Semi-analytic Lattice Integration

of a Markov Functional Term

Structure Model

�...

Christ Church College

University of Oxford

A thesis submitted for the degree of

Master of Science in Mathematical Finance

Hilary 2009

Abstract

One common use of Markov functional models is to approximate LIBOR market

models and to avoid complications the terminal forward measure is typically used. If

this method is applied to long term structures (ten or more years), the distribution

of the early LIBORs in the term structure has a very large tail, which is normally

not completely captured by common numerical techniques (either Monte Carlo or

grid-based methods).

A numerical method that is frequently applied to Markov functional models is

known as the semi-analytic lattice integrator (Sali) tree. This thesis examines the

implications of the long tails on the Sali tree. Adequate boundary conditions and

grid sizes are derived in order to capture the effect of the long tails.

It turns out that this method either exhibits stability problems or demands for a

relatively small lattice spacing. The reason for this is examined in detail and several

variations of the Sali tree to avoid this effect are suggested and analysed. Furthermore

the optimisation of the grid parameters is considered in order to reduce the necessary

computation time.

2

Contents

1 Introduction 1

2 Preliminaries 3

2.1 Markov Functional Models . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Sali-Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Stability of Cubic Splines . . . . . . . . . . . . . . . . . . . . 6

3 Markov Black-Derman-Toy Model 7

3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Analytical Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Approaches to Improve Convergence 18

4.1 Other Spline Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.1 Akima Interpolation . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.2 Tension Splines . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1.2.2 Selection of Tension Factors . . . . . . . . . . . . . . 21

4.1.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Splitting off the Asymptotic Behaviour . . . . . . . . . . . . . . . . . 24

5 Optimising the Grid 29

5.1 Grid Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Lattice Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

i

6 Conclusion 37

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A Analytical solution of the MBDT-Model 39

A.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A.2 Estimation of the Tails . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.2.1 Non-Asymptotic Contributions . . . . . . . . . . . . . . . . . 40

A.2.2 Integral over the Tail . . . . . . . . . . . . . . . . . . . . . . . 41

B Tension Splines 42

C Expectation Values of Splines 44

C.1 Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

C.2 Tension Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

D Combined Approaches to Improve Convergence 46

Bibliography 49

ii

List of Figures

3.1 Analytic L − Li for different tenors and levels of volatility . . . . . . 9

3.2 It(x) analytic according to eq. (3.1) . . . . . . . . . . . . . . . . . . . 11

3.3 comparison of ai and ai . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Effect of h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Oscillations of the cubic spline approximation . . . . . . . . . . . . . 17

4.1 Effect of h with Akima interpolation . . . . . . . . . . . . . . . . . . 20

4.2 Effect of h with splines under tension . . . . . . . . . . . . . . . . . . 23

4.3 Tension factor σ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 Effect of h on L − L with spilt asymptotics . . . . . . . . . . . . . . . 26

4.5 f1 for n = 20 and ψ = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . 27

4.6 Sali L − Li for different tenors and levels of volatility . . . . . . . . . 28

5.1 Effect of different tail extrapolations on L − L with spilt asymptotics 32

5.2 Grid size needed for calibration in figure 5.1 . . . . . . . . . . . . . . 33

5.3 Sali L − Li for different tenors and levels of volatility with optimised

grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

D.1 Effect of h on L − L with spilt asymptotics and Akima splines . . . . 47

D.2 Effect of h on L − L with spilt asymptotics and splines under tension 48

iii

Chapter 1

Introduction

Markov functional models, suggested by Hunt, Kennedy and Pelsser [11, 9] as tools

for pricing exotic derivatives, have the characteristic property that the discount bond

prices are at any time a function of some low dimensional Markov process.

With LIBOR market models they share the easy calibration to market prices,

but due to the low dimension of the random process they allow for a much more

efficient implementation. So one common use of Markov functional models is to

approximate LIBOR market models. For simplicity, Markov functional models are

typically formulated in the terminal forward measure.

If this method is applied to long term structures (ten years or longer), the dis-

tribution of the early LIBORs in the term structure has a very large tail, which is

typically not captured completely by common numerical methods. Either the grid or

tree is too small or a vast number of Monte Carlo steps would be necessary to capture

these contributions.

A method that is often used to implement Markov functional models is the semi-

analytic lattice integrator (Sali) tree [6]: The backward integration is done on a grid

using exact formulae to integrate piecewise defined functions against the propagation

kernel of the driving stochastic process, interpolating the resulting function at the

prior time.

The goal of this thesis is to analyse the effect of the large tails observed for long

term structures on the accuracy of the Sali tree. The determination of an adequate

grid-size will be examined as well as a proper treatment of the semi-infinite intervals

beyond the grid.

This thesis is structured as follows: In chapter 2 basic concepts are introduced.

Markov functional models are defined and the Sali tree will be outlined. Special

attention is payed to cubic splines that are normally used as basic functions.

1

This is followed by the definition of a model that is still analytically solvable but

complex enough to show the main characteristics of a realistic Markov functional

model in chapter 3. The analytical calibration of the model is presented as well

as a Sali approach. In contrast to the standard Sali tree, the contributions of the

semi-infinite intervals beyond the grid are taken into account in order to keep the

effects of the long tails and to derive realistic boundary conditions for the cubic

spline interpolation on the finite grid.

In chapter 4 three variations of the Sali approach are introduced with the goal

to achieve better results at larger lattice spacing. Two are based on different spline

types, the third is based on splitting off the exponential behaviour that is responsible

for the long tails and apply the Sali approach to “well behaved” functions.

The purpose of chapter 5 is to optimise the grid in order to get precise results

with a minimum of computation time while chapter 6 summarises the results.

2

Chapter 2

Preliminaries

2.1 Markov Functional Models

In practice, exotic derivatives are priced by calibrating a model to the market prices

of liquid simple derivatives and then using this model to price the exotic derivative.

As such the role of the model could be described as an ’extrapolation tool’. As the

model should describe prices in an efficient market it must be arbitrage free. For a

practical applications two other features are important. The model should

• be well-calibrated, i.e. correctly price a large class of liquid instruments without

over-fitting

• allow for an efficient implementation

Hunt, Kennedy and Pelser [11, 9] suggested a Markov functional model for this pur-

pose, where the randomness comes through a low dimensional Markov process and

the interest rates are a functional of this random process.

In a single currency economy a Markov functional model can be described as

follows: Let DtT be the value at time t of a zero coupon bond maturing at time T

with DTT = 1. The underlying assets should be a finite number of these bonds with

T ∈ T = {Ti|i = 1, . . . , n}. This is enough for the present purpose, but the treatment

can be generalised to an infinite number of underlyings.

Let Ft be the filtration representing the information available at time t. Any

trading strategy in the market should be self-financing. The value of a portfolio

generated by such a trading strategy is called a price process and any price process

that is positive almost surely is called a numeraire.

Given a numeraire N we assume there is a measure N equivalent to the natural

measure P such that the process (DtT /Nt)T∈T is an {Ft} martingale.

3

We assume that any derivative can be replicated by a self financing portfolio.

If we further assume that there is a time limit ∂∗, at which the value of a derivative

is determined purely by the asset prices, the value of this derivative at any earlier

time is given by

Vt = NtEN[V∂∗/N∂∗|Ft]

= NtEN[VT /NT |Ft] (2.1)

for 0 ≤ t ≤ T ≤ ∂∗.

2.1.1 Definition

Let Xt be a time inhomogeneous Markov process under N and the boundary function

∂S : [0, ∂∗] → [0, ∂∗] with ∂S ≤ S a real function defining the boundary times up to

which the following assumptions on the discount bonds are valid. We assume that

• the prices of the pure discount bonds are a function of the random process:

DtS = DtS(xt) for 0 ≤ t ≤ ∂S (2.2)

• the same should apply to the numeraire N :

Nt = Nt(xt) for 0 ≤ t ≤ ∂∗ (2.3)

Then a Markov functional interest rate model is completely determined by

1. the law of the process X under N

2. the functional form of the numeraire Nt(Xt) for 0 ≤ t ≤ ∂∗

3. because of equation (2.1) we do not need the form of DtS(xt) for all times. It

is sufficient to know the functional at the boundary t = ∂S.

The calibration of the model demands the determination of these three elements.

2.2 Sali-Trees

A numerical method which is well suited for the treatment of Markov functional

models is the semi-analytic lattice integrator (Sali) tree [3, 6]. Let T = {t1, t2 . . . tn}be a set of discrete time steps with ti < tj for i < j. Assume that the conditional pdf

f(xt′|xt) is available for all t, t′ ∈ T , t < t′.

