A model for interest rates in repressed markets...2019/04/13 · bank commitments to easy monetary...

A model for interest rates inrepressed markets

Alexey ErekhinskiyChrist Church College

University of Oxford

A thesis submitted in partial fulfillment of the MSc in

Mathematical Finance

18th April 2013

We consider an extension of the Vasicek short-rate model aimed to capturecentral bank actions to repress fixed income markets by keeping rates low for ex-tended period of time. We also incorporate this model into a Libor Market Model.

i

Contents

1 Introduction 11.1 Economic preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The model 32.1 Short-rate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Model interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Short rate volatility as logistic function . . . . . . . . . . . . . . . . . 4

3 No-arbitrage bond pricing 63.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Forward rate volatility as a function of model parameters . . . . . . . 83.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Principal components 164.1 Business cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Libor Market Model with DMRV forwards 245.1 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 LMM and HJM volatility . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Calibration 316.1 Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3 Calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7 Conclusion 40

ii

A 41A.1 Libor drifts in LMM-DMRV . . . . . . . . . . . . . . . . . . . . . . . . 41A.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Bibliography 53

iii

Chapter 1

Introduction

1.1 Economic preview

Target federal funds rate has been near zero1 from 16th of December 2008 andat the time of writing it is still not clear when monetary policy in the United Stateswill become aggressive again2. Bernanke (2010) stated two examples of centralbank commitments to easy monetary policy conditional on certain event occurringin the future3. Such central bank actions reduce uncertainty in the future path ofshort-term interest rates, which can be used in mathematical modelling of interestrates and pricing of fixed income products.

1.2 Modelling

In terms of the yield curve volatility, easy monetary policy results in low volatility ofits short end during the period when this policy is maintained. In the present paperwe study how calendar dependency of the volatility structure can be modelled in theHeath-Jarrow-Morton (HJM) framework. We use affine short rate model to predictforward rates in a financial scenario when loose monetary policy is tightened aftercertain period of time.

1 Target range 0-25 basis points.2 Although on 12th December 2012, the Federal Open Market Committee (FOMC) released

the notes of its October 2012 meeting where it voted to keep the target federal funds rate low untilunemployment falls to 6.5 per cent (conditional on inflation expectations). The minutes are availablein Federal Open Market Committee press release (2012).

3 The Bank of Canada in April 2009 committed to low policy rate until the second quarter of2010 (conditional on the inflation outlook), and the Bank of Japan in March 2001 committed tomaintain its policy rate at zero until consumer prices stabilise or increase on a year-on-year basis.

1

1.3 Strategy

We construct a Libor market model (LMM) with low Markov dimensionality of theshort rate by starting from an affine model with plausible dynamics of forward rates,calculating their instantaneous volatilities and then using this volatility structure tocalculate volatilities of discrete Libor rates for calibration and pricing purposes.

Structure of the paper

Section 2 introduces the short-rate affine term-structure model we will use through-out the rest of the paper. In Section 3 we calculate no-arbitrage bond prices, de-termine basic probabilistic properties of short and forward rates and show howinstantaneous forward rate volatility depends on model parameters. Section 4 isdevoted to study principal components of the model and also how can it be usedin modelling business cycles, long yields and central bank actions. In section 5we build a Libor Market Model consistent with the affine model introduced above.We introduce a special probability space on which the approximate LMM with Liborrates with Gaussian dynamics is correctly defined. Section 6 justifies approxima-tions we use when calibrating the LMM to at-the-money USD volatilities in 2012.We conclude on the model in section 7. Libor drifts are derived in Appendix A.1and Appendix A.2 has proofs of all propositions used in the paper.

2

Chapter 2

The model

2.1 Short-rate model

Dynamic mean-reverting Vasıcek (DMRV) model is an extension of the Vasıcekmodel, where mean reversion level is dynamic and follows an Ornstein-Uhlebeckprocess:

drt = κ(θt − rt)dt+ σrtdWrt , (2.1)

dθt = α(β − θt)dt+ σθdW θt , (2.2)

d〈r, θ〉t = ρ dt, (2.3)

where d〈r, θ〉t is the quadratic cross-variation process for processes rt and θt.Here rt is the short rate, θt is its dynamic mean-reversion level, κ and α are the

constant mean reversion speeds, β is the reversion level of the short rate meanreversion level; σθ is the mean reversion level’s constant volatility and σrt is thetime-dependent deterministic volatility of the short rate.

2.2 Model interpretation

Initial short rate r0 is assumed to be low to represent low level of short-term USDinterest rates at the time of writing. Short rate normal volatility σr0 is also low initiallyto represent central bank’s commitment to maintain easy monetary policy for anextended period of time. Eventual exit and tightening of short-term interest ratesis represented by a combination of higher level of mean reversion than initial shortrate:

β > r0,

3

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

Time to expiry (years)

BP

vol

ATM Caplet Vols

Caplet fwd vol (mkt)Instantaneous calibrated volRMS calibrated vol

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80

90

100

Time to expiry (years)B

P v

ol

ATM Caplet Vols

Caplet fwd vol (mkt)Instantaneous calibrated volRMS calibrated vol

Figure 2.1: At-the-money USD caplet normal volatility curve and DMRV instan-taneous and RMS forward volatilities (with constant σrt on the left-hand side andmonotonically increasing σrt on the right). Market data as of 13-Sep-2012.

and monotonically increasing σrt as a function of t. Typical instantaneous and root-mean squared (RMS) forward normal volatilities for constant σrt and for monotoni-cally increasing σrt are plotted on Figure 2.1 alongside market implied caplet normalvolatilities.

2.3 Short rate volatility as logistic function

We consider short rate volatility σrt in the following form:

σrt = σr0 +σrmax − σr0

1 + ek(T ?−t), k > 0, (2.4)

which monotonically increases from σr0 at t = 0 to σrmax at t→∞. T ? is the inflectionpoint, and interpreted as the moment when policy rate will no longer be repressedand be set freely (that is to say to be raised, we will assume σrmax > σr0 throughoutthe rest of the paper). Typical graph of function (2.4) is plotted on Figure 2.2.

4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 520

40

60

80

100

120

140

Calendar time (years)

BP

vol

Short rate volatility as Fermi function

k = 2k = 9

Figure 2.2: Short rate volatility as logistic function. σr0 = 20 bps, σrmax =120 bps, T ? = 2, k = 2, 9. At k → ∞, logistic function converges to the Heavi-side function.

5

Chapter 3

No-arbitrage bond pricing

3.1 Derivation

DMRV model is a particular case of a more general Gaussian model with threefactors, well described by James and Webber (2000) in Chapter 7. As such, it isan affine model, that is a model in which zero-coupon T -bond prices are given by

P (t, T ) = e−A(t,T )−B(t,T )rt−C(t,T )θt . (3.1)

By Ito’s formula, stochastic differential equation for bond price (3.1) is

dP (t, T ) =∂P

∂t−B(t, T )P (t, T ) drt − C(t, T )P (t, T ) dθt +

1

2B2(t, T )P (t, T ) d〈r〉t

+1

2C2(t, T )P (t, T ) d〈θ〉t +B(t, T )C(t, T )P (t, T ) d〈r, θ〉t ,

where 〈x〉t is the quadratic variation of a process xt and 〈x, y〉t is the quadraticcross-variation process of xt and yt.

We know that bond price is driftless in the risk-neutral measure:

0 = −At −Btrt − Ctθt −Bκ(θt − rt)− Cα(β − θt)

+1

2B2(σrt )

2 +1

2C2(σθ)2 +BCρσrtσ

θ,(3.2)

where for brevity A = A(t, T ), At = ∂A∂t

(t, T ) with equivalent expressions for B andC.

Equation (3.2) must be true for all values of rt and θt, therefore we obtain thefollowing system of ordinary differential equations for A, B and C:

At =1

2B2(σrt )

2 − Cαβ +1

2C2(σθ)2 +BCρσrtσ

θ, (3.3)

Bt = Bκ, (3.4)

Ct = −Bκ+ Cα. (3.5)

6

Boundary conditions are:

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

because bond prices pull to par at maturity: P (T, T ) = 1.ODEs for B(t, T ) and C(t, T ) can be solved analytically:

B(t, T ) =1

κ

(1− e−κ(T−t)

), (3.6)

andC(t, T ) =

1

α

(1− κ

κ− αe−α(T−t)

)+

1

κ− αe−κ(T−t). (3.7)

We give expression for A(t, T ) for the case when σrt is constant in Appendix A.Solution for A(t, T ) when σrt is a logistic function (2.4) involves confluent hy-

pergeometric functions for which fast, accurate and robust numerical procedure iscurrently not known to the author. However, it is easy to solve ODE for A(t, T )

numerically, for example with Matlab’s routine ode45.

Proposition 3.1.1. In DMRV model (2.1)—(2.3), strong solutions to short rate andmean reversion levels are

θt = θ0e−αt + β(1− e−αt) + σθ

∫ t

0

eα(s−t)dW θs , (3.8)

and

rt = r0e−κt +

κ(θ0 − β)

κ− α(e−αt − e−κt

)+ β(1− e−κt)

+κσθ

κ− α

∫ t

0

(eα(s−t) − eκ(s−t))dW θs +

∫ t

0

eκ(s−t)σrs dW rs .

(3.9)

Proof. In Appendix A.2.

Corollary 3.1.2. Short rate rt in DMRV model is a Gaussian normal variable withmean

Ert = r0e−κt +

κ(θ0 − β)

κ− α(e−αt − e−κt

)+ β(1− e−κt), (3.10)

and variance

var (rt) =

(κσθ

κ− α

)2 [1

2α(1− e−2αt) +

1

2κ(1− e−2κt)− 2

κ+ α(1− e−(κ+α)t)

]+

∫ t

0

e2κ(s−t)(σrs)2ds+

ρκσθ

κ− α

∫ t

0

(e(κ+α)(s−t) − e2κ(s−t))σrs ds.

(3.11)

7

If, in particular, short rate volatility is constant σrt = σr,

var (rt) =

(κσθ

κ− α

)2 [1

2α(1− e−2αt) +

1

2κ(1− e−2κt)− 2

κ+ α(1− e−(κ+α)t)

]+

(σr)2

2κ(1− e−2κt) +

ρκσθσr

κ− α

[1

κ+ α(1− e−(κ+α)t)− 1

2κ(1− e−2κt)

].

