A Two-Regime Markov-Chain
Model for the Swaption Matrix
�
Dr Mathias Thorsten Keil
Christ Church College
University of Oxford
A thesis submitted in partial fulfillment of the MSc in
Mathematical Finance
August 16, 2006
Abstract
The most commonly used models for pricing complex interest rate products are
LIBOR Market Models. These models describe the evolution of a set of forward
LIBOR rates. For a given set of LIBOR rates such a model is defined by the term
structures of their volatilities and the correlations among these rates. It is therefore of
vital importance to use a parameterization for these quantities that is able to capture
the important features of the market the model is calibrated to. Since this is usually
the swap market a good parameterization of the LIBOR Market Model should also
be a good model for the volatility term structure of forward rates, i.e. the swaption
matrix which contains all available information about the volatility term structure in
the swap market.
It is a well known feature of the interest rate markets that in times of market
turmoil the volatility term structure undergoes sudden changes in shape when the
market enters these excited regimes. After a short period the market is normal again
which also switches the shape of the volatility term structure back to normal. Re-
bonato and Kainth proposed an approach to capture this market feature in a LIBOR
Market Model. They use the most common parameterization of the volatility term
structure to describe the normal market situation and the excited regime each with
one set of stochastic parameters. In the present work a more simplistic approach is
taken by keeping the parameters constant. The model captures the switches between
the two regimes by transition probabilities.
After introducing the setup of a LIBOR Market Model the parameterization of
the forward rate volatility term structure is presented and it is shown how this relates
to swap rates. The algorithms for calibrating to caplets as well as swaptions are
discussed. The sensitivities of the model parameters are systematically analyzed. By
allowing the fit to vary various settings of parameters the quality of the calibration to
swaption data is assessed. Finally, a very promising and simple parameterization of
the model that captures the regime switches is fitted with four parameters to a series
of monthly market data spanning a period of four years that includes major financial
events. This model yields a dramatic improvement over the model without regime
switches in the description of the swaption matrix without increasing the number of
fitting parameters. While the latter approach yields an average deviation of 76 basis
points in implied volatility the new model obtains 63. Dramatic improvements are
obtained during and after periods of market turbulence. In addition the fit parameters
are a lot more stable over time which implies lower re-hedging costs when using this
model.
2
To Meike
3
Acknowledgment
First of all I would like to thank my supervisor, Dr Riccardo Rebonato, for his out-
standing support and his patience throughout the project. By giving me the oppor-
tunity to participate in the Oxford program, my employer, d-fine GmbH Frankfurt,
made this work possible. I benefited form discussions with Dr Laurent Hoffmann and
Dr Matthias Mayr. For pointing out the minpack-algorithm to me I thank Dr Gotz
Rienacker.
4
Contents
1 Introduction 6
1.1 Market Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 LIBOR Market Models With a Smile . . . . . . . . . . . . . . . . . . 10
2 Instantaneous Volatility 12
2.1 Caplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Volatility Term Structure . . . . . . . . . . . . . . . . . . . . 18
2.3.2 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Calibration to Swaption Prices 28
3.1 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 The Fitting Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Some Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Results for Fits with Regime Switches 32
4.1 Regime Switches for a “Normal” and an “Excited” Market . . . . . . 32
4.2 Fits for Many Trading Days . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Necessary and Possible Future Steps . . . . . . . . . . . . . . . . . . 46
5 Conclusions 47
A Bootstrapping 49
B The Levenberg-Marquardt Method 50
5
Chapter 1
Introduction
In the past years, with the trading of more and more complex interest-rate deriva-
tive instruments, the modelling of the underlying rates received enormous attention.
There is some feedback reaction in modelling and market behaviour, so as the models
became more complex also the markets evolved. The observed changes in the market
influence the models and the changes in modelling standards influence the markets.
Complex products include optionalities on LIBOR as well as swap rates. Therefore
the models used for pricing and hedging such products should be able to describe the
markets of caps and floors as well as swaption markets in a satisfactory manner. The
smiles observed in implied volatility surfaces of cap, floor and swaption markets have
evolved during the years. While in earlier models (like the one by [6]) the occurrence
of a smile was sometimes considered as a side effect, by now the modelling of the
complex shapes takes up maybe most of the effort.
In pricing interest-rate derivatives the market standard for modelling the term
structure of interest rates are LIBOR and swap market models. There is a close
connection between the two markets so that models that describe the dynamics of
LIBOR rates and the ones describing swap rates can be used rather interchangeably.
However, either model assumes a log-normal distribution in the changes of the un-
derlying rates which cannot be the case for both models at the same time and not
even for forward rates of different tenors. But the discrepancy is small [15] and does
not matter in the common applications of the models. The market models are so
successful because most interest rate derivatives depend on a finite number of points
on the yield curve and can therefore be modelled by simulating the dynamics of the
corresponding set of rates. In agreement with market conventions (as long as one
neglects smiles) the rates are log-normal and Black’s [1] formulae can be applied.
Usually in market models a set of forward LIBOR rates or a set of co-terminal swap
rates or both is considered. The set of traded instruments that should be used to
6
calibrate the model is therefore a rather natural choice, which makes the sensitivity
to market prices very transparent.
On the other hand the setup also has some disadvantages, which are due to mod-
elling a finite set of rates. And the sampling of a set of rates is a high dimensional
problem, but the dimensionality can be reduced by principal component analysis.