4

The calculation of the expectation value of a quantity V at a time step ti from

the distribution at ti+1

Vi(xi) = Ei[Vi+1|xi] =

∫f(xi+1|xi)Vi+1(xi+1)dxi+1 , (2.4)

where xi is a short notation for xti , is done in two steps:

1. The integral is evaluated at time ti for a finite number of grid points xt,k,

k = 1 . . . ni. So the method relies on the fact that the function Vi+1 is such that

the integral (2.4) can be evaluated either analytically or by efficient numerical

integration. This is further discussed in the next paragraph.

2. Then the function Vi(xi) is approximated by fitting a spline or another piecewise

polynomial function to the grid {xi,k}k=1...ni, leading to a function Vi(xi).

To do the next time-step, we have to integrate Vi(xi)f(xi|xi−1). As Vi is piecewise

a linear combination of the base functions bk(x), this can be done if the integral∫ xi,j+1

xi,jbk(xi)f(xi|xi−1)dxi can be evaluated efficiently. This is certainly the case if

Xt is a Brownian motion and cubic splines or any other set of piecewise polynomial

interpolation functions are used. The integrals relevant for this case are given in

Appendix C.1.

To get the iteration started, the payoff VT is approximated by the base functions

bk(x) as well.

In general a much smaller number of grid points is needed compared with conven-

tional trees to get a similar precision. A notable strength of the Sali-tree comes with

payoff functions that are non-continuous either themselves or in their first derivative.

It is possible to define domains of integration where the payoff is well-behaved. This

major advantage of the Sali-tree is described in [6]. As it is not needed for the cali-

bration of the model described in the next chapter this point will not be elaborated

further.

2.3 Cubic Splines

The following section summarises properties of cubic splines that are relevant for

this thesis. A detailed introduction can be found in most textbooks about numerical

mathematics such as [14]

Let ∆ = {a = x0 < x1 < . . . < xn = b} be a partition of the interval (a, b) and

f : (a, b) → R a real function. The cubic spline for f on ∆ is a function S∆ : (a, b) → R

with the following properties:

5

• S∆(xi) = f(xi) for all i ∈ {1, . . . , n}

• S∆ ∈ C2(a, b), i.e. f is twice continuously differentiable

• On each subinterval (xi, xi+1) S∆ corresponds to a polynomial of order three.

To make the cubic spline unique, two further equations are needed which are usu-

ally chosen to be a condition imposed on the spline’s derivatives at the boundaries.

Common boundary conditions are either vanishing second derivative or a fixed first

derivative. The first case is usually referred to as a ’natural spline’.

Determining the parameters of the polynomials boils down to a linear equation.

The implementation used in this work is taken from [15].

2.3.1 Stability of Cubic Splines

Cubic splines are guaranteed to converge towards the original function with dimin-

ishing distance between the grid points. To be more precise:

Let f ∈ C4(a, b), assume an L ∈ R exists, so that f (4)(x) ≤ L ∀x ∈ [a, b]. Given

a sequence of grids ∆m = {a = x(m)0 < x

(m)1 < . . . < x

(m)nm = b} with maximal lattice

spacing

‖∆m‖ = maxi

(x(m)i+1 − x

(m)i ) (2.5)

and

supm,i

||∆m||x

(m)i+1 − x

(m)i

≤ K (2.6)

for some number K ∈ R, then for i ∈ {0, 1, 2, 3} constants Ci ≤ 2 independent of ∆m

exist, so that for all x ∈ [a, b]

|f (i)(x) − S(i)∆m

(x)| ≤ CiLK‖∆m‖4−i . (2.7)

This ensures that S∆mwill eventually converge to f . As the derivatives of f enter

the right hand side of (2.7) via L, the lattice spacing necessary to obtain a decent

precision might be prohibitively small. Note that (2.6) is always fulfilled as long

as equidistant grids are used. Hall and Mayer [8] were able to prove the following

estimates for ci = CiK: c0 = 5/384, c1 = 1/24, c2 = (K + K−1)/2, where c0 and c1

are optimal.

6

Chapter 3

Markov Black-Derman-Toy Model

3.1 The Model

To demonstrate the effects of the long tails we choose a model that is still analyti-

cally tractable but shows the main features of a Markov functional model for a term

structure of LIBOR rates. It can be considered as the Markov functional version

of the Black-Derman-Toy model [7]. In this work it will be referred to as ‘Markov

Black-Derman-Toy model’ or MBDT model.

Let Li be the LIBOR-rate from time Ti to Ti+1 and δi the accrual factor for

that period. In the terminal measure, i.e. using the last discount bond DTi,Tnas a

numeraire, we assume

Li = Li

1 +

exp(−qiψ(Ti)

∫ Ti

0g(t)dWt − 1

2q2i ψ(Ti)

2∫ Ti

0g(t)2dt

)− 1

qi

(3.1)

where ψ and g are positive, real functions of time, 0 < qi ≤ 1, Li positive, real

numbers and Wt is a standard Brownian motion under the terminal measure.

We consider only the lognormal case, i.e. qi = 1. The integral WGi=

∫ Ti

0g(t)dWt

is a normal variable with variance Gi =∫ Ti

0g(t)2dt and can thus be considered as

a time changed Brownian motion. So (3.1) with qi = 1 depends only on Gi and

ψi = ψ(Ti):

Li = Li exp

(−ψiWGi

− 1

2ψ2

i Gi

)= LiEi(−ψiWGi

) (3.2)

where E is Dolean’s exponential of a contiuous martingale Xt:

Et(X) = exp(Xt −1

2var(Xt)) . (3.3)

7

and we use the shorthand Ei(X) = Eti(X). Note that Es[Et(X)] = Es(X) for all s ≤ t.

We use the short form Di,j = DTi,Tj(WGi

) with i < j for the discount bonds and

Di,j =Di,j

Di,nfor the numeraire adjusted discount bonds in the terminal measure.

3.2 Analytical Calibration

Assume that ψi and Gi are given. So for the model calibration we only need to

determine the Li from the market values at time t = 0, i.e. D0,i. With

Di,i+1Di,n = Di,i+1 = (1 + δiLi(Ti))−1 (3.4)

we arrive at

Dj,i =Dj,i

Dj,n

= Ej

[ 1

Di,n

](3.5)

= Ej

[Di,i+1(1 + δiLiEi(−ψiWG·

))]

(3.6)

where j < i and Ei[·] = E[·|FTi] is the expectation value given the filtration at time

Ti. For j = 0 in particular we get

D0,i = D0,i+1 + δiLiE0[Ei(−ψiWG·)Di,i+1] (3.7)

which allows to calculate Li from the initial prices of the zero bonds D0,i and D0,i+1

and the distribution of Di,i+1(WGi). Starting with

Dj,n = 1 ∀j ≤ n (3.8)

we can determine Li by induction. In Appendix A.1 it is shown that Di,i+1 has the

form

Di,i+1 =2n−i−1−1∑

j=0

Xi.jEi(−Yi,jWG·) (3.9)

with constants Xi,j and Yi,j given by (A.3,A.4) that depend only on those Lj with

j > i. We can thus get Li using (3.7):

Li =D0,i − D0,i+1

δiE0[Ei(−ψiWG·)Di,i+1]

(3.10)

From (3.6) with i = j we get the functional form of the numeraire Di,n. Together

with the definition of the Markov process W and the form of the discount bonds at

8

the boundary Dn,n = 1 the definition of the Markov functional model is complete

according to section 2.1.1.

Below results for L are shown with the following parameters: The value of the

discount bonds at time T0 are chosen to get a flat initial LIBOR rate of L0 = 5% and

different tenor structures with yearly resets and annual compounding are used. For

simplicity we choose Gi = Ti. Figure 3.1 shows the convexity adjustment L0 − L for

different time independent levels of the volatility ψ.

0 2 4 6 80

1

2

3

4

50.70.60.50.40.30.20.1

0 5 10 15 200

1

2

3

4

5

0.50.40.30.20.150.1

0 5 10 15 20 25 300

1

2

3

4

5

0.40.30.20.180.150.140.130.1

Figure 3.1: L0 − L with L calculated analytically according to (3.10), a flat initialLIBOR rate of L0 = 5%, a tenor of (from top left) 10, 20 and 30 years and differentvalues of the volatility ψ, which is assumed to be time-independent

9

Now we have a closer look on the functional form of Di,i+1(xt). As can be seen

from equation (3.9) it is the sum of terms that each are proportional to exp(−Yi,jx).

