Piterbarg’s FL-TSS vs. SABR/LMM: A comparative study 674088 sub... · Piterbarg’s FL-TSS vs....

Piterbarg’s FL-TSS vs.SABR/LMM: A comparative

study

University of Oxford

A thesis submitted in partial fulfillment of the MSc in

Mathematical Finance

September 29, 2014

A life dedicated to the pursuit of knowledge is a life worth living

Acknowledgements

I would like to express my gratitude to Dr Jeff Dewynne for his helpful

advice and comments on my dissertation.

Abstract

The LIBOR market model (LMM) is an established method for encoding

all at-the-money volatility information for swaptions across all swaption

expiries and maturities, however it is not able to recover the volatility

smile. As a result, many extensions to the LMM have been proposed

with the aim of extending it to encode volatility smile information. Two

models of interest are Vladimir Piterbarg’s forward LIBOR model with

time-dependent skew (FL-TSS), and the SABR/LIBOR market model

(SABR/LMM) developed by Riccardo Rebonato et al.

FL-TSS assumes that forward rates follow a shifted log-normal diffusion.

Forward rate volatility is modelled as a mean reverting process, and

no correlation is assumed between the volatility and the forward rate

dynamics.

SABR/LMM uses the industry standard SABR model to provide ac-

curate analytic prices for European options. By combining SABR and

LMM, it’s able to bring both forward rates and forward rate stochastic

volatilities under the same measure, such that for all underlyings the dy-

namics are simultaneously valid and complex derivatives can be priced.

By constructing both models in this study, we examine and compare

various features such as their ability to calibrate to different market

states. We investigate how well both models predict future volatility

smiles, we explore their implementation details and we make observations

on their computational aspects.

We find the time-homogeneity that’s inherent to the SABR/LMM model

allows for better fits and future predictions of swaption implied volatility

smiles overall. We find the calibration of SABR/LMM to be more stable

and more computationally efficient.

Contents

1 Introduction 1

1.1 Markets & Pricing of Caplets & Swaptions . . . . . . . . . . . . . . . 2

1.2 Notations & Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The SABR/LIBOR market model 5

2.1 The LIBOR market model . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Volatility functions . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Splitting Correlation and Volatility . . . . . . . . . . . . . . . 8

2.1.3 Forward-Forward Correlation . . . . . . . . . . . . . . . . . . 9

2.2 The SABR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 The SABR/LMM Model Definition . . . . . . . . . . . . . . . . . . . 12

2.4 The SABR/LMM Model for Caplets . . . . . . . . . . . . . . . . . . 13

2.5 The SABR/LMM Model for Swaptions . . . . . . . . . . . . . . . . . 14

2.6 Implementing SABR/LMM . . . . . . . . . . . . . . . . . . . . . . . 16

2.6.1 Calibrating g(·) . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6.2 Calibrating h(·) . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6.3 Selecting β and ρ . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6.4 Approximating Σ0 and V . . . . . . . . . . . . . . . . . . . . . 18

2.6.5 Approximating B . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6.6 Approximating RSABR . . . . . . . . . . . . . . . . . . . . . . 19

3 Piterbarg’s term structure of skew forward LIBOR model 20

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 A stochastic volatility forward LIBOR model for European Swaptions 21

3.2.1 A simple model . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 A forward LIBOR market model with stochastic volatility . . 22

3.3 Piterbarg’s term structure of skew LIBOR market model . . . . . . . 24

3.3.1 The FL-TSS model set-up . . . . . . . . . . . . . . . . . . . . 24

3.4 Pricing European swaptions via parameter averaging techniques . . . 27

3.4.1 Effective Skew . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.2 Effective Volatility . . . . . . . . . . . . . . . . . . . . . . . . 28

i

3.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5.1 Finding λmktn,m and bmkt

n,m . . . . . . . . . . . . . . . . . . . . . . 30

3.5.2 Finding β(t;n), t ≥ 0n and σk(t;n), t ≥ 0n,k . . . . . . . . 32

4 Methodology 34

4.1 Smile Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Smile Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 The Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.1 The Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.2 SABR/LMM . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.3 FL-TSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Theoretical Analysis 37


5.1.1 Under normal market conditions . . . . . . . . . . . . . . . . 37

5.1.2 Under excited market conditions . . . . . . . . . . . . . . . . . 39


5.2.1 normal ⇒ normal . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.2 normal ⇒ excited . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.3 excited ⇒ excited . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Results and discussion 41

6.1 Intermediate Calibrations . . . . . . . . . . . . . . . . . . . . . . . . 41

6.1.1 SABR/LMM . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.1.2 FL-TSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


6.2.1 Under normal market conditions . . . . . . . . . . . . . . . . 44

6.2.2 Under excited market conditions . . . . . . . . . . . . . . . . . 45

6.2.3 Computational considerations . . . . . . . . . . . . . . . . . . 46


6.3.1 normal ⇒ normal . . . . . . . . . . . . . . . . . . . . . . . . 47

6.3.2 normal ⇒ excited . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.3.3 excited ⇒ excited . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Conclusion & further research 53

Bibliography 58

Appendix A Data 59

Appendix B Matlab code listing 61

ii

List of Figures

2.1 A Doust correlation surface that shows convexity . . . . . . . . . . . 10

6.1 Caplet calibration fits for 23-Nov-2006 . . . . . . . . . . . . . . . . . 42

6.2 Caplet calibration fits for 4-Mar-2008 . . . . . . . . . . . . . . . . . . 42

6.3 Filon-Levy fits for 23-Nov-2006 . . . . . . . . . . . . . . . . . . . . . 43

6.4 Fits for expression (3.27) . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.5 Implied volatility smiles generated by calibrated SABR/LMM and

FL-TSS models, compared to the market for 23-Nov-2006 . . . . . . . 44

6.6 Implied volatility smiles generated by calibrated SABR/LMM and

FL-TSS models, compared to the market for 4-Mar-2008 . . . . . . . 46

6.7 Predicted volatility smiles vs. Actual smile for 1Y expiry and different

maturities from a normal market to a normal market . . . . . . . . . 48


maturities from a normal market to a normal market . . . . . . . . . 48

6.9 Predicted volatility smiles vs. Actual smile for 10Y expiry and differ-

ent maturities from a normal market to a normal market . . . . . . . 49


maturities from a normal market to an excited market . . . . . . . . 49


maturities from a normal market to an excited market . . . . . . . . 49


ent maturities from a normal market to an excited market . . . . . . 50


maturities from a excited market to an excited market . . . . . . . . 50


maturities from a excited market to an excited market . . . . . . . . 51


ent maturities from a excited market to an excited market . . . . . . 51

iii

List of Tables

A.1 Market Caplet data for 23-Nov-2006 . . . . . . . . . . . . . . . . . . 59

A.2 Market Caplet data for 04-Mar-2008 . . . . . . . . . . . . . . . . . . 59

A.3 g(·) calibrated parameters for 23-Nov-2006 . . . . . . . . . . . . . . . 59

A.4 h(·) calibrated parameters for 23-Nov-2006 . . . . . . . . . . . . . . . 59

A.5 g(·) calibrated parameters for 4-Mar-2008 . . . . . . . . . . . . . . . . 59

A.6 h(·) calibrated parameters for 4-Mar-2008 . . . . . . . . . . . . . . . 60

A.7 bmktn,m for 23-Nov-2006 found using Filon-Levy approximation . . . . . 60

A.8 λmktn,m for 23-Nov-2006 found using Filon-Levy approximation . . . . . 60

A.9 Parameters found for (3.63), for swaption expiry 10Y on 23-Nov-2006 60

A.10 Parameters found for (3.64), for swaption expiry 10Y on 23-Nov-2006 60

A.11 Parameters found for (3.63), for swaption expiry 5Y on 4-Mar-2008 . 60

A.12 Parameters found for (3.64), for swaption expiry 5Y on 4-Mar-2008 . 60

iv

Chapter 1

Introduction

Before volatility smiles existed in the interest rate market, the LIBOR market model

[ABM97] was the de facto standard for pricing complex interest rate derivatives.

Since then there has been a considerable amount of work done to create models

that are able to obtain non-monotonic smiles. These models come in a variety of

forms, for example Levy market models [Klu05] or two-state Markov-chain volatility

models [RK03] that take into account the overall regime of a market. The largest

body of work however is in the area of extending the current LIBOR market model

to incorporate stochastic volatility.

Stochastic volatility models also come in various flavours, some assume that the

underlying forward rate volatility follows a CEV process, others assume a displaced-

diffusion process. It is argued in [RRW09] that stochastic volatility LIBOR market

models are fairly troublesome to implement, and there are a number of variables

that allow tweaking of the model that don’t have any real economic meaning.

Another way to obtain non-monotonic smiles has been proposed by Riccardo

Rebonato in [Reb07] and further extended by Rebonato et al in [RRW09]. The idea is

to marry together two industry standard frameworks – the LIBOR market model and

the SABR model [PSHW02]. The SABR model provides analytical approximations

to the true price of European options, it produces stable fitted parameters and it’s

easy to calibrate. The smile dynamics it describes are fundamentally correct as

stated in [RRW09]. The drawback of just using SABR is that it doesn’t take into

account correlation between forward rates and hence views European options in

isolation.

This is where the LIBOR market model comes in. On its own it brought forward

rates under a single measure so that dynamics were valid simultaneously for all

underlyings. Combining it with SABR as in [RRW09] allows both forward rates and

forward rate stochastic volatilities to be brought under the same measure so that

complex derivatives can be priced. The SABR/LMM model now has to describe

many correlation terms such as the forward rate/forward rate volatility correlation

1

structure or the volatility/volatility correlation structure. It can be argued that this

provides too many degrees of freedom.

The aim of this study is to compare and contrast this SABR/LMM model with

a stochastic volatility forward LIBOR model developed by Vladimir Piterbarg in

[Pit03] and [Pit05a]. The model, known in this dissertation as FL-TSS, assumes

that forward rates follow a shifted log-normal diffusion. It models volatility as a

mean reverting process and assumes no correlation between volatility and the rate

dynamics. Using generalised time-dependant parameters, FL-TSS allows for fast and

accurate European option prices. The model links these time-dependent parameters

to effective (constant) parameters that describe the smile for each swaption in the

swaption cube (expiry × maturity × strike). Using this link the model can be

calibrated accurately to swaptions.

We compare the models’ abilities to calibrate to caplets and swaptions in both

normal and excited market conditions (the definition for normal and excited mar-

ket conditions can be found in [RRW09]). We look at how the models reproduce

future implied volatility smiles in different market conditions and we investigate

computational aspects and implementation details of both models.

In Chapter 2 we describe in detail the SABR/LMM model, and in Chapter 3

we do the same for the FL-TSS model. Chapter 4 then describes the tests that are

run and how the tests will be set up. Chapter 5 provides a theoretical comparison

of both models which we use to make predictions on the outcomes of the tests. In

Chapter 6 we analyse and discuss the results, referring back to Chapter 5 to see if

our predictions are correct, and we attempt to explain observations made. Finally in

Chapter 7 we give conclusions based on the results of the tests, and suggest further

research.

1.1 Markets & Pricing of Caplets & Swaptions

A caplet is a European call option on an interest rate, and a floorlet is a European

put. Market participants may pay or receive cash flows based on a floating interest

rate such as LIBOR – caplets or floorlets can be used to hedge the risk on these cash

flows. They’re not traded directly, instead strings of caplets or floorlets are chained

together into products called a cap or a floor. The cap(floor)lets in a cap (floor)

typically have the same strike price but different expiries. In most cases the expiries

match the underlying LIBOR rate, so for example a ten year cap on three-month

LIBOR will generate a portfolio of 39 caplets which will have expiries that range

from six months to ten years in steps of 3 months.

A derivative known as a swap allows two counterparties to exchange cash flows on

instruments they own, where each cash flow is known as a leg. Swaps can be based

2

on different types of instruments such as bonds, interest rates or foreign exchange

rates. Specifically interest rate swaps allow two counterparties to exchange the

interest payments on a fixed rate and variable rate loan.

A swaption is an option on a swap. The swaption expiry denotes when the

option itself expires and the swap starts. The swaption maturity denotes when the

underlying swap makes/receives its final payment from the time it started. They are

often referred to in the format expiry x maturity. So a 5Y x 10Y swaption expires in

5 years and contains an underlying swap which has a maturity of 10 years. Swaptions

can be exercised as European, Bermudan or American and can come in two basic

forms – the payer swaption and the receiver swaption.

Options on the market are quoted by their implied volatility. Hagan et al

[PSHW02] show that the implied Black volatility σ(K,T ) of an option can be found

using

σ(K, f, T ) = A

(z

X(z)

)B (1.1)

A =σT0

(fK)(1−β)2

2

[1 + (1−β)2

24ln2 f

K+ (1−β)4

1920ln4 f

K+ · · ·

] (1.2)

B =

[1 +

((1− β)2

24

(σT0 )2

(fK)1−β +ρβνσT0

4(fK)1−β2

+2− 3ρ2

24ν2

)T + · · ·

](1.3)

z =ν

σT0(fK)

1−β2 ln

f

K(1.4)

X(z) = ln

(√1− 2ρz + z2 + z − ρ

1− ρ

)(1.5)

where K is the strike, T is the expiry, f is the initial forward rate, σT0 is the initial

volatility, ν is the vol-vol of the underlying rate, ρ is the correlation between the

forward price process and the volatility process, and β is the skewness parameter.