(3.12)

Proof. Immediate consequence of the fact that integrands in stochastic integrals in(3.9) are deterministic functions and Ito’s isometry:

var (rt) = E((rt − Ert)

2)

= E 〈r〉t .

Corollary 3.1.3. Long rate r∞ in DMRV model with constant short rate volatility isa Gaussian normal variable with mean

Er∞ = β, (3.13)

and variance

var (r∞) =

(κσθ

κ− α

)2 [1

2α+

1

2κ− 2

κ+ α

]+

(σr)2

2κ+ρκσθσr

κ− α

[1

κ+ α− 1

2κ

].

(3.14)

Proof. Setting t→∞ in (3.12).

3.2 Forward rate volatility as a function of model pa-rameters

Bond prices can be expressed via forward rates:

P (t, T ) = exp

(−∫ T

t

f(t, s) ds

),

where f(t, T ) is the instantaneous forward rate set at t for dealing at T .Forward rates are then:

f(t, T ) = −∂ lnP (t, T )

∂T.

8

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time to maturity (years)

dB/dT

kappa = 0.6kappa = 0.3

Figure 3.1: ∂BTt∂T

as a function of time to maturity.

Instantaneous forwards in DMRV model are:

f(t, T ) = aTt + bTt rt + cTt θt, (3.15)

whereaTt =

∂A

∂T(t, T ),

bTt =∂B

∂T(t, T ),

andcTt =

∂C

∂T(t, T ).

Equations for bTt and cTt have closed-form solutions:

bTt = e−κ(T−t), (3.16)

cTt =κ

κ− α(e−α(T−t) − e−κ(T−t)

). (3.17)

Equation (3.15) is linear in rt and θt and C∞-differentiable in t, therefore we canapply Ito’s formula:

df(t, T ) = O(dt) + bTt drt + cTt dθt

= O(dt) + bTt σrtdW

rt + cTt σ

θdW θt ,

(3.18)

where O(dt) is the drift term.

9

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


dC/dT

kappa = 0.6, alpha = 0.03kappa = 0.6, alpha = 0.3kappa = 0.3, alpha = 0.03

Figure 3.2: ∂CTt∂T

as a function of time to maturity.

Quadratic variation of f(t, T ) as a process in t is then

d〈f(t, T )〉 = (bTt )d〈r〉t + 2bTt cTt d〈r, θ〉t + (cTt )2d〈θ〉t

= (bTt σrt )

2 dt+ 2bTt cTt ρσ

rtσ

θ dt+ (cTt σθ)2 dt.

(3.19)

Let us denote

σTt =√

(bTt σrt )

2 + (cTt σθ)2 + 2bTt c

Tt ρσ

rtσ

θ, (3.20)

then another way of writing equation (3.19) is

d〈f(t, T )〉 = (σTt )2dt,

so instantaneous total volatility σTt of f(t, T ) can be seen as the “derivative” ofquadratic covariation w.r.t. time.

Apart from having plausible dynamics of short rate (which is automatic in DMRVmodel because it is built in), we would like to preserve humped shape of capletvolatility curve. As a first approximation to caplet volatilities we take instantaneousvolatilities of forwards (3.20).

Figures 3.1 and 3.2 show graphs of ∂B∂T

and ∂C∂T

as functions of time to maturityT − t with fixed maturity T = 10 and moving calendar time t.

Naturally, ∂B∂T

is just an exponential decay and thus no value of reversion speedκ can help us to achieve humped-shaped σTt . Behaviour of ∂C

∂Tis richer: it starts at

zero, reaches its maximum and then exponentially decays with speed dictated by

10

0 1 2 3 4 5 6 7 8 9 1070

80

90

100

110

120

130

140


BP

vol

Instantaneous forward rate volatility

kappa = 0.6, alpha = 0.03, short rate vol = 80 bpskappa = 0.6, alpha = 0.03, short rate vol = 140 bpskappa = 0.3, alpha = 0.03, short rate vol = 80 bps

Figure 3.3: Volatility of the instantaneous forwards σTt as a function of time tomaturity. Short rate volatility is constant. Volatility of the reversion level is 120basis points per year, correlation ρ is 0.4.

α, the reversion speed of the reversion level (we assume κ > α). With this complexbehaviour, it is plausible that one can find a combination of values κ and α, thatlead to humped-shaped instantaneous volatility curve.

Figures 3.3 and 3.5 show specimen instantaneous forwards’s volatility curvesfor various combinations of parameters. A small, but visible dip in σTt near originis caused by a particular combination of parameters in (3.20). Attempts to removeit by increasing short rate volatility σr greatly change the shape of the curve, so isnot available option if one wants to calibrate to at-the-money market volatilities (ofcaps and swaptions). Increasing σθ, the volatility of the mean reversion level doesremove the dip (Figure 3.4), but it also changes the shape of the instantaneousvolatility curve σTt . There are three major ways to combat this dip:

• reduce short rate volatility σr to a low level, for example 20 basis points peryear,

• use root-mean squared averages of instantaneous volatilities σT given byequation (3.21), this will “smooth out” the dip,

• use calendar time-dependant (deterministic) instantaneous volatility functionσrt , such as the logistic function (2.4).

11

0 1 2 3 4 5 6 7 8 9 1070

80

90

100

110

120

130

140


BP

vol


reversion level vol = 120 bpsreversion level vol = 140 bpsreversion level vol = 160 bps

0 1 2 3 4 5 6 7 8 9 1020

30

40

50

60

70

80

90

100

110


BP

vol


short rate vol = 80 bpsshort rate vol = 40 bpsshort rate vol = 20 bps

Figure 3.4: Dependency of the instantaneous forwards σTt on volatility of the rever-sion level σθ and volatility of the short rate σr. Short rate volatility σr is constant 80bps per year on the left-hand side graph and reversion level volatility σθ is constantat 120 basis points per year on the right-hand side, correlation ρ is 0.4 on bothcharts.

Recall the definition of root-mean square average of volatility σTT :

σ2T =

1

T

∫ T

0

(σTs )2ds. (3.21)

First approach is not very attractive because we are effectively “wasting” one statevariable. However, since market observable quantities are cap and swaption volatil-ities correspond to RMS volatilities (3.21), second approach is more natural. In fact,because the dip only affects short-tenor forward rates near origin, we can combineall three approaches together: use low short rate volatility near t = 0, use logisticfunction to bridge it with higher short rate volatility after initial “repressed” periodand calculate RMS average of such obtained σTt . As we shall see later in section6, this approach would give very good fit to at-the-money market volatilities (fromcaps and swaptions).

Figure 3.5 displays RMS volatility curves with various parameters.

12

0 1 2 3 4 5 6 7 8 9 1020

30

40

50

60

70

80

90

100

Maturity (years)

BP

vol

Root−mean squared forward rate volatility

kappa = 0.6, alpha = 0.03, short rate vol = 20 bpskappa = 0.6, alpha = 0.03, short rate vol = 20 bps−−80 bpskappa = 0.6, alpha = 0.03, flat short rate vol = 80 bps

Figure 3.5: RMS volatility of the instantaneous forwards σT as a function of maturityT . Short rate volatility given by logistic function (2.4). Volatility of the reversion levelis 121 basis points per year, correlation ρ is 0.4. Blue solid line corresponds to theblue line on Figure 3.3. Red dashed line corresponds to short rate volatility being“repressed” for 2 years at 20 basis points per year. The effect of this repressionon forward rates with long maturity T is muted. The dotted line represents RMSvolatility with constant short rate volatility of 20 basis points per year. Logisticfunction acts as a bridge between the two curves.

3.3 Correlation

Instantaneous volatilities of forward rates σTt in DMRV model have a rich depen-dency on correlation coefficient ρ. On Figures 3.6 and 3.7 we have graphs for σTtas a function of residual time to maturity T − t (and also as a function of calendartime t on Figure 3.7). Note that function σTt is time homogeneous when short ratevolatility σrt = const, that is map T 7→ σT+tt does not depend on t. It is not so whenσrt has calendar time dependency and we showed this as s series of charts withvarying t.

We shall see later in Section 5.3 how this correlation structure affects the corre-lation surface between Libor rates in LMM-DMRV model and how it brings calendartime dependency into it.

13

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120


BP

vol

Instantaneous forward rate volatility (flat short rate vol at 80 bps)

rho = 0.99rho = 0.5rho = 0rho = −0.5rho = −0.99

Figure 3.6: Effect of correlation coefficient ρ on instantaneous volatility σTt . Shortrate volatility is constant at 80 basis points and volatility of the reversion level is121 basis points per year.

14

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120


BP

vol

Instantaneous fwd rate vol (short rate vol islogistic fn 20−−80 bps, t = 0 years)


0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120


BP

vol

Instantaneous fwd rate vol (short rate vol islogistic fn 20−−80 bps, t = 1 year(s))


0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

Time to maturity (years)B

P v

ol

Instantaneous fwd rate vol (short rate vol islogistic fn 20−−80 bps, t = 1.5 year(s))


0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120


BP

vol

Instantaneous fwd rate vol (short rate vol islogistic fn 20−−80 bps, t = 1.75 year(s))


0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120


BP

vol



0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120


BP

vol



Figure 3.7: Dynamics of the σTt as a function of calendar time t and residual time tomaturity T−t when short rate volatility is a logistic function (2.4). All parameters areidentical to that of Figure 3.6. Different charts show snapshots taken at calendartime points tstart ∈ 0, 1, 11/2, 13/4, 2 and 4 years, and the functions are σT+tstarttstart withT − tstart on the horizontal axis.

15

Chapter 4

Principal components

To establish more knowledge of the dynamics of the yield curve in the DMRVmodel, we shall analyse its principal components. Principal component analysis,or PCA is a traditional tool in fixed income modelling to study the driving factorsbehind yield curve moves, typically parallel shifts, tilts and changes of curvature.1

In other words, since short-rate models are specified in terms of dynamics of theshort-rate, which is not directly observable one needs to transform it into move-ments of a more observable quantity, for example bond yields.