The expectation values of these terms are

Et [exp(−Yi,jx)] =1√

2π(Gi+1 − Gt)

∫ ∞

−∞

exp

(−1

2

((x − xt)

2

Gi+1 − Gt

+ 2Yi,jx

))dx

=1√

2π(Gi+1 − Gt)exp

(1

2Y 2

i,j(Gi+1 − Gt) − Yi,jxt

)

∫ ∞

−∞

exp

(−1

2(Gi+1 − Gt)

(x − xt

Gi+1 − Gt

+ Yi,j

)2)

dx . (3.11)

For a given xt the main contribution to the integral is at x = xt − Yi,j(Gi+1 − Gi)

where, according to (A.4) Yi,j could get as large as∑n

i+1 ψi, leading to contributions

far from the central value x = 0. This behaviour is most striking in the denominator

of (3.10), where t = 0. Figure (3.2) shows the integrand

Ii(x) = Ei(−ψiWGi)Di,i+1 · n(x; Gi) . (3.12)

For e.g. i = 4 there are significant contributions at x ≈ −12, six times the standard

deviation from the central value. The observation described above will be of interest

as soon as we use numerical methods to calibrate the model. Trees, finite differences

and Monte Carlo methods all have some kind of finite cut-off for x. The first two

explicitly by the grid size, the later implicitly, as a finite number of runs will lead

to a negligible probability several standard deviations from the central value. For a

Monte Carlo integration that uses a uniform sampling of the distribution WGiover

a region including six standard deviations from the central value, about 1011 Monte

Carlo steps would be necessary. The effect for Monte Carlo simulation has already

been investigated by Merrill Lynch Quantitative Risk Management [10].

3.3 Numerical Approach

The expense to calculate the Li numerically using the above analytic solution is of

order 2n, which starts to become prohibitively large for longer tenors. So even in

this case of an analytically solvable model a numerical approach like the Sali-tree is

necessary. As there are no discontinuities involved in D, we do not have to care about

different domains of integration.

Let xi,k be the grid for the underlying stochastic process WG at time-step i and

Di,j the Sali approximation for Di,j. Then the Sali step i → (i − 1) is:

10

-30 -20 -10 0 10 20 300

0,2

0,4

0,6

0,8

18161412108642

-30 -20 -10 0 10 20 300

0,1

0,2

0,3

0,4

18161412108642

Figure 3.2: It(x) (eq. 3.12) for a tenor of 20 years, for several t and ψ = 0.15 (above)and ψ = 0.3 (below). The other parameters are chosen as in fig. 3.1. While we seeonly a slight asymmetry for ψ = 0.15, the contributions far from the central value aresignificant for the larger volatility.

11

1. Starting with Di,i+1 calculate Li using (3.10).

2. For each xi−1,k evaluate

(Di−1,i)k = E[(1 + δiLi exp(−ψixi − ψ2

i Gi/2))Di,i+1

∣∣WGi= xi−1,k

](3.13)

3. From these (Di−1,i)k determine Di−1,i by fitting the splines.

Here we consider cubic splines that are continuous in their second derivative as

described in section 2.3.

Apart from the question which splines to use, the main decisions necessary for

applying this method are the boundary conditions and the placement of grid points.

For the time being we assume equally spaced grid points (−x,−x + h, . . . , x) with

h = 2x/N , but we have a closer look at the boundary conditions:

As the single summands in (3.9) grow exponentially for x < 0, natural boundary

conditions, i.e. vanishing second derivative at the boundaries, are clearly inappro-

priate. Instead, we try to get a reasonable estimate for the first derivative at the

boundaries. As the method that is developed here should not be limited to the sim-

ple case of a model that is in principle solvable analytically, we will not use detailed

knowledge about the analytical form of Di,n(xi).

Instead, estimates for the asymptotical behaviour of Di,n(xi) for xi ≫ 0 and

xi ≪ 0 are needed.

For xi → ∞ from (3.2) we get Lj → 0 a.s. ∀j > i. So Di,n(xi) → 1 for xi → ∞As Lj is monotonous in xi, this is also true for Di,n(xi) and we may assume a zero

first derivative at the upper boundary.

To determine the boundary conditions for xi → −∞, define ∆j = xj−xi for j > i.

From (3.4) and (3.2) we get for x ≪ 0

Dj,j+1(xj) = (1 + δiLi(xi + ∆j))−1

∼ exp(ψj(xi + ∆j)) (3.14)

and therefore

E[Dj,j+1|WGi= xi] ∼ exp(ψjxi) (3.15)

With (3.5)

Di,i+1 = E

[ 1

Di+1,n

∣∣∣WGi= xi

]

∼ exp

(−xi

n−1∑

j=i+1

ψj

)(3.16)

12

So we assume that Di,i+1(xi) has the asymptotic form

Di,i+1(xi) ∼ bi · exp(−aixi) for xi ≪ 0 (3.17)

and the boundary condition at xi = −x is D′i,i+1 = −aibi · exp(−aixi). To determine

ai and bi we could either set ai to ai =∑n−1

j=i+1 ψj and determine bi from the first grid

point bi = exp(aix)(Di,i+1)0 or we could determine both ai and bi from the first two

grid points:

ai = ln((Di,i+1)0/(Di,i+1)1)/h (3.18)

bi = (Di,i+1)0 exp(aix) . (3.19)

For a correct estimation of the asymptotic behaviour it is important to choose x

large enough so that the asymptotic behaviour dominates all other terms for |x| > x.

An estimation of Di,i+1(xi)/(bi exp(−aixi)) for |x| > x would be helpful but in most

practical cases unrealistic to achieve. Instead, the difference ai − ai will be used as a

consistency check to see if x is chosen large enough so that the asymptotic behaviour

can be assumed for x < −x. x will be chosen so that ai − ai stays within a fixed

interval (−δa, δa) for all i.

In this special case of an analytically solvable model, the role of the asymptotic

behaviour can be seen by comparing it to the analytic solution (3.9). The exponential

with maximal coefficient is singled out as the leading term for x → −∞. It depends

on the factors Xi,j what value of x is necessary for this exponential to dominate the

others. For the derivation of an upper limit to the non-asymptotic terms see appendix

A.2. The result justifies the use of ai − ai as an indicator for an adequate grid size.

The boundary effects on the first grid points have to be taken into account. When

taking the expectation value (3.13) it is insufficient to calculate the integral from −x

to x as this will lead to wrong results for xk close to ±x and to wrong boundary

conditions. At each further time-step these errors will cause deviations closer to the

centre of the grid. Instead, the integral for the semi-infinite intervals (−∞,−x) and

13

(x,∞) will be approximated by the integral over the asymptotic behaviour:

(Di−1,i)k =

∞∫

−∞

(1 + δiLi exp(−ψix − ψ2i Gi/2))Di,i+1(x)n(x − xk; Gi − Gi−1)dx

≈ bi−1

−x∫

−∞

e−ai−1xn(x − xk; Gi − Gi−1)dx

+N−1∑

j=0

(xi)j+1∫

(xi)j

(1 + δiLie

−ψix−ψ2i Gi/2

)Di,i+1(x)n(x − xk; Gi − Gi−1)dx

+

∞∫

x

n(x − xk; Gi − Gi−1)dx (3.20)

=N−1∑

j=0

(xi)j+1∫

(xi)j

(1 + δiLie

−ψix−ψ2i Gi/2

)Di,i+1(x)n(x − xk; Gi − Gi−1)dx

+bi−1

2e−ai−1xk+(Gi−Gi−1)a2/2

(1 + erf

(−x − xk + a(Gi − Gi−1)√

2(Gi − Gi−1)

))

+1

2

(1 − erf

(x − xk√

2(Gi − Gi−1)

)). (3.21)

As Di,i+1 is a polynomial of third order in each of the intervals (xi,j, xi,j+1), the

integral can be solved analytically.

To verify, whether x is large enough for the asymptotic behaviour described above

to be a good approximation, we compare the theoretical value ai to the value calcu-

lated from the to first grid points using (3.18). Figure 3.3 shows the numerical value

for ai compared to the theoretical value ai for n = 20, h = 0.25, ψ ∈ {0.15, 0.3} and

several values of x as a function of the time-step i.

As we can see, the asymptotic behaviour described above is a good approximation

for these parameters if x > 50. For ψ = 0.3 the effect of the contributions far from

the central value is clearly visible, as even with x = 30 the asymptotic behaviour can

be observed for large i, but large deviations from the asymptotic behaviour can be

observed, as soon as the additional peak shown in figure 3.2 becomes significant.

In the present and the next chapter numerical examples are with a tenor of 20

and ψ ∈ (0.15, 0.3). Figure 3.3 illustrates that a grid size given by x = 60 is fully

appropriate. In chapter 5.1 the choice of an optimal grid size will be investigated

further.

14

0 5 10 15 200

0,5

1

1,5

2

2,5

3

analytical102030405060

i

ai

0 5 10 15 200

1

2

3

4

5

6

a0

1020304050

ai

i

Figure 3.3: ai and ai according to (3.18) for n = 20, h = 0.25, different grid sizes xand ψ = 0.15 (left) and ψ = 0.3 (right)

After determining the boundary conditions and the grid size, the spacing between

the grid points must be set. For the time being we stick with equally spaced points

and vary h. Figure 3.4 illustrates that h has to be chosen relatively small to get

reliable results. For any larger grid spacing the numerical solution is good up to some

point i. For any j < i Lj is practically zero.