1.2 Notations & Definitions

The notations and definitions here will be used throughout this dissertation (un-

less otherwise stated) and describe some of the basic building blocks of both the

SABR/LMM and FL-TSS models.

Take the following tenor structure, where the sequence of N tenors is approxi-

mately equally spaced

0 = T0 < T1 < ... < TN , τi = Ti+1 − Ti (1.6)

3

where τi is the time in between each tenor.

Next consider a forward rate indexed by i at an arbitrary time t denoted by f it .

f it = f(t, Ti, Ti+1), i = 1, 2, ..., N (1.7)

This represents a forward rate at time t resetting at time Ti and then paying at time

Ti + τi where, in the case of spanning forward rates, Ti + τi = Ti+1. The payment is

made upon reset of the next forward rate. Reset times are denoted by (1.6).

Both FL-TSS and SABR/LMM make use of a discrete set of zero coupon bonds

P it for time t indexed by i. Each bond has an associated f it . The numeraire used to

discount the forward rates can be chosen to be the zero coupon bonds upon which

the forward rate is based. This is beneficial as the zero coupon bond does not pay

coupons/dividends and its price is strictly positive. The forward rate written in

terms of its corresponding zero coupon bond is shown in (1.9)

P it = P (t, Ti) (1.8)

f it =

(P it

P i+1t

− 1

)1

τi, τi = Ti+1 − Ti (1.9)

P it can be further defined in terms of just forward rates and τi+1 in

P it =

i−1∏j=0

1

1 + τj+1fjt

(1.10)

Next, the swap rate is defined as the following:

Sn,m(t) =P nt − Pm

t∑mi=n+1 τiP

it

(1.11)

where n is the first fixing date of the swap (and hence the expiry of the swaption)

and m is the last payment date of the swap (hence the maturity of a swaption on

this swap).

All other definitions and assumptions will be stated in the relevant sections.

4

Chapter 2

The SABR/LIBOR market model

2.1 The LIBOR market model

The LIBOR Market Model (LMM) is based on a series of no arbitrage conditions

between discount bonds or forward rates in a deterministic volatility setting. The

model itself depends upon the chosen numeraire, that is, the basic standard by which

instrument values are measured. Constructing these no arbitrage conditions means

eliminating the covariance between the instrument and numeraire – The payoff must

be completely independent of the way the instrument has been discounted so that

one can obtain a price that doesn’t depend on the numeraire that was used. This

follows the argument and definitions laid out in [RRW09].

Here, we use the definitions made in Section 1.2 and use further notation to

describe LMM and SABR/LMM. We start by defining instantaneous volatility which

can be written as

σ(t, Ti) = σit (2.1)

The forward rate i and the forward rate j have an instantaneous correlation which

can be written as

ρ(t, Ti, Tj) = ρti,j i, j = 1, 2, ..., N (2.2)

As discussed in Section 1.2, the numeraire chosen to discount the forward rates f it in

(1.9) is the zero coupon bond in (1.8). These pieces of the framework now need to be

tied together to create an expression describing the evolution of forward rates in a

deterministic volatility setting. It’s important to note that this framework describes

a discrete collection of forward rates referenced by a continuous time index.

df itf it

= µi(ft,σt,ρ, t)dt+ σi(t, Ti)dzit (2.3)

and

5

E[dzitdz

jt

]= ρ(t, Ti, Tj)dt (2.4)

In (2.3) a vector of spanning forward rates and their associated volatilities are de-

noted by ft and σt respectively, and ρ represents the matrix of correlations between

the forward rates. As [RRW09] explains, (2.3) allows the possibility of different

volatility functions for σi(t, Ti) by inclusion of the superscript i. σi allows for the

definition of a different volatility function per forward rate. If, however, the volatil-

ity functions are the same, then the model is said to be time-homogeneous. For

this to be the case the volatility function must be of the form in (2.5) where the i

superscript is dropped.

σ(t, Ti) = σ(Ti − t) (2.5)

This defines σ as being a function of Ti − t. As in [RRW09] the i superscript is

now reintroduced with a different meaning where it represents the dependence on

a specific reset time Ti, rather than representing a specific volatility function as it

denoted previously in (2.1).

σ(Ti − t) = σit (2.6)

This notation emphasises the importance of the expiry Ti in the time-homogeneous

volatility function – it means that for two forward rates, the volatilities are different

only because of their differing times to expiry (the value of (Ti − t)).The drifts µi(ft,σt,ρ, t) that are stated in (2.3) are defined later in Section 2.4.

2.1.1 Volatility functions

[Reb02] and [Reb04] both explain that (2.7) is a good functional form which satisfies

(2.6), where a, b, c and d are constants. There are several justifications for this as

outlined in [RRW09].

σit = (a+ bτi)) exp(−cτi) + d τi = Ti − t (2.7)

(2.7) is square-integrable over the interval 0 and Ti. This is important because it

allows for closed-formed solutions to be found for the integral of its square, which is

used to price caplets and swaptions. The parametrisation also picks out key features

of the volatility curve such as

limτi→0

σit = a+ d (2.8)

As τi goes to zero, a+ d represents the instantaneous volatility of the forward rate.

Looking at the case where τi →∞

6

limτi→inf

σit = d (2.9)

which allows control over the instantaneous volatility for very long expiries, and

shows that the function converges at these high values of τi. By differentiating to

find the maximum of the function, which is found at a value of τi equal to τi, we get

τi =1

c− a

b(2.10)

which shows that the position of the hump in the instantaneous volatility curve can

be controlled by a, b and c. This shows that (2.7) allows two types of volatility

functions to be modelled – humped shape and monotonically decreasing. [Reb02]

and [Reb04] explain that a humped shape is important for normal market conditions

and a monotonically decreasing volatility correctly models excited conditions.

The function (2.7) is able to achieve humped shaped curves and monotonically

decreasing curves by using a, b and c to adjust the location of the peak. A monoton-

ically decreasing curve will have a peak at very short expiries. Note that d doesn’t

effect the position of the peak and therefore only effects the volatility close to 0 and

as τi →∞.

[RRW09] and [Reb04] detail the financial justification for having a humped or

monotonically decreasing instantaneous volatility curve. For very short expiries,

monetary authorities tend to indicate their intentions well before any rate decisions.

This effects short-term deposit rates, and as a consequence pins the prices of futures

contracts. Therefore in these normal market situations, volatility is low as there

aren’t any surprises.

Looking at longer expiries, the driving factor behind changing rates is long-term

inflation. This is also controlled by monetary authorities, and therefore the expec-

tation is that the banks and authorities will be influencing inflation to hit a specific

target. Therefore the most uncertainty in rates for normal market conditions, as

[Reb04] explains, is in the region of 6 to 18 months (hence for expiries in this period,

instantaneous volatilities will be higher).

In excited conditions, the lack of any consensus in the decisions that monetary

authorities are going to make regarding rates will effect the earliest expiring forward

rates, and therefore the volatility for short expiries is high. Hence at the short end

the curve is steep, causing the hump to disappear and generating a monotonically

decreasing curve.

7

2.1.2 Splitting Correlation and Volatility

It is now useful to split the stochastic term in (2.3) into two parts – a term relating

only to the volatility which can be related to caplet prices, and a term relating only

to correlation. It’s possible to rewrite (2.3) as

df itf it

= µi(ft,σt, t)dt+m∑k=1

σikdzk (2.11)

where

E [dzjdzk] = δjkdt (2.12)

This has essentially split the volatility of the Brownian increments up into a series

m factors, where m ≤ N . Another way to look at this is that σik represents the

weight of the kth factor on the ith forward rate. δjk represents a value known as

the Kronecker delta where δjk = 0 for j 6= k and δjk = 1 otherwise.

This relationship between σit and loadings σik for time t (denoted by the addition

of the t subscript in (2.13)) is given by the caplet pricing condition

σit =

√√√√ m∑k=1

(σik,t)2 (2.13)

In order for the caplet prices to be correctly calculated, the implied Black volatilities

σi must be related to the volatility functions σit by∫ Ti

0

(σit)2dt = σ2

i Ti (2.14)

Taking this further, (2.11) needs to be split into a volatility term and a correlation

term. By multiplying and dividing the stochastic term by the volatility σi of the

ith forward rate (notice we have dropped the t subscript and will also be dropping

the t subscript from σik,t because their dependence on t is not important for this

derivation) we get

df itf it

= µi(ft,σt, t)dt+ σim∑k=1

σikdzkσi

(2.15)

By substituting the caplet pricing (2.13) condition into (2.15) the following expres-

sion is obtained

df itf it


σikdzk√m∑k′=1

(σik′)2

(2.16)

By defining bik to be

8

bik ≡σik√m∑k′=1

σ2ik′

(2.17)

we can write

df itf it


bikdzk (2.18)

If we take b to be a matrix of size N x m (remembering that m is the number of

volatility factors and N is the total number of forward rates), where each element

of b is (2.17), [RRW09] states that it can be readily shown that

bbT = ρ (2.19)

where ρ is the same as in (2.3).

Both (2.18) and (2.19) show that there is an expression for ρ in (2.3) that is

independent of volatility, and an expression for σi which has no references to the

correlation. This is important as it allows hedging with caplets. This definition can

now be used to link the implied volatility with caplet prices.

2.1.3 Forward-Forward Correlation

As stated in [RRW09], in a deterministic volatility setting the LMM has never placed

as much emphasis on the correlation structure as it does the volatility function.

When moving to SABR/LMM, more care must be taken to correctly specify the

forward-forward correlation matrix. A simple form for the correlation matrix is

ρ(t, Ti, Tj) = exp[−β|Ti − Tj|] t ≤ min(Ti, Tj) (2.20)

where β is a positive constant, and the ith and jth forward rates have reset times

denoted by Ti and Tj. Expression (2.20) shows that the correlation is a function of

the time distance between forward rate reset times. The further apart they are, the

less correlated they will be. Although this is a desirable feature, the simple model

assumes that 1Y and 2Y forwards will have the same correlation as 25Y and 26Y

forwards. [RRW09] shows using empirical results that this is simply not the case.

We must ensure that the correlation matrix is positive definite and all elements

are between 1 and −1. One way to achieve this is shown in [RRW09] using L

constants with values as where s = 1, 2, ..., L. Given a 5 x 5 matrix, using 4 constants

a1, a2, a3 and a4 all between 1 and −1, it’s possible to form the correlation matrix

9

1 a1 a1a2 a1a2a3 a1a2a3a4

a1 1 a2 a2a3 a2a3a4

a1a2 a2 1 a3 a3a4

a1a2a3 a2a3 a3 1 a4

a1a2a3a4 a2a3a4 a3a4 a4 1

(2.21)

[RRW09] suggests selecting as via the following expression

as = exp(−βs∆T ) (2.22)

This allows complete flexibility in selecting the right correlation shape by choosing

an appropriate βs for each factor in the correlation matrix. An example of a Doust

correlation function that will be used in this dissertation can be seen in Figure 2.1.

It’s clear that when looking down the peak of the surface, it’s convex towards the

front and concave towards the rear. This means that forwards with shorter reset

times will be more correlated than forwards with longer reset times.

[RRW09] shows that this is what empirical evidence indicates.

Figure 2.1: A Doust correlation surface that shows convexity

2.2 The SABR Model

The SABR model is described fully in [PSHW02] and here we will state results and

cover the key aspects which are useful to this study.

10

Given a forward rate f it , the SABR model defines its dynamics as

df it = (f it )βiσitdz

Qit (2.23)

dσitσit

= νidwQit (2.24)

EQi [dzitdwit] = ρi (2.25)

Here νi,βi and ρi are constants and are not functions of time. They are all specific

to a forward rate and hence have been given the superscript i. Once f i0 and σi0 are

added to expressions (2.23) – (2.25), the CEV model with stochastic volatility is

fully specified.

The forward rate is working under the terminal measure Qi. Under this measure

both the volatility and the forward rate have no drift (is a martingale). This is the

case when looking at forward rates in isolation, however a different forward rate

and it’s volatility under this same measure would not be driftless. These drifts are

calculated when deriving the SABR/LMM model. We should also note that dzQi

t

and dwQit are increments of Brownian motion in the Qi measure.

Within the SABR model framework, it’s impossible for forward rates to interact

with each other. This means that the payoff for a path-dependent option cannot be

calculated using SABR. Each forward rate works in its own measure, and therefore

the dynamics of a yield curve are unable to be described by this model.

The following bullets briefly describe the purpose of the 3 constants defined

in this model. The i superscript is dropped as the same definition applies to the

constants for all forward rates.

β : Is an exponent that [RRW09] refers to as chosen by the market. It has 3

common effects on the volatility smile

• As β goes from 0 to 1, a steepening of the smile can be observed.

• As β increases, the level of the smile is lowered.

• The curvature is increased as β goes from 0 to 1.

ρ : Is the correlation between the forward rate process and the volatility process.

As ρ is reduced, the smile obtains a more negative slope. It also has the small

effect of decreasing its curvature.

ν : It is observed that increasing ν increases the curvature of the smile.

SABR is modelled on a stochastic CEV process which has the specific advantage

that negative rates are avoided. However this creates subtle issues for low rates and

high volatilities. [RRW09] shows the need to create a zero rate as an absorbing

11

barrier and this has implications on the choice of β in the CEV process – once a

rate hits zero, it stays at zero.

