Swaption pricing approximations for LIBOR market models

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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: AlexYueciLi@outlook.com

Date: July 15, 2018

Supervisor: prof. dr. ir. M.H. (Michel) Vellekoop

Second reader: dr. S.U. (Umut) Can

Supervisor: MSc R. (Remco) Stam

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Statement of Originality

This document is written by Student Yueci Li who declares to take full responsibility for the con-tents 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.

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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

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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 possi-ble. 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.

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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-DerGold-man-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, 2016Yoshino (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 marketHull (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 com-mon 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

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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 pric-ing uspric-ing 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 com-plicated 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 back-grounds 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 for-ward LIBOR model (LFM), and also, the two approximation methods for the pricing of European swaptions. And in Chapter 5, the detailed numerical implementation meth-ods 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.

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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 pric-ing 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 P (t, T ) = h 1 +Rn(t, T ) n in(T −t) , (2.2)

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 prac-tice, 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.

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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, Fn(t, T1, T2) = n × " P (t, T₁) P (t, T2) 1 n(T2−T1) _{− 1} # (2.5) 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 datesVeronesi (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

contractSaunders 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,

1 − S n × M X j=i+1 P (Ti, Tj) + 1 × P (Ti, TM) (2.6)

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,

S = n × 1 − P (t, T_PM M)

j=1P (t, Tj)

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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.

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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 priceBlack(1976). In continuous form, the value of a caplet

at the current moment is,

Caplet_i=F (t, Ti−1, Ti)N (d1) − KN (d2)P (t, Ti−1) (2.8)

d1 = log F (t,Ti−1,Ti) K + v2 2 Ti−1 v√Ti−1 (2.9) d2 = d1− vpTi−1 (2.10)

where Caplet_i 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−1and paid out at Ti:

Cap = M X

i=1

Caplet_i (2.11)

Caplet_i=F (t, Ti−1, Ti)N (d1) − KN (d2)P (t, Ti−1) (2.12)

d1 = log F (t,Ti−1,Ti) K + v2 2 Ti−1 v√Ti−1 (2.13) d2 = d1− vpTi−1 (2.14)

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2.4 European Swaptions

After setting up both the vanilla swaps and the caps, it leads us directly to the combi-nation 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 in-termediate 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 instru-ments, 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.

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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 devel-opment 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).

dF (t, T ) = σ(t, T )

Z T

t

σ(t, u)dudt + σ(t, T )dWt (3.1)

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) andHull (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

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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

com-pared 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)

Z(t, Ti+1) = 1 + τiFi(t) (3.3)

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

dFi(t) = (· · · )1dt +

p X q=1

σi,q(t)Fi(t)dZq (3.6)

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

instanta-neous 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,

dZ(t, Ti) = (· · · )2dt +

p X q=1

si,q(t)Z(t, Ti)dZq (3.7)

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

the expected growth rate of the market price of risk byPp_q=1σi,qsm(t),q(t) − si+1,q(t)

Hull (2012) 3 and using Ito’s Lemma,

dFi(t) Fi(t) = p X q=1 σi,q[sm(t),q(t) − si+1,q(t)]dt + p X q=1 σi,q(t)dZq (3.8)

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

d log[1 + τiFi(t)] = (· · · )dt + τi 1 + τiFi(t) p X q=1 σi,q(t)dZq (3.9)

1_{We follow the convention of}_{Hull and White} ₍₂₀₀₀_{) and}_Hull₍₂₀₁₂_{). This part will not affect the}

following derivations thus is not given.

2

Same as above

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

d log Z(t, ti)

Z(t, ti+1)

= (· · · )dt + [s_m(t),q(t) − si+1,q(t)]dZq (3.10)

Combining (3.5), (3.9) and (3.10), we have

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

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 inHull and White (2000) when they set up the model.

From their experiments, when the interest rate goes as low as 1%, the stationary as-sumption 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 dis-placed 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.

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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 approxi-mation 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 calibra-tion 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 Et β X i=α+1 P (t, Ti)τi Sα,β(Tα) − K+ (4.2)

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 prod-ucts of the numeraires based on forward rates of spot rate form.