The reason for this lies in a well known problem with cubic splines (see e.g.

[14, 2]): Cubic splines are a global interpolation method, which means that every

grid point affects the parameters for the spline in every single interval. This can lead

to oscillations in the whole domain. For a series of lattices ∆m on a finite interval

[−x, x] with lattice spacing ‖∆m‖ → 0, a function f ∈ C(4)[−x, x] and corresponding

cubic spline S∆mthe convergence theorem for cubic splines (see section 2.3.1) states

that S∆mdoes converge uniformly to f . But this convergence is influenced by the

fourth derivative of f :

|f(x) − S∆m(x)| ≤ c0L‖∆m‖4 (3.22)

with f (4)(x) ≤ L ∀x ∈ [a, b] and c0 = 5/384. In our case we have a series of

functions fi that grow exponentially. At the lower boundary fi(x) ≈ bi exp(−aix).

So L ≥ f(4)i (−x) ≈ fi(−x)a4

i and as soon as c0(aih)4 > 1 the error might even be

larger than the function value itself. Figure 3.5 shows Di,i+1(x) for i = 11, 10, 9, 8.

It can clearly be seen that first even with ‘well behaved’ grid points the oscillations

set in and at a later step these oscillations lead to implausible grid points.

In principle this problem could be solved by choosing h small enough, but this

would lead to excessive need of computing time. The following chapter will show

alternative approaches.

15

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2

3

4

5

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0 5 10 150

1

2

3

4

5

210.50.25analytic

i

Figure 3.4: The effect of h on the quality of the Sali approach to L0− Li with n = 20,x = 60, ψ = 0.3 (above) and ψ = 0.2 (below)

16

-50 -48 -46 -44 -42 -400

1×1024

2×1024

3×1024

xi = 11

-50 -48 -46 -44 -42 -40-1×1028

0

1×1028

xi = 10

-50 -48 -46 -44 -42 -40-5×1031

0

5×1031

xi = 9

-50 -48 -46 -44 -42 -40-5×1033

0

5×1033

xi = 8

Figure 3.5: Di,i+1(x) for i = 11, 10, 9, 8 for n = 20, x = 60, ψ = 0.2 and h = 1. Thedotted line connects the values at the grid points, the solid line shows the cubic splinedefined by these points. It can clearly be seen that the oscillations of the spline thatset in at i = 11 lead to inconsistent values at the grid points for smaller i.

17

Chapter 4

Approaches to Improve

Convergence

To handle the oscillations that were observed for the spline approximation, basically

two approaches can be used. Either the method used to calculate the splines can be

varied or the function that is approximated can be changed. Both ideas are further

investigated in the following sections.

4.1 Other Spline Types

Though cubic splines are much less prone to over-oscillation than e.g. fitting of a

polynomial, the phenomenon is well known in the literature (see e.g. [14, 2]). This

is in part due to the cubic spline’s lack of locality. A local change in the input data

will modify the curve even far away from this point. Several other versions of splines

have been presented to handle this problem. We will use Akima interpolation [1] and

tension splines [13, 14, 4].

Let again be ∆ = {a = x0 < x1 < . . . < xN = b} a partition of the interval

(a, b) and y1 . . . yN ∈ R. As for cubic splines we search a function y : (a, b) → R with

y(xi) = yi ∀1 ≤ i ≤ N , but the additional conditions that make y unique differ for

each type of splines.

4.1.1 Akima Interpolation

Like cubic splines, Akima interpolation is based on cubic polynomials. The condition

of a continuous second derivative is abandoned and instead the first derivative at a

given grid point is estimated using the neighbouring points. With

qi =yi − yi−1

xi − xi−1

for i ∈ {1, . . . N} (4.1)

18

the first derivative at xi is estimated as

y′i =

qi|qi+2 − qi+1| + qi+1|qi+2 − qi+1||qi+2 − qi+1| + |qi+2 − qi+1|

for i ∈ {3, . . . N − 2} (4.2)

If qi+2 = qi+1 and qi+2 = qi+1 then y′i can not be derived from (4.2). In this case

y′i = (qi + qi+1)/2.

The derivative at the first two points and the last two points has to be chosen by

other means. For Dj,j+1(xj) we use the asymptotic behaviour described in section 3.3

and assume

D′j,j+1(xj,i) =

{−ajbj · exp(−ajxj,i for i ∈ {1, 2}0 for i ∈ {N,N − 1} (4.3)

with aj and bj from (3.18) and (3.19).

Considering an interval (x0, x1) with y1, y2, y′1, y

′2 given, the polynomial can be

expressed as

y(x) = p0 + p1(x − x1) + p2(x − x1)2 + p3(x − x1)

3 (4.4)

where

p0 = y1 (4.5)

p1 = y′1 (4.6)

p2 = [3(y2 − y1)/(x2 − x1) − 2y′1 − y′

2]/(x2 − x1) (4.7)

p3 = [y′1 + y′

2 − 2(y2 − y1)/(x2 − x1)]/(x2 − x1)2 . (4.8)

(4.9)

The integrations needed to determine the expectation values of the polynomials can

again be done using appendix C.1

Figure 4.1 shows L0 − Li calculated using a Sali tree with Akima interpolation.

Compared to figure 3.4 it shows no abrupt transition to the over-oscillating behaviour

and an overall better convergence.

4.1.2 Tension Splines

Splines under tension, first suggested by Schweikert in 1966 [13], are a common tool

for shape preserving interpolation, i.e. for interpolation that should avoid the spurious

oscillations observed for cubic splines. The name comes from the fact that the curve

can be interpreted as a very light and flexible bar, that is not only constrained to run

through certain points, but is also ’pulled’ at the ends. The strength of this tension

is described by a tension factor σ.

19

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1

2

3

4

5

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i

0 5 10 15 200

1

2

3

4

5

210.50.25analytic

i

Figure 4.1: The effect of h on the quality of the Sali approach with Akima interpola-tion instead of cubic splines. The graphs show L0 − Li with n = 20, x = 60, ψ = 0.3(above) and ψ = 0.2 (below)

20

4.1.2.1 Definition

For σ = 0 no tension is applied, which should lead to normal cubic splines. For

σ → ∞ the tension should minimise the length of the curve, leading to straight lines

connecting the given function values, thus avoiding spurious oscillations but loosing

smoothness and any non-spurious extrema.

The derivation of the formulation of splines under tension used here can be found

in [4]. It is summarised in appendix B. The basic assumptions are a continuous

second derivative of y and piecewise linearity of the term y′′(x) − σ2y(x) in each

interval (xi, xi+1). This leads to the following form of the interpolation:

y(x) =y

(2)i+1 sinh(σ′(x − xi)) + y

(2)i sinh(σ′(xi+1 − x))

sinh(σ′(xi+1 − xi))

+(yi+1 − y

(2)i+1)(x − xi) + (yi − y

(2)i )(xi+1 − x)

xi+1 − xi

(4.10)

for x ∈ (xi, xi+1), where

σ′ = σ · xN − x1

n − 1(4.11)

is used instead of σ and the parameters y(2)i are determined by a system of linear

equations very similar to cubic splines. Boundary conditions are needed either for

y′(xj) or y′′(xj) with j ∈ {1, N}. The same conditions as in section 3.3 are used. An

algorithm to determine y(2)i is given in [5].

The use of σ′ instead of σ is to avoid scaling effects when the grid size is changed.

The integrations that are needed to determine the expectation values of the tension

splines can still be done analytically. The results are given in appendix C.2.

4.1.2.2 Selection of Tension Factors

Special care has to be taken when choosing the tension factor σ′. If it is too small,

the same problems as with cubic splines will be observed. If it is chosen too large,

smoothness will be lost and extrema will be underestimated. If additional conditions

are know about the function, like convexity or bounds on a derivative, it is in principle

possible to determine the set of σ′ ≥ 0 for which the constraint is satisfied [12].

In this work no such conditions are used as they would limit the applicability of

the results to more complex models for which no analytical solution is known. So

another way has to be found to rule out values of σ′ that lead to unstable results. To

avoid spurious oscillations as observed in figure 3.5, σ′ will be chosen so that within

21

any two adjacent intervals (xi−1, xi), (xi, xi+1) there is at most one inflexion point.