Each specification of a LIBOR Market Model results in a multi-factor model for
the simultaneous evolution of a given number of forward rates. As I briefly show in
the following section all the drifts are given by the no-arbitrage condition and the
only remaining degrees of freedom are the volatility and correlation term structures.
1.1 Market Models
Today, market models are well known and detailed descriptions can be found in text
books and other publications. This overview is based on [3], [16], [17], [22]. The
dynamics of the forward rates are given by the equations
dfi
fi
= µ({σj}, {fj})dt + σi(Ti) dzi (1.1)
dzidzj = ρij(Ti, Tj)dt (1.2)
where all variables are at time t and fi stands for the i-th forward LIBOR rate starting
at time Ti and maturing after one period of length τi, i runs from 1 to the number of
considered rates. The corresponding σi are the volatility term structures dependent
on the starting times Ti, and dzi represent the Wiener increments for each rate. The
Wiener increments are correlated via the matrix ρij . We will see that the crucial
parameters in the model are the σi and ρij . Once these quantities are determined the
functional form of the µ({σj}, {fj}) depends on the choice of numeraire only. The
actual functional form for a given choice of numeraire is fixed by the no-arbitrage
requirement. I will use the notation µki for the drift of fi under the measure k as
discussed below.
A natural choice of numeraire is to use one of the discount bonds P (Tk) with
maturity Tk. Under this numeraire one of the equations (1.1) has a zero drift term
since the corresponding forward rate is a martingale. In order to determine the other
drift terms I consider a forward rate agreement (FRA) which has the underlying rate
that starts at time Ti and matures at time Ti + τi = Ti+1. Its value is then given by
τifiP (Ti). Since FRAs are tradable, the expression
τifi
P (Ti)
P (Tk)
7
is a Martingale in the chosen measure for any i. The special choice i = k shows that
under this measure also fk is a martingale and therefore has zero drift as stated before.
In addition, also the discount bonds are tradable and P (Ti)/P (Tk) is a Martingale
as well. Using Ito’s lemma we can now look at the differential of the FRA value and
adjust the drift terms in Eqs. (1.1) such that the sde for the FRA has zero drift.
d
(
fi
P (Ti)
P (Tk)
)
=P (Ti)
P (Tk)dfi + fid
(
P (Ti)
P (Tk)
)
+ dfid
(
P (Ti)
P (Tk)
)
(1.3)
The ratio of the two discount bonds depends on the forward rates in the following
way if i < k
P (Ti)
P (Tk)=
k∏
j=i+1
(1 + τjfj) (1.4)
For the differential of this ratio I need to consider the product rule.
d
(
P (Ti)
P (Tk)
)
= d
(
k∏
j=i+1
(1 + τjfj)
)
(1.5)
=k∑
l=i+1
1
1 + τlfl
k∏
j=i+1
(1 + τjfj) τl dfl
+k∑
l=i+1
l∑
m=i+1
1
1 + τlfl
1
1 + τmfm
k∏
j=i+1
(1 + τjfj) τl dfl τm dfm
=P (Ti)
P (Tk)
k∑
l=i+1
(
τl
1 + τlfl
dfl +l∑
m=i+1
τl
1 + τlfl
τm
1 + τmfm
dfl dfm
)
By using Eq. (1.1) and Ito’s lemma I can simplify this expression by keeping terms
of order dt and less:
d
(
P (Ti)
P (Tk)
)
=P (Ti)
P (Tk)× (1.6)
k∑
l=i+1
(
τlfl
1 + τlfl
(µldt + σldzl) +l∑
m=i+1
τlfl
1 + τlfl
τmfm
1 + τmfm
σlρlmσmdt
)
=P (Ti)
P (Tk)
k∑
l=i+1
τlfl
1 + τlfl
σldzl
In the last step I used the fact that the ratio of discount bonds is a martingale and
therefore the sde must not have a drift term. In other words, the µl in the second
line have the form that the first and second dt terms cancel.
Now, I consider the drift terms in Eq. (1.3). From the left hand side it is clear
that the total drift is zero. Since the second term on the right hand side is just fi
8
times the differential of the ratio of discount bond this term has no drift. So the drifts
in the first and last terms need to cancel. The first term is
P (Ti)
P (Tk)dfi =
P (Ti)
P (Tk)fi(µidt + σidzi) (1.7)
Keeping only terms to order dt and using Ito again the second term is
dfid
(
P (Ti)
P (Tk)
)
= fiσidzi
P (Ti)
P (Tk)
k∑
l=i+1
τlfl
1 + τlfl
σldzl (1.8)
=P (Ti)
P (Tk)fiσi
k∑
l=i+1
τlfl
1 + τlfl
σlρildt
The drift term µi can now be read off, by demanding that the drifts of the rhs of
Eq. (1.7) and Eq. (1.8) add up to zero:
µi = −σi
k∑
l=i+1
τlfl
1 + τlfl
σlρil (1.9)
These drift terms are for i < k and having chosen the discount bond P (Tk) as a
numeraire.
The same steps can be carried out for i > k where the ratio of discount bonds is
given by
P (Ti)
P (Tk)=
k∏
j=i+1
1
1 + τjfj
(1.10)
The use of the product rule finally yields the solution
µi = σi
i∑
l=k+1
τlfl
1 + τlfl
σlρil (1.11)
With µk = 0 all the drift terms in the model where we use P (Tk) as a numeraire are
now determined. For other numeraires similar results can be obtained.