Recall that yield y(t, T ) at time t of a zero-coupon bond maturing at T is

y(t, T ) = − lnP (t, T )

T − t, (4.1)

that is to say the yield is the discount rate which reprices the bond today (or at agiven point in the future before bond’s maturity).

For PCA, we need to fix a set of bond maturities, say 6 months, 1 year, 2 yearsand so on:

T = Ti : i = 1, 2, . . . , n.

We then discretise calendar time t, simulate short rates rt and reversion levelsθt, calculate bond yields, calculate covariance matrix of changes in yields and thenrun principal component analysis on this matrix (for covariance matrices this isequivalent to finding its eigenvalues and eigenvectors).

Figures 4.1 and 4.2 show simulated short rates with weekly time steps for 30years (and with daily steps in Figure 4.2).

1 These movements can be interpreted as frequencies in Fourier series expansion, with parallelshift being lowest frequency and curvature highest. Even higher order frequencies can also bestudied, but often ignored as noise. This is arguable for fourth and fifth principal components, butis certainly true for higher order terms.

16

0 5 10 15 20 25 30−4

−2

0

2

4

6

8

10

12

14Short rate


Per

cen

t

0 5 10 15 20 25 30−5

0

5

10

15

20

25Short rate

Calendar time (years)P

er c

ent

Figure 4.1: Simulation of short rates. Model parameters: short-rate volatility is alogistic function with σr0 = 20 bps, σrmax = 120 bps, k = 4 and T ? = 1.8 years, mean-reversion level volatility is constant σθ = 133 bps, initial short rate r0 = 1.25% andinitial reversion level θ0 = 3.5%, reversion level of the mean reversion β = 9.5%,reversion speeds: for short rate κ = 0.3, for reversion level α = 0.0561. Theseparameters are typical when calibrating to USD market data of late 2012. Bothgraphs on the left- and right-hand side are simulated using Brownian motion driverswith the same seed.

We take 10 maturities for our PCA:

T = 6m, 1y, 2y, 3y, 5y, 7y, 10y, 15y, 20y, 30y,

and discretise calendar time in weekly time steps:

0 = t0 < t1 < . . . < tN = 30y.

On each time step tj we calculate bond yields for bonds with constant maturity:

y(tj, tj + Ti) = Atj+Titj +B

tj+Titj rtj + C

tj+Titj θtj , ∀Ti ∈ T ,

and calculate changes in yields:

Y = y(tj, tj + Ti)− y(tj−1, tj−1 + Ti)|Ti ∈ T , 1 6 j 6 N , (4.2)

where Y is the matrix with bond maturities Ti as rows and calendar time steps tjas columns.

17

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

−1

0

1

2

3

4

5

6

7Short rate


Per

cen

t

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4

−2

0

2

4

6

8

10Short rate


er c

ent

Figure 4.2: The same simulation as in Figure 4.1, zoomed in to 5 years calendartime. One can clearly see the effect of the logistic function on short-rate volatilityin about 2 years’s time.

Of course, (4.2) is just a discretisation of the stochastic differential equation: 2

dyit =

(∂at+Tit

∂t+∂bt+Tit

∂trt +

∂ct+Tit

∂tθt

)dt+ bt+Tit drt + ct+Tit dθt,

where for brevity

aTt =ATtT − t

,

bTt =BTt

T − t,

and

cTt =CTt

T − t.

General form of the equation above may be complicated, but since we areonly interested in covariances between changes in yields, we shall just calculatequadratic cross-variations between yields as stochastic processes:

d⟨yi, yk

⟩t

= bTi bTkd〈r〉t + cTi cTkd〈θ〉t + bTi cTkd〈r, θ〉t + cTi bTkd〈θ, r〉t , (4.3)

where bt+Tt and ct+Tt were shortened to bT and cT , since they only depend on resid-ual time to maturity, and not on calendar time.

2 Which is an application of the Ito’s formula to equation (4.1). We can use Ito’s formula becauseexpression for yields y(t, T ) is linear in rt and θt and is continuously differentiable in t.

18

The matrix we are interested in, Y , consists of integrated cross-variances:

Yik =1

T

∫ T

0

d⟨yi, yk

⟩t. (4.4)

For example, for standard Vasıcek model, this matrix is

Yik = bTi bTj(σr)2,

or in matrix form,Y = bb> (σr)2,

where b is the column-vector with values bTi.For this matrix, the single non-zero eigenvalue is

λ = ‖b‖2 (σr)2,

where‖b‖2 = b>b.

Corresponding eigenvector is:b

‖b‖,

which has exponentially decaying shape.To find the eigenvalues of (4.4) for DMRV, we need to arrive to instantaneous

cross-variances d⟨yi, yk

⟩t

differently. First, let us orthogonalise Brownian motionsin the formula for dyit:

dyit = O(dt) +

(bTiσrtcTiσθt

)>d

(W rt

W θt

)= O(dt) +

(bTiσrtcTiσθt

)>(1 0

ρ√

1− ρ2

)d

(W 1t

W 2t

)= O(dt) +

(bTiσrt + ρcTiσθtcTiσθt

√1− ρ2

)>d

(W 1t

W 2t

),

where Brownian motions W 1 and W 2 are independent:

d⟨W 1,W 2

⟩t

= 0.

Define c as a column-vector with CTi0

Ti, then instantaneous quadratic cross-variances

for yields is thus (in formal vector form)

d⟨y,y>

⟩t

=(σrt b + ρσθc

√1− ρ2σθc

)(σrt b> + ρσθc>√1− ρ2σθc>

)dt,

19

where instantaneous covariation matrix is square and has the same dimensions asthe number of maturities we are interested in. Because of its particular form, theeigenvalues are easy to calculate analytically (although expressions involved arerather cumbersome):

λ1,2 =1

2

(b>b + c>c±

√(b>b)2 + (c>c)2 − 2b>bc>c + 4(b>c)2

)dt, (4.5)

where for simplicityb = σrt b + ρσθc,

andc =

√1− ρ2σθc.

Interpreting these factors in general form is difficult, but we can illustrate themvia a numerical experiment.

On Figure 4.3 we have eigenvalues and corresponding eigenvectors for DMRVmodel with the same parameters as in simulation on Figure 4.1, which are typi-cal when calibrating LMM-DMRV model3 to at-the-money volatilities of USD capand swaption markets of late 2012. Second principal component (represented bygreen line in figure 4.3) distributes shocks to the curvature of the yield curve andin this particular example controls approximately 14 per cent of the total variationof changes in yields. In other words, DMRV can be though of as a “1.5” factormodel, rather than full two factor model, such as additive 2-factor Hull-White or2-factor Vasıcek G2++ in terminology of Brigo and Mercurio (2006). Let this notupset the reader because DMRV explains and is good at incorporating businesscycles and central bank actions as we shall see in the next subsection. Also, inorder to get a hump-shaped cap volatility curve, we do not have to impose a rathercounter-intuitive negative correlation between factors. As we shall see, correlationρ in DMRV stays between 0.2 and 0.5 to give good fit to USD cap vols.

Balduzzi et al. (1996) and Chen (1996) study three-factor models, similar toDMRV but richer to include stochastic variance as a third driving Brownian mo-tion.4 Unfortunately, neither model admits closed-form solutions for bond prices (infunctions that are fast to compute) and also pose restrictions on model parameters(such as correlation between short rate and its dynamic reversion level must bezero).

3 That is the LMM model with forward rate volatilities given by DMRV model, equation (3.20).We study this model in section 5.

4 Balduzzi et al use Gaussian family of affine models, while Chen’s model is from the CIR(Cox-Ingersoll-Ross) family.

20

0 5 10 15 20 25 30−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Maturity (years)

Principal components for changes in yields

1st2nd3rd

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−6 Eigenvalues

Figure 4.3: Eigenvectors and eigenvalues in DMRV model. All parameters arethe same as in Figure 4.1. Eigenvalues from third are zeroes and so the thirdeigenvector is already noise, rather than curvature.

4.1 Business cycles

Models with dynamic mean reversion level can be used to model business cycles.James and Webber (2000) have a good overview of various models available forthis purpose. In particular, in Section 11.3.2 they consider generalised Vasıcekmodel:

drt = κ(θt − rt)dt+ σrdW rt ,

dθt = α(prt + (1− p)β − θt)dt+ σθdW θt ,

they also show how this can lead to chaotic behaviour in short rate even in deter-ministic case (σr = 0 and σθ = 0).

Shiryaev (1998) in §4a.4 (p. 340) derives a similar model (although non-linear)considering short rate as the conditional mathematical expectation of an unob-servable “economy internal state” process Xt, consisting of a Markov chain withtwo values: “expansion” and “contraction”, and a noise term. See also Diebold andRudebusch (1994) for a survey of economic papers on business cycle modelling.

Beaglehole and Tenney (1991) and Dai and Singleton (2000) have a very gooddescription of dynamic mean-reversion models, including non-linear ones.

Modern developments in this field include papers by Diebold and Li (2006) forextension of 3-factor Nelson-Siegel model controlling parallel moves, changes in

21

0 5 10 15 20 25 300.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45Eigenvectors of DMRV with 0 short rate vol

Maturity (years)

Figure 4.4: Eigenvector of DMRV with 0 short rate vol. Random shocks are deliv-ered mostly from middle to the long end of the yield curve. Volatility of the meanreversion level is constant at 133 basis points. Reversion speeds: for short rateκ = 0.3, for reversion level α = 0.0561.

slope and curvature of bond yield curve, and Christensen et al. (2008) for non-arbitrage version of it.

Andersen and Piterbarg (2010b) in Chapter 12.1.4, p. 497, suggest that whencentral bank actions become predictable, this can be incorporated in a DMRVmodel by setting short rate volatility to 0. Note that in this case, bond yield co-variance matrix (4.4) is simple and equation (4.5) has a single eigenvalue: 5

λ0 = ‖c‖2 (σθ)2dt,

where‖c‖2 = c>c.

Corresponding eigenvector is justc

‖c‖,

it has a humped shape, peaked around 10-15 years (naturally, dependent on modelparameters) and its specimen is shown on Figure 4.4. Figure 4.5 shows simulationof short rates with no short rate volatility, one can clearly see long periods of similarvalues in rt with no short-term “noise”.