The derivatives of y in Ii = (xi, xi+1) are:

y′(x) = σ′[y

(2)i+1 cosh(σ′(x − xi)) − y

(2)i cosh(σ′(xi+1 − x))

]/ sinh(σ′hi)

+[yi+1 − yi − y

(2)i+1 + y

(2)i

]/hi (4.12)

y′′(x) = σ′2[y

(2)i+1 sinh(σ′(x − xi)) + y

(2)i sinh(σ′(xi+1 − x))

]/ sinh(σ′hi)(4.13)

with hi = xi+1 − xi. As all sinh terms in y′′ are positive for xi < x < xi+1, y′′ has

no zeros if y(2)i+1y

(2)i > 0. Now assume y

(2)i+1y

(2)i < 0. As y′′(xj) = σ′2y

(2)j , the second

derivative has at least one zero in Ii.

y′′(x) = 0

⇔ |y(2)i+1| sinh(σ′(x − xi)) = |y(2)

i | sinh(σ′(x+1 − x)) . (4.14)

As the left hand side is zero for x = xi and strictly increasing in Ii and the right

hand side is zero for x = xi+1 and strictly deceasing in Ii, exactly one x ∈ Ii exists

with y′′(x) = 0. With the same reasoning we see that if y(2)i = 0, there are no further

inflexion points in (xi−1, xi+1), though y′′ might be constantly zero, this is still at

most one inflexion point.

To summarise: If y(2)i+1y

(2)i > 0 there is no inflexion point in Ii, with y

(2)i+1y

(2)i < 0

there is exactly one inflexion point in Ii and with yi = 0 there is at most one inflexion

point in (xi−1, xi+1).

When the calibration of the splines is started, σ′ is set to a positive, but small

value. When two inflexion points are observed within two adjacent intervals, σ′ is

increased by a fixed amount ∆σ′ until no two such intervals exist.

Though y′′(x) → 0 ∀x ∈ Ii, this convergence is not uniform. So it can not be

guaranteed that the condition can be fulfilled for any σ′ < ∞. Due to this fact σ′ will

only be increased up to a maximum value σ′max. Then σ′

max will be used as tension

factor. The performance can be optimised by choosing ∆σ′ relatively large and then

optimising σ′ by nested intervals.

4.1.2.3 Results

Fig. 4.2 shows the result of the model calibration for n = 20 and ψ ∈ {0.2, 0.3}.Though the results are much better than for the simple cubic splines, the splines

under tension show no improvement compared to Akima interpolation.

Figure 4.3 shows the values of σp determined by the algorithm described above

for the same cases. The value of σ′max = 1000 is not reached, so the condition of no

22

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1

2

3

4

5

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0 5 10 15 200

1

2

3

4

5

210.50.25analytic

i

Figure 4.2: The effect of h on the quality of the Sali approach with splines undertension instead of cubic splines. The graphs show L0 − Li with n = 20, x = 60,ψ = 0.3 (above) and ψ = 0.2 (below)

23

oscillations is always fulfilled. The deviation of the Sali result using splines under

tension is not due to oscillations, but due to a loss of smoothness caused by the high

tension factor.

0 5 10 15 200

50

100

150

200

δx = 2

δx = 1

δx = 0.5

δx = 0.25

i

σ’

0 5 10 15 200

100

200

300

δx = 2

δx = 1

δx = 0.5

δx = 0.25

i

σ’

Figure 4.3: Tension factor σ′ as a function of the time-step i for n = 20, x = 60,ψ = 0.3 (above) and ψ = 0.2 (below)

4.2 Splitting off the Asymptotic Behaviour

In the last chapter we saw that cubic splines are unsuitable to handle exponential

growth. The basic idea discussed in the present section is to keep the cubic splines

and split of the exponential asymptotic behaviour instead. Consider the two terms

that have to be integrated during a Sali step (3.10, 3.13). Using

gs,i(xi) = (1 + b(s)i−1 exp(−ai−1xi)) s ∈ {1, 2} (4.15)

we define fs,i for s ∈ {0, 1} by

g0,i(xi) · f0,i(xi) = exp(−ψixi − ψ2i Gi/2)Di,i+1(xi) (4.16)

g1,i(xi) · f1,i(xi) = (1 + δiLi exp(−ψixi − ψ2i Gi/2))Di,i+1(xi) (4.17)

where the b(s)i−1 are chosen so that fs,i(−x) = 1. If b

(s)i−1 ≤ 0 the asymptotic behaviour

will not be split off, a normal Sali-step will be performed instead. Using the asymp-

totic behaviour of D discussed in Section 3.3, we get:

f ′s,i(xi) ≈ 0 for |xi| ≫ 0 . (4.18)

Natural boundary conditions, i.e. f ′′s,i(xi) = 0 at x = ±x will be used for fs,i . The

exponential term that has been split off can be handled analytically:

Ej[exp(−ψixi − ψ2i Gi/2)Di,i+1(xi)]

= Ej[(1 + b(0)i−1 exp(−ai−1xi)) · f0,i(xi)]

24

=

∞∫

−∞

(1 + b(0)i−1 exp(−ai−1xi)) · f0,i(xi) · n(xi; Gi − Gj) dxi

=

∞∫

−∞

f0,i(xi) · n(xi; Gi − Gj) dxi

+ b(0)i−1

∞∫

−∞

exp(−ai−1xi) · f0,i(xi) · n(xi; Gi − Gj) dxi

=

∞∫

−∞

f0,i(xi) · n(xi; Gi − Gj) dxi (4.19)

+ b(0)i−1e

a2i−1

2(Gi−Gj)

∞∫

−∞

f0,i(xi − ai−1(Gi − Gj)) · n(xi; Gi − Gj) dxi

and

Ej[(1 + δiLi exp(−ψixi − ψ2i Gi/2))Di,i+1(xi)|xj]

=

∞∫

−∞

f1,i(xi) · n(xi − xj; Gi − Gj) dxi (4.20)

+ b(1)i−1e

a2i−1

2(Gi−Gj)

∞∫

−∞

f1,i(xi − ai−1(Gi − Gj)) · n(xi − xj; Gi − Gj) dxi

From here on we can proceed like in section 3.3 with the sole exception that the

cubic spline interpolation is used on fn,i which should lead to a much more stable

behaviour.

The tails x > x and x < −x are less problematic in this case, as fs,i(x) ≈ 1 for

x < −x and fs,i(x) ≈ s for x > x. Nevertheless they will be taken into account and

fs,i(x) will be assumed to be constant outside (−x, x).

Figure 4.4 shows Li for different values of h and the same parameters as in figure

3.4 with the only difference that the asymptotics are split off here. As it can clearly

be seen the stability is improved significantly. The deviations observed for h = 2 are

caused by oscillations as those observed in figure 3.5. This can be seen in figure 4.5.

Figure 4.6 shows the result of the Sali-approach using cubic splines with split-off

exponential behaviour for the same set of tenors and volatilities ψ as in figure 3.1 using

the analytical solution. With h = 1 the analytical solution can be reproduced in most

cases except for long tenor and very high volatility. To get an impression whether

x = 60 is large enough so that the asymptotic behaviour is a good approximation for

25

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4

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0 5 10 150

1

2

3

4

5

210.50.25analytic

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Figure 4.4: The effect of h on the quality of the Sali approach with split off asymptoticbehaviour. The graphs show L0−Li with n = 20, x = 60, ψ = 0.3 (above) and ψ = 0.2(below)

26

-20 -10 0

0

500

1000

1500

2000 1086421

x

Figure 4.5: f1,i according to (4.17) for different time-steps i (n = 20, ψ = 0.2, h = 2).

all x < −x, the limit to the non-asymptotic terms c0 (A.14) is monitored. For all sets

of tenor and volatility considered and every time step it is smaller than 0.1.

The combination of the methods described above, splitting off the exponential

behaviour and using a spline type that is less prone to overoscillations does show an

improvement compared to the simple use of the other spline types but is not as good

as using the normal cubic splines after splitting off the exponential. As long as no

overoscillations occur, the simple cubic splines seems to be best suited among the

examined spline types. For completeness, figures showing these results are given in

appendix D.

When applying this method to other models it is not necessary to identify the

exact asymptotic behaviour. It suffices to identify a function gs,i(x) so that fs,i(x) is

‘well behaved’ in the sense that it shows less spurious oscillations than the original

function gs,i(x) · fs,i(x) and the integrations involved can still be performed in an

efficient manner.

27

0 2 4 6 80

1

2

3

4

50.70.60.50.40.30.20.1

0 5 10 15 20

0

1

2

3

4

5

0.50.40.30.20.150.1

0 5 10 15 20 25 300

2

40.40.30.20.180.150.140.130.1

Figure 4.6: L0 − L with L calculated using a Sali tree with split off asymptoticbehaviour, a tenor of (from top left) 10, 20 and 30 years and different values of ψwith x = 60 and h = 1 (symbols). The lines show the analytic results from figure 3.1for comparison

28

Chapter 5

Optimising the Grid

Up to now no attention has been paid to the structure of the underlying grid. In

principle, three aspects can be considered to optimise the grid :

• The size of the grid given by the maximal value x. If this is chosen too small,

essential contributions far from equilibrium might get lost. If x is chosen too

large, computation time will be higher than necessary. In addition this may

increase stability problems at the lower end of the grid.