P (Ti−1, Ti) = 1

1 + τiFi(t) (4.3)

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:

Et h Xβ i=α+1 P (t, Ti)τi Fi(t) − K i (4.4) 10

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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,

Et h Xβ i=α+1 P (t, Ti)τi Fi(t) − K i = 0 Et h _Xβ i=α+1 P (t, Ti)τiFi(t) − K β X i=α+1 P (t, Ti)τi i = 0 (4.6) Since, Et h P (t, Ti)τiFi(t) i = Et h P (t, Ti−1) − P (t, Ti) i (4.7) Thus, Et h _Xβ i=α+1 P (t, Ti)τiFi(t) i = Et h _Xβ i=α+1 P (t, Ti−1) − β X i=α+1 P (t, Ti) i = EthP (t, Tα) − P (t, Tβ) i (4.8)

We can replace the corresponding terms in (4.6) and get,

Sα,β(t) = K = Et " P (t, Tα) − P (t, Tβ) Pβ i=α+1P (t, Ti)τi # (4.9)

To facilitate the derivations following, we set up a new concept, the forward discount

factor D(t; Tα, Ti) that,

D(t; Tα, Ti) = P (t, Ti)

P (t, Tα) (4.10)

By dividing both nominator and denominator of (4.9), and plug in (4.3), we get,

Sα,β(t) = 1 − D(t; Tα, Tβ) Pβ i=α+1τiD(t; Tα, Ti) = 1 −Qβ_j=α+1 _1+τ1 jFj(t) Pβ i=α+1 Qi j=α+11+τj1Fj(t) . (4.11)

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

computa-tional 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.

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4.2.1 Swaption Dynamics and Volatilities

As introduced inBrigo and Mercurio(2007) 1, a log-normal swaption pricing dynamics

can be assumed:

dSα,β(t) = σα,β(t)Sα,β(t)dW_tα,β (4.12)

where σα,β(t) is the instantaneous swaption volatility at t.

Applying Ito’s Lemma,

d log Sα,β(t) = σα,β(t)dW_tα,β− 1

2σ

2

α,βdt (4.13)

The Black-style swaption volatility2 is,

vα,β· (Tα) 2 = Z Tα 0 σ2_α,β(t)dt (4.14) Notice that by (4.13), d log Sα,β(t) d log Sα,β(t) = σα,β(t)2dt (4.15)

Combining (4.14) and (4.15), we finally arrive at the general formula of Black-style

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

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 methodBrigo and Mercurio(2007) is based directly on the derivations

introduced above. We start with equation (4.6),

β X i=α+1 P (t, Ti)τiFi(t) − β X i=α+1 P (t, Ti)τiK = 0 K = Pβ i=α+1P (t, Ti)τiFi(t) Pβ j=α+1P (t, Tj)τj = Pβ i=α+1D(t; TαTi)τiFi(t) Pβ j=α+1D(t; TαTj)τj = β X i=α+1 D(t; TαTi)τi Pβ j=α+1D(t; TαTj)τi Fi(t) (4.17)

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):

ωi(t) = D(t; TαTi)τi Pβ j=α+1D(t; TαTj)τj (4.18) 1

Chapter 6, page 240 ofBrigo and Mercurio(2007)

2_{Note that this swaption volatility is squared and is multiplied by T} α

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

Sα,β(t) =

β X i=α+1

ωi(t)Fi(t) (4.19)

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, dSα,β(t) = β X i=α+1 ωi(0)dFi(t) = β X i=α+1 ωi(0)(· · · )dt + σi(t)Fi(t)dZi(t) (4.20)

Since the following relation between the Brownian motions holds,

dZi(t)dZj(t) = ρi,jdt (4.21) We have, dSα,β(t)dSα,β(t) ≈ β X i,j=α+1

ωi(0)ωj(0)Fi(t)Fj(t)ρi,jσi(t)σj(t)dt (4.22)

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

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,

Sα,β(t) =

β X i=α+1

ωi(0)Fi(0) = Sα,β(0) (4.24)

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

vα,β· (Tα) 2 = β X i,j=α+1 ωi(0)ωj(0)Fi(0)Fj(0)ρi,j Sα,β(0)2 Z Tα 0 σi(t)σj(t)dt (4.25)

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 rea-sonably accurate. Another similar approximation method will be introduced in the next section.

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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),

dSα,β(t) = β X i=α+1 ωi(t)dFi(t) + Fi(t)dωi(t) + (· · · )dt = β X i=α+1 ωh(t)Ih,i+ Fi(t) ∂ωi(t) ∂Fh dFh(t) + (· · · )dt (4.26)

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),

¯ ωh(t) = ωh(t) + β X i=α+1 Fi(t) ∂ωi(t) ∂Fh (4.27)

where the partial derivative term can be derived directly by differentiating ωi(t)’s as in

(4.18) with regard to Fh, ∂ωi(t) ∂Fh = − τh 1 + τhFh(t) 2 1 Pβ k=α+1τk Qk j=α+11+τj1Fj(t) 2 × I_i,h0 − β X k=α+1 τk k Y j=α+1 1 1 + τjFj(t) τi i Y j=α+1 1 1 + τjFj(t) 1 + Fh(t) (4.28)

where I_i,h0 is the indicator function that is true only when h ≤ i.