Lastly in this section, let us look at the dynamics of long-term yields:5 We consider DMRV with constant short-rate volatility here: σrt = 0 for all t > 0.

22

0 5 10 15 20 25 300

2

4

6

8

10

12Short rate


Per

cen

t

0 5 10 15 20 25 30−5

0

5

10

15

20Short rate


er c

ent

Figure 4.5: Simulation of short rates with no short rate volatility. Volatility of themean reversion level is constant at 133 basis points. Reversion speeds: for shortrate κ = 0.3, for reversion level α = 0.0561. Reversion level of the mean reversionβ = 9.5%.

Proposition 4.1.1. The long yield y∞ in DMRV model, that is the limit in (4.1) withT →∞ is

y∞ = β − (σrmax)2

2κ2− (σθ)2

2α2− ρσrmaxσ

θ

κα. (4.6)


The fact that long yield is a constant is of course no surprise. Dybvig et al.(1996) have proven it in a very general setting.

23

Chapter 5

Libor Market Model with DMRVforwards

5.1 Basic setup

Our main model is LMM-DMRV model, that is a hybrid of Libor market model, butwith forward rate volatilities given by short-rate model DMRV, formula (3.20).

We take a filtered probability space (Ω,F ,F,P), where Ω is the sample space,F the σ-algebra of measurable sets in Ω interpreted as “events”, P the probabilitymeasure and F the filtration, or the set of non-decreasing sub-σ-fields:

F = (Ft)t>0, Fs ⊆ Ft ⊆ F ,

for 0 6 s 6 t <∞. We assume the usual conditions for F.Measure P is called the objective measure, but it will not be used directly any-

where in this paper (apart from its existence and full mass: P(Ω) = 1, we do notrequire anything from it).

Key modelling objects in the Libor market model are tenor dates1,

0 = T0 < T1 . . . < TN <∞,

and Libor rates, discrete forward rates set in arrears at Ti and paid in advance atTi+1:

Li(t) =1

τi

(P (t, Ti)

P (t, Ti+1)− 1

), (5.1)

here τi is the coverage, or day-count fraction between Ti and Ti+1 (with floating legconvention, typically “Act/360”).

1 Libor fixing dates, reset times, . . .

24

In a general LMM setup, Libor rates L1(t), L2(t), . . . , LN(t) are individually mod-elled as stochastic processes:

dLi(t) = O(dt) + σi(t, Li)>dW i

t, (5.2)

here O(dt) are drifts (we will calculate them later), σi(t, Li) is the vector of Li-bor volatilities and W i

t is the vector of driving P-Brownian motions adapted to thefiltration F. Initial conditions for equations (5.2) are to reproduce the initial (deter-ministic) yield curve:

Li(0) = L?i .

We will be working with two types of local volatility functions:

• shifted log-normal, σi(t, Li) = (1 + τiLi)σf (t, Ti),

• Gaussian, σi(t, Li) = σf (t, Ti),

where σf (t, T ) is the vector of volatilities of instantaneous forward rates f(t, T ).These two sub-models are similar and with a relatively high level of shift in the

former, both produce distributions for Libor rates which are close to Gaussian. Ourcalibration experiment is done on the Gaussian model and therefore we need totheoretically justify it first. Let us begin with a link between short-rate models andLibor market model.

5.2 LMM and HJM volatility

Recall a useful formula (14.11) on p. 598 in Andersen and Piterbarg (2010b),linking Libor market model and instantaneous forward volatilities (Heath-Jarrow-Morton framework):

σi(t, Li) =1

τi(1 + τiLi(t))

∫ Ti+1

Ti

σf (t, u)du, (5.3)

where σf (t, T ) is the vector of volatilities of instantaneous forward rates f(t, T ).Recall formula (3.15) for instantaneous forward rates in DMRV model:

f(t, T ) = aTt + bTt rt + cTt θt,

where xTt is a symbolic shorthand for xTt = ∂X∂T

(t, T ).

25

By Ito’s formula, the dynamics of f(t, T ) is:

df(t, T ) = O(dt) + bTt drt + cTt dθt

= O(dt) + bTt σrtdW

rt + cTt σ

θdW θt ,

(5.4)

where O(dt) is the drift term which we don’t need now.We now can proceed to finalise the set up of LMM-DMRV model. Because

we know explicitly how forward rates evolve, and we know that they are driven bytwo factors, our version of LMM is going to have two factors. This naturally limitspossible shapes of correlation matrices we can have in our model, but instead wegain knowledge about financial side (rate repression and eventual exit).

Vector σf (t, T ) isσf (t, T ) = (bTt σ

rt , c

Tt σ

θ)>.

The first approximation we make is to treat function σf (t, T ) piecewise-constantbetween two fixing dates, in which case equation (5.3) becomes

σi(t, Li) = (1 + τiLi(t))σf (t, Ti).

This is the shifted log-normal formulation of local volatility. Andersen and Piter-barg (2010b) prove in their Lemma 14.2.5 that this formulation of LMM has strongsolutions Li(t) and that forward measures QTi are theoretically sound (that is theyare equivalent martingale measures, or that bond prices are positive almost surely).

For calibration we will use the Gaussian formulation of σi(t, Li), that is we aregoing to approximate discrete forward rate with instantaneous forward rate of thesame expiry,

σi(t, Li) = σf (t, Ti). (5.5)

Assuming forward rates below 10%2 and 3-month3 floating frequency, shiftedlog-normal distribution is close to Gaussian, with advantage of the Gaussian modelin calibration to swaptions. However, because in a model with constant local volatil-ity, rates are unbounded from below, we need to say a few words about the proba-bility space where this model operates.

2 After all, we are modelling very low interest rates.3 Default frequency of floating rate payments in USD markets is quarterly and hence default

tenor for Libor rates is 3 month.

26

A note on the sample space

We need to make sure that bond prices in LMM-DMRV model with Gaussian volatil-ity are positive so that we can use discount bonds as numeraire assets. Unfortu-nately, this is not the case in general, so a restriction of the sample space is re-quired: with Gaussian model we shall be working in the sub-space (Ω0, F , F, P),where

Ω0 =

ω ∈ Ω : Li(t, ω) > − 1

τi+ ε0, ∀t,∀i

, (5.6)

where ε0 is a small positive constant, Li(t, ω) is the Libor rate, and τi is its coverage,or day-count fraction.4 The other parameters are restrictions of their respectivecounterparts to Ω0:

F = A ∈ F : A ⊆ Ω0 ,

F = (Ft)t>0 Ft = Ft

⋂F ,

P(A) = P(A|Ω0).

Proposition 5.2.1. Space (Ω0, F , F, P) is a filtered probability space satisfying theusual conditions.


Proposition 5.2.2. Assume coordinate-wise Lipschitz continuity and linear growthfor Gaussian volatility function

σi = σf (t, Ti),

for all i. Then the system (5.2) has unique, non-explosive and t-continuous so-lutions. Moreover, when solutions Li(t) are considered on restricted probabilityspace (Ω0, F , F, P), then bond prices are strictly positive and solutions exist in allforward measures QTi.


5.3 Correlation

The vector W = (W 1,W 2) of driving Brownian motions in the LMM-DMRV modelis two-dimensional, with correlations given as part of the DMRV model setup:⟨

W 1,W 2⟩t

= ρt. (5.7)

27

0 2 4 6 8 10

0

5

100.7

0.75

0.8

0.85

0.9

0.95

1

Correlation matrix for Libor rates

Time to maturity (years)0 0.5 1 1.5 2 2.5 3

0

1

2

30.75

0.8

0.85

0.9

0.95

1

Correlation matrix for Libor rates


Figure 5.1: Specimen of correlation structure ρij(0) in LMM-DMRV model. Right-hand side chart is the same structure zoomed in to 3 years to maturity. “Long”Libor rates are nearly perfectly correlated at t = 0, while rates with shorter expiriesdecorrelate quite fast. Model parameters: σθ = 133 bps, σrt is a logistic functionwith σr0 = 20 bps, σrmax = 120 bps, k = 4 and T ? = 1.8 years. Reversion speeds: forshort rate κ = 0.3, for reversion level α = 0.0561.

Consider the Gaussian version of LMM-DMRV model. First, let us calculateinstantaneous covariances between two Libor rates, Li(t) and Lj(t). This can bedone easily with the usual rules on quadratic cross-variation between Ito integralsfrom equations (5.2), (5.5) and (5.7):

d〈Li(t), Lj(t)〉 = bitbjt(σ

rt )

2dt

+(bitc

jt + citb

jt

)ρσrtσ

θdt

+ citcjt(σ

θ)2dt

def= Λij(t)dt,

(5.8)

where

bit = bTit =∂BTi

t

∂T= e−κ(Ti−t),

and

cit = cTit =∂CTi

t

∂T=

κ

κ− α(e−α(Ti−t) − e−κ(Ti−t)

).

4 This restriction for USD markets turns out to be rather loose: in this market typically τ = 1/4,so the additional requirement on Libor rates: Li > −400% is not very restrictive.

28

Instantaneous correlation coefficient between the two Libor rates is therefore:

ρij(t) =Λij(t)√

Λii(t)Λjj(t). (5.9)

Figures 5.1 and 5.2 show charts of ρij(t) with typical model parameters forvarious t. Note how long-term Libor rates initially are almost perfectly correlated,but become decorrelated with time because of strong calendar time dependencyof σrt .

29

02

46

810

02

46

8100.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Correlation matrix for Libor rates, t = 0m

Time to maturity (years)0

0.51

1.52

2.53

0

1

2

30.7

0.75

0.8

0.85

0.9

0.95

1

Correlation matrix for Libor rates, t = 0m


24

68

10

24

68

10

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Correlation matrix for Libor rates, t = 1y


1.52

2.53

3.54

1

2

3

40.7

0.75

0.8

0.85

0.9

0.95

1



24

68

10

24

68

10

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Correlation matrix for Libor rates, t = 1y6m

Time to maturity (years)1.5

22.5

33.5

44.5

1.52

2.53

3.54

4.50.7

0.75

0.8

0.85

0.9

0.95

1

Correlation matrix for Libor rates, t = 1y6m


24

68

1012

24

68

10120.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1



2.53

3.54

4.55

2

3

4

50.7

0.75

0.8

0.85

0.9

0.95

1



46

810

12

46

810

12

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1



3.54

4.55

5.56

3

4

5

60.7

0.75

0.8

0.85

0.9

0.95

1



Figure 5.2: Dynamics of correlation matrix ρij(t) with t = 0, 1, 11/2, 2 and 3 years.We use rolling maturity so that time to maturity remains fixed on all charts. Modelparameters are the same as on Figure 5.1.