From this sketch of a derivation of the drift terms in the specific setup of the LI-
BOR Market Model it is immediately clear that the complete set of model parameters
is given by today’s forward rates and the covariance term structure for these rates.
A similar statement is true for swap market models. The only difference is that a
set of co-terminal swap rates and their volatility term structures need to be specified.
Instead of discount bonds the natural numeraire for a swap market model are swap
annuities
B(Ti, Tn) =
n−1∑
j=i
τjP (Tj+1) (1.12)
9
where Ti is the start date of the swap and P (Ti+1) is a discount bond maturing at
the end of the period τi. Thus the fixed leg of a swap starting at Ti has the value
SRN×MB(Ti, Tn) with the swap rate SRN×M , where the starting date is N = Ti and
the swap matures after M = Tn − Ti years.
The sde for the forward swap rates can be written down in the same way as for the
forward LIBOR rates. Choosing a swap annuity as a numeraire and by considering
the forward value of the fixed leg of a swap it is possible to do very similar steps
as detailed above in the case of the FRA. Keeping in mind that both, the fixed leg
and the annuity are martingales, one finds the drift terms of the co-terminal swap
rates. The expressions are more complicated than for the forward LIBOR rates.
Since I will consider LIBOR Market Models in the following sections I will not go
into any further detail for the swap market models. As stated before, there is a close
connection between the two types of models. In the next chapter I show how to
calibrate a LIBOR Market Model using swap rates. But before that I present some
approaches to account for the smile observed in the Plain Vanilla Markets.
1.2 LIBOR Market Models With a Smile
As pointed out earlier, most of the current research in the field of LIBOR Market
Models goes into modelling the shape of the volatility surface (or cube) and the
correlations. Initially, when a kind of smile was first observed in JPY fixed income
markets the volatility was a monotonically decreasing function of the moneyness. The
models for plain-vanilla options started taking these features into account in the late
1990s and it was then observable in most currencies. Major financial events like the
Russia default and the LTCM crisis in late 1998 triggered market turbulences and
thereby also led to more complex volatility shapes. A historical review of the various
approaches and a nice discussion can be found in e.g. [18].
The approach that I will be using throughout this work is a constant elasticity of
variance (CEV) approach where a very common re-parameterization is applied (see
e.g. [16]). The standard form of a CEV process is given by
df = µ(f, σ)dt + σ(τ)fβdz (1.13)
where β is a constant parameter which leads to a normal process for β = 0 and a
log-normal process for β = 1 but it can also take on other values in the interval (0, 1).
Analytic solutions to some typical problems exist for special choices of β only (0, 12,23,
10
and 1). It is therefore very useful to work with a displaced rate
d(f + α)
f + α=
df
f + α= µ(α)(f, σ)dt + σ(α)(τ)dz (1.14)
with the constant α. The notion of a displaced diffusion process was introduced in [21]
in the context of modelling the firm value. This process offers a lot more analytical
tractability (see [16] for detailed calculations) and in addition for a wide range of rate
values the displaced diffusion process is a re-parameterization of the CEV process
[11]. The approximate re-parameterization
α = f01 − β
β(1.15)
will be used in section 2.3.2 (f0 represents the forward rate as of today).
So far, the volatility term structure is a deterministic function of time and the
forward rate. The CEV or displaced diffusion approach can account for monotonically
decreasing smiles. An obvious extension would be to introduce jumps, as such models
are well known from equities. However, such a model is very complex and it can be
shown that for most of these models prices of the same quality can be obtained by
deterministic coefficients ([16] and references therein). A more flexible way to obtain
complex shapes is to make the volatility a function of one or more stochastic processes
other than the forward rates, e.g. [8].
In what follows a different approach is chosen which is mainly inspired by market
observation. The idea was first presented in [20]. It is based on the observation that
the instantaneous volatility term structure for forward rates and also for swap rates
has different shapes in “normal” and “excited” market situations. Changes between
normal and turbulent market situations are more or less instantaneous and therefore
modelled by a Markov chain. The following chapter explains how to calibrate such a
model and shows the possible volatility term structures that can be obtained.
11
Chapter 2
Instantaneous Volatility
As I pointed out in the previous chapter the crucial quantities that need to be chosen
in order to use a LIBOR or swap market model are term structures of the volatilities
and correlations. In order to calibrate the model to market data the term structure of
the instantaneous volatility needs to be parameterized and the parameters need to be
adjusted to market data. There is a number of possible parameterizations available
on the market from which I use a very common and rather general one (see e.g. [3],
[16]). In addition the volatility term structure contains a stochastic component as it
is discussed in [19].
2.1 Caplets
Our starting point is the stochastic process
df
f= µ(σ, f)dt + σ(t, T )dz (2.1)
where f is the forward rate, dz is a Wiener increment. For the percentage volatility
σ I choose the well-known parameterization ([3], [16])
σ(t, T ) = σ(τ) = (a + bτ) exp(−cτ) + d (2.2)
with τ = T − t and a, b, c, and d real valued constants. In order to use this parame-
terization in the displaced-diffusion set up
df
f + α= µ(σα, f)dt + σα(t, T )dz (2.3)
I transform the parameters of the forward-rate volatility to the volatility of the dis-
placed forward rate as follows:
aα = af
f + α
12
bα = bf
f + αcα = c
dα = df
f + α(2.4)
In order to use the Black formalism I consider the root mean square σ of the
instantaneous volatility which is defined by the following equation:
σ(T, t)2(T − t) =
∫ T
t
σinst(T, u)2du . (2.5)
Equivalently for the volatility of the displaced forward rates
σα(T, t)2(T − t) =
∫ T
t
σαinst(T, u)2du (2.6)
With the obtained volatility I can then use the Black formula in order to obtain the
price of the caplet. With the parameterization of the instantaneous volatility as given
in Eq. (2.2) the integration can be done analytically as given in e.g. [16].