We can arrive at the final formula for ∂ωi(t)

∂Fh , ∂ωi(t) ∂Fh = τhωi(t) 1 + τhFh(t) Pβ k=hτkQkj=α+11+τj1Fj(t) Pβ k=α+1τk Qk j=α+11+τj1Fj(t) − I_i,h0 . (4.29)

Now dSα,β(t) has the same structure as the Rebonato’s method. And by freezing

the ¯ωi(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),

vα,β· (Tα)2 ≈ β X i,j=α+1 ¯ ωi(0)¯ωj(0)Fi(0)Fj(0)ρi,j Sα,β(0)2 Z Tα 0 σi(t)σj(t)dt (4.30)

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.

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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:

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

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 rep-resents 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, fromBrigo 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 envi-ronment, a displaced diffusion term should be added to the forward rate Fi(t). This is

1_{See Appendix A}

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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,

dFk(t) = σk(t)Fk(t) i X j=m(t) ρk,jτjσj(t)Fj(t) 1 + τjFj(t) dt + σk(t)Fk(t)dZk(t) (5.2)

with k = 1, 2, . . . β − 1 and dZk(t) the Brownian motion term with correlation matrix

{ρ_i,j}, i, j = 1, 2, . . . , β − 1.

Applying Ito’s Lemma,

d log Fk(t) =0 + 1 Fk(t)σk(t)Fk(t) k X j=m(t) ρk,jτjσj(t)Fj(t) 1 + τjFj(t) − 1 2 1 F_k2(t)σ 2 k(t)Fk2(t)dt + 1 Fk(t)σk(t)Fk(t)dZk(t) = σk(t) k X j=m(t) ρk,jτjσj(t)Fj(t) 1 + τjFj(t) dt − 1 2σ 2 k(t)dt + σk(t)dZk(t) (5.3)

In order to implement in practice, I transform the continuous formula (5.3) to its

discrete version Brigo and Mercurio (2007):

log Fk(t+∆t) = log Fk(t)+σk(t) k X j=α+1 ρk,jτjσj(t)Fj(t) 1 + τjFj(t) ∆t− σk(t)2 2 ∆t+σk(t)Zk(t+∆t)−Zk(t) (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−1and 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

and the payment dates are thus T1, T2, . . . , T19. And the time step ∆t is set to be 1₆.

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:

2

The 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.

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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 agree-ments. 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

{ρk,j} = PDP0 (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 =

P0P = I19. The columns of matrix P are composed of the eigenvalues with the same

order as in the diagonal of matrix D.

D =      e1 0 · · · 0 0 e2 · · · 0 .. . ... . .. 0 0 0 · · · (e19)      (5.7)

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Since all the elements in D are non-negative, we can take square root of D and define a new diagonal matrix T:

T =      √ e1 0 · · · 0 0 √e2 · · · 0 .. . ... . .. 0 0 0 · · · (√e19)      (5.8)

And by setting A = PT, we see that the following equation holds:

AA0 = P(TT0)P0 = PDP0 = {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

BB0 ≈ {ρk,j} according toRebonato(1999).

By lowering the rank of the correlation matrix from α to N in this way, we can rewrite the Brownian motion term. Now,

(B dW)(B dW)0 = B (WW0)B0 = BB0dt = { ˆρk,j}dt (5.10)

where { ˆρk,j} is the parameterised correlation matrix that we try to make as close to

{ρ_k,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 j=1

b2_i,j = 1 (5.11)

This coincides with a polar coordinate system. Take N = 2 for example, for i = 1, 2, 3, . . . , 19, take

bi,1= cos θi

bi,2= sin θi (5.12)

Since obviously

cos2θi+ sin2θi = 1 (5.13)

which is equivalent to PN_j=1b2_i,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

bi,j = cos θi,j

j−1 Y k=1 sin θi,k j = 1, 2, · · · , β − 1 bi,j = j−1 Y k=1 sin θi,k j = α (5.14)

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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 byRebonato

(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:

{ˆρk,j} =      1 cos(θ1− θ2) · · · cos(θ1− θ19) cos(θ2− θ1) 1 · · · cos(θ2− θ19) .. . ... . .. cos(θ18− θ19) cos(θ19− θ1) cos(θ19− θ2) · · · 1      (5.15)

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

{ˆρk,j}.