2.3 The SABR/LMM Model Definition

The complete SABR/LMM model is displayed here. Given that i = 1, 2, ..., N , the

joint dynamics for N forward rates and their instantaneous volatilities are defined

as

df it = µit(f , s,ρ)dt+ (f it )βisit

NF∑j=1

bijdzj (2.26)

dsit = g(t, Ti)dkit (2.27)

dkitkit

= µkit (f , s)dt+ hit

NV∑j=1

cijdwj (2.28)

NV and NF represent the number of factors driving the volatility and the forward

rates respectively where NF ≤ N and NV ≤ N .

sit is the volatility process where g(t, Ti) is the instantaneous volatility. βi is

defined in the SABR model and is a constant value for a given forward rate. bij is

defined in (2.17) and is the correlation between forward rates. ρ is the same value

as defined in (2.19).

kit is the stochastic process driving the volatility, and hit represents the volatility

of volatility function. cij is the correlation between the instantaneous volatilities.

µit(f , s) and µkit (f , s) both represent the fact that no-arbitrage forward rate drifts

and no-arbitrage volatility drifts are both dependent on all of the forward rates and

volatilities. The model specification is completed by setting

NF∑j=1

b2ij = 1 (2.29)

NV∑j=1

c2ij = 1 (2.30)

(2.31)

and defining

E[dzjdzk] = δjkdt (2.32)

E[dwjdwk] = δjkdt (2.33)

E[dzjdwk] = xjkdt (2.34)

12

where dzj and dwk are independent Brownian increments and δjk is the Kronecker

delta as defined earlier. xjk is correlation between the forward rate and volatility

Brownian increments.

These definitions now mean that each forward rate f it will have an instanta-

neous CEV volatility sit. It also means that each volatility factor kit will have an

instantaneous log-normal volatility hit.

2.4 The SABR/LMM Model for Caplets

Here we present a model for caplets which re-expresses the SABR model for f it

under the SABR/LMM framework. The forward rate resets at time Ti and pays at

Ti+1 = Ti + τi as defined previously. The forward rate dynamics are specified under

its terminal measure Ti as

df it = (f it )βisitdz

it (2.35)

dsit = gitdkit (2.36)

dkitkit

= µikdt+ hitdwit (2.37)

E[dzitdz

jt

]= ρijdt |ρij| ≤ 1 (2.38)

E[dwitdw

jt

]= rijdt |rij| ≤ 1 (2.39)

E[dwitdz

jt

]= Rijdt |Rij| ≤ 1 (2.40)

Here βi is the same as in the SABR framework. sit and kit are both defined in Section

2.3. µik is the volatility drift term and is non-zero for any arbitrary numeraire for

all forward rate volatilities.

ρij is the correlation between forward rates i and j. rij is the correlation between

forward rate volatilities, and Rij is the correlation between forward rates and forward

rate volatilities. However, caplets only depend on a single forward rate, its volatility,

and the correlation between its forward rate and its forward rate volatility.

git = g(Ti − t) and hit = h(Ti − t) are time-homogeneous. Their only dependence

is the time left until expiry. [RRW09] explains that this is an important property

for normal market conditions due to the belief that, in this situation, the market is

self-similar and autonomous.

13

What about periods of market stress? Both SABR and SABR/LMM assume

that the market is in one of two states – excited or normal and the market can be

described only in one of these states at a time. The model doesn’t support regime

switching between pricing of options with different expiries. In times of market

excitement short expiry volatilities will be high and medium expiry volatility will

be higher than normal. If this model state is then applied in a normal market, short

expiry volatilities will be a lot higher than expected, the hump in the volatility curve

will be closer to lower expiries. This is a shortcoming of time-homogeneous models.

Riccardo Rebonato and Dherminder Kainth propose a solution to this problem,

described further in [RK03], which suggests having a two stage Markov-chain with

a transition probability that switches the model from one regime into another.

The same time-homogeneous functional form described in (2.7) is a good defi-

nition for the functions g(·) and h(·) above for the reasons discussed in Subsection

2.1.

g(τi) = (a+ bτi) exp(−cτi) + d, τi = Ti − t (2.41)

and

h(τi) = (α + βτi) exp(−γτi) + δ (2.42)

Completing the definition, using the time-homogeneous forms above, the process for

kTt and sTt can be written as

kTt = kT0 exp

[∫ t

0

(−1

2h(T − s)2ds+ h(T − s)dws

)](2.43)

and

sTt = gTt kTt = gTt k

T0 exp

[∫ t

0

(−1

2h(T − s)2ds+ h(T − s)dws

)](2.44)

2.5 The SABR/LMM Model for Swaptions

The forward rate dynamics for the full SABR/LMM model is presented here.

df it = µitdt+ (f it )βisitdz

it (2.45)

dsit = gitdkit (2.46)

dkitkit

= µkit dt+ hitdwit (2.47)

where the following expectation are repeated again here for ease of reference:

14

E[dzitdz

jt

]= ρijdt (2.48)

E[dwitdw

jt

]= rijdt (2.49)

E[dzitdw

jt

]= Rijdt (2.50)

As previously stated i and j act as indices to the appropriate forward rates, dz is the

process driving the forward rate and dw is the process driving the forward volatility

and ρ, r and R are part of the super correlation matrix P .

P =

[ρ RRT r

](2.51)

The price of a caplet depends on the following 3 things – A single forward rate, its

volatility and the correlation between the forward rate and its volatility.

Swaptions however depend on multiple forward rates and multiple volatilities,

therefore the model needs to capture the correlations between these forward rates

(ρij) and the correlation between these volatilities (rij). Correlations across forward

rates and volatility also need to be captured (Rij where i 6= j). SABR market prices

of caplets provide data for the diagonal of Rij.

The matrices ρ, r and R can be parametrised in the following way as discussed

extensively in [RRW09]

ρij = η1 + (1− η1) exp[−λ1(|Ti − Tj|)] (2.52)

rij = η2 + (1− η2) exp[−λ2(|Ti − Tj|)] (2.53)

Rij = sign(Rii)√|RiiRjj| exp

[−λ3(Ti − Tj)+ − λ4(Tj − Ti)+

](2.54)

where η1, η2, λ1, λ2, λ3 and λ4 are all constants.

Next recall the definition for the swap rate Sn,m(t) in (1.11) with expiry n and

maturity m. [RRW09] assumes the SABR dynamics of Sn,m(t) are

dSn,m(t) = Sn,m(t)BΣtdZt (2.55)

dΣt

Σt

= V dWt (2.56)

E[dZtdWt] = RSABRdt (2.57)

If we are able to find the initial values in the dynamics above, then we can use the

SABR model to give prices of swaptions.

15

2.6 Implementing SABR/LMM

The following subsections explain how to calibrate g(·) and h(·) for caplets. Then

they explain how to approximate the four SABR values for swaption based prices

(shown in (2.55) and (2.57)) in terms of the forward rate parameters git(τi), βi, the

super correlation matrix P , hit(τi) and f i0 – for i = n, ...,m − 1. The four SABR

values are:

• The exponent B

• Σ0, the initial swap rate volatility

• RSABR the correlation between the swap rate and the volatility of the swap

rate

• V the vol-vol of the swap rate

2.6.1 Calibrating g(·)

In order to find the parameters for the deterministic function in (2.41), we have to

determine an equivalent value in the SABR model to calibrate to. To do so we write

the caplet pricing equation (2.14) as

σ2TT = (kT )2

∫ T

0

g(T − u)2du (2.58)

where the T superscript on k implies taking the forward rate volatility (k) for the

forward rate with expiry T . By defining the root mean square of g(·) as

g(T ) =

√1

T

∫ T

0

g(T − u)2du (2.59)

Then (2.58) can be rewritten as

kT =σTg(T )

(2.60)

Expression (2.60) doesn’t hold in a stochastic volatility situation, so in order to

determine kT0 it’s important to note that

E[σ(t)] = σ0 (2.61)

The reason for this is because, for SABR, the stochastic volatility is equal to

σ(t) = σ0 exp

[1

2

∫ t

0

σ2sds+

∫ t

0

σ(s)dzs

](2.62)

So for (2.61) to be true, we must have

16

E[exp

(1

2

∫ t

0

σ2sds+

∫ t

0

σ(s)dzs

)]= 1 (2.63)

Hence it follows that g(·) should be chosen such that it matches as closely as possible

σi0 at time t = 0 for the SABR volatility σis. Therefore the sum of the squared

discrepancy X2 should be minimised

X2 =N∑i

[σi0 − g(Ti)

]2(2.64)

Where the minimisation is over all caplet expiries and the g parametrisation is

g(Ti) =

√1

Ti

∫ Ti

0

[(a+ bτi) exp(−cτi) + d]2dτi (2.65)

2.6.2 Calibrating h(·)

Note here that the superscript T or Ti denotes a parameter for a specific forward

rate with reset time T or Ti.

As stated in [RRW09] it was previously suggested that the volatility of the volatil-

ity hTt could be parametrised as (2.42) and could simply be calibrated to νTi by

minimising its squared difference with the root mean squared value of hTt . Where

the root mean square of ht is defined as

ht =

√1

T

∫ t

0

(hs)2ds (2.66)

[RRW09] shows that it is incorrect to calibrate ht in this way because it actually

matters when the vol-vol occurs. To say when the vol-vol occurs means that it

makes a difference to the option price when high or low periods of vol-vol occur.

The terminal distribution of the forward rates isn’t uniquely determined by the

terminal volatility or the average vol-vol. For instance if the volatility is low and the

vol-vol is high, [RRW09] suggests that this stochasticity goes to waste. For another

example, imagine a flat volatility, a high concentration of vol-vol would essentially

add curvature at the point where it occurs, creating skew and having a small effect

on the option price.

Chapter 5 in [RRW09] shows a different way to match SABR and SABR/LMM

parameters. The matching of the two models assumes that, since the same errors

are made in both approximations, then the errors will cancel out to give accurate

relationships. The proof detailed in [RRW09] yields the expression

νTSABR =k0

σ0T

(2

∫ T

0

g(t)2h2t tdt

)1/2

(2.67)

17

To help with the implementation of this calibration, (2.67) can be modified to

νTSABR =1

g(T )T

(2

∫ T

0

g(t)2h2t tdt

)1/2

(2.68)

Looking at (2.68) it’s clear that in the integral, the time when the vol-vol occurs

is taken into account. This allows the implementation to control when the vol-vol

occurs so that an accurate representation of νTSABR can be made.

2.6.3 Selecting β and ρ

Specifically for caplets it can be assumed that ρLMM ≈ ρSABR. The exponent β is a

parameter which is chosen by the market and it is therefore reasonable to assume

that βLMM = βSABR.

2.6.4 Approximating Σ0 and V

Approximations for Σ0 and V both rest on an approximation of the instantaneous

swap rate volatility St. This can achieved by using a freezing strategy described

in [Reb02] and [RRW09] where further details and derivations can be found. The

resulting expression is

St =

√ ∑k,m=1,nj

W 0kW

0ms

kt smt ρk,m (2.69)

where

W tk = ωk

(fkt )βk

(Sn,m)B(2.70)

and the swap-rate between two points n and m is denoted by Snm and defined in

(1.11). ωi represents a weighting for a particular forward rate i.

ωi(t) =τiP

i+1t∑m−1

i=n τiPi+1t

(2.71)

where τi is the time spacing between forward rates. In this thesis, 6M LIBOR

forwards are used, therefore τi = 0.5. Using these definitions, the initial swap rate

volatility is defined by (2.72) and the volatility of volatility is defined by (2.73).

Again, the proof of these can be found in [RRW09].

Σ0 =

√√√√ 1

T

∑i,j

(ρijW 0

i W0j k

i0k

j0

∫ T

0

gitgjtdt

)(2.72)

V =1

Σ0T

√√√√2∑i,j

(ρijrijW 0

i W0j k

i0k

j0

∫ T

0

gitgjt hij(t)

2tdt

)(2.73)

18

2.6.5 Approximating B

B represents the swap rate exponent. It is defined by

B =∑k=1,nj

ωkβk (2.74)

[RRW09] suggests that reason for this is heuristic because the sum of CEV values

with exponent β is generally not a CEV value with the same exponent. However

[RRW09] suggests that for a log-normal case this approximation is generally good,

and for the normal case it’s exact. This suggests that for a value of CEV β between

0 and 1 the approximation should be as good as the log-normal case, and it will get

better as β → 0.

2.6.6 Approximating RSABR

RSABR is the correlation between the swap-rate and its volatility. Proof can be

found in [RRW09] that

RSABR =∑i,j

ΩijRij (2.75)

where

Ωij =2ρijW

0i W

0j k

i0k

j0

∫ T0gigjhij(t)

2tdt

(V Σ0T )2(2.76)

19

Chapter 3

Piterbarg’s term structure of skewforward LIBOR model

3.1 Introduction

The log-normal forward LIBOR model was the first model that made it easy to

calibrate ATM swaption volatilities for all expiries and maturities. It became the

de facto standard, but (as mentioned in the introduction) a major issue was that it

was unable to reproduce the volatility smile – all swaptions of the same maturity

and expiry but of different strikes had the same Black volatility – which isn’t what

is seen in the market.

There have been many attempts to encode volatility skew information, but the

major building block of Piterbarg’s model has been the Stochastic Volatility Forward

LIBOR model (from here on known as FL-SV). [AA02] and [ABR01] both present

the FL-SV model.