In order to account for the observed changes of the volatility term structure in the
market I then introduce another stochastic parameter that determines whether the
market makes the transition from a normal state into an excited state or vice versa.
The probabilities for these transitions are
[
λnn λxn
λnx λxx
]
(2.7)
Then the function Eq. (2.2) needs to be extended to
σ(t, Ti) = yt σn(t, Ti) + (1 − yt) σx(t, Ti)
⇔ σ(t, Ti)2 = yt σn(t, Ti)
2 + (1 − yt) σx(t, Ti)2 . (2.8)
The variable y assumes values 0 or 1 and follows a two-state Markov-chain process
and σn(t, Ti) and σx(t, Ti) represent the parameterizations of the normal and excited
states respectively. For a particular realization of yt on the considered time interval I
obtain a chain of normal and excited pieces in the volatility term structure squared.
The integration of this given chain then yields the variance and therefore the root
mean sqaure.
13
2.1.1 The Algorithm
As input parameters I use the current forward rate f0 and the parameterizations
for σn(t, Ti) and σx(t, Ti), i.e. an, bn, cn, dn, ax, bx, cx, and dx with the transition
probabilities λnx and λxn and the initial state (excited or normal). I also need the
displacement in the diffusion α and the expiry and strike of the caplet. The insight
is that, conditional on a particular realization of the stochastic parameter y, the
setting is exactly that of a deterministic-volatiliy LIBOR Market Model. The price
under regime switches can therefore be obtained by integrating the conditional prices.
For a given number Nsteps of subintervals I integrate the squares of both volatilities
σn(t, Ti)2 and σx(t, Ti)
2 by splitting up the integral in Eq. (2.5). I perform a Monte-
Carlo simulation where I determine for a number Nsim of simulations whether these
subintervals are in normal or excited states by drawing uniform random numbers
and using the probabilities from Eq. (2.7). In each simulation I add up the already
integrated variances for the subintervals and obtain the Black volatility by dividing
through the time interval and taking the square root. This volatility I can then use
in the Black formula and obtain the caplet price. I take the average over the prices of
all simulations and, from the unconditional price thus obtained, compute the implied
volatility.
2.2 Swaptions
A swaption is the option to enter into a swap with fixed rate K that starts at a given
time Tn and matures at time Tm. The underlying swap pays/receives a fixed rate
of the underlying nominal value and in return receives/pays a floating rate at some
initially fixed dates. Let τi denote the period following Ti, fi(t) ≡ f(t, Ti, Ti + τi)
the forward LIBOR rate spanning that period, Ni the nominal value relevant for
the payments at the end of that period, and P (t, Ti+1) a discount bond maturing at
Ti+1. In order to obtain the par rate of the swap both legs need to have the same
present value. The corresponding forward par swap rate can be viewed as a linear
combination of forward rates (see e.g. [14]):
SRk(t) =m−1∑
i=n
wi(t)fi(t) =
∑m−1i=n Niτifi(t)P (t, ti+1)∑m−1
i=n NiτiP (t, ti+1)(2.9)
where k labels the par rate of a swap starting at Tn and maturing at time Tm. I
denote the percentage instantaneous volatility of a swap rate expiring N = Tn − t
14
years from time t and maturing another M = Tm − Tn years after expiry by σN×M (t)
or also by σSRk(t) for short. By applying Ito’s Lemma I obtain
(σN×M (t))2 ≡ (σSRk(t))2 =
∑
i
∑
j
∂SR∂fi
∂SR∂fj
fifjρijσiσj
(∑
l wlfl)2(2.10)
where σi is the instantaneous volatility of the i-th forward swap rate and ρij is the
correlation of the i-th and the j-th rate. The sums over i, j, and l start with the
expiry and run over all fixing dates.
Using the approximation of static weights together with Eq. (2.9) and as in the
case of the forward rates Eq. (2.5) we calculate the Black volatility which is the root
mean square of the instantaneous volatility of the k-th swap rate:
(σSRk(t))2(Tn − t) =
∫ Tn
t
(σSRk(u))2 du (2.11)
I obtain
(σSRk(t))2(Tn − t)SRk
2 ≈n+m−1∑
i=n
n+m−1∑
j=n
wiwjfifj
∫ Tn
t
ρijσiσjdu (2.12)
and the weights have a value wi(t) independent of the integration time. The σi are
a short notation for instantaneous volatility σ(t, Ti) of the i-th forward rate with
the parameterization of Eq. (2.2). Correlations are taken into account by the simple
function
ρij = exp(−β|Ti − Tj |) (2.13)
and can therefore be pulled out of the integral. For the given parameterization of σi
the integral∫ Tn
t
σiσjdu (2.14)
can be done analytically (see e.g. [16]). The explicit result is
∫ t
0
σiσjdu =1
4c3ec(Ti+Tj)×
[
2c2(
a2(e2ct − 1) + 2ad(ect − 1)(ecTi + ecTj) + 2cd2ec(Ti+Tj)t)
+
b2(
−1 − 2c2TiTj − c(Ti + Tj)+
e2ct(
1 + 2c2(t − Ti)(t − Tj) + c(−2t + Ti + Tj)))
−2bc(
a(
1 + e2ct(−1 + c(2t − Ti − Tj)) + c(Ti + Tj))
+
2d(
ecTject(−1 + c(t − Ti)) + ecTj (1 + cTi)+
ecTiect(−1 + c(t − Tj)) +ecTi(1 + cTj)))]
(2.15)
15
As in the case of caplets I use the displaced-diffusion ansatz Eq. (2.3). The step
that is more complicated for swaptions is due to the correlations between the various
forward swap rates.