• The value of the lattice spacing h. For each interpolation technique used up to

now, the effect of changing h has been demonstrated (fig. 3.4, 4.1, 4.2 and 4.4).

The conflict between performance and stability should be solved such that h is

small enough to get an accurate result but within this restriction as large as

possible.

• The concept of equidistant points can be abandoned in favour of a lattice that

is adapted to the structure of the function. The convergence theorem for cubic

splines clearly favours a homogeneous grid, as this case allows for a minimal

factor K (2.6) compared with other grids with the same size and number of

grid points. So within this work only homogeneous grids will be considered.

The optimisation should be done with the goal to minimise computation time while

keeping an acceptable level of precision. As the integration (3.21) has to be done

for each single grid point and the integration implies a sum over all grid points, the

computing time is of order O(N2) as long as the number of grid points is the same

for each time-step.

But the Sali method does not rely on a constant grid for all time steps. So all of

the above adjustments can be done during the calibration for each single time-step.

29

The optimisation will be done on the basis of cubic splines with split-off asymptotic

behaviour from section 4.2, as these led to the most reliable results.

5.1 Grid Size

According to equation (4.19) two integrations are done over f0,i. The first with a

normal distribution centred at x = 0, the other centred at x = −ai−1(Gi −Gj), both

with variance Gi − Gj. So it must be guaranteed that a reasonable vicinity of both

points is covered by the grid.

In the following a range of ±3√

Gi − Gj is considered to be sufficient. The validity

of this assumption is confirmed a posteriori using the bound on non-asymptotic terms

derived in appendix A.2.1.

As in the first part of a Sali-step the integration (4.19) is done with j = 0, the

grid should at least cover the interval

Gi = (x0, xN) =(−ai−1 Gi − 3 ·

√Gi, 3 ·

√Gi

). (5.1)

The last step i = 0 earns special attention. All other values of x are integrated

over, averaging out small oscillations, but for i = 0 the limit of zero variance is reached

and the two integrals in equation (4.19) evaluate as f0,0(0). To obtain maximum

precision for this single point, the grid is chosen so that x = 0 is a grid point for all

time-steps. So x0 and xN are chosen as

x0 = −m0 · h < −ai−1 Gi − 3 ·√

Gi (5.2)

xN = mN · h > 3 ·√

Gi (5.3)

with minimal m0,mN ∈ N.

It is not guaranteed that Di,i+1 already shows the asymptotic behaviour at x0 =

−ai−1 Gi − 3 ·√

Gi. So neither ai nor bi can be determined from the first two grid

points. ai from section 3.3 will be used. b(s)i will be determined as follows: From

equations (4.16,4.17) it can be seen that

b(0)i−1 = exp

(−1

2Giψ

2i

)bi (5.4)

b(1)i−1 = δiLib

(0)i−1 (5.5)

where bi defined in equation (3.17) can be determined from b(1)i by using equation

(4.20): As f1,i(x) ≈ 1 for x ≪ 0, both integrals in equation (4.20) converge to 1 for

30

xj → −∞. So bi = b(1)i exp(a2

i (Gi − Gj)/2). From this and bN = 1 the following

expressions for b(s)i are derived:

b(0)i = exp

(−1

2Giψ

2i

) n∏

j=i+1

δjLj exp

(−1

2ψ2

j Gj +1

2ψj(Gj − Gn)2

)(5.6)

b(1)i = δiLib

(0)i . (5.7)

The above considerations are valid for the integration over f0,i in equation (4.19).

For equation (4.20) the situation is basically different as the Gaussians are centred

around x = xj and x = −ai−1(Gi − Gj) + xj, where xj can be any grid-point. So

vicinities of these points can not be covered by any finite grid and the tails still have

to be taken into account.

As fs,i(x0) can be far from 1, the tail x < x0 has to be handled in a different

manner than in section 4.2. As fs,i(x) → 1 for x → −∞, assuming fs,i(x) = fs,i(x0)

for x < x0 clearly overestimates the tail, though fs,i(x) = 1 clearly underestimates it.

As an estimation of the functional form for x < x0 an extrapolation is used that

fits with the known values for x ≥ x0 and x → −∞: An exponential decay

fs,i(x) = 1 + c · exp(d · x) (5.8)

if fs,i(x0) < fs,i(x1) and a Gaussian

fs,i(x) = 1 + c · (−d(x − m)2) (5.9)

if fs,i(x0) > fs,i(x1). The parameters are chosen so that the extrapolation fits with

values at the first two or three grid points respectively leading to

c =log(y1 − 1) − log(y0 − 1)

x1 − x0

(5.10)

d = (y0 − 1) · exp(ax0) (5.11)

for the exponential decay and

m = −log(y0−1)−log(y1−1)log(y0−1)−log(y2−1)

(x20 − x2

2) − x20 + x2

1

2(

log(y0−1)−log(y1−1)log(y0−1)−log(y2−1)

(x2 − x0) + x0 − x1

) (5.12)

d =log(y1 − 1) − log(y0 − 1)

(x0 − c)2 − (x1 − c)2(5.13)

c = (y0 − 1) exp(b(x0 − c)2) (5.14)

for the Gaussian.

31

The above estimation of the tails can still lead to inconsistent results if d < 0

in equation (5.9). In this case either an other estimate for the tail is needed or the

grid has to be chosen larger. It must be emphasised that the importance of the tails

does not come from equation (4.19), where the tails are far from the central value of

the Gaussian, but from equation (4.20), where the Gaussian is centred around any

grid-value.

Choosing a more or less arbitrary functional form for x < x0 is in principle not

different from choosing a cubic polynomial for each interval inside the grid. The

different functional form comes from the fact that the known behaviour for x → −∞has to be taken into account.

For consistency reasons the same approach should be used on the upper interval

(xN ,∞) though the numerical impact is negligible.

0 5 10 15 200

0,5

1

1,5no tailf(x)=f(x

0) for x<x

0

f(x)=1 for x<x0

exponentialanalytical solution

i

Figure 5.1: Effect of different tail extrapolations on the quality of the Sali-approachwith spilt asymptotics. The graphs show L0 − Li with n = 20, ψ = 0.2 and h = 1. Itis obvious that the exponential tails defined in equations (5.8–5.14) are necessary toreproduce the analytical results.

Figure 5.1 illustrates the effect of the different approaches to the interval (−∞, x0).

All estimations of a constant tail clearly over- or underestimate the integral, while

the exponential extrapolation reproduces the analytical solution with high accuracy.

32

The number of grid points that were used for each single step is displayed in figure 5.2

The total number of integrations done is 1368 compared to 4678 integrations needed

in the previous chapters with a fixed −x0 = xN = 60.

0 5 10 15 2010

20

30

40

50

i

N

Figure 5.2: The Grid size needed forthe calibration in figure 5.1 as a func-tion of the time-step i.

Note that the driving factor on the grid

size is basically different from the previous

chapters. While in the previous chapters it

was important to choose x large enough so

that the asymptotic behaviour takes over,

after splitting off the exponential behaviour

the approximated function is ’well behaved’

so that the grid can be limited to a much

smaller interval.

This approach relies on the fact that the

coefficients of the asymptotic exponential

can be determined analytically (i.e. ai and

bi) as they are needed in (4.19) and (4.20).

Otherwise a larger grid would be needed to

determine them numerically.

As both ai and bi are known analytically, the estimation of the influence of non-

asymptotic terms used previously can be applied in a straightforward manner to get

a boundary on the integral over the lower tail x < x0 as shown in appendix A.2.2.

The integration is done for f0,i with σ =√

Gi, x0 = −ai−1Gi − 3σ and both µ = 0

and µ = −ai−1Gi, so according to (A.17) the error from the lower tail is bounded by

Θx0=

c

2

[erfc

(3 + ψ

√Gi√

2

)+ erfc

(3 + (ai−1 + ψ)

√Gi√

2

) ]

· exp

(Giψ

2

√2

− ai−1Gi

)(5.15)

with time-independent volatility ψ. If ψi varies with i, ψ has to be replaced by

minj≥i ψi. The ratio between Θx0and the integral (4.19) will be monitored to verify

the adequacy of the choice of the grid size.

This estimate is rather rough as it assumes an approximation f0,i ≈ 1 for x < x0,

whereas the functional form introduced earlier is supposed to be much more precise.

On the other hand this approximation can be used to verify the functional form used

as an approximation for the tail.

33

5.2 Lattice Spacing

The basic idea for an automatic selection of the lattice spacing h is the same as for

the selection of the tension factor σ′ for splines under tension in section 4.1.2.2. If

oscillations are observed, h is reduced by a constant factor w and the time-step is

repeated.