Its main features are the application of a local stochastic volatility function,

which is time-independent, that is applied to all forward LIBOR rates. Higher

volatility of this stochastic variance process means greater curvature of the volatility

smile. This stochastic volatility component is the same for all LIBOR forwards

regardless of its maturity. This leads to the issue that FL-SV generates volatility

smiles with very similar curvatures which again is not consistent with the market,

therefore FL-SV is unable to match all volatility smiles for the whole swaption grid.

In [Pit05a] Piterbarg proposes a new model which from here on will be called

FL-TSS (term structure of skews). This model allows skews to vary across forward

LIBOR rates and times, however the same stochastic volatility process is used for

all forwards. Piterbarg explains that it is possible to use swaption-specific volatility

drivers, however he argues in [Pit05a] that this is not necessary.

In [Pit05a] Piterbarg goes on to relate time-dependent volatilities and skews to

model term volatilities and skews. This relation is important as it provides a direct

20

relationship between the time-dependent slope of a local volatility function and the

total amount of skew that the model itself produces.

3.2 A stochastic volatility forward LIBOR model

for European Swaptions

Here we use the definitions in Section 1.2, in particular that of the swap rate Sn,m(t)

in (1.11). Reminding the reader that n is the swaption expiry, m is the swaption

maturity and i is an index to an arbitrary forward rate. Tn is therefore the first

fixing date and Tm is the last payment date.

Looking at the swap rate measure Qn,m, equation (3.1) is the numeraire. Under

this measure the swap rate Sn,m(t) is a martingale, and Brownian motions under

this measure are denoted by dW n,m.

Nt =m∑

i=n+1

τiPit τi = Ti+1 − Ti (3.1)

where P it is defined by function (1.8). Under this swap rate measure, Qn,m, a

European swaption can be seen as a European option on the underlying rate Sn,m.

The price of the swaption is

Swaptionn,m(t) = NtE[

(Sn,m(Tn)−K)+

NTn

](3.2)

where K is the strike.

3.2.1 A simple model

A possible method to price Swaptionn,m is to only look at the dynamics of Sn,m. The

model is coined “simple” because it only describes the term distribution of a single

swap rate, and not the evolution of the rate curve as a whole. Here it’s modelled as

a displaced diffusion process with stochastic volatility.

Piterbarg’s paper [Pit05a] follows the method used in [AA02] which is to describe

the distribution of the swap rate with 3 parameters:

• b: The swap rate skew

• λ: The stochastic volatility

• η: The volatility of variance

The swap rate is expected to follow the process

21

dSn,m(t) = λ(bSn,m(t) + (1− b)Sn,m(0))√z(t)dW n,m

t

dz(t) = θ(z0 − z(t))dt+ η√z(t)dVt

z(0) = z0

〈dV, dW 〉 = 0 (3.3)

The dynamics of the swap rate through time are described by this model, however

the only relevant aspect is its terminal distribution. dVt is a scalar Brownian motion

that is also working under the swap rate measure Qn,m. The zero correlation is there

to maintain analytic tractability and to ensure that the distribution of z(t) is the

same under all annuity measures. To ensure z(t) > 0 we make the requirement that

θ, z0, η > 0.

The model is calibrated to each swap rate Sn,m which results in a grid of triplets

which is generally referred to as the swaption grid. Each Sn,m has associated with

it a triplet λn,m, bn,m, ηn,mNn,m=1 where the relevant grid points are at n+m < N .

Since the model is calibrated to each smile across all strikes, expiries and maturities,

the swaption grid encodes all available market information in each triplet grid point.

Taking a closer look at the parameters of the model, [Pit05a] states that λ is

similar to, but not equal to the Black volatility. They do become equal, however,

when η = 0 and b = 1; η controls the curvature of the smile and b controls the slope

specifically of the at-the-money volatility smile.

θ is a global parameter and is defined as the strength of mean reversion of

variance. It controls how fast the smile flattens out as time increases. A good choice

of θ means that ηn,m can be chosen as constant η, therefore rather than having

triplets for each grid point, they will be tuples λn,m, bn,m with constant η.

The FL-TSS model developed later forms a relation between these parameters

and time dependent skew and volatility, which is why these parameters are impor-

tant. They will be used for calibration of the resulting FL-TSS model.

3.2.2 A forward LIBOR market model with stochastic volatil-ity

The “simple” model only describes each swap rate separately, but many exotic

interest rate derivatives require the full term structure to be present in order to

price and adequately risk-manage. Here the LIBOR market model with stochastic

volatility (FL-SV) will be formally introduced.

A definition for spanning LIBOR rates is first constructed using a zero coupon

bond with price P it at time t and pays $1 at time Ti. The same definitions in Section

22

1.2 for the forward rate f it (indexed by i) and zero coupon bonds are used here. We

model the LIBOR rates under some measure P which is further detailed below.

The dynamics of the spanning LIBOR rates can be described as

dfnt = (βfnt + (1− β)fn0 )√z(t)

K∑k=1

σk(t;n)(√

z(t)µk(t;n) + dW kt

)(3.4)

n = 1, ..., N − 1

with the stochastic variance process z(t) defined by

dz(t) = θ(z0 − z(t))dt+ η√z(t)dVt (3.5)

z(0) = z0

Here:

• (µk(t;n), t ≤ Tn)k=1:Kn=1:N−1 are K-dimensional drifts that are specific to the mea-

sure P and ensures that there is no arbitrage present in the model. In this

forward measure P under numeraire P n+1t we know that µk(t;n) = 0;

• dWt = (dW 1t , ..., dW

Kt ) is a K-dimensional Brownian motion which is in the

forward measure P and is completely independent of dVt;

• (σk(t;n), t ≤ Tn)k=1:Kn=1:N−1 are the instantaneous volatility functions;

Under the FL-SV model, Sn,m approximately follows the dynamics

dSn,m(t) = σ(t)(βSn,m(t) + (1− β)Sn,m(0))√z(t)dUt (3.6)

where σ(t) is a volatility function dependent on time and Ut is a Brownian motion

under the swap rate measure Qn,m. Although this looks very similar to (3.3) it’s

crucial to remember that (3.3) is for a specific swap rate, therefore β and η in (3.6)

are global parameters which means all swap rates will have the same values.

It may be realistic to assume η is the same for all swap rates given a good choice

of θ, but it’s not realistic to assume β is the same for all swap rates. This means

that FL-SV cannot reproduce volatility smiles for all maturities and expiries. The

model needs to be improved to take into account the change of the swaption skews

between different maturities and expiries.

23

3.3 Piterbarg’s term structure of skew LIBOR

market model

To give some intuition behind this, the main goal is to make the FL-SV model more

flexible so that the changeability of swaption skews are accounted for. This is done

by assuming a time-dependent skew which is implemented by taking (β(t;n), t ≥0)N−1

n=1 .

3.3.1 The FL-TSS model set-up

The dynamics of FL-TSS given this new time dependent β is

dfnt = (β(t;n)fnt + (1− β(t;n))fn0 )√z(t)

K∑k=1

σk(t;n)(√

z(t)µk(t;n) + dW kt

)(3.7)

n = 1, ..., N − 1

The dynamics of dfnt are in the forward measure for which P n+1t is the numeraire,

therefore (3.7) can be simplified. fnt is a martingale under the forward measure

hence the drift µk(t;n) becomes zero yielding the expression

dfnt = (β(t;n)fnt + (1− β(t;n))fn0 )√z(t)

K∑k=1

σk(t;n)dW k,n+1t (3.8)

n = 1, ..., N − 1

In [Pit05a] Piterbarg develops an approximation to European option values that is

accurate and speedy to calculate which helps with calibration of the FL-TSS model.

Calibration involves calibrating to the market-inferred parameters λmktn,m, b

mktn,m. This

means the model parameters λmodn,m , b

modn,m need to be estimated, therefore a relation

to connect them to the time dependent β(t) and σk(t) in (3.8) is required.

The first step is to derive an approximation for the dynamics of the swap rate

Sn,m(t) under the swap measure Qn,m that causes Sn,m(t) to be a martingale. As

defined in Section 1.2 n is the expiry of the swaption and m is the maturity. The

dynamics are stated as

dSn,m(t) = (β(t;n,m)Sn,m(t) + (1−β(t;n,m))Sn,m(0))√z(t)

×K∑k=1

σk(t;n,m)dW k,n,mt (3.9)

where

24

σk(t;n,m) =m∑

i=n+1

qi(n,m)σk(t, i) (3.10)

β(t;n,m) =m∑

i=n+1

pi(n,m)β(t, i) (3.11)

qi(n,m) =f i0

Sn,m(0)

∂Sn,m(0)

∂f i0(3.12)

pi(n,m) =

∑k σk(t; i)σk(t;n,m)∑

k σ2k(t;n,m)

(3.13)

where σk(t;n,m) are the volatility factors for the swap rate Sn,m, and β(t;n,m) is

a time dependent skew for a swap rate.

It’s interesting to note here that the swap rate approximation (3.9) is of the same

form as the SDE for LIBOR (3.8). It’s also similar to the simple model (3.3), with

the only difference being the time-dependent skew.

The proof for (3.9) is as follows. We can ignore all dt terms since Sn,m(t) is a

martingale under the swap measure Qn,m. Writing the SDE for dSn,m(t)

dSn,m(t) =m−1∑i=n

∂Sn,m(t)

∂f itdf it

=m−1∑i=n

∂Sn,m(t)

∂f it(β(t; i)f it + (1− β(t; i))f i0)

√z(t)

K∑k=1

σk(t; i)dWk,n,mt

(3.14)

The aim of this proof is to write (3.14) in the autonomous form that’s shown in

equation (3.9). To allow us to go from (3.14) to (3.9) we need to impose 2 conditions.

Firstly the R.H.S should agree with each other along the forward path

Sn,m(t) = Sn,m(0) (3.15)

f it = f i0 (3.16)

and secondly we must ensure that the slopes of the R.H.S agree. Using the first

condition we get

m−1∑i=n

f t0∂Sn,m(0)

∂f i0

(K∑k=1

σk(t; i)dWk,n,mt

)= Sn,m(0)

K∑k=1

σ(t;n,m)dW k,n,mt (3.17)

By defining qi to be (3.12) then for each k = 1, ..., K the standard swaption volatility

approximation weights can be expressed by (3.10). Substituting (3.10) into (3.9) we

get

25

dSn,m(t) = (β(t;n,m)Sn,m(t) + (1− β(t;n,m))Sn,m(0))√z(t)

×m−1∑i=n

qi

K∑k=1

σk(t; i)dWk,n,mt (3.18)

We now differentiate the R.H.S of (3.18) with respect to the forward rate f jt . This

gives us

∂

∂f jt

((β(t;n,m)Sn,m(t) + (1− β(t;n,m))Sn,m(0))

√z(t)

×m−1∑i=n

qi

K∑k=1

σk(t; i)dWk,n,mt

)

= β(t;n,m)∂Sn,m(t)

∂f jt

√z(t)

m−1∑i=n

qi

K∑k=1

σk(t; i)dWk,n,mt (3.19)

Now differentiating the R.H.S of (3.14) with respect to f jt we get

∂

∂f jt

(m−1∑i=n

∂Sn,m(t)

∂f it(β(t; i)f it + (1− β(t; i))f i0)

√z(t)

K∑k=1

σk(t; i)dWk,n,mt

)

=m−1∑i=n

∂2Sn,m(t)

∂f jt ∂fit

(β(t; i)f it + (1− β(t; i))f i0)√z(t)

K∑k=1

σk(t; i)dWk,n,mt

+∂Sn,m(t)

∂f jtβ(t; j)

√z(t)

K∑k=1

σk(t; j)dWk,n,mt (3.20)

We can now equate (3.19) and (3.20) along the path of the forward. By ignoring

the second-order terms we get

∂Sn,m(0)

∂f j0β(t;n,m)

√z(t)

m−1∑i=n

qi

K∑k=1

σk(t; i)dWk,n,mt

=∂Sn,m(0)

∂f j0β(t; j)

√z(t)

K∑k=1

σk(t; j)dWk,n,mt

j = 1, ..., N (3.21)

We can therefore cancel ∂Sn,m(0)

∂fj0

√z(t) on both sides. By using (3.10), and by equat-

ing the diffusion coefficients for each W k,n,mt on both sides of (3.21) we get

β(t;n,m)σk(t;n,m) = β(t; j)σk(t; j) (3.22)

26

where

j = 1, ..., N

k = 1, ..., N

We’re not able to solve (3.22) for all j, so a least-squares solution is formulated

∑j,k

(β(t;n,m)σk(t;n,m)− β(t; j)σk(t; j)

)2 → min (3.23)

This is then solved to yield (3.11).

3.4 Pricing European swaptions via parameter av-

eraging techniques

The next step is to now link the parameters in (3.9) to the models specified earlier.

If expressions can be found that link σk(t; i) and β(t; i) to the effective constant

λn,m and bn,m, then fittings can be done directly to the market, instead of having to

perform inverse or direct option valuations during calibration.

Looking back at equation (3.9), each swap rate follows an SDE of the same

form as (3.24) under it’s own appropriate measure. Where dU is the infinitesimal

increment of Brownian motion. Since we’re looking at the form of an SDE for a

specific forward rate, the n and m indices are dropped.

dS(t) = σ(t)(β(t)S(t) + (1− β(t))S(0))√z(t)dUt (3.24)

The following two steps show how to approximate 3.24 by using constant parameters.

dS(t) = σ(t)(β(t)S(t) + (1− β(t))S(0))√z(t)dUt

⇓ β(t) replaced with an average skew b

dS(t) = σ(t)(bS(t) + (1− b)S(0))√z(t)dUt (3.25)

⇓ σ(t) replaced with an average volatility λ

dS(t) = λ(bS(t) + (1− b)S(0))√z(t)dUt (3.26)

(3.24) is first approximated by (3.25) which still has time-dependent volatility but

a constant “effective” skew. Then (3.25) is approximated by (3.26) which has both

a constant “effective” volatility and a constant “effective” skew.