2.2.1 The Algorithm
In order to obtain the implied swap-rate volatility I do the following steps. First,
I initialize the parameters of our model. For the parameterization of the volatility
term structure an, bn, cn, dn, ax, bx, cx, and dx, the transition probabilities λnx and
λxn, and the correlation parameter β. The displacement parameter α needs to be
set and for the simulation I set the number of simulations Nsim and the number of
possible regime switches Nsteps. The discount factors are obtained by bootstrapping
(see Appendix A) from the swap rates that are interpolated from 0.5 and 1 year
deposit rates and the par rates of 2, 5, 10, 20, and 30 years swaps. Here and in what
follows the tenor is set to half a year.
For given strike and expiry of the swaption Tn and maturity of the swap Tm I
then compute the volatility term structure in the following way. First I transform
the volatility parameters a, b, and d from the given swap-rate volatility values to
the displaced-diffusion values by multiplying the forward swap rate SRk and dividing
by the displaced swap rate SRk + α as given for caplet case, Eq. (2.4). For Nsteps
intervals I compute two arrays of variances, one for the exited market and one for
the normal state. For each of the Nsteps intervals I use the integration in Eq. (2.12)
in the displaced diffusion setup, however, with different integration boundaries. Let(
σαSRk
(t))2
abdenote the root mean square volatility of the displaced swap rate in the
interval that runs from ta to tb then I get the following equation
(
σαSRk
(t))2
ab≃ 1
Tn(SRk + α)2
n+m−1∑
i=n
n+m−1∑
j=n
wiwj(fi + α)(fj + α)ρij
∫ tb
ta
σαi σα
j du (2.16)
The two arrays of excited-state and normal-state variances are each filled with Nsteps
of the(
σαSRk
(t))2
abwith the parameterizations for excited and normal resp. For the
first entry in both arrays the lower boundary of the time interval is ta = t (ta = t = 0
in our simulation) and the last entry in both arrays the upper boundary of the interval
is tb = Tn. Each interval spans the time dt = Tn−tNsteps
.
Now, I am in a position to perform a large number of simulations using the Markov-
chain process. In each of the simulations I start with the same initial state at time t,
e.g. for the normal state y from Eq. (2.8) is equal to zero in the first interval. Using
the probabilities λnx and λxn I decide for each of our Nsteps − 1 intervals following
16
the initial one by drawing uniform random numbers whether I stay in the same state
for the next interval or whether I switch to the other state. Finally, I end up with a
structure of y = 0 or y = 1 for each interval of the simulation under consideration.
Now, I just need to pick the corresponding entries from the two variance arrays.
Adding up all the entries that I picked for one of the simulations leaves me with one
realization of the integrated volatility term structure. After Nsim simulations I have
a Monte-Carlo sample of the σαSRk
(t).
The next goal is to obtain the Monte-Carlo sample of prices for the swaptions
corresponding to the realizations of the obtained rms displaced swap-rate volatilities.
Using the displaced swap rate SRk +α, the displaced strike K +α, and the volatilities
from our Monte-Carlo simulation σαSRk
(t) I compute the prices of a payer’s swaption
that correspond to the sample of volatilities via the well-known Black formula
Vswpt = ((SRk + α) N(h1) − (K + α) N(h2))n+m−1∑
i=n
0.5 P (t, ti+1) (2.17)
where N(h1,2) is the cumulative normal and
h1,2 =ln(SRk+α
K+α) ± 1
2
(
σαSRk
(t))2
(Tn − t)
σα√
Tn − t. (2.18)
The factor 0.5 is the tenor. I obtain a set of prices and take the average.
I then use this average price to plug it into an implied-volatility algorithm in order
to get the Black implied volatility without the displacement.
2.3 Variation of Parameters
After the model has been explained in detail it is now a good point to get some
intuition about the model parameters. In order to get a feeling for the effect that
the variation of the various parameters has I varied one of the input parameters
at a time in a reasonable range and plotted the curves for the chosen samples. In
the title of each plot I give the name of the parameter that I varied, the range of
variation [a, b], and the width of each step. The interval for the volatility σ (either
instantaneous or implied) gives the values [σ(a), σ(b)] in order to determine the way
that the lines change when increasing the parameter. If σ(a) > σ(b) then the upper
most curve corresponds to the parameter under consideration equal to a. In each
section the choice of all the model parameters is always given at the beginning. I
begin with the parameterization of the forward-volatility term structure and then
study the parameterization of the swaption matrix. All volatilities in the plots are
given in percent.
