0.9334 0.9806 0.9949 0.7889 0.9943 0.8334 0.9925 0.7738 0.6717 0.9952 0.9240 0.8837 0.8196 0.9007 0.8144 0.6145 0.6707 1.0000 Swaption Pricing Approximations for LMMs — Yueci Li 37 i,j σ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 12 0.1803 3 0.14244 0.1235 0.2303 5 0.1099 0.15676 0.1031 0.1291 0.2534 7 0.1212 0.1118 0.16388 0.1255 0.1010 0.1314 0.2650 9 0.1004 0.1202 0.1095 0.1668 0.1396 0.1269 0.1001 0.1287 0.2697 0.0981 0.1216 0.1086 0.1634 0.1240 0.1012 0.1276 0.2642 0.1188 0.1099 0.1620 0.0989 0.1291 0.2619 0.1074 0.1638 0.1262 0.2650 0.1601 0.2589 1011 0.121612 0.0000 0.138213 0.2851 0.1246 0.097214 0.0000 0.0000 0.1416 0.122815 0.1035 0.2895 0.1005 0.0995 0.117616 0.0000 0.0000 0.0000 0.1143 0.1258 0.098017 0.0000 0.1062 0.2861 0.1021 0.0803 0.1205 0.106318 0.0000 0.0000 0.0000 0.0000 0.1160 0.1015 0.1004 0.124919 0.0109 0.0000 0.1095 0.2935 0.1009 0.0816 0.0973 0.1089 0.1586 0.0555 0.0000 0.0000 0.0000 0.0000 0.1147 0.1031 0.0810 0.1280 0.2564 0.0112 0.0000 0.1126 0.3027 0.1035 0.0806 0.0988 0.0879 0.1624 0.0000 0.0000 0.0000 0.0000 0.1177 0.1019 0.0822 0.1033 0.2627 0.0000 0.1156 0.3111 0.1067 0.0827 0.0976 0.0892 0.1311 0.0000 0.0000 0.0000 0.1213 0.1045 0.0813 0.1049 0.2120 0.1180 0.3195 0.1097 0.0853 0.1001 0.0882 0.1331 0.0000 0.0000 0.1247 0.1078 0.0834 0.1037 0.2152 0.3260 0.1127 0.0876 0.1032 0.0905 0.1315 0.0000 0.1281 0.1108 0.0860 0.1063 0.2127 0.1150 0.0900 0.1061 0.0933 0.1350 0.1307 0.1138 0.0884 0.1097 0.2182 0.0918 0.1090 0.0959 0.1391 0.1161 0.0908 0.1127 0.2250 0.1112 0.0985 0.1430 0.0926 0.1158 0.2313 0.1005 0.1469 0.1181 0.2376 0.1499 0.2424 Appendix C: Instantaneous Forward Rate Volatility Appendix D: Volatility vα,β for Black’s Formula

1 2 3 4 5 6 7 8 9 10 1 0.164 0.158 0.146 0.138 0.133 0.129 0.126 0.123 0.120 0.117 2 0.177 0.156 0.141 0.131 0.127 0.124 0.122 0.119 0.117 0.114 3 0.176 0.155 0.139 0.127 0.123 0.121 0.119 0.117 0.115 0.113 4 0.169 0.146 0.129 0.119 0.116 0.114 0.113 0.111 0.110 0.108 5 0.158 0.139 0.124 0.115 0.111 0.109 0.108 0.107 0.105 0.104 7 0.145 0.129 0.116 0.108 0.104 0.103 0.101 0.099 0.098 0.096 10 0.135 0.115 0.104 0.098 0.094 0.093 0.091 0.088 0.086 0.084

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