In more detail, after performing a time-step i with lattice-spacing h, Di−1,i is split

into an asymptotic part and a non-asymptotic part as defined in equation (4.16). The

non-asymptotic part f0,i−1 is checked for oscillations as in the case of splines under

tension in section 4.1.2.2. In the case of cubic splines it is trivial that a change of

the sign of y′′(xi) indicates a point of inflexion. If within any two adjacent intervals

two points of inflexion are observed, the calculation of Di−1,i is repeated with lattice

spacing h/w.

Figure 5.3 shows the result of the Sali approach using cubic splines with split-

off exponential behaviour and optimised grid size given by equations (5.2,5.3) and

a lattice spacing determined by the algorithm described in the present section with

starting value h0 = 2 and w = 1.5. In the case n = 30 and ψ = 0.1 the estimation

of the tail (5.9) fails which is indicated by an ‘upside-down’ Gaussian with d < 0.

Here it is necessary to revert to a larger grid size. The relative effect of the lower tail,

Θx0divided by (4.19) can get as large as 3% for the last non-trivial step i = 1, but

otherwise stays well below 0.2%.

Comparing figure 5.3 to the matching results for a fixed grid in figure 4.6 we see

that the adjustment of the lattice spacing leads to a good convergence even with very

high values of ψ.

Table 5.2 shows the number of integrations needed for the calibration with fixed

and optimised grid respectively. As each integration contains a sum over all grid

points, the total number of summands is given as well, indicating the computation

cost. For short tenor the improvement is significant. For longer tenor a smaller lattice

spacing than used previously for the fixed grid is necessary to avoid the errors for high

volatility. This leads to a higher computational cost.

The actual computation time on an average desktop system (Athlon64 X2 4200+)

is given in table 5.2. For long tenor the computation time of the analytical solution

gets out of hand, making it necessary to use the SALI approach even for this simple

model with a known analytical solution.

34

0 2 4 6 80

1

2

3

4

50.70.60.50.40.30.20.1

0 5 10 15 200

1

2

3

4

5

0.50.40.30.20.150.1

0 5 10 15 20 25 300

1

2

3

4

5

0.40.30.20.180.150.140.130.1

Figure 5.3: L0 − L with L calculated using a Sali tree with split off asymptoticbehaviour, a tenor of (from top left) 10, 20 and 30 years, different values of ψ withoptimised grid size and lattice spacing (symbols). The lines show the analytic resultsfrom figure 3.1 for comparison

35

fixed grid, x = 60, h = 1 optimised gridn Integrations Summands Integrations Summands10 15,386 1,861,706 4,160 135,78620 27,828 3,367,188 18,298 22,34,41030 56,624 6,851,504 60,594 14,498,006

Table 5.1: The number of integrations and the number of summands necessary forthe calibration using the Sali approach with a fixed grid (figure 4.6) and optimisedgrid (figure 5.3)

n Fixed grid, x = 60, h = 1 optimised grid Analytical solution10 5 1 < 120 9 7 230 17 37 1,693

Table 5.2: The computation time in seconds necessary for the calibration using theanalytical solution (figure 3.1) or the SALI tree with fixed or optimised grid (figures4.6 and 5.3 respectively) on an average desktop system.

36

Chapter 6

Conclusion

6.1 Summary

The large tails that are observed for Markov functional models for long term struc-

tures have been investigated using an analytically solvable model based on the Black-

Derman-Toy model. The analytical solution has been used to demonstrate notable

effects on the model calibration by the large tails.

For the application of the Sali approach to the MBDT model the asymptotic be-

haviour of the price function of the discount bonds had to be investigated in order

to get realistic boundary conditions and an appropriate approximation of the contri-

bution beyond the grid. Even in this case of an analytically solvable model, the Sali

approach proved to be necessary to get results for long tenor within an acceptable

time.

The difference between the parameters of the asymptotic behaviour determined

from the outer grid points and the parameters determined analytically can be used

as an indicator whether the grid size is adequate to assume asymptotic behaviour

beyond the grid.

The cubic splines used in this attempt turned out to be rather unstable for high

volatilities and long tenor. This is consistent with the stability theorem for cubic

splines which indicates that a very small lattice spacing may be needed to get reason-

able results when modelling exponential behaviour with cubic splines. This depends

on the factor inside the exponential which in our case is determined by volatility and

time to maturity.

Three approaches to improve the quality of the numerical results were investi-

gated. Two were based on different spline types that are supposed to be less prone

to over-oscillations. Some improvement can be seen, but this is limited by a lack of

smoothness that comes with these new spline types.

37

The third approach is based on splitting off the exponential behaviour and ap-

plying the semi-analytic integration to the remaining ‘well behaved’ function. This

improved the numerical results significantly.

Based on the third approach an optimisation of the underlying grid was investi-

gated:

As the asymptotic behaviour to split off can completely be determined analytically,

it is possible to restrict the lattice to a much smaller vicinity of two ’centres’ of the

probability density. So the calculation time can be reduced significantly.

A good value of the lattice spacing can be determined by dynamical adjustment

during the calculation, which makes sure that no over-oscillations occur.

6.2 Outlook

As a next step one can be either apply the methods described above to more complex

Markov functional models or calibrate the MBDT model to real market data and

compare the prices to these obtained by a LIBOR market model.

The application of the methods derived within this thesis to more complex models

relies on the determination of the asymptotic behaviour. This could either be done

giving an exact analytical formula or by fitting a parameterised function to a small

number of grid points close to the boundary.

In the latter case, the choice of the grid size will be a non-trivial issue. It has to

be chosen large enough so that the asymptotic behaviour is determined correctly and

is a good approximation outside the grid, while in the earlier case the grid can be

chosen much smaller.

38

Appendix A

Analytical solution of the

MBDT-Model

A.1 Calibration

First we show that Di,i+1 has the form

Di,i+1 = Ei

2n−i−1−1∑

j=0

Xi,jEi(−Yi,jWG·)

(A.1)

with constants Xi,j and Yi,j. This is certainly true for i = n− 1 with Xn−1,0 = 1 and

Yn−1,0 = 0. Now assume that (A.1) has been proven for some i < n. Then we get by

using (3.6) and the martingale property of E

Di−1,i =2n−i−1−1∑

j=0

Ei−1

[Xi,jEi(−Yi,jWG·

)(1 + δiLiEi(−ψiWG·)]

=2n−i−1−1∑

j=0

(Xi,jEi−1(−Yi,jWG·

) + δiLiXi,jEi−1

[Ei(−Yi,jWG·

)Ei(−ψiWG·)])

=2n−i−1−1∑

j=0

(Xi,jEi−1(−Yi,jWG·

)

+δiLiXi,jEi−1

[e((Yi,j+ψi)

2−ψ2i −Y 2

i,j)Gi/2Ei(−(Yi,j + ψi)WG·)] )

=2n−i−1∑

j=0

(Xi−1,jEi−1(−Yi−1,jWG·

))

(A.2)

39

with

Xi−1,j =

{δiLie

(Yi,j−2n−1−i+ψi)

2−ψ2i −Y 2

i,j−2n−1−i)Gi/2Xi,j−2n−1−i for 2n−i > j ≥ 2n−1−i

Xi,j for j < 2n−1−i(A.3)

Yi−1,j =

{Yi,j for j < 2n−1−i

Yi,j−2n−1−i + ψi for 2n−i > j ≥ 2n−1−i (A.4)

A.2 Estimation of the Tails

A.2.1 Non-Asymptotic Contributions

The present section will establish an estimation of the influence of non-asymptotic

terms for the lower tail x < x0. This is to show in which cases the approximation

used in 3.21 is justified. From the analytical solution we will use the fact that Di−1,i

is a sum of exponential terms and that the coefficient Yi−1,j is a sum of ψi or in the

case of homogenous volatility just a multiple of ψ. The whole derivation is done at a

fixed time-step, so the notation can simplified.