30

Chapter 6

Calibration

Let us first describe calibration instruments and their LMM-DMRV model impliedprices and volatilities.

6.1 Caps

In LMM, model price of a T -cap struck at K is1

n−1∑i=1

P (0, Ti+1)τiEi+1(Li(Ti)−K)+, (6.1)

where τi is the day-count fraction (or coverage) of the i-th caplet2 and Li is theLibor rate set at Ti and paid at Ti+1 (summation starts from 1 because there isalmost no variation in L0: T0 is typically only 2 business days from today), n is suchthat Tn = T and Ei+1 is the mathematical expectation operator under Ti+1-forwardmeasure (that is the measure with Ti+1-bond as numeraire).

Andersen and Piterbarg (2010b) show in Proposition 14.4.1 that Li is driftlessunder QTi+1

, or Ti+1-forward measure, and has the dynamics:

dLi(t) = ‖σi(t, Li)‖ dY i+1t , (6.2)

where Y i+1 is a Brownian motion in measure QTi+1.

Because of Gaussian dynamics of Libor rates, we can’t use Black’s formula forexpectation in (6.1). However, Bachelier (1900) provides us with such a formula:

E(F + σWT −K)+ = (F −K)Φ (d) + σ√Tϕ (d) , (6.3)

1 See for example Proposition 14.4.1 in Andersen and Piterbarg (2010b), p. 615.2 That is τi is the day-count fraction between Ti and Ti+1 with floating payment day-count

convention, “Act/360” in USD markets.

31

where F is forward, d is option’s moneyness:

d =F −Kσ√T,

and ϕ and Φ are respectively standard normal probability density and cumulativedistribution functions:

ϕ(x) =1√2π

e−x2

2 ,

andΦ(x) =

∫ x

−∞ϕ(u)du.

Proposition 6.1.1. In LMM-DMRV model, Ti-caplet price struck at K is given bythe formula:

Ei+1(Li(Ti)−K)+ = (Li(0)−K)Φ (d) + σi√Tiϕ (d) , (6.4)

whered =

Li(0)−Kσi√Ti

,

and

σ2i =

1

Ti

∫ Ti

0

(σTis )2 ds.


RMS volatilities in LMM-DMRV with calendar-time dependent instantaneousforward volatilities σTt can be computed using a numerical scheme, for exampleGaussian quadrature. When this dependency is a logistic function (2.4), closed-form solutions are available but they involve confluent hypergeometric functionsand therefore have limited practical use.

For our calibration we are going to minimise the following functional

Ccapdef=

N∑i=2

(σi − σMKTi )2 → min, (6.5)

where N is such that TN is the maturity of the longest market quoted cap andσMKTi is the market implied Bachelier volatility3 of i-th caplet. Caplet volatilities are

obtained by the usual bootstrap procedure described in Section 6.4.3 by Brigo andMercurio (2006).

3 Also known as basis point vol, Black (normal) vol or Gaussian vol.

32

6.2 Swaptions

Let us begin with a quick introduction of swap rate and its decomposition into alinear combination of Libor rates.

Coupon rate for a swap starting at Tα with final payment at Tβ is defined as

Sα,β(t) =P (t, Tα)− P (t, Tβ)

Aα,β(t),

where Aα,β(t) is the annuity4

Aα,β(t) =

β∑j=α+1

δjP (t, Tj).5

By definition of Libor rate, formula (5.1):

P (t, Tα)− P (t, Tα + τα) = ταLα(t)P (t, Tα + τα),

that is to say:

P (t, Tα)− P (t, Tβ) =

β−1∑i=α

τiLi(t)P (t, Ti+1).6,7

Now swap rates allow simple decomposition as weighted sum of Libor rates:

Sα,β(t) =

β−1∑i=α

wi(t)Li(t), (6.6)

wherewi(t) =

τiP (t, Ti)

Aα,β(t).

Although wi(t) is stochastic, its variation 〈wi〉t is assumed to be small. Forcalibration purposes we will make this approximation:

Sα,β(t) ≈β−1∑i=α

wi(0)Li(t). (6.7)

4 Level, swap PV01, swap duration, . . .5 By round deltas δj we shall denote year fractions under the swap’s fixed leg convention:

day-count rule, business day convention and payment frequency. τi will refer to floating leg’s yearfractions.This can cause some nuisance, but is necessary to get right if one wants to implementaccurate calibration.

6 We allow for slight abuse of notation here: Ti denote payment dates of the floating leg (withindexing of the floating leg), while Tj are the fixed leg payment dates, so T1 when i = 1 is not thesame as T1 when j = 1. We shall be clear on which date schedule is used where lest confusionarises.

7 Also with slight abuse of terminology, β shall indicate the payment index of the last paymentof the swap, potentially different on the fixed and floating sides.

33

This function is linear in Li(t) so we can apply Ito’s formula to both sides of theequation above to get approximately:

dSα,β(t) ≈β−1∑i=α

wi(0)dLi(t),

with approximate quadratic variation

d〈Sα,β〉t ≈β−1∑i,k=α

wi(0)wk(0)d〈Li(t), Lk(t)〉 .

We know (Lemma 4.2.4 by Andersen and Piterbarg (2010a)) that there exists aP-equivalent martingale measure8 in which Aα,β is a numeraire asset and thereforeSα,β is a martingale (as a combination of traded assets discounted by numeraire).Denote this measure QA and let EA be the corresponding mathematical expectationoperator.

A European payer swaption has payoff

(Sα,β −K)+,

so it’s an option to pay fixed rate K and receive floating (Libor).By the change of numeraire technique, no-arbitrage price of such option is

Aα,β(0)EA(Sα,β −K)+. (6.8)

To value this integral we need know the distribution of Sα,β(Tα) (we already knowit has mean zero, thanks to martingale property). This is a complex and potentiallynon-elementary distribution, but we know that approximate process Sα,β(t) hasGaussian marginals:

dSα,β =

β−1∑i,k=α

wi(0)dLi(t),

this is due to Gaussian distribution of increments Li(t+ s)−Li(t) and deterministicvolatility coefficient of dSα,β (note that neither of this is true for the original dSα,β).

The last approximation we are going to make in order to be able to quicklycalibrate to the swaption volatility surface is that Sα,β is also a martingale underQA. In this case, the QA-distribution of Sα,β(Tα) is Gaussian with mean 0 andvariance

⟨Sα,β

⟩Tα

.

8 Here as always, P is the objective probability measure.

34

We can now use Bachelier’s formula to approximate integral (6.8):

Aα,β(0)EQA(Sα,β(Tα)−K)+

≈ Aα,β(0)(

[Sα,β(0)−K]Φ(d)− σα,β√Tαφ(d)

),

(6.9)

whereσα,β = 〈Sα,β〉Tα ,

andd =

Sα,β(0)−Kσα,β√Tα

.

Of course, for analytical tractability we are going to use approximate variance⟨Sα,β

⟩Tα

, rather than exact 〈Sα,β〉Tα.9

Volatility for purposes of Bachelier formula is then

σα,β =

√1

Tα

⟨Sα,β

⟩Tα, (6.10)

with ⟨Sα,β

⟩Tα

=

β−1∑i,k=α

wi(0)wk(0)

∫ Tα

0

Λik(s)ds, (6.11)

where Λik(s) is the instantaneous covariance defined in equation (5.8).This is the volatility we are going to use for minimisation of the calibration func-

tional Cswpn(S ):Cswpn(S )

def=∑i∈S

(σi − σMKT

i

)2 → min, (6.12)

where S is the set of calibration swaptions, and σMKTi their market implied basis

point vols.10

Another way of computing the calibration functional is to take model impliedBlack (log-normal) volatilities for swaptions and compare them with market impliedBlack volatilities. One could argue that this would introduce unnecessary numericalerror from the conversion (from model’s natural quantity, Gaussian vol, into log-normal vol); we do not pursue this approach in our experiment.

9 It would be desirable to know 〈Sα,β〉Tαexactly, of course, because in this case the only approx-

imation we will be making is about marginal distributions of Sα,β and not about its variance. Optionsat or near the money depend on variance more than on the shape of the probability distribution.

10 These are the volatilities that if used in Bachelier formula, would give current market swaptionprices.

35

6.3 Calibration results

We calibrate LMM-DMRV model to at-the-money US Dollar caps and swaptions on7-July-2012, 13-Sep-2012, 7-Mar-2013 and 8-Mar-2013.

Our strategy for calibration problem is to use multidimensional Levenberg-Marquardtmethod to minimise cap calibration functional Ccap from a cold start11, then alsousing Levenberg-Marquardt method proceed to minimisation of Cswpn(S ) from thepoint obtained on the cap calibration step.

More formally, we proceed as follows

1. With multidimensional Levenberg-Marquardt method, find

(κc, αc, βc, σc0, σ

cmax, σ

θc , ρc, T

?c ) = arg min Ccap(κ, α, β, σr0, σ

rmax, σ

θ, ρT ?, k),

2. fix a set of calibration swaptions S , we used the following set of at-the-moneyswaptions: 12

S =

expiry\tenor 1y 2y 3y 4y 5y 7y 10y1y x x x x x x x2y x x x x x3y x x x x x4y x x x x x5y x x x x x7y x x x x x10y x x x x

3. using this point as initial guess, find

(κs, αs, βs, σs0, σ

smax, σ

θs , ρs, T

?s ) = arg min Cswptn(S , κ, α, β, σr0, σ

rmax, σ

θ, ρT ?, k).

Note that we do not calibrate k, parameter of the logistic function responsiblefor the “speed of loosening”, that is how fast short rate volatility rises from σr0 toσrmax around T ?. We prefer to keep this parameter fixed, since it is a purely modelparameter with no direct impact on the dynamics of Libor rates. The risk to thisparameter however, is useful and should be calculated, because it shows sensitiv-ity of swaption value (or any other fixed income product we wish to price with thuscalibrated LMM-DMRV) to how fast central bank policy will be changed.