17
2.3.1 Volatility Term Structure
In this subsection I show how the volatility term structure Eq. (2.2) changes when
three of the parameters a, b, c, and d are kept fix and one parameter is varied. Except
the parameters that I varied all the other parameters were set to the best fit result for
the data of 05.01.1998. The a, b, c, and d parameters are summarized in the following
table:
an 0.00057399bn 0.00255834cn 0.32406904dn 0.00519951
The dependence of the instantaneous volatility on the four parameters is rather
simple. From Eq. (2.2), which is
σ(t, T ) = σ(τ) = (a + bτ) exp(−cτ) + d
it is obvious that a variation in d just yields a parallel shift of the curve (see
Fig. 2.4). At the same time
limτ→∞
σ(τ) = d
and therefore in all cases except the one where d is varied all the curves asymptotically
go to dn = 0.00519951.
The level of the volatility at τ = 0 is additionally influenced by a:
limτ→0
σ(τ) = a + d
as shown in Fig. 2.1. A maximum exists for a positive τ if b is not zero and 1c
> ab. If
a maximum is present it is at position
τmax =1
c− a
b.
In Fig. 2.2 only the lowest line where bn was set to zero shows no maximum. The
height of the maximum is given by
σ(τmax) =b
cexp(−1 +
ac
b) + d .
In a normal market situation the maximum of the curve was found to be at 12 to 18
months [19]. An explanation can be suggested along the following lines. At the long
end of the volatility term structure the level of uncertainty is given by the expected
variation of the long term inflation. Monetary authorities usually act in such a way as
to keep these variations low which is the reason for the low volatility at the long end.
18
At the short end the variation in the rates directly depends on the steps taken by
the monetary authorities. These steps are usually well anticipated by the market due
to the policy of the authorities to indicate rate changes in advance. The mid range
therefore has the most uncertainty in normal market situations. In market turmoil
the situation changes dramatically within a short period of time. At the short end
rates are very uncertain and volatility rises drastically. The effect at the long end is
less pronounced which in total leads to a decaying term structure. (See also Fig. 3.2)
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0 2 4 6 8 10 12 14 16
σ i
τ
an in [0.0, 0.0025] step 0.0005 and σi in [ 0.63, 0.80] for τ = 0.5
Figure 2.1: Possible shapes of the instantaneous forward-volatility term structure.The parameter an was varied as given in the title of the plot. Which choice of an eachline corresponds to can be inferred from the interval for σi.
19
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0 2 4 6 8 10 12 14 16
σ i
τ
bn in [0.0, 0.005] step 0.001 and σi in [ 0.57, 0.74] for τ = 0.5
Figure 2.2: Same as 2.1 with variation of the parameter bn.
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
0 2 4 6 8 10 12 14 16
σ i
τ
cn in [0.1, 1.1] step 0.2 and σi in [ 0.70, 0.64] for τ = 0.5
Figure 2.3: Same as 2.1 with variation of the parameter cn.
20
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12 14 16
σ i
τ
dn in [0.0, 0.01] step 0.002 and σi in [ 0.16, 0.96] for τ = 0.5
Figure 2.4: Same as 2.1 with variation of the parameter dn.
21
2.3.2 Swaptions
After presenting the dependence of the volatility term structure of forward rates on
the variation of the relevant parameters we now turn to the implications of these
parameter variations for the swaption matrix. In this section I present a systematic
study of the dependence of the whole swaption matrix under consideration on the
parameters a, b, c, and d in the normal and exited states and α. I use the same values
for the set of normal and excited parameters with the exception of the one parameter
that I vary. The following table summarizes all model parameters that are used:
an 0.00057399 λxn 20.0bn 0.00255834 λxn 2.0cn 0.32406904 α 1.18013071dn 0.00519951 β 0ax 0.00057399 NSim 1000bx 0.00255834 NSteps 60cx 0.32406904 Initial State 0.0dx 0.00519951
The a, b, c, and d parameters correspond to the four-parameter best-fit result
of 05.01.1998. The swaptions used in the following figures are 0.5, 1, 3, 5, 10 years
expiry into 1, 2, 3, 5, 7, 10 years maturity swaps. The series always starts with the
0.5 year swaption into all the underlying swaps and continues with the 1 year expiry
swaption and so on. I plot the implied volatility in percent for all these swaptions.
In the plot for various choices of α (Fig. 2.13 the implied volatility is normalized to
the value for the 0.5 × 0.5 swaption. The reason is that varying the α parameter
corresponds to first order to multiplying the implied volatility by a factor of SRk+α
SRk
for the N ×M swaption. The fact that the implied volatilities differ for the different
choices of α is due to the swap curve which is on the 05.01.1998 basically a rising
function of the maturity (Fig. 2.14). This dependence is of course only present if
the a, b, c, and d parameters are the parameterization of the instantaneous volatility
in the displaced model. In order to obtain a reasonable distribution of the plotted
volatility term structures I use equally spaced steps in the CEV exponent βCEV and
then transform this into α by the approximate relation Eq. (1.15)
α = f01 − βCEV
βCEV
.
22
12.5
13
13.5
14
14.5
15
15.5
16
16.5
17
17.5
18
0 5 10 15 20 25 30
σ im
p
Swaption
an in [0.0, 0.002] step 0.0005 and σimp in [12.93, 16.19] for Swaption 0
Figure 2.5: Variation of the parameter an.
10
11
12
13
14
15
16
17
18
19
0 5 10 15 20 25 30
σ im
p
Swaption
bn in [0.0, 0.004] step 0.001 and σimp in [11.90, 14.98] for Swaption 0
Figure 2.6: Variation of the parameter bn.