27

3.4.1 Effective Skew

The effective skew for (3.24) is the average parameter b over [0, T ] and is denoted

below.

bn,m =

∫ Tn

0

β(t;n,m)ωn,m(t)dt (3.27)

n,m = 1, ..., N − 1

where ω(t) represents a series of weightings given by

ω(t) =υ2(t)σ2(t)∫ t

0υ2(u)σ2(u)du

(3.28)

and

υ(t) = z20

∫ t

0

σ2(u)du+ z0η2e−θt

∫ t

0

σ2(u)eθu − e−θu

2θdu (3.29)

All of the terms in (3.29) are represented in (3.3) and all refer to the process (dz)

driving the volatility. Equations (3.10) to (3.13) along with the above fully represents

everything needed to calculate and calibrate the effective skew – hence it provides

a connection between β(t;n), t ≥ 0n in model (3.8) to bn,m in the simple model

(3.3). The full proof of this can be found in [Pit05a].

3.4.2 Effective Volatility

Subsection 3.4.1 shows how to approximate the SDE (3.24) with a constant skew

parameter b, however there is still a time-dependent volatility component present –

see (3.25). With the process (3.25) in mind, in order to derive a constant effective

volatility parameter, Piterbarg changes the representation of the European option

price to the integral of a deterministic function and the stochastic z(t) process

E[(S(T )− S0)+

]= E

[E[(S(T )− S0)+|z(.)

]](3.30)

Note that this derivation applies to options at-the-money (i.e. where K = S0).

The underlying Brownian motions that drive both z(t) and S(T ) are independent,

therefore S(T ) conditioned on a specific path of z(t)Tt=0 is shifted log-normal. The

expectation inside (3.30) can be easily evaluated.

E[(S(T )− S0)+

]= E

[g

(∫ T

0

σ2(t)z(t)dt

)](3.31)

where

28

g(x) =S0

b

(2N(b

√x/2)− 1

)(3.32)

N(y) =1√2π

y∫−∞

e−y2/2dt (3.33)

Remember (3.31) is the call option price for the process (3.25). Going through

the same procedure for (3.26) to change the representation of the option price, and

equating to (3.31), both processes are assumed to yield the same option prices at-

the-money, the following expression is obtained

E[g

(∫ T

0

σ2(t)z(t)dt

)]= E

[g

(λ2

∫ T

0

z(t)dt

)](3.34)

The procedure to find the effective variance would be to find λ in (3.34), however

as they are, the expectations are very difficult to calculate. The Laplace transform

of∫ T

0σ2(t)z(t) is however easy to calculate numerically (see Appendix E of [Pit03]

for the details). Application of the transform yields the following expressions.

ϕ0

−g′′(z0

∫ Tn0σ2n,m(t)dt

)g′(z0

∫ Tn0σ2n,m(t)dt

) λ2n,m

= ϕ

−g′′(z0

∫ Tn0σ2n,m(t)dt

)g′(z0

∫ Tn0σ2n,m(t)dt

) (3.35)

where

σ2n,m(t) =

K∑k=1

σ2k(t;n,m) (3.36)

n,m = 1, ..., N − 1

ϕ0(µ) =E[exp

(−µ∫ T

0

z(t)dt

)](3.37)

ϕ(µ) =E[exp

(−µ∫ T

0

σ2(t)z(t)dt

)](3.38)

As explained in [AA02] and [Pit05a], the function ϕ(µ) for process z(·) can be

represented as

ϕ(µ) = exp(A(0, T )− z0B(0, T )) (3.39)

Both A(t, T ) and B(t, T ) satisfy a system of Riccati ODEs

A′(t, T )− θz0B(t, T ) = 0 (3.40)

B′(t, T )− θB(t, T )− 1

2η2B2(t, T ) + µσ2(t) = 0 (3.41)

29

with the terminal conditions

A(T, T ) = 0 (3.42)

B(T, T ) = 0 (3.43)

These ODEs are easy to solve numerically. ϕ0(µ) satisfies the same Riccati ODEs

above with σ(t) ≡ 1. For this case it can be solved explicitly

ϕ0(µ) = exp(A0(0, T )− z0B0(0, T )) (3.44)

A0(0, T ) =2θz0

η2log

(2γ

θ + γ(1− e−γT ) + 2γe−γT

)− 2θz0

µ

θ + γT (3.45)

B0(0, T ) =2µ(1− e−γT )

(θ + γ)(1− e−γT ) + 2γe−γT(3.46)

γ =√θ2 + 2η2µ (3.47)

3.5 Calibration

The first step in the calibration process is to parametrise the market implied volatil-

ity to the parameters λmktn,m and bmkt

n,m. Once this is done, expressions (3.27) and (3.35)

can then be calibrated to these market parameters to obtain β(t;n) and σk(t;n).

3.5.1 Finding λmktn,m and bmkt

n,m

Recalling (3.3), the “Simple Model” follows the swap rate under the swap measure

with the following dynamics.

dSn,m(t) = λ(bSn,m(t) + (1− b)Sn,m(0))√ztdW

n,mt

dzt = θ(z0 − zt)dt+ η√ztdVt

0 = 〈dV, dW 〉 (3.3 revisited)

Under this model the expectation C0 = EQn,m [(ST −K)+] needs to be computed.

C0 represents the value of the swaption – quoted in the market as the implied Black

volatility, and the expectation is taken in the swap rate measure Qn,m. This can be

written as

C0 =1

bEQn,m [(ζT −K ′)+

](3.48)

Where

30

ζt = bSt + (1− b)S0

K ′ = bK + (1− b)S0

b > 0 (3.49)

Based on this transformation, an SDE can be derived for the dynamics of ζt.

dζt = ζ√ztdWt (3.50)

dzt = θ(z0 − zt)dt+ η√ztdVt (3.51)

Where

z0 = (bλ)2z0

η = |b|λη (3.52)

By making the changes above, (ζt, zt) is seen to follow the dynamics of the Heston

model. Therefore it is possible to use the characteristic function of the log spot

which is known, along with transformation based equations. In this instance the

Lewis formula (see [Lew01]) can be used.

C0 =1

b

[ζ0 −

√ζ0K ′

π

∫ ∞0

<(eiu log( ζ0K′ )φT

(u− i

2

))du

u2 + 14

](3.53)

where

φT (u− i

2) = E

[ei(u− i

2) log

(ζTζ0

)](3.54)

Since there is no correlation between V and W φT is in R. This leads to the following

solution for φT .

φT (u− i

2) = exp(A(0, T ) +B(0, T )z0) (3.55)

In (3.55), A(t, T ) and B(t, T ) satisfy the Riccati equations

A′(t, T ) = −κz0B(t, T )

B′(t, T ) = κB(t, T ) +1

2(u2 +

1

4)σ2(t)− 1

2η2B2(t, T )

A(T, T ) = B(T, T ) = 0

(3.56)

There are however several numerical issues with this approach

31

• When K 6= S0 the integrand in (3.53) is oscillatory and the frequency of the

oscillations increase as the strike K moves further from the forward S0. This

has the effect of causing large numerical errors when pricing options that are

away from the money.

• Expression (3.53) is not defined at b = 0, and there could be a significant loss

of accuracy for small values of b.

• Since the integral in (3.53) is over the infinite domain, it needs to be truncated.

This must be done such that errors introduced are an acceptable size.

Andrew Dickinson in [Dic11] shows an alternative way to price options under a

displaced-lognormal Heston model. From this point on in this dissertation, the

model in [Dic11] will be known as the Filon-Levy option pricing model. This model

is used to generate a grid of λn,m and bn,m for the swaption matrix that FL-TSS

will be calibrated to as described in Subsection 3.5.2. The code that was written

to implement the Filon-Levy model in MATLAB is based on the code referenced in

[Dic11].

3.5.2 Finding β(t;n), t ≥ 0n and σk(t;n), t ≥ 0n,kThe next step is to use λmkt

n,m and bmktn,m obtained in Subsection 3.5.1 for the swaption

grid. The goal of this calibration is to find the model parameters σk(t;n), t ≥ 0n,kand β(t;n), t ≥ 0n such that λmod

n,m in (3.35) is as close as possible to λmktn,m, and

bmodn,m in (3.27) is as close as possible to bmkt

n,m.

This calibration will be fast as the expressions relate the model skews and volatil-

ities directly and there is no use of European option valuation formulas.

Calibrating to a full set of market data is further sped up by treating volatility

calibration and skew calibration separately. It could be perceived that this model

is liable to over fitting because of the large number of parameters that need to be

calibrated, however treating volatility and skew separately mitigates this risk.

Piterbarg in [Pit05a] proposes these separate fitting procedures as first fitting the

term structure of swaption skews to bn,mn,m and then fitting the term structure

of swaption volatilities λn,mn,m. However (3.27) (for bn,m) contains references to

σk(t;n), t ≥ 0n,k and (3.35) (for λn,m) contains references to β(t;n), t ≥ 0n.

To get around this issue, Piterbarg proposes calibrating via the following steps

(the third step is optional).

1. First set all of the model skews β(t;n), t ≥ 0n to the same value β. This

value can be selected to be the mean bn,m over the full swaption grid. The

32

model volatilities σk(t;n), t ≥ 0n,k can then be found by the following fitting∑n,m

(λmodn,m − λmkt

n,m)2 → min

over σk(t;n), t ≥ 0n,k (3.57)

2. Using the values of σk(t;n), t ≥ 0n,k obtained in Step 1, the model skews

β(t;n), t ≥ 0n are found using the following.∑n,m

(bmodn,m − bmkt

n,m)2 → min

over β(t;n), t ≥ 0n (3.58)

3. The third step is to then use the skews found in Step 2 to recalibrate σk(t;n), t ≥0n,k using (3.57). Piterbarg states that Steps 1 and 2 alone provide a very

good fit.

To aid calibration, an optimisation can be made. Taking a look at (3.27) and (3.11),

the equation for the model skew bmodn,m can be changed to

bmodn,m =

∫ T

0

(m∑

i=n+1

pi(n,m)β(t, i)

)ωn,m(t)dt (3.59)

where ωn,m(t) are independent of β(t, i), which means the weightings ωn,m(t) can be

precomputed.

3.5.2.1 Parametrising the skew and volatility

Once the values for β(t;n), t ≥ 0n and σk(t;n), t ≥ 0n,k have been found, they

need to be tied together in some parametrised form. Piterbarg suggests parametri-

sation using time/offset

β(t, i) = β∗(t, T ) (3.60)

σk(t, i) = σ∗k(t, T ) (3.61)

T = Ti − t (3.62)

A time/offset grid is chosen where (3.60) and (3.61) are defined. For all other points

(ti, Tj) bilinear interpolation is used.

The following functions describe the parametrisation that will be used for the

skew and volatility.

β(t, i) = (W (Ti − t) +X)e−Y (Ti−t) + Z (3.63)

σk(t, i) = (W (Ti − t) + X)e−Y (Ti−t) + Z (3.64)

Here W ,X,Y ,Z,W ,X,Y and Z are all constants.

33

Chapter 4

Methodology

This chapter will describe the methodology behind comparing the two models. Al-

though both models assume that the market is always in a single volatility regime

at any one time, [RRW09] suggests that in actual fact this is not the case. The

market switches between normal and excited states (and potentially a third quiet

state [RC09]). The methods have been devised to capture differences that will arise

or could be accentuated due to assumptions made on the current market regime.

As discussed previously time-homogeneity in a LIBOR model is highly favourable.

Both of these models deal with time-homogeneity in different ways, and these tests

have been devised to expose differences.

4.1 Smile Calibrations

Initially the models will both be calibrated to caplet/swaption matrices on different

dates to see how well they fit. Although this is a required first step for both models, it

will provide data on the speed and computational effort required in the calibrations.

The swaption matrices chosen will consist of swaptions that have both short and

long expiries and maturities to assess the flexibility of the models.

Both models will be calibrated to caplet/swaption data from the market in a

normal state and from the market in an excited state to compare the differences, if

any. This will provide an insight into the models’ abilities to capture the extremes

of market state.

4.2 Smile Predictions

Both SABR/LMM and FL-TSS are dynamic, therefore a defining feature is how well

they both predict the future evolution of swaption volatility smiles, so as to provide

accurate estimations of future swaption prices.

Using calibrated model parameters from specific days, the volatility curves for a

variety of swaptions in the matrix are projected forward using LIBOR forward data

34

obtained for these specific days. The resulting curves are then compared with the

actual smiles for the future days to make observations. Again a range of swaption

expiry and maturities are selected.

Future smile estimation will be done in the following way

• Both models are calibrated to the swaption matrix on a date when the market

is in a normal state. The models will then be rolled forward to a date when

the market is also in a normal state. This test is referred to as normal ⇒normal.

• Both models are calibrated on a normal day, but then rolled forward to a day

when the market is excited. This is known as normal ⇒ excited.

• Finally the third scenario is excited ⇒ excited.