When doing the minimisations, we also enforce a number of relationships andconstraints:

11 That is without a good initial guess.12 There is nothing particularly special about this set of instruments except that it is reasonably

global, that is representative of the entire swaption volatility surface up to 10 years of expiry, andcovers most of liquid points.

36

• κ > α,

• σr0 < σrmax,

• σr0 < σθ,

• 0.2 6 ρ 6 0.5.

This is done with model’s explanation power in mind. Our goal is not goodness of fit(standard LMM with piecewise constant local volatility with parametric correlationwill probably calibrate a lot better), but having control over “financial side” of things.

Note that we do not use constrained optimisation, but rather use functionalrelationships:

x > y ⇔ x = y + z2,

anda 6 x 6 b⇔ x = a+ (b− a) cos2 ϕ.

As with any other LMM, computational bottleneck of the calibration routine isthe calculation of covariance integrals∫ Tα

0

Λij(s)ds,

and for LMM-DMRV this is made worse by having to calculate them numerically(recall again that as it stands, there is no fast and robust way of calculating conflu-ent hypergeometric functions that are necessary to integrate the logistic function).Obviously, parts of these integrals that are common for different swaptions can becached in memory, but unfortunately on each iteration we will have to empty thecaches because the model parameters would change.

Non-global algorithm, such as the “cascade” described by Brigo and Mercurio(2006) in section 7.4, is obviously not suitable for LMM-DMRV because it has toofew parameters and each parameter has non-local impact.

By and large, computation time for LMM-DMRV calibration is acceptable andfrom a cold start on a modern laptop computer, it takes about 2 minutes to calibrateto the set of swaptions described above.13

13 Needless to say that calibration to the cap volatilities is almost instant, because of linearcomputational complexity.

37

6.4 Data

We used end-of-day quotes14 for at-the-money USD cap/floor and swaption volatil-ity as published by ICAP on their Reuters screen (VCAP23 for caps and floors andVCAP21 for swaptions). USD swap rates for initial yield curve construction wereobtained from the end-of-date database of one investment bank in London.

Calibrated model parameters are presented below in Table 6.1. A few thingsstand out: firstly, calibration is quite robust (even unaided by special methods suchas Tikhonov regularisation) and parameters are quite stable over time. Secondly,repression time horizon shrinks over time, which only adds support to the model’sability to explain things, not just reproduce market data. Thirdly, we included a veryshort period of one day between 7th and 8th of March 2013 when USD marketanticipated an important labour statistic announcement: non-farm payroll reporton the 8th of March 2013. The report came particularly optimistic and somewhatreduced US unemployment 15, which brought, according to the model, market ex-pectation of a possible “exit” date from short rate repression almost 3 months closer(from about 2 years and 2 month to 1 year and 11 months).

Market date κ α β σr0 σrmax σθ ρ T ? k

6 -Jul-2012 0.2369 0.0505 9.50 0.20 0.83 1.38 0.4719 2.3187 413-Sep-2012 0.1573 0.0789 9.50 0.20 0.80 1.75 0.4523 2.2401 47-Mar-2013 0.2414 0.0610 9.50 0.20 1.04 1.43 0.4004 2.1649 48-Mar-2013 0.2429 0.0632 9.50 0.20 1.02 1.45 0.4141 1.9068 4

Table 6.1: Calibrated model parameters. All columns except for β, σr0, σrmax and σθ

are in units, unscaled. Aforementioned columns are in per cent.

14 Snapshots were taken around 21 p.m. London time each day, to include as much of New Yorktrading hours as possible.

15 Recall that Federal Open Market Committee tied monetary policy to unemployment and infla-tion.

38

Expiry \Tenor 1Y 2Y 3Y 5Y 7Y 10Y1Y 4.0987 10.6167 8.7321 2.7828 −3.4937 −11.61332Y 4.7916 6.3448 2.4611 −3.9022 −5.9246 −10.51573Y 5.2758 0.3281 −0.3879 −3.2267 −4.8194 −5.32125Y −3.5097 −4.6805 −1.6920 −2.0600 0.0360 −0.58567Y 0.9954 1.0473 0.3212 3.2225 3.5115 2.7505

10Y 5.0787 8.2367 8.7169 7.4360 7.4113 5.9300

Table 6.2: Calibration errors for swaptions, in basis points. Volatility surface from13-Sep-2012.

Expiry \Tenor 1Y 2Y 3Y 5Y 7Y 10Y1Y 30.2669 31.6864 40.6199 57.2382 70.5528 84.20182Y 40.0084 45.4308 55.1823 70.3975 78.0412 86.68653Y 57.4644 67.0851 71.8554 80.7340 85.8169 88.19395Y 86.5049 90.1151 89.1965 92.0986 90.9380 90.67337Y 90.3680 91.5012 93.1470 91.0878 90.3534 88.8343

10Y 90.3671 87.6183 87.1328 87.6913 86.1965 84.3071

Table 6.3: Market implied basis point volatilities of swaptions. Volatility surface from13-Sep-2012.

1Y3Y5Y7Y10Y

15Y20Y

25Y30Y

1Y3Y

5Y7Y

10Y

15Y

20Y

25Y

30Y

30

40

50

60

70

80

90

100

Expiry

Swaption ATM VSurf (Normal)

Tenor

BP

vol

1Y

3Y

5Y

7Y

10Y

1Y

3Y

5Y

7Y

10Y

−10

−5

0

5

10

Tenor

Accuracy of model ATM swaption vols (in bps)

Expiry

BP

vol

Figure 6.1: Market implied swaption volatility surface (left-hand side) and calibra-tion errors in basis points (right-hand side). Market as of 13-Sep-2012. The dataare the same as in Tables 6.2 and 6.3.

39

Chapter 7

Conclusion

Main motivation for introducing affine term-structure model into LMM is to be ableto use explicit dynamics of forward rates that can explain future changes in interestrates. Monetary policy makers in the US have committed to start raising short-terminterest rates sooner or later and a model that is built for this purpose is a usefultool for price discovery and risk-management. We saw in Section 6.4 an exampleof how this model can interpret economic facts, and also give reasonable qualityfit to market data (stressing again that goodness of fit is not our goal, but having amodel that can explain future dynamics of the yield curve is).

40

Appendix A

A.1 Libor drifts in LMM-DMRV

Proposition A.1.1. In LMM-DMRV model, drift of Li(t) under Ti-forward measureis:

dLi(t) = (1 + τiLi(t))σf (t, Ti)> (τiσf (t, Ti)dt+ dW i

t

), (A.1)

or in shifted log-normal formulation, the drift is deterministic

dLi(t)

1 + τiLi(t)= σf (t, Ti)

> (τiσf (t, Ti)dt+ dW it

).

Here W i is a Ti-Brownian motion.

Proof. Recall from Heath et al. (1992) that discount bond price has risk-neutraldynamics:

dP (t, T ) = O(dt)− P (t, T )

∫ T

t

σf (t, s)>ds dW t,

where we omit the drift term for now and W is a Brownian motion under the risk-neutral measure.

Libor rates in LMM are discretely compounded forward rates:

Li(t) =1

τi

(P (t, Ti)

P (t, Ti+1)− 1

),

with dynamics by Ito’s formula,

dLi(t) = O(dt) +1

τiP (t, Ti+1)dP (t, Ti)−

P (t, Ti)

τiP (t, Ti+1)2dP (t, Ti+1).

From this formula, simple algebra gives us

dLi(t) = O(dt) +1

τi(1 + τiLi(t))

∫ Ti+1

Ti

σf (t, s)>ds dW t.

41

We agreed to use piecewise-constant volatilities between two fixing dates, sowe can integrate the above expression into:

dLi(t) = O(dt) + (1 + τiLi(t))σf (t, Ti)> dW t.

Li(t)’s natural numeraire is P (t, Ti+1), hence in Ti+1-forward measure it is amartingale as a linear combination of tradable assets, and therefore the drift iszero:

dLi(t) = (1 + τiLi(t))σf (t, Ti)> dW i+1

t . (A.2)

To establish Ti-dynamics of Li(t), we need to use the Girsanov theorem andchange of numeraire technique.

Let Zt be the process

Zt =P (t,Ti)/P (0,Ti)

P (t,Ti+1)/P (0,Ti+1),

then by Theorem 1 in Geman et al. (1995), there exists measure Qi, the Ti-forwardmeasure and Radon-Nikodym derivative between it and Ti+1-forward measure is

dQi

dQi+1

∣∣∣∣Ft = Zt, Qi+1-a.s.

We need to find a process Xt, such that

Zt = E(X)t, Qi+1-a.s.

where E(ξ)t is the stochastic exponent:

E(ξ)t = exp

(ξt −

1

2〈ξ〉t),

and then by Girsanov theorem (Theorem 5.1 in Karatzas and Shreve (1991)),

W i = W i+1 −⟨X,W i+1

⟩t,

would be a Qi-Brownian motion.Rearranging Zt in terms of Libor rate Li(t), we get

Zt =1 + τiLi(t)

1 + τiLi(0),

and by Ito’s formuladZt =

τi1 + τLi(0)

dLi(t),

42

ordZtZt

= τiσf (t, Ti)> dW i+1

t .

This SDE admits explicit solution, a geometric Brownian motion:

Zt = Z0 exp

(−τ

2i

2

∫ t

0

(σTis )2 ds+ τi

∫ t

0

σf (s, Ti)> dW i+1

s

),

where(σTt )2 = σf (t, T )>σf (t, T ).

That is to say,Zt = E(Xt),

with

Xt = τi

∫ t

0

σf (s, Ti)> dW i+1

s .

Now calculate quadratic cross-variation of this process with W i+1:

d⟨X,W i+1

⟩t

= τiσf (t, Ti)dt

Using Girsanov theorem, rewrite equation (A.2) with W i:

dLi(t) = (1 + τiLi(t))σf (t, Ti)> (d⟨X,W i+1

⟩t+ dW i

t

)= (1 + τiLi(t))σf (t, Ti)

> (τiσf (t, Ti)dt+ dW it

).