23
10
12
14
16
18
20
22
24
26
28
0 5 10 15 20 25 30
σ im
p
Swaption
cn in [0.1, 0.9] step 0.2 and σimp in [14.28, 13.05] for Swaption 0
Figure 2.7: Variation of the parameter cn.
4
6
8
10
12
14
16
18
20
22
0 5 10 15 20 25 30
σ im
p
Swaption
dn in [0.0, 0.008] step 0.002 and σimp in [ 4.63, 19.24] for Swaption 0
Figure 2.8: Variation of the parameter dn.
24
13
14
15
16
17
18
19
0 5 10 15 20 25 30
σ im
p
Swaption
ax in [0.0, 0.016] step 0.004 and σimp in [13.78, 17.19] for Swaption 0
Figure 2.9: Variation of the parameter ax.
12
13
14
15
16
17
18
19
20
21
0 5 10 15 20 25 30
σ im
p
Swaption
bx in [0.0, 0.016] step 0.004 and σimp in [13.71, 14.94] for Swaption 0
Figure 2.10: Variation of the parameter bx.
25
12.5
13
13.5
14
14.5
15
15.5
16
16.5
17
0 5 10 15 20 25 30
σ im
p
Swaption
cx in [0.1, 0.9] step 0.2 and σimp in [13.89, 13.80] for Swaption 0
Figure 2.11: Variation of the parameter cx.
12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
17
0 5 10 15 20 25 30
σ im
p
Swaption
dx in [0.0, 0.008] step 0.002 and σimp in [13.32, 14.38] for Swaption 0
Figure 2.12: Variation of the parameter dx.
26
90
95
100
105
110
115
120
0 5 10 15 20 25 30
σ im
p
Swaption
βCEV in [0.2, 1.0] step 0.2 and σimp in [ 94.44, 98.65] for Swaption 29
Figure 2.13: Variation of the parameter α by stepping through a range of the CEVexponent. In this plot I normalized all the swaption matrices to the 0.5×0.5 volatility.More details are given in the text.
6
6.05
6.1
6.15
6.2
6.25
6.3
6.35
6.4
0 5 10 15 20 25 30
SR
[%]
swap
Figure 2.14: The swap rates that were used for obtaining Fig.2.13.
27
Chapter 3
Calibration to Swaption Prices
3.1 The Data
For the data analysis I used samples of swaption prices from a data collection of 05
January 1998 through 31 May 2002, a total of 34,500 values. The collection of data
comprises for each of the 1,150 trading days the swaption expiries of 0.5, 1, 3, 5,
and 10 years were considered each expiry into swaps of lengths 1, 2, 3, 5, 7, and 10
years. For the same trading days I used the 6-months and 1-year deposit rates as
well as the 2-, 5-, 10-, and 20-years swap rates in order to obtain swap rates and the
corresponding discount factors by boot strapping as described in Appendix A. The
swap rates and discount factors for all dates relevant to the pricing of the swaptions
that lie in between the given dates were obtained by linear interpolation.
The data include some major financial events like the Russia default in August
1998, the LTCM crisis one month later, the unexpected rate cuts by the Fed in early
2001, and the market turmoil in the aftermath of the terror attacks on September
11 in 2001. As it has been shown in [19] it is crucial to capture these excited states
of the market in a model for the volatility term structure. Fig. 3.1 shows the first
month of data. For each day all the rows of the swaption matrix are plotted in a
row as in the previous chapter, i.e. starting with the half-year expiry swaption and
all possible underlying swap maturities in increasing order and then doing the same
for the one-year maturity swaption and so on. So the x-axis represents the swaption
matrix, the y-axis the trading days and the z-axis the implied volatility. During a
“normal” market period as in Fig. 3.1 there are basically no noticeable changes in the
shape of the matrix. Once there are turbulences in the market as in August through
October 1998 the picture changes completely (Fig. 3.2). During the last days of July
1998 and the first days of August the picture is qualitatively the same as in Fig. 3.1.
28
05.01.199813.01.1998
21.01.199829.01.1998
Date0.5x10.5x101x10
3x105x10
10x10Swaption
0.12
0.15
0.18
σimp
Figure 3.1: The term structure of the swaption matrix during January 1998, a “nor-mal” period.
With the Russia default the turmoil starts and the “slope” gets steeper. After the
LTCM crisis there is even a large jump between two trading days.
3.2 The Fitting Algorithm
In order to adjust the model parameters a least-squares fit is carried out. I use
an implementation of the minpack [13] package. The underlying algorithm is the
Levenberg-Marquardt Method as briefly described in Appendix B.
The calibration is carried out on a day by day basis. For a given trading day I use
the 30 swaption implied volatilities σmkt impi×j as market data and compute the implied
volatilities for the same swaptions by the model σimpi×j . The function that needs to be
minimized is
f(x) =∑
i,j
(σmkt impi×j − σimp
i×j )2 (3.1)
where x represents the vector of the parameters that are varied, i runs over all swap-
tion maturities and j over all swap expiries.
29
20.07.199809.08.1998
29.08.199818.09.1998
08.10.1998
Date0.5x10.5x101x10
3x105x10
10x10Swaption
0.09
0.13
0.17
σimp
Figure 3.2: The term structure of the swaption matrix during the Russia default andthe LTCM crisis.