Consider a sum of exponential terms

f(x) =k∑

i=0

βi exp(−αix) (A.5)

with βi > 0 and αi > αj > 0 for all k ≥ j > i ≥ 0. For x → −∞ clearly the term for

i = 0 gives the asymptotic behaviour, so a = α0. For x0 < x1 < 0 the approximated

coefficient a is

a =1

hlog

(f(x0)

f(x1)

). (A.6)

Choose β > 0 so that

f(x1) = β0 exp(−α0x1) + β exp(−α1x1) . (A.7)

Thenk∑

i=1

βi exp(−αix) < β exp(−α1x) ∀x < x1 (A.8)

Let

g0(x) = β0 exp(−α0x) and (A.9)

g1(x) = β exp(−α1x) . (A.10)

40

g0(x) is the asymptotic behaviour and g1(x) limits the non-asymptotic terms for

x < x0. As g1(x)/g0(x) < g1(x0)/g0(x0) for all x < x0, it is sufficient to get an upper

limit for

c = g1(x0)/g0(x0) (A.11)

for an estimation of the non-asymptotic terms. From

exp(ah) = f(x0)/f(x1) ≤g0(x0) + g1(x0)

g0(x1) + g1(x1)(A.12)

with h = x1 − x0, we arrive at

exp((a − a)h) = exp(ah)g0(x1)

g0(x0)≤ 1 + g1(x0)/g0(x0)

1 + g1(x1)/g0(x1)

=1 + c

1 + c exp(h(α0 − α1))(A.13)

and so

c ≤ c0 =exp((a − a)h) − 1

exp((a − α1)h) − 1(A.14)

Where the difference a − α1 can be derived from (A.4). In the case of a time inde-

pendent volatility ψi = ψ, we have a − α1 = ψ.

A.2.2 Integral over the Tail

This section establishes an estimation of the integral over the lower tail to be used in

section 5.1 where both a and b are known analytically and the integration is performed

after splitting off the asymptotic behaviour. Let

f0(x) =f(x)

β0 exp(−α0x)(A.15)

with f from equation A.5. Obviously

1 ≤ f0(x) ≤ 1 + c0 exp((a0 − a1)x) (A.16)

for x < x0. For simplicity we assume a time-independent volatility, so α0 − α1 = ψ.

Integrating (A.16) leads to an estimate of the error made by replacing the tail by 1.

∫ x0

−∞

(f0(x) − 1)n(x − µ; σ)dx ≤ c0

2

(1 + erf

(x0 − µ − ψσ2

√2σ

))

· exp

((σψ)2

2+ µψ

)(A.17)

41

Appendix B

Tension Splines

A derivation of splines under tension can be found in [4]. For completeness it is

summarised here. Let again ∆ = {a = x0 < x1 < . . . < xN = b} be a partition of

the interval (a, b) and f : (a, b) → R with f(xi) = yi. The tension spline for f with

tension factor σ is a function y : (a, b) → R with

y(xi) = yi ∀1 ≤ i ≤ N , (B.1)

that is continuous in its second derivative and where for each interval (xi, xi+1) the

quantity y′′(x) − σ2y(x) is linear in x:

y′′(x) − σ2y(x) = (y′′(xi) − σ2yi) ·xi+1 − x

hi

+(y′′(xi+1) − σ2yi+1) ·x − xi

hi

(B.2)

for xi < x < xi+1 with hi = xi+1 − xi. Solving (B.2) and using (B.1) results in

y(x) =y′′

i

σ2· sinh(σ(xi+1 − x))

sinh(σhi)+

(yi −

y′′i

σ2

)xi+1 − x

hi

+y′′

i+1

σ2· sinh(σ(x − xi))

sinh(σhi)+

(yi+1 −

y′′i+1

σ2

)x − xi

hi

(B.3)

for xi < x < xi+1 and y′′i = y′′(xi). Taking the second derivative and equating left-

and right-hand derivatives at xi leads to(

1

hi−1

− σ

sinh(σhi−1)

)· y′′

i−1

σ2(B.4)

+

(σ coth(σhi−1) −

1

hi−1

+ σ coth(σhi) −1

hi

)· y′′

i

σ2

+

(1

hi

− σ

sinh(σhi)

)· y′′

i+1

σ2

=yi+1 − yi

hi

− yi − yi−1

hi−1

, (B.5)

42

for 1 < i < N . Like in the case of simple cubic splines two more equations are

needed to make the splines unique. As boundary conditions the first derivative at the

boundary are chosen as in section 3.3. Taking the first derivative of (B.3) leeds to

(σ coth(σh1) −

1

h1

)· f ′′

1

σ2+

(1

h1

− σ sinh(σhi)

)· f ′′

2

σ2=

y2 − y1

h1

− y′1 (B.6)

and(

1

hN−1

− σ sinh(σhN−1)

)· f ′′

N−1

σ2

(σ coth(σhN−1) −

1

hN−1

)· f ′′

N

σ2

= y′N − yN − yN−1

hN−1

. (B.7)

As coth(x) > 1/x for x > 0, the matrix representing the above system of linear

equations is strictly diagonal dominant and thus nonsingular. Due to the tridiagonal

structure it can be solved numerically by LU decomposition in O(N) operations

[5, 15].

43

Appendix C

Expectation Values of Splines

C.1 Cubic Splines

The integrals of polynomials times the normal distribution are a textbook matter.

In the form they are presented here they can e.g. be found in [6] and [9]. They are

included in this work for completeness and to emphasise a detail that can for longer

tenors cause large errors in the numerical evaluation.

We define

Ip(a, b, c) =

∫ b

a

(x + c)pn(x)dx (C.1)

with the normal distribution n(x) = (2π)−1/2 exp(−x2/2). For the treatment of cubic

splines only the cases p ∈ {0, 1, 2, 3} are relevant:

I0(a, b, c) = N(b) − N(a) (C.2)

I1(a, b, c) = c(N(b) − N(a)) + n(a) − n(b) (C.3)

I2(a, b, c) = (c2 + 1)(N(b) − N(a)) + (2c + a)n(a) − (2c + b)n(b) (C.4)

I3(a, b, c) = (c3 + 3c)(N(b) − N(a)) + (2c2 + ca + (c + a)2 + 2)n(a)

−(2c2 + cb + (c + b)2 + 2)n(b) , (C.5)

where

N(x) =

∫ x

−∞

n(x′)dx′ (C.6)

=1

2(1 + erf(x/

√2)) (C.7)

=1

2(2 − erfc(x/

√2)) (C.8)

is the cumulative normal distribution. The difference N(b)−N(a) can cause numerical

problems, as for a, b ≫ 1 both terms are very close to 1. Above some x ∈ R N(a) =

44

N(b) = 1 for all a, b > x within numerical precision. So, if the terms we want to

evaluate numerically have substantial contributions far from the central value of zero,

these contributions are not correctly taken into account.

For these cases the complementary error function provides a numerically stable

form of the cumulative normal distribution. Assuming |a− b| ≤ 1 the following terms

are used:

N(b) − N(a) =

(erfc(−b/√

2) − erfc(−a/√

2))/2 for a/√

2 ≤ −2

(erf(b/√

2) − erf(a/√

2))/2 for − 2 < a/√

2 < 2

(erfc(a/√

2) − erfc(b/√

2))/2 for 2 ≤ a/√

2

(C.9)

C.2 Tension Splines

The tension splines consist of a term that is linear in x and another term that is of the

form sinh(σ′(x−x0)). The expectation values of both can be determined analytically.

The integration of the linear term can be taken from (C.3), for the other term we

define

B±(a, b, c) =

∫ b

a

exp(±(x + c))n(x)dx . (C.10)

So ∫ b

a

sinh(x + c)n(x)dx =1

2(B+(a, b, c) − B−(a, b, c)) (C.11)

with

B±(a, b, c) =1√2π

∫ b

a

exp

(±(x + c) − 1

2x2

)dx

=1√2π

∫ b

a

exp

(−1

2(x ∓ 1)2 +

1

2+ c

)dx

= exp

(1

2+ c

)[N(b ∓ 1) − N(a ∓ 1)] (C.12)

45

Appendix D

Combined Approaches to Improve

Convergence

The two methods used to improve convergence in sections 4.1 and 4.2, using a spline

type that is less prone to over-oscillations and splitting off the exponential behaviour,

can be easily combined.

As Figures D.1 and D.2 indicate, this does not lead to a further increase in stability.

Instead, larger deviations from the analytical solution can be observed than in the

case of ordinary cubic splines and split off exponential behaviour.

This hints to the tradeoff that comes with the other spline types which is a lack

of smoothness. It becomes most evident when considering splines under tension with

a high tension factor. Then the spline under tension converges to straight lines con-

necting the grid-points, cutting off any maximum of fs,i that does not happen to lie

on a grid-point.

46

0 5 10 15 200

1

2

3

4

5

210.50.25analytic

i

0 5 10 15 200

1

2

3

4

5

210.50.25analytic

i

Figure D.1: The effect of h on the quality of the Sali approach with split off asymptoticbehaviour and Akima splines. The graphs show L0− Li with n = 20, x = 60, ψ = 0.3(above) and ψ = 0.2 (below)

47

0 5 10 15 200

1

2

3

4

5

210.50.25analytic

i

0 5 10 15 200

1

2

3

4

5

210.50.25analytic

i

Figure D.2: The effect of h on the quality of the Sali approach with split off asymptoticbehaviour and splines under tension. The graphs show L0 − Li with n = 20, x = 60,ψ = 0.3 (above) and ψ = 0.2 (below)

48

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