4.3 The Experimental Set-up

This section describes the implementation details of both models. From the data

that’s used to how decisions are made about specific model constants to use.

4.3.1 The Dataset

The market data that is used in these experiments is USD LIBOR implied volatility

data for swaptions ranging from 6M x 1Y to 10Y x 10Y. It was recommended that

into 1 year swaptions (swaptions with a maturity of 1 year) could be used as proxies

to caplet prices. The implied volatility data is decomposed into SABR parameters,

so implied volatilities can easily be found using equation (1.1).

6M LIBOR forward data was obtained from the Bank of England website [oE14].

4.3.2 SABR/LMM

Probably the largest consideration when implementing SABR/LMM for swaptions is

how to select the correct shapes for the correlation structures that the SABR/LMM

model requires - ρij, rij and Rij. Calibrating these correlation structures is a whole

subject matter in itself, it was decided that estimates of the correlation structure

would be made using empirical results found in [RRW09].

It was decided that both the forward rate/forward rate correlations (ρij) and the

forward-rate volatility/forward rate volatility correlations (rij) should have a Doust

like correlation scheme as depicted in Figure 2.1.

For the correlations between the forward-rate and their volatilities (Rij), the

elements i = j are chosen to be the correlations specified by the market. For the

elements i 6= j expression (2.54) is used.

35

4.3.3 FL-TSS

As described in Subsection 3.5.1, in order to calibrate FL-TSS there needs to be

an intermediate calibration that’s performed to essentially convert swaption data

obtained for this experiment, from SABR parameters, into parameters of skew and

volatility. Once a swaption matrix is generated, where each element is a tuple

bn,m, λn,m, FL-TSS can be calibrated to it.

The expression (3.36) suggests K volatility factors. For the tests in this study it

is decided to only have a single volatility factor driving rates, hence K = 1.

The main two constants that need to be decided on are θ, the mean reversion of

variance, which is set to 15% as described in [Pit05a] and η, the volatility of variance,

which is set to 130%. During the experiments however, these will potentially be

adjusted to improve fits if required.

36

Chapter 5

Theoretical Analysis

This section looks at how the models are expected to differ from a theoretical point of

view and makes predictions on the outcomes of the experiments described in Chapter

4. Once the models are run and the results obtained, they will be compared to the

analysis presented here to see how different or similar the two models are.


Static fits for both SABR/LMM and FL-TSS are both expected to be good since

the model parameters are chosen for each particular day of interest. The swaption

matrix contains swaptions going from 1Y x 1Y to 10Y x 10Y which should accentuate

subtle differences between the two models, and give some indication on how flexible

they both are.

5.1.1 Under normal market conditions

5.1.1.1 SABR/LMM

Calibrating the full swaption matrix in SABR/LMM hinges on how well caplets

are calibrated. The parametrisation of g(·) and h(·) shown in (2.41) and (2.42)

respectively allow multiple shapes to be captured. In normal market conditions

g(·) is expected to have a humped shape, where the hump appears approximately 2

years after the spot date and then gradually declines, as shown in [RRW09] and in

Subsection 2.1.1.

Both [Reb02] and [Reb04] explain why this shape is expected for g(·). They ex-

plain that actions by monetary authorities can mean that the value of rates has max-

imum uncertainty in the intermediate maturity range. This g(·) starts off low and

tails off slowly to a low value. Both g(·) and h(·) are chosen with time-homogeneity

in mind as mentioned in Chapter 2.

The correlation structures chosen for forward-forward, forward-volatility and

volatility-volatility correlations will determine the quality of the static fits. They

37

provide the model with many degrees of freedom which suggests a good fit can be

reached, however care should be taken not to make the model too specific.

From a computational standpoint calibrations should be fairly fast. (2.67) shows

that νTSABR requires an integral within an integral to be evaluated, where the upper

limit of the inner integral depends upon the value of t passed in from the outer.

This means that if the calculation is implemented as is, the algorithmic complexity

is O(n2), where n is the number of integration steps. However this problem has

the classic hallmarks of Schlemiel the Painter’s algorithm1, and can therefore be

optimised to O(n) by storing previous results to avoid multiple recalculations.

5.1.1.2 FL-TSS

This model requires the whole swaption matrix to be used in its calibration. As ex-

plained in Subsection 4.3.3, the data that is being used is swaption implied volatility

data encoded in SABR parameters. However as discussed previously, FL-TSS re-

quires implied volatility data to be encoded in the λ and b parameters as shown

in (3.26). Successful fits are therefore dependent on two calibrations, which means

features of both models will be observed.

FL-TSS model parameters are calculated per expiry, therefore calibrated values

for a given swaption expiry allows recovery of swaption prices of all maturities for

this expiry. A volatility structure σk(t;n) is also calibrated, and therefore the model

could be susceptible to overfitting. The model provides many degrees of freedom so

a good fit should be reached.

Since Dickinson’s numerical approximation in [Dic11] is used to generate the

base data for FL-TSS, its behaviour needs to be considered. One such effect is that,

as [Dic11] shows, error increases slightly when far from the money. This error could

be pulled into the final FL-TSS fits.

The FL-TSS model assumes forward rates follow a shifted lognormal diffusion

with a stochastic volatility process that is mean-reverting. This means negative

rates are also a possibility.

Calibration of (3.59) will be the most computationally costly, the reason is be-

cause of the multiple embedded integrals that need to be evaluated. The algorithmic

complexity of the implementation is O(n2) (again where n is the number of integra-

tion steps) but some intelligent caching can improve this.

1Joel Spolsky uses a Yiddish joke to convey bad programming practice. Schlemiel is tasked withpainting lines on a road, and each day he paints fewer and fewer. When asked why, he complainsthat each day it takes longer and longer to walk to the paint can.

38

5.1.2 Under excited market conditions

5.1.2.1 SABR/LMM

In excited market conditions, naturally h(·) is expected to be higher, especially at

short expiries, due to the uncertainty present.

The hump in g(·) described above should be steeper and peak at shorter expiries.

The peak could move to a zero expiry in which case the curve becomes monotonically

decaying. The model allows for this variation therefore it’s expected that static fits

for SABR/LMM in an excited market will still be good.

5.1.2.2 FL-TSS

Dickinson’s approximation [Dic11] could generate a larger error which feeds into

the FL-TSS fits, so it’s important to calibrate to points on the smile curve close to

at-the-money. This can effect the level of the fits.

Otherwise it is anticipated that fits will still be of good quality due to the large

number of degrees of freedom that the model provides.


The quality of the smile predictions is expected to vary considerably. The experiment

is set up such that it tests the models at their boundaries where differences in

behaviour are more likely to become apparent.

5.2.1 normal ⇒ normal

The time-homogeneity of the SABR/LMM model is crucial for a good quality fore-

cast of the smiles. Since the market in which the model is calibrated, and the

market in which the smile is forecast are both in normal states, then it can be as-

sumed that the market parameters of the model will be very similar. The value

of β in the forecast market should be similar and the forward-forward correlation,

the forward-volatility correlation and the volatility-volatility correlation should have

similar values due to analogous forward dynamics of the market states on both dates.

It is estimated that smile curvature will be very similar, however the volatility levels

could vary.

The FL-TSS model on the other hand has a strict trade-off between time-

homogeneity and quality of fit. Piterbarg in [Pit05a] suggests using a non-time

homogeneity penalty (described in the paper), but this can impede the quality of

the market fit. It is expected that the time dependence of the skew and the volatility

will generate error in the forecast implied volatility curve. This error will be higher

in short to medium term swaption expiries due to the higher levels of volatility. It

39

is likely this error will be observed both by difference of overall volatility level, and

by a difference in skew.

5.2.2 normal ⇒ excited

Since the source market and the forecast market are now in different states, fits are

expected to differ considerably. The normal market used to calibrate the SABR/LMM

model is highly likely to have different forward-forward, forward-volatility and volatility-

volatility correlations due to the differing market dynamics. The market may also

command a different value of β. As mentioned previously SABR/LMM doesn’t sup-

port this kind of regime change, but the two-state Markov-chain process suggested

in Subsection 2.5 can be used, however this is beyond the scope of this study.

The same goes for FL-TSS. The time dependence in the volatility and skew, and

the fact that the market dynamics in the calibration market will differ considerably

leads to the prediction that there could be large errors in both the skews and volatil-

ity levels in the forecast curves. Estimating that the market could be in a different

regime could lead the implementer to select a different value of θ or η for example,

but obviously this would require a different model that defines their dynamics.

5.2.3 excited ⇒ excited

The quality of this forecast is expected to be better than the normal ⇒ excited case,

but worse than the normal ⇒ normal case.

It is predicted that the issues that come with forecasting in a different market

regime won’t exist, however the issues associated with calibrating both models in an

excited market, as described in Section 5.1.2, will. Therefore it is expected that this

will be the cause of observed differences between a predicted smile for both models

and the actual smile. Moreover the difference could be accentuated by cumulative

errors due to the calibration market being in a more volatile state.

40

Chapter 6

Results and discussion

Here results are presented and discussed, and findings will be linked back to the the-

ory presented in order to make valid conclusions for the similarities and differences

between SABR/LMM and FL-TSS.

Due to the large amount of data that was generated, selected tabular data and

calibration results can be found in Appendix A.

6.1 Intermediate Calibrations

This section presents and analyses the results of the intermediate calibrations. That

is, the initial calibrations that are performed to obtain model constants.

6.1.1 SABR/LMM

Figure 6.1 shows the model fit for g(·) and h(·) on 23-Nov-2006. Data from this

date can be considered as being from a normal day. The hump shape in Figure 6.1a

can be clearly observed, which shows that the caplet expiry for which there is the

greatest volatility is around 3 years.

Looking at the vol-vol chart in Figure 6.1b, the model fits very well to the market

data and the expected shape is obtained.

Figure 6.2 shows the model fit for g(·) and h(·) on 4-Mar-2008, considered a date

when the market is in an excited state.

Immediately it can be observed that the hump is almost non-existent in Figure

6.2a and the graph is monotonically decreasing. The peak would be more prominent

if more data was available for expiries in the 1 month to 6 month range which would

have allowed for a higher resolution in this part of the chart. Essentially the model

knows that although it is in an excited state, it will eventually settle down. The

model is able to replicate this behaviour which is why the long tails are observed in

both Figures 6.1a and 6.2a.

41

(a) σTi0 data and g(Ti) fit (b) νSABR data and νLMMSABR fit

Figure 6.1: Caplet calibration fits for 23-Nov-2006

The vol-vol chart in Figure 6.2b shows a steep decline as the expiry increases

and then levels out. Both Figures 6.1b and 6.2b show very high vol-vol as T → 0.

These kinds of large values are not typical for ν and the reason is because these

large values take into account surprises that might occur in the market in the very

short term. The SABR model is unable to reproduce the smile curvature for such

short expiries, so in practice ν values are quoted for expiries no less than a couple

of weeks.

As expected, the calibrated vol-vol curve for 4-Mar-2008 starts off a lot steeper

than the curve for 23-Nov-2006 which reflects the high level of immediate uncer-

tainty. The model calibrates well for vol-vol in both market states.

Tables A.1 and A.2 show the caplet market data that was used (the into 1

year swaption prices as mentioned earlier). Tables A.3, A.4, A.5 and A.6 show the

resulting calibrated model parameters for the dates mentioned in this section.

(a) σTi0 data and g(Ti) fit (b) νSABR data and νLMMSABR fit

Figure 6.2: Caplet calibration fits for 4-Mar-2008

42

6.1.2 FL-TSS

The first step in calibrating FL-TSS is to find λmktn,m and bmkt

n,m. Two Filon-Levy fits

that were generated can be seen in Figure 6.3. Both charts show a good fit with the

market data even though both swaptions are quite different.

(a) 1Y x 2Y Swaption (b) 5Y x 10Y Swaption

Figure 6.3: Filon-Levy fits for 23-Nov-2006

Tables A.7 and A.8 show values obtained for λmktn,m and bmkt

n,m respectively for the

full swaption matrix on 23-Nov-2006. FL-TSS is then calibrated to these values.

Figures 6.4a and 6.4b are two examples of fits that were generated for (3.27) on two

different dates and for two different expiries.

(a) bn,m fit for swaptions with 10Y ex-piry on 23-Nov-2006

(b) bn,m fit for swaptions with 5Y expiryon 4-Mar-2008

Figure 6.4: Fits for expression (3.27)

The fits in Figure 6.4 show the versatility of the function (3.27) to fit quite

different shapes of data. The graphs have a jagged shape because the maturity data

is discrete as it is denominated in multiples of 6M LIBOR forwards. The fit in Figure

6.4a is from a normal market and the fit in Figure 6.4b is from an excited market.

As estimated both fits are good due to the large number of degrees of freedom the

model has.

43

Tables A.9 and A.10 show the calibrated values obtained for expressions (3.63)

and (3.64) respectively for the fit in Figure 6.4a. Tables A.11 and A.12 show the

calibrated values for Figure 6.4b.


The rest of this chapter will follow a similar layout to Chapter 5. Using the models

that have been built, this chapter will show results and aim to prove or disprove the

predictions made in Chapter 5 about the differences and similarities of SABR/LMM

and FL-TSS that should be observed. An explanation will be attempted for any

unexpected differences/similarities.

6.2.1 Under normal market conditions

Figure 6.5 shows the volatility smiles generated using the calibrated FL-TSS and

SABR/LMM models, both plotted against the smile given by the market data. The

first column of charts show the into 1 year swaptions, as discussed before, these are

proxies for caplets.