Corollary A.1.2. Libor rate under Ti-forward measure is

Li(t) =

(Li(0) +

1

τi

)exp

(τ 2i2

∫ t

0

(σis)2ds+ τi

∫ t

0

σf (s, Ti)>dW i

s

)− 1

τi, (A.3)

where σTt = ‖σf (t, T )‖2 is the total volatility of instantaneous forwards in DMRV asgiven by equation (3.20).

Proof. Note that,

τidLi(t)

1 + τiLi(t)=

d(

1τi

+ Li(t))

1τi

+ Li(t).

So that the process ξt = 1τi

+ Li(t) is a geometric Brownian motion with drift:

ξt = ξtµtdt+ ξtσ>t dW ξ

t ,

and as such can be integrated directly:

ξt = ξ0 exp

[∫ t

0

(µs −

1

2σ>s σs

)ds+

∫ t

0

σ>t dW ξt

].

43

Definition A.1.1. Spot measure QB is the measure defined by numeraire B(t):

B(t) = P (t, Tηt)

ηt−1∏i=0

(1 + τiLi(Ti)),

where ηt is the first Libor fixing time after t.

Corollary A.1.3. In LMM-DMRV under the spot measure, Li(t) dynamics is

dLi(t)

1 + τiLi(t)=

i∑j=ηt

τjΛij(t)dt+ σf (t, Ti)dWB,

where Λij(t) is the instantaneous covariation between Li(t) and Lj(t) as given byequation (5.8) and WB is a 2-dimensional Brownian motion under QB.

Proof. Immediate consequence of Lemma 14.2.3 by Andersen and Piterbarg (2010b).For convenience of the reader, the idea of the proof is to split the numeraire assetunder QB

B(t) = P (t, Tηt)

ηt−1∏i=0

(1 + τiLi(Ti)),

into random part, the discount bond P (t, Tηt) and the remainder. Then we applyProposition A.1.1 iteratively to get shifted log-normal dynamics of Li(t) in Ti-, Ti−1-,. . ., Tηt-forward measures, adding an extra drift term on each step:

τjσf (t, Ti)>σf (t, Tj)dt,

for j “rolling” backwards: j = i, i− 1, . . . , ηt.

Corollary A.1.4. In LMM-DMRV under the terminal measure, or TN -forward mea-sure, shifted log-normal dynamics of Li(t) is

dLi(t)

1 + τiLi(t)= −

N−1∑j=i+1

τjΛij(t)dt+ σf (t, Ti)dWTN .

Proof. Use the same idea as in the proof of previous corollary.

Corollary A.1.5. Libor rate under the terminal measure is

Li(t) =

(Li(0) +

1

τi

)exp

[∫ t

0

(µis −

τ 2i2

Λii(s)

)ds+ τi

∫ t

0

σf (s, Ti)>dW i

s

]− 1

τi,

(A.4)where

µit =N−1∑j=i+1

τiτjΛij(t).

44

The fact that in LMM-DMRV model, Libor rates have shifted log-normal distri-bution with deterministic drift has many advantages over the usual log-normal orshifted log-normal Libor Market Model. In a Monte Carlo implementation one candirectly simulate terminal distributions (without predictor corrector or similar dis-cretisation schemes) which increases the performance and accuracy. Assumingforward rates below 10%, the shift of 1 is fairly high and leads to a distribution thatis close to a normal distribution. The implied skew is therefore similar to the skewof a normal model (e. g. Vasıcek or Hull-White).

A.2 Proofs

Proposition A.2.1. In DMRV model with constant parameters, A(t, T ) is given by:

A(t, T ) = −(σr)2

2κ2τ +

(σr)2

κ2(1− e−κτ )− (σr)2

4κ2(1− e−2κτ )(

−αβ +ρσrσθ

κ

)(− 1

ατ +

κ

α2(κ− α)(1− e−ατ )− 1

κ(κ− α)(1− e−κτ )

)(σθ)2

2

(− 1

α2τ +

2κ2

α2(κ− α)(1− e−ατ )− κ2

2α2(κ− α)2(1− e−2ατ )− 1

2κ(κ− α)2(1− e−2κτ )

)+ρσrσθ

κ2α(1− e−κτ )− ρσrσθ

α(κ2 − α2)(1− e−(κ+α)τ )− ρσrσθ

κ2(κ− α)(1− e−κτ ),

(A.5)

where τ = T − t for brevity.

Proof. By direct integration of ∂A(t,T )∂t

.

Proof of Proposition 3.1.1.

Proof. Existence of strong solutions and their properties (uniqueness, continuity)are given by the general theory of stochastic calculus (for example, Theorem 5.2.1by Øksendal (1998)), as long as diffusion coefficients are well behaved (Lipschitzcontinuous and grow no faster than linear). Coefficients of (2.1)—(2.3) satisfy boththese conditions (with finite k in the definition of σrt , that is we do not allow the stepfunction as short rate volatility because it will fail to be Lipschitz continuous).

We apply Ito’s formula to eαtθt

d(eαtθt) = αeαtθtdt+ eαtdθt

= αβeαtdt+ eαtσθdW θt .

45

The latter equation can be integrated:

eαtθt = θ0 + β(eαt − 1) + σθ∫ t

0

eαsdW θs .

Dividing both sides by eαt, we obtain (3.8).We can now apply Ito’s formula to eκtrt:

d(eκtrt) = κeκtrtdt+ eκtdrt

= κeκtθtdt+ eκtσrtdWrt .

We already know θt, so can integrate d(eκtrt) too:

eκtrt = r0 + κ

∫ t

0

eκs[θ0e−αs + β(1− e−αs) + σθ

∫ s

0

eα(u−s)dW θu

]ds+

∫ t

0

eκsσrsdWrs

= r0 +κ(θ0 − β)

κ− α(e(κ−α)t − 1) + β(eκt − 1)

+κσθ∫ t

0

∫ s

0

eκs+α(u−s)dW θuds+

∫ t

0

eκsσrsdWrs .

By stochastic Fubini’s theorem (for example, Theorem 65 in Protter (2004)), wecan interchange the order of integration:∫ t

0

∫ s

0

eκs+α(u−s) dW θu ds =

∫ t

0

∫ t

u

eκs+α(u−s) ds dW θu

=1

κ− α

∫ t

0

eαu(e(κ−α)t − e(κ−α)u) dW θu .

So, dividing both sides of expression for eκtrt by eκt we get the desired result:

rt = r0e−κt +

κ(θ0 − β)

κ− α(e−αt − e−κt) + β(1− e−κt)

+κσθ

κ− α

∫ t

0

(eα(u−t) − eκ(u−t)) dW θu +

∫ t

0

eκsσrsdWrs .


Proof. Integrating ∂A∂t

taking into account boundary condition A(T, T ) = 0, we get

A(t, T ) = αβ

∫ T

t

C(s, T )ds− 1

2

∫ T

t

B(s, T )2(σrs)2ds

− (σθ)2

2

∫ T

t

C(s, T )2ds− ρσθ∫ T

t

B(s, T )C(s, T )σrsds.

(A.6)

46

Let us fix t, T and find t1 such that

σrt1 > σrmax −ε

T,

where ε is a constant which we will determine later.Then ∫ T

t

B(s, T )2(σrs)2ds =

(∫ t1

t

+

∫ T

t1

)B(s, T )2(σrs)

2ds.

Due to monotonicity of σrt , we can calculate upper and lower bounds for the latterintegral:∫ T

t1

B(s, T )2(σrt1)2ds 6

∫ T

t1

B(s, T )2(σrs)2ds 6

∫ T

t1

B(s, T )2(σrmax)2ds.

In other words,∫ T

t1

B(s, T )2(σrmax −

ε

T

)2ds 6

∫ T

t1


∫ T

t1

B(s, T )2(σrmax)2ds.

We can integrate B(t, T )2:∫ T

t

B(s, T )2ds =1

κ2(T − t)− 2

κ(1− e−κ(T−t)) +

1

2κ(1− e−2κ(T−t)).

Now the bounds are explicit:(σrmax −

ε

T

)2 [ Tκ2

+ O(1)

]6∫ T

t1


(σrmax)2T

κ2+ O(1),

where O(1) is a constant plus terms infinitesimal to T . We can take ε = 1.Now we can calculate the limit

limT→∞

1

T − t

∫ T

t

B(s, T )2(σrs)2ds.

Calculate both bounds first:

limT→∞

1

T − t

(σrmax −

ε

T

)2 [ Tκ2

+ O(1)

]=

(σrmax)2

κ2,

andlimT→∞

1

T − t

((σrmax)

2T

κ2+ O(1)

)=

(σrmax)2

κ2.

Both bounds exist in the limit T →∞, limits are identical, therefore by the sandwichtheorem1, we have

limT→∞

1

T − t

∫ T

t1

B(s, T )2(σrs)2ds =

(σrmax)2

κ2.

1 Also known as the two policemen theorem, or two gendarme theorem.

47

Finally,

limT→∞

1

T − t

∫ T

t

B(s, T )2(σrs)2ds = lim

T→∞

1

T − t

(∫ t1

t

+

∫ T

t1

)B(s, T )2(σrs)

2ds

= limT→∞

1

T − t

∫ T

t1

B(s, T )2(σrs)2ds

=(σrmax)

2

κ2.

Using the same argument,

limT→∞

1

T − t

∫ T

t

B(s, T )C(s, T )σrsds =σrmax

κα.

We are now ready to calculate the limit in (A.6), integration of other terms isstraightforward:

limT→∞

A(t, T ) = β − (σrmax)2

2κ2− (σθ)2

2α2− ρσrmaxσ

θ

κα.


Proof. We need to check the definition. There are no restrictions on the samplespace, so Ω0 is fine. First, show that Ω0 ∈ F , that is Ω0 is measurable set inthe original probability space. By Theorem 5.2.1 on existence and uniquenessof solutions to SDEs in Øksendal (1998), a strong, unique, non-explosive and t-continuous (for fixed ω ∈ Ω) solution to (5.2) exists as long as the coefficients ofdrift and volatility σi(t, Li) satisfy conditions of linear growth and Lipschitz continuity(separate for each component of the vector σi(t, Li)). In LMM-DMRV, volatility isgiven as vector

σi(t, Li) = (1 + τiLi(t))

(bTit σ

rt

cTit σθ

),

where bTit and cTit are given by equations (3.16) and (3.17). Both functions bTit

and cTit are Lipschitz continuous (they are even C∞-differentiable). Clearly, bothcomponents of σi(t, Li) is growing linearly in Li(t), so the conditions of Theorem5.2.1 are met. Solution Li(t) is strong in that Li(t) is defined on the same probabilityspace (Ω,F ,P). Processes Li(t) have progressively measurable modifications asIto integrals with cadlag sample paths (the paths are actually continuous on bothsides), this is due to for example Proposition 1.13 by Karatzas and Shreve (1991).