Rebonato (2005) compared a few market models of correlation coefficients, and the most practical one is,

ρi,j = LongCorr + (1 − LongCorr) expγ | Ti− Tj |

(5.16) where

γ = d1− d2max(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,

χ2_k,j= 19 X k,j=1 ρk,j− ˆρik,j 2 (5.21)

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 inBrigo 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,

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

Since P (t, Tα) is observed at present, it is actually deterministic and can be calculated directly using the initial forward rate values.

The rest of the pricing numeraire term, Pβ_i=α+1P (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.

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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 byBrigo 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) d1 = logSα,β(0) K + 1 2v2α,βTα vα,β √ Tα d2 = d1− v_α,βpTα (5.23)

where vα,βis 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 d1 = 1 2vα,β p Tα (5.24)

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, ¯ωh(t) is

constructed specifically. Except for this difference, all the other structures as well as approximations made remain exactly the same.

The ωi(t) and ¯ωh(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

for-mula for the volatilities is,

vα,β· (Tα)2 ≈ β X i,j=α+1 ωi(0)ωj(0)Fi(0)Fj(0)ρi,j Sα,β(0)2 Z Tα 0 σi(t)σj(t)dt (5.25)

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

RTα

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

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

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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 Figure5.2.

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

ac-cording 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

and ∆t is taken to be 1₆6.

The simulated swaption prices are shown in Table 5.3and Figure5.3.

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.

5

To make the figures clearer, Figure5.1and Figure5.2only include 100 generated forward rate paths with ∆t =1

6. 6

InBrigo and Mercurio(2007), the number of simulations is 200000 and ∆t = 1₄. It is very surpris-ing that ∆t = 1

4 would satisfy the accuracy requirements of the calculations. Considering the heavy

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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 Table5.4 and Figure5.4:

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.

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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:

Deviation = Simulation Results − Black’s Formula Results

Simulation Results (5.27)

We choose to calculate the deviation with regard to simulation results instead of

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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

imple-mentation 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 calcu-lating 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: ¯ ωh(t) = ωh(t) + β X i=α+1 Fi(t)∂ωi(t) ∂Fh (5.28)

while term ωh(t) is exactly the weight term for Rebonato’s method.

We freeze both ωi(t) and ¯ωi(t) at t = 0. ωi(0) as calculated directly from (4.18) is

always larger than 0:

ωi(0) = _Pβ D(0; TαTi)τi

j=α+1D(0; TαTj)τj

> 0 _(5.29)

The partial derivative term ∂ωi(t)

∂Fh according to (4.29): ∂ωi(t) ∂Fh = τhωi(t) 1 + τhFh(t) Pβ k=hτk Qk j=α+1 1+τj1Fj(t) Pβ k=α+1τk Qk j=α+1 1 1+τjFj(t) − I_i,h0 (5.30)

For ¯ωi(t), its component ∂ω_∂Fi(t)_h can be negative occasionally, for example when α+1 =

1, β = 2, and i = h = 2: the Indicator function I_i,h0 equals 1 while the fraction of

summations Pβ k=hτkQkj=α+1 1 1+τj Fj (t) Pβ k=α+1τk Qk j=α+1_{1+τj Fj (t)}1 < 1. The term ∂ωi(t)

∂Fh in this case is negative.

8

The values taken fromBrigo and Mercurio(2007) are only rounded to 3 digits, which is not con-sidered accurate enough.

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It should be noted that in most cases, even with some negative ∂ωi(t)

∂Fh terms involved,

the ¯ωi(0) does not necessarily go negative. In fact, only ¯ω2(0) is observed to be negative

in our implementation. The other ¯ωi(0)’s remain positive because we make summation

Pβ

i=α+1Fi(t) ∂ωi(t)

∂Fh , with longer underlying swap lengths, more positive components are

added up, the summations remain positive, and thus the Hull-White weights ¯ωi(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:

Deviation = Hull-White Results − Rebonato’s Results

Hull-White Results (5.31)

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,βare shown in Table5.6and Table5.7respectively:

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 Table5.8:

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

v2,1is 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)

∂Fh = 0, ∀i for these swaptions according to equation (5.30). All

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

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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:

Deviation = Approximation Results − Simulation Results

Simulation Results (5.32)

The deviations of the two methods are shown in Table 5.11and 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

9

It 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.

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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

cal-culate the Mean Squared Errors (MSE). We denote Xi,j as the model (simulated)

swaption prices and ˆXi,j the approximated swaption prices where i = 1, 2, . . . , 7 and

j = 1, 2, . . . , 10. There are in total 70 Xi,j and ˆXi,j respectively. The corresponding

swaption prices are given in Table5.3,5.9 and 5.10.

MSE = 1 70 7 X i=1 10 X j=1 Xi,j− ˆXi,j2 (5.33)

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.

Swaption pricing approximations for LIBOR market models