Figure 6.5: Implied volatility smiles generated by calibrated SABR/LMM andFL-TSS models, compared to the market for 23-Nov-2006

44

As predicted, both models provide good fits. Looking closely, the SABR/LMM

smile curvature is almost identical to the market’s in all cases. This isn’t the case

with FL-TSS where the curvature and slope varies. In some cases, take for example

the 5Y x 5Y swaption, FL-TSS seems to match the curvature only as the option

goes out of the money.

This difference in FL-TSS curvature could be attributed to the errors introduced

by the initial calibration to the Filon-Levy model to generate b and λ for the full

swaption matrix.

The reason why SABR/LMM smiles match the market smile curvature so closely

is because they both use β in the same way. Since the market data is in SABR form,

and the SABR/LMM uses the same underlying model, then it’s no real surprise that

the curvatures are the same.

Looking at 5Y x 2Y or 1Y x 10Y it can be seen that although the SABR/LMM

smile curvature matches that of the market, it’s volatility level is different. It is

differences in Σ0 that causes this, and shows that there are some inaccuracies that

have been introduced when evaluating (2.72). The most likely errors could be in the

correlation shapes. A better selection of forward-forward correlations might improve

this fit.

6.2.2 Under excited market conditions

Figure 6.6 shows the volatility smiles generated for 4-Mar-2008 using calibrated FL-

TSS and SABR/LMM models. Some of the curves look very different to the smiles

shown in Figure 6.5. Again the first column shows the proxy caplets.

What is immediately apparent is the low quality of the 1Y expiry swaption fits.

SABR/LMM shows a good fit for 1Y x 1Y and 1Y x 2Y swaptions, but then the

quality of the fit gets worse as the maturity for the 1Y expiry swaptions increases.

Curvature, slope and level of the SABR/LMM curve is different to the market.

FL-TSS fits badly to all 1Y expiry swaptions. As with SABR/LMM, the FL-

TSS fits for 1Y x 1Y and 1Y x 2Y swaptions show similar curvature to the market,

however the volatility level is very different.

A possible explanation for the bad fits of the 1Y expiry swaptions is that there

is a large amount of uncertainly that the market’s facing. As expiry is increased, it

can be seen that the smiles start to behave as expected and the fits for the higher

expiry options are generally good. This shows that both models have an inherent

feature built in - that although immediate vol and vol-vol is high, the market will

level out and hence the pricing of longer dated products is more stable.

In Figure 6.5 it can be seen that SABR/LMM accurately captured smile curva-

ture information. In this excited state, SABR/LMM has captured the smile curva-

ture accurately in most cases, however there are cases such as 5Y x 10Y swaption

45

Figure 6.6: Implied volatility smiles generated by calibrated SABR/LMM andFL-TSS models, compared to the market for 4-Mar-2008

where the curvature is slightly different. The FL-TSS curve for the 5Y x 10Y swap-

tion however fits well close to at-the-money. A possible explanation for this could

be inaccuracies in the chosen forward-forward correlation matrix (ρ). It makes sense

that this would effect longer maturity/expiry swaptions because the correlation ma-

trix determines the amount forwards further away are affected by other forwards.

The other differences observed can be attributed to the same explanations pro-

vided in Subsection 6.2.1. The FL-TSS curves show that the model isn’t able to as

accurately capture the curvature as well as SABR/LMM in this experiment.

6.2.3 Computational considerations

Both models were run on a quad core Intel i7 machine. The parallel framework

features in MATLAB were used to make use of the 4 cores and speed up calibrations.

This helped immensely.

Fitting the SABR/LMM models to caplets took around 45 seconds. Calculating

g(·) parameters was extremely fast, and h(·) took the rest of the time. Subsequent

generation of the implied volatility curves for the swaption matrix was also quick

due to the analytic solutions.

46

FL-TSS, however, was highly compute intensive. Finding the parameters for

σk(t, i) in (3.64) meant that a solution to ϕ(µ), which can be solved via the Riccati

system of equations represented in (3.39), needed to be found. This was done

numerically using the ODE solvers built into MATLAB, and required a fair amount

of compute power.

Calibration of bn,m also required a lot of compute power due to the multiple

embedded integrals as shown in expression (3.27). The implementation could be

improved to make use of intelligent result caching so the same calculation isn’t

repeated. This would help hugely during the fitting process.

The resulting implied volatility curves for the swaption matrix from the FL-TSS

model were generated using the Filon-Levy algorithm described in [Dic11]. This

was also very quick once the model had been calibrated and the correct skew and

volatility had been obtained for the desired maturity.


The results of the static fits have been presented and discussed. In this section the

results of future implied volatility smile predictions will be presented and discussed.

The charts have been grouped in order of increasing maturity whilst keeping expiry

the same.

6.3.1 normal ⇒ normal

Figure 6.7 shows that both SABR/LMM and FL-TSS predicted curves agree with

each other, and the actual market implied vol on 23-Nov-2007 is a little higher. The

curvature of the market smile for the short maturity swaption is slightly different to

the predicted.

As the maturity increases, the predicted curves and the market curves seem

to move closer together - This observation, along with the fact that FL-TSS and

SABR/LMM curves agree with each other, means that the difference in the predicted

and market curves is down to the higher volatility of the 1Y expiry swaptions. Figure

6.9 shows that the longest dated swaption in the matrix, the 10Y x 10Y, has the

best prediction because it is the least volatile.

Figures 6.8 and 6.9 show that as the expiry and maturity increases, the curves

move closer together. This is down to the reduction in volatility for the longer

dated swaptions. Shorter maturity swaptions at long expiries also don’t seem to be

predicted as well by both FL-TSS and SABR/LMM - take the 10Y x 2Y for example

in Figure 6.9.

As observed with the static fits, SABR/LMM in general does a better job (in

this experiment) at predicting the curvature and slope of the curves. Since the

47

Figure 6.7: Predicted volatility smiles vs. Actual smile for 1Y expiry and differentmaturities from a normal market to a normal market


market in which the prediction was made is in the same regime as the market for

which the curves are calibrated, it can be expected that SABR/LMM will accurately

estimate smile curvature. Both the lack of time-homogeneity in FL-TSS and errors

introduced in the Filon-Levy fits as discussed in Section 6.2 have contributed to

FL-TSS not predicting implied volatility curvature as well as SABR/LMM.

This was hypothesised in Chapter 5, and this experiment has shown the impor-

tance of time-homogeneity built into a model. It was also estimated that the FL-TSS

would show larger differences in both skew and volatility for short to medium ex-

piries and maturities. Although this is correctly observed, the skew is still quite

different even for long dated swaptions such as the 10Y x 10Y.

6.3.2 normal ⇒ excited

As estimated in Chapter 5, Figure 6.10 shows that both models are not able to

estimate volatility smiles when the market has undergone a regime change. Both

the FL-TSS and SABR/LMM curves roughly agree with each other however the

actual smile on the forecast date is very different.

Figure 6.11 shows that even for a longer expiry, neither FL-TSS or SABR/LMM

forecast curves match the actual market curve observed on the forecast date. The

curves start to line up however in-the-money, but due to the complete difference in

48


Figure 6.10: Predicted volatility smiles vs. Actual smile for 1Y expiry and differentmaturities from a normal market to an excited market

curvature the market smile rapidly moves away from the estimated curves as strike

increases.

Figure 6.11: Predicted volatility smiles vs. Actual smile for 5Y expiry and differentmaturities from a normal market to an excited market

Looking at Figure 6.12 the estimated smiles are starting to look more like the

actual observed smile. This is an effect of the models believing that the market will

eventually settle, and therefore although there is a regime change, the expiry is high

enough not to let these long dated swaptions feel the effect of large immediate and

medium term volatility.

It was hypothesised that these issues would be seen. The market essentially com-

mands a different value of β in the SABR/LMM model, and since market dynamics

49

Figure 6.12: Predicted volatility smiles vs. Actual smile for 10Y expiry anddifferent maturities from a normal market to an excited market

are very different, the 3 correlation shapes will need to be adjusted.

For FL-TSS θ and η will also need to be adjusted due to the change in market

dynamics. A model that is able to estimate the change in these values would be

much better at estimating curves across a market regime change.

6.3.3 excited ⇒ excited

Figure 6.13 shows that implied volatility curves for 1Y expiry swaptions aren’t pre-

dicted well. FL-TSS estimates the correct smile curvature for a 1Y x 2Y swaption

that is in the money. The curvature of the 1Y x 10Y seems to be correctly estimated

by both SABR/LMM and FL-TSS for in the money swaptions.

This can be explained by referring back to Subsection 6.2.2 where the issues sur-

rounding calibrating both models in an excited market can be observed. Although

there are parts of the implied volatility curves that look correct, overall the predic-

tions aren’t great, although better than the normal ⇒ excited case as predicted in

Chapter 5.

Figure 6.13: Predicted volatility smiles vs. Actual smile for 1Y expiry and differentmaturities from a excited market to an excited market

Looking at higher expiry swaptions in Figure 6.14 it can be seen that the predic-

tions are much better. SABR/LMM’s predicted smile almost perfectly follows the

market smile in the money for all 5Y expiry swaptions, but they diverge as strike

50

increases. In all 3 cases FL-TSS seems to roughly follow the curvature of the market

smile, however it has a different slope.

Figure 6.14: Predicted volatility smiles vs. Actual smile for 5Y expiry and differentmaturities from a excited market to an excited market

For the 10Y expiry swaptions shown in Figure 6.15 the implied volatility curves

are much flatter, both FL-TSS predicted curves and SABR/LMM predicted curves

correctly follow the same curvature as the market in the money. Both predicted

curves seem to diverge from the market curve as the swaption goes out of the money.

Figure 6.15: Predicted volatility smiles vs. Actual smile for 10Y expiry anddifferent maturities from a excited market to an excited market

As estimated in Chapter 5 the implied volatility smile predictions given by FL-

TSS and SABR/LMM in the excited ⇒ excited case is better than the predictions

made for the normal ⇒ excited case, but not as good as the predictions made in

the normal ⇒ normal case. The change of market regime is detrimental to the

performance of both models, and the higher volatility in an excited market makes

calibrations for short dated swaptions more difficult.

SABR/LMM is able to better predict the skew than FL-TSS in these experi-

ments. Since SABR/LMM is calibrated only to caplets (in this experiment), but

FL-TSS is calibrated to the whole skew and volatility structure in a swaption ma-

trix, in excited markets, the high volatility of caplet prices could cause errors in the

model. FL-TSS will not be effected as much by this high volatility of short maturity

swaptions (proxy to caplet prices) since it’s effective skew and volatility calibrations

51

are viewing the swaption matrix as a whole. This could explain why FL-TSS is able

to predict the market smile curvature for the 1Y x 2Y swaption in Figure 6.13.

A way to improve the SABR/LMM predictions is to calibrate the global corre-

lation matrix P in (2.51) to the whole swaption matrix, rather than making esti-

mations on what this matrix should look like.

52

Chapter 7

Conclusion & further research

In this thesis we have described in detail Piterbarg’s forward LIBOR term struc-

ture of skew model (FL-TSS) and the SABR/LIBOR market model (SABR/LMM).

We have built both models in order to compare and contrast their performance in

calibrating to market data and reproducing swaption implied volatility curves.

We devised tests that would allow us to observe differences in the performance

of both models. The first test is to ascertain the ability of both models to calibrate

to market data. We picked two dates, one where the market is in a so-called normal

state and another date where the market is in a so-called excited state. We esti-

mated that for the normal market, both models should calibrate well and accurately

reproduce swaption implied volatility smiles. We also estimated that for an excited

market there could be errors introduced in the Filon-Levy [Dic11] calibrations to

generate λn,m and bn,m which is the precursor to calibrating the FL-TSS model.

We showed, using the models that were built, that in a normal market both

models calibrate well to market data and accurately reproduce swaption implied

volatility smiles for a range of expiries and maturities. We found that SABR/LMM

was better at reproducing the smile curvature, and attributed errors in the FL-TSS

curves to errors accumulated when fitting the market data initially to the Filon-Levy

model.

We found both models didn’t reproduce swaption implied volatility smiles in

an excited market as well as they did in a normal market. The high volatility in

this market for short and medium term rates means that the models would need to

be adjusted further to capture the excited dynamics. For example different shapes

could be used that represent the forward-forward, forward-volatility and volatility-

volatility correlation matrices in the SABR/LMM model. For FL-TSS further ex-

periments could be performed to select optimum η and θ values.

Next we tested the models’ ability to predict future swaption volatility smiles.

We did this by calibrating the models to a specific date, and then rolling the model

forward through time by multiples of 6 months as we are using 6M LIBOR forward

53

rates. We then compared these predicted curves to the actual implied volatility

curves that were recorded on the forecast dates. To accentuate similarities and

differences between the two models we looked at 3 scenarios:

1. Calibrating the models in a normal market, and making predictions on swap-

tion implied volatility curves for a date considered to have normal market

dynamics.

2. Calibrating the models in a normal market and forecasting curves for a date

considered to have excited market dynamics.

3. Calibrating and forecasting the models on dates both considered to be excited.

This test highlights issues around time-homogeneity and regime switching in

markets and gives an insight into how both models deal with these. We made

predictions based on the theory and found that for the normal ⇒ normal case, the

predictions are fairly accurate. We found that the time-homogeneity built into the

SABR/LMM gave it the advantage over FL-TSS as it performed better overall.