48

Then the level setLi(s, ω) > c ,

is B([0, t]) × F -measurable for all i, here B([0, t]) is the Borel sigma-algebra ofopen sets in [0, t].

And because we have a finite set of Libor rates, the union set

Li(s, ω) > c, i = 0, 1, . . . , N

is B([0, TN ]) ×F -measurable. Now because all Libor rates become deterministicafter their respective fixing date, we have a finite time horizon in Ω0. The set ofsingle points

t : 0 6 t 6 TN

is a Borel set, and hence

Li(t, ω) > c, 0 6 t 6 TN , i = 0, 1, . . . , N

is F -measurable. Now we take constants ci = −1/τi + ε0 (there is no loss ofgenerality in the above arguments to take index dependent constant ci, rather thanglobal c).

We showed that Ω0 ∈ F , and hence Ω0 ∈ F .For F to be a σ-algebra, it needs to satisfy:

• ∅ ∈ F , obviously because ∅ ∈ F and ∅ ⊂ Ω0.

• A ∈ F ⇒ Ac ∈ F . Let A ∈ F , then its Ω0-complement set Ω0 \ A ⊂Ω0 and Ω0 \ A ∈ F because of the sigma-ring closedness under relativecomplementation and due to F -measurability of Ω0. But then by definitionΩ0 \ A ∈ F .

• Countable additivity, A1, A2, . . . ∈ F ⇒⋃∞n=1An ∈ F . Clearly,

⋃∞n=1An ∈ F

and we just need to show that this union lies in Ω0. Suppose this isn’t true andthere exists a point ω0 ∈

⋃∞n=1An such that ω0 /∈ Ω0. Form a sequence of sets

Bn =⋃ni=1Ai, then by definition of the limit, ∃N ∀n > N ω0 ∈ Bn. But Bn ⊂ Ω0

because it is a finite union of sets, and hence we have a contradiction. Itmeans that

⋃∞n=1An ⊂ Ω0.

Filtration F inherits the usual conditions (right-continuity and completeness)from F.

49

Probability P is defined as Bayesian conditional probability,

P(A)def= P(A|Ω0), A ∈ F ,

such that the axioms of probability measure are met: P(∅) = 0, P(Ω0) = 1 and forAi ∈ F , i = 1, 2, . . .

P

(∞⋃i=1

Ai

)=

P (⋃∞i=1Ai)

P(Ω0)

=

∑∞i=1 P(Ai)

P(Ω0)

=∞∑i=1

P(Ai),

where the infinite union and summation commute because P is a probability mea-sure.2


Proof. Existence of solutions on the original space (Ω,F ,F,P) is proved in theproof of Proposition 5.2.1, we just need to remove the local volatility function(1 + τiLi(t)) (neither linear growth condition nor Lipschitz continuity are violatedby doing so).

Then by definition of Libor rate,

P (t, Ti+1) =P (t, Ti)

1 + τiLi(t),

in other words, for all ω ∈ Ω0, P (t, Ti) > 0 as long as P (0, 0) > 0. Since bonds pull topar, P (0, 0) = 1, all bond prices with maturities on fixing dates are positive. Henceby Theorem 1 in Geman et al. (1995), we can define measures QTi associated withappropriate discount bond numeraires, and that each Li(t) can be redefined in anyother measure QTj .

2 Probability P of course could have been defined as conditional probability on sigma-algebraF ,

P(A) = P(A|F )def= E(1A|F ),

and for sets A ∈ F , this is justP(A) = 1A.

But in this case the system (Ω0, F , P) would fail the axioms of probability space. This “conditionalstochastic calculus” is a topic of original research.

50


Proof. By definition of stochastic integral,

Li(Ti) = Li(0) +

∫ Ti

0

dLi(s).

Formula (2.19) on p. 138 by Karatzas and Shreve (1991) states that Li(Ti) isa continuous, square-integrable martingale and gives us that quadratic variation ofLi(Ti) is:

〈Li(Ti)〉 =

∫ Ti

0

d〈Li(s)〉

=

∫ Ti

0

‖σi(s, Li)‖2d⟨Y i+1

⟩s,

where the last equality comes from (6.2).We know that Y i+1 is a Brownian motion under Qi+1, therefore by Levy’s char-

acterisation of Brownian motion,

〈Li(Ti)〉 =

∫ Ti

0

‖σi(s, Li)‖2ds.

By definition,σi(t, Li) = σf (t, Ti),

where σf (t, Ti) is the vector of instantaneous forward volatilities from DMRV model.Define a continuous martingale

Xt :=Li(t)

σt,

where

σ2t =

1

t

∫ t

0

‖σf (s, Ti)‖2 ds.3

Quadratic variation of Xt is thus:

〈Xt〉 =〈Li(t)〉σ2t

= t, (A.7)

and by Levy’s characterisation of Brownian motion (Theorem 3.16 in Karatzas andShreve (1991)), Xt is a Brownian motion. Therefore, all its marginal distributionsare Gaussian and in particular Li(Ti) − Li(0) is also Gaussian with zero and vari-ance 〈Li(Ti)〉.

3 By l’Hospital rule, σ0 = 1.

51

Note that we can only take σt out of quadratic variation operator in (A.7), be-cause it is deterministic.

For total volatility of forward rate from DMRV model, or ‖σf (s, Ti)‖ we have byformula (3.20):

‖σf (s, Ti)‖ = σTis .

Now we can apply Bachelier’s formula for a call option on Li(Ti) with forwardF = Ei+1Li(Ti) = Li(0) and volatility:√

1

Ti

∫ Ti

0

(σTis )2 ds.

52

Bibliography

Leif B. G. Andersen and Vladimir V. Piterbarg. Interest Rate Modeling. Volume 1:Foundations and Vanilla Models. 2010a.

Leif B. G. Andersen and Vladimir V. Piterbarg. Interest Rate Modeling. Volume 2:Term Structure Models. 2010b.

Leif B. G. Andersen and Vladimir V. Piterbarg. Interest Rate Modeling. Volume 3:Products and Risk Management. 2010c.

Louis Bachelier. Theorie de la speculation. Annales Scientifiques de l’Ecole Nor-male Superieure, 3(17):21–86, 1900.

Pierluigi Balduzzi, Sanjiv Ranjan Das, Silverio Foresi, and Rangarajan Sundaram.A simple approach to three-factor affine term structure models. Journal of FixedIncome, 12:43–53, 1996.

David R. Beaglehole and Mark S. Tenney. General solutions of some interest rate-contingent claim pricing equations. Journal of Fixed Income, 1(2):69, 1991.

Ben S. Bernanke. Fedspeak. January 2004.http://www.federalreserve.gov/boarddocs/speeches/2004/200401032/default.htm.

Ben S. Bernanke. The economic outlook and monetary policy. August 2010.http://www.federalreserve.gov/newsevents/speech/bernanke20100827a.htm.

Damiano Brigo and Fabio Mercurio. Interest Rate Models — Theory and Practice.With Smile, Inflation and Credit. 2nd edition, 2006.

Lin Chen. Stochastic mean and stochastic volatility — a three-factor model of theterm structure of interest rates and its application to the pricing of interest ratederivatives. Financial Markets, Institutions, and Instruments, 5:1–88, 1996.

53

Jens H. E. Christensen, Francis X. Diebold, and Glenn D. Rudebusch. Anarbitrage-free generalized nelson-siegel term structure model. NBER WorkingPaper, (14463), 2008.

Qiang Dai and Kenneth J. Singleton. Specification structure of affine term structuremodels. Journal of Finance, 55(5), 2000.

Francis X. Diebold and Canlin Li. Forecasting the term structure of governmentbond yields. Journal of Econometrics, (130):337–364, 2006.

Francis X. Diebold and Glenn D. Rudebusch. Measuring business cycles: A mod-ern perspective. NBER Working Paper, (4643), 1994.

Philip H. Dybvig, Jonathan E. Ingersoll Jr., and Stephen A. Ross. Long forwardand zero-coupon rates can never fall. Journal of Business, 69:1–25, 1996.

Alexey Erekhinsky. On negative forward rates in interest rates modelling. Availableby request from the author, 2011.

Federal Open Market Committee press release. December 2012.http://www.federalreserve.gov/newsevents/press/monetary/20121212a.htm.

Helyette Geman, Nicole El Karoui, and Jean-Charles Rochet. Changes ofnumeraire, changes of probability measure and option pricing. Journal of Ap-plied Probability, 32(2):443–458, 1995.

Alan Greenspan. Monetary policy under uncertainty. August 2003.http://www.federalreserve.gov/boarddocs/speeches/2003/20030829/default.htm.

David Heath, Robert Jarrow, and Andrew Morton. Bond pricing and the term struc-ture of interest rates: A new methodology for contingent claims valuation. Econo-metrica, 60(1):77–105, 1992.

Jessica James and Nick Webber. Interest rate modelling. Wiley, 2000.

Farshid Jamshidian. Libor and swap market models and measures. Finance andStochastics, 1:293–330, 1997.

Ioannis Karatzas and Steven E. Shreve. Brownian Motion and Stochastic Calculus.2nd edition, 1991.

54

Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applica-tions. Springer Verlag, 5th edition, 1998.

Philip E. Protter. Stochastic Integration and Differential Equations. 2nd edition,2004.

Riccardo Rebonato, Kenneth McKay, and Richard White. The SABR/LIBOR MarketModel: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives.Wiley, 2009.

Albert Nikolaevich Shiryaev. Essentials of Stochastic financial mathematics (inRussian), volume 1. Fazis, Moscow, Russia, 1998. [Osnovy stohasticheskojfinansovoj matematiki. Tom 1. Fakty i Modeli (1998)].

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