In the normal⇒ excited case the predictions for the shorter dated swaptions were

not great, however they got better as the swaption expiry and maturity increased.

We put this down to the fact that neither model is able to model this type of regime

change between market dynamics, and make recommendations to adjust many of

the model constants.

We estimated that in the excited ⇒ excited scenario, the predictions would be

better than the normal⇒ excited scenario but not as good as the normal⇒ normal

scenario. Again we found that SABR/LMM made better predictions overall, how-

ever we also found that FL-TSS could perform well by averaging out the effects of

high volatility in shorter dated rates in an excited market, since FL-TSS calibrates

to the whole swaption matrix. SABR/LMM is more susceptible to the higher volatil-

ities seen in caplet prices which create errors that can propagate into the forecast

volatility smiles. For SABR/LMM in this experiment, the correlation matrices were

estimated, however SABR/LMM can be improved by calibrating correlations to the

whole swaption matrix.

Finally, we found the FL-TSS model to be far more compute intensive to calibrate

and gave reasons as to why this was the case. SABR/LMM was very quick to

calibrate. We recommend ways to optimise the algorithm used to calculate νTSABRin the SABR/LMM model.

An interesting extension to this study is to investigate ways to allow both models

to handle regime switching of markets, and compare their performance in pricing

swaptions and predicting future volatility curves. Further information about regime

switching can be found in [RRW09].

54

An investigation could also be made into choosing (calibrating) better correlation

shapes for SABR/LMM, and more accurately generating skew and volatility for the

swaption matrix that FL-TSS calibrates to. It would also be very interesting to

investigate how both models can be used in hedging. For example, the models’

ability to calculate the greeks, and their hedging accuracy and performance can be

compared.

55

Bibliography

[AA02] Leif B.G. Andersen and Jesper Andreasen. Volatile volatilities. Risk,

15(12), 2002.

[ABM97] Dariusz Gatarek Alan Brace and Marek Musiela. The market

model of interest rate dynamics. http://users.aims.ac.za/~davidw/

BGMlibor.pdf, April 1997.

[ABR01] Leif B.G. Andersen and Rupert Brotherton-Ratcliffe. Extended LIBOR

market models with stochastic volatility. working paper. 2001.

[DB06] Fabio Mercurio Damiano Brigo. Interest Rate Models Theory and Prac-

tice. Springer, 2006.

[Dic11] Andrew Samuel Dickinson. Numerical approximation of option premia in

displaced-lognormal heston models. http://papers.ssrn.com/sol3/

papers.cfm?abstract_id=1833438, May 2011.

[Dic13] Andrew Dickinson. Filon-levy option pricer. http://sourceforge.net/

projects/filonlevyoption/, April 2013.

[E.S00] Steven E.Shreve. Stochastic Calculus for Finance II. Springer, 2000.

[FALS01] Pedro Santa-Clara Francis A. Longstaff and Eduardo S. Schwartz. The

relative valuation of caps and swaptions: Theory and empirical evidence.

The Journal of Finance, LVI:2067–2109, 2001.

[Hes93] Steven Heston. A closed-form solution for options with stochastic volatil-

ity and applications to bond and currency options. The review of Fi-

nancial Studies, 6(2):327–343, 1993.

[HL07] Pierre Henry-Labordere. Unifying the BGM and SABR models: a short

ride in hyperbolic geometry. http://papers.ssrn.com/sol3/papers.

cfm?abstract_id=877762, January 2007.

[HL08] Pierre Henry-Labordere. Combining the SABR and LMM models. RISK,

August 2008.

56

http://users.aims.ac.za/~davidw/BGMlibor.pdf

http://users.aims.ac.za/~davidw/BGMlibor.pdf

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1833438


http://sourceforge.net/projects/filonlevyoption/

http://sourceforge.net/projects/filonlevyoption/



[Klu05] Wolfgang Kluge. Time-inhomogeneous levy processes in interest

rate and credit risk models. http://www.freidok.uni-freiburg.de/

volltexte/2090/pdf/diss.pdf, July 2005.

[Kre99] Erwin Kreyszig. Advanced Engineering Mathematics. Wiley, 1999.

[Lew01] A Lewis. A simple option formula for general jump-diffusion and other

exponential levy processes. http://optioncity.net/pubs/ExpLevy.

pdf, 2001.

[oE14] Bank of England. LIBOR forward data. http://www.bankofengland.

co.uk/statistics/Pages/yieldcurve/default.aspx, September

2014.

[Pit03] Vladimir Piterbarg. A stochastic volatility forward LIBOR model with

a term structure of volatility smiles. http://papers.ssrn.com/sol3/

papers.cfm?abstract_id=472061, October 2003.

[Pit05a] Vladimir Piterbarg. Stochastic volatility model with time-dependent

skew. Applied Mathematical Finance, 12(2):147–185, 2005.

[Pit05b] Vladimir Piterbarg. Time to smile. Risk, May 2005.

[PSHW02] Andrew S. Lesniewski Patrick S. Hagan, Deep Kumar and Diana E.

Woodward. Managing smile risk, 2002.

[RC09] Riccardo Rebonato and Jian Chen. Evidence for state transition and

altered serial correlation in US$ interest rates. Quantitative Finance,

9(3):259–278, 2009.

[Reb02] Riccardo Rebonato. Modern Pricing of Interest-Rate Derivatives: The

LIBOR Market Model and Beyond. Princeton University Press, 2002.

[Reb04] Riccardo Rebonato. Volatility and Correlation: The perfect Hedger and

the Fox, 2nd edition. Wiley, 2004.

[Reb07] Riccardo Rebonato. A time-homogeneous, SABR-consistent extension

of the LMM. RISK, November 2007.

[RK03] Riccardo Rebonato and Dherminder Kainth. A two-regime,

stochastic-volatility extension of the LIBOR market model.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.

1.9.4702&rep=rep1&type=pdf, June 2003.

57

http://www.freidok.uni-freiburg.de/volltexte/2090/pdf/diss.pdf

http://www.freidok.uni-freiburg.de/volltexte/2090/pdf/diss.pdf

http://optioncity.net/pubs/ExpLevy.pdf

http://optioncity.net/pubs/ExpLevy.pdf

http://www.bankofengland.co.uk/statistics/Pages/yieldcurve/default.aspx

http://www.bankofengland.co.uk/statistics/Pages/yieldcurve/default.aspx



http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.9.4702&rep=rep1&type=pdf

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.9.4702&rep=rep1&type=pdf

[RRW08] Andrey Pogudin Riccardo Rebonato and Richard White. Delta and vega

hedging in the SABR and LMM-SABR models. RISK, December 2008.

[RRW09] Kenneth McKay Riccardo Rebonato and Richard White. The

SABR/LIBOR Market Model. Wiley, 2009.

[RW09] Riccardo Rebonato and Richard White. Linking caplets and swaptions

prices in the LMM-SABR model. The Journal of Computational Fi-

nance, 13(2):19–45, 2009.

[Wan07] Zaizhi Wang. Effective parameters for stochastic volatility models.

https://www.eurofidai.org/Wang.pdf, September 2007.

58

https://www.eurofidai.org/Wang.pdf

Appendix A

Data

Caplet T σ0 β ρSABR νSABR f0

6M 0.4958904 0.031056 0.5 -0.36 0.625 0.049373

1Y 1 0.035999 0.5 -0.4 0.48 0.047615

2Y 2.0054795 0.038482 0.5 -0.35 0.38 0.047544

5Y 5.0027397 0.0394 0.5 -0.38 0.33 0.049996

10Y 10.008219 0.035208 0.5 -0.38 0.2775 0.052425

Table A.1: Market Caplet data for 23-Nov-2006

Caplet T σ0 β ρSABR νSABR f0

6M 0.5041096 0.091658 0.5 -0.505 0.54 0.022788

1Y 1 0.085779 0.5 -0.49 0.46 0.025161

2Y 2 0.071312 0.5 -0.41 0.405 0.033707

5Y 5.0027397 0.050497 0.5 -0.43 0.3375 0.048837

10Y 10.008219 0.038386 0.5 -0.41 0.29 0.056696

Table A.2: Market Caplet data for 04-Mar-2008

a b c d

-0.23% 0.0251 0.584 2.81%

Table A.3: g(·) calibrated parameters for 23-Nov-2006

α β γ δ

104.17% -0.9938 0.9194 -19.05%

Table A.4: h(·) calibrated parameters for 23-Nov-2006

a b c d

6.93% 0.1305 1.427 2.16%

Table A.5: g(·) calibrated parameters for 4-Mar-2008

59

α β γ δ

123.67% -21.5264 13.1304 45.65%

Table A.6: h(·) calibrated parameters for 4-Mar-2008

Maturity

1Y 2Y 3Y 5Y 7Y 10Y

Expiry

6M -0.86765 -0.92112 -0.90239 -0.83689 -0.79645 -0.75819

1Y -0.71579 -0.66864 -0.62693 -0.55172 -0.54025 -0.5134

2Y -0.36918 -0.37056 -0.37048 -0.36613 -0.37635 -0.36471

5Y -0.25449 -0.2696 -0.27233 -0.27233 -0.26858 -0.26667

10Y -0.15416 -0.15029 -0.15104 -0.15112 -0.14346 -0.13103

Table A.7: bmktn,m for 23-Nov-2006 found using Filon-Levy approximation

Maturity

1Y 2Y 3Y 5Y 7Y 10Y

Expiry

6M 0.23939 0.22691 0.22347 0.21933 0.22087 0.223

1Y 0.21444 0.21244 0.21085 0.20725 0.20701 0.20664

2Y 0.20702 0.20689 0.20652 0.20569 0.20465 0.20333

5Y 0.20864 0.20804 0.20762 0.20694 0.20566 0.20376

10Y 0.19982 0.19914 0.19862 0.19755 0.19578 0.1931

Table A.8: λmktn,m for 23-Nov-2006 found using Filon-Levy approximation

W X Y Z

0.006762 -0.05627 0.2395 -0.001398

Table A.9: Parameters found for (3.63), for swaption expiry 10Y on 23-Nov-2006

W X Y Z

-0.0408 0.0258 0.1395 0.0355

Table A.10: Parameters found for (3.64), for swaption expiry 10Y on 23-Nov-2006

W X Y Z

0.1984 -0.09409 1.01 0.001944

Table A.11: Parameters found for (3.63), for swaption expiry 5Y on 4-Mar-2008

W X Y Z

-0.0166 0.0104 0.1458 0.0246

Table A.12: Parameters found for (3.64), for swaption expiry 5Y on 4-Mar-2008

60

Appendix B

Matlab code listing

Due to the large amount of code written for this project, it can be downloaded from

http://goo.gl/TH7qgM. The tables below provide short descriptions of the code.

Source file name Notes

caplet implied vol.m calls into the SABR/LMM g(Ti) function (2.65)and νTSABR function 2.68 given calibrated parame-ters and cubes containing implied volatility infor-mation. Returns the implied volatilities calculatedby expression 1.1 along with the market impliedvolatilities for comparison.

caplet implied volatility.m This implements expression (1.1).

DickinsonApproximation.m This is the main implementation in MATLAB ofthe Filon-Levy option pricer. Note that this isbased on the code that can be found here [Dic13].It returns the implied volatility given, amongstother things, a Heston volatility λ and a skew b.

DickinsonFit.m This invokes the Filon-Levy and solves for λ andb given implied volatilities and strikes.

FilonScheme.m A class used by the Filon-Levy option pricer.

GaussScheme.m A class used by the Filon-Levy option pricer.

lmm SABR caplet calibrate g.m Fits the g(·) function (2.65) to given data.

lmm SABR caplet calibrate v.m Fits the νTSABR function (2.68) to given data.

lmm SABR caplet gRMS.m Implementation of function (2.65).

lmm SABR caplet h.m Implementation of function (2.42).

lmm SABR caplet v SABR.m Implementation of function (2.68).

marketImpliedVolLoader.m This parses and loads swaption data and LIBORforward data which it stores in internal memorystructures for easy use in other functions.

61

http://goo.gl/TH7qgM

Source file name Notes

NewtonDividedDifference.m A class used by the Filon-Levy optionpricer.

piterbarg implied vol.m Invoke FL-TSS given calibrated parame-ters and returns implied volatility, alongwith strikes used and the market impliedvolatility for comparison.

pitSkewFunc.m This is the implementation of the func-tion (3.27).

pitSkewGenerator.m Invokes the calibration of the function(3.35) to find the parameters in (3.63).

pitVolFunc.m Implementation of the function (3.35).

pitVolRunner.m Calibrates the function (3.27) to find theparameters in (3.64).

run implied swaptions single strike.m Given all discovered calibration parame-ters, this runs both the FL-TSS and theSABR/LMM model for a single strike.The implied volatility is returned alongwith the market value for comparison.

run implied vol pit sabr swaptions.m This does the same asrun implied swaptions single strike.m

but for many strike values.

run implied vols for future.m This does the same asrun implied vol pit sabr swaptions.m

but is able to roll the date forward. Itreturns FL-TSS and SABR/LMM pre-dictions for a future date, along withthe actual market volatility smile for theforecast date for comparison.

setGenericFilonGlobals.m A utility class for Filon-Levy optionpricer.

swaptionImpliedVol.m Implementation of SABR/LMM.

results.xlsx This spreadsheet helps to manage thelarge amount of data generated by theexperiments.

62

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