3.2.1 Some Checks
In order to test the algorithm I performed various fit runs with no regime switches
and the four fit parameters an, bn, cn, dn. The stability of the fit was tested by
varying the initial parameters for the test cases. The fits always yielded the same
very good results. As Fig. 3.3 shows the agreement with the fits obtained by [19] is
very good considering the different fit algorithms. The upper panel shows the fit to
the data from 05.01.1998 where the market was in a “normal” state. The solid line
represents the result of the model fit, the dotted line which only slightly differs for
long dated swaptions is the fit result from [19] and the dashed line shows the market
data of that day. Another consistency check is the same setup but for an excited
market situation. The lower panel shows the same as the upper panel for 05.11.2001,
where the market was in an excited state. As before both fits agree rather well. The
sum of squared deviations to market data, however, almost doubles. The difference
in relative deviations is of course smaller with the higher level of volatility. In the
following chapter I move on to the results with regime switches.
30
12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
0 5 10 15 20 25 30
σ im
p
Swaption
10
15
20
25
30
35
0 5 10 15 20 25 30
σ im
p
Swaption
Figure 3.3: A comparison of the market data (dashed) and the fit from [19] (dotted)and this fit (solid). The upper panel shows market data and fits for the first tradingday 05.01.1998 in the time series where the market was in a normal state and thelower one a fit on 05.11.2001 where the market was excited. The sum of squares ofthe deviations is in the lower case twice as high as in the upper.
31
Chapter 4
Results for Fits with Regime
Switches
In this chapter the main results are presented. The full model with regime switches
can have up to 13 parameters that need to be chosen. The goal is to have the smallest
possible number of free parameters, while still allowing for a good description of the
swaption matrix. Parameters that are not varied by the fit will be kept at reasonable
values as described in the text. In the fits that are discussed in the following only
combinations of the a, b, c, d parameters are varied for the normal and excited shapes.
As a first approach the next section gives details about varying various combinations
of these parameters for one day where the market was in a normal state and one day
in an excited day. After this in section 4.2 the best parameterization that was found
is applied to a large number of trading days spanning the period from January 1998
until January 2002. Finally, possible improvements and further steps are discussed.
4.1 Regime Switches for a “Normal” and an “Ex-
cited” Market
As an example for a day with a normal market I chose the first day of the data set
05.01.1998. I start by varying one additional parameter as compared to the fit without
regime switches. In principle I expect that an additional fit parameter should reduce
the sum of squared deviations. If the financial motivation of the model is true, i.e. it
is necessary to capture regime switches between normal and excited market states,
I expect to have an exponentially decaying shape for the excited forward-volatility
term structure and therefore choose bx = 0 as a simple assumption (see Sec. 2.3).
This is a rather crude choice and a different choice might well lead to a better fit
result, but in this study the focus is on keeping the model as simple as possible. In
32
addition to the normal parameters I varied one of the remaining excited parameters
ax, cx, dx and kept the other two excited parameters fixed at the values found for
the four-parameter fit. The resulting sum of squares of all deviations, however, is
hardly affected. This is true for any of the fits, including ax, cx, dx. Even for a fit
that includes an overall scaling factor for the excited term structure, i.e. I fit the
parameter kx and replace ax and dx by kxax and kxdx, there is only an improvement
of a few percent. In order to present an example, the following table gives the whole
parameter set for the fit of dx in addition to the normal parameters.
an 0.00095014 λxn 20.0bn 0.00274649 λxn 2.0cn 0.29944074 α 1.18013071dn 0.00449303 β 0ax 0.00098784 NSim 1000bx 0.0 NSteps 300cx 0.29788851 Initial State 0.0dx 0.00859404
Fig. 4.1 shows the resulting volatility term structures for excited (upper line)
and normal (lower line) parameters. The shapes are roughly as expected, i.e. the
excited line is decaying (which is trivially true after I chose bx = 0) and lies above the
humped normal shape. In order to meet the empirical data it is, however, necessary
that both functions asymptotically go to about the same level for large τ . Obviously,
the independent variation of parameters of the normal and excited functions does not
yield to the shapes observed in the market. Since the sum of squared deviations does
not improve much this choice of parameters does not improve the ability of the model
to describe the market. It does at the same time also not worsen the model.
As another test I keep the four normal parameters at their best fit values from the
four-parameter fit and let the solver change all four excited parameters, i.e. this time
including bx. Also in this case there is only very minor improvement in the overall
fit. Maybe,the weakness of this fit is, that the set of normal parameters that is kept
fix was a best fit in the four-parameter case and should therefore not be a “purely”
normal set of parameters. Both, the excited shape and the normal shape show a
hump in this fit which, because bx differs from zero quite a bit.
As a next stage I analyze the fit to the excited market situation on 05.11.2001 in
a similar way. The best fit results from the four-parameter cases are taken for the
normal and excited a, b, c and d respectively with the exception of bx which is again
set to zero. As a first try I fit an overall factor to each of the forward-volatility shapes,
33
i.e. kn and kx. It turns out that it is not possible to reach the accuracy of the fit
with just the four excited parameters. There is however a clear tendency to push the
excited shape up and the normal shape down (kn = 0.7800282 and kx = 1.84407039).
In order to give the fit some flexibility at the long end (large τ) of both curves
as well as on the short end I perform a four-parameter fit with the following setup.
The normal parameters are initially set to the best fit values from the normal four-
parameter fit (of 05.01.1998) and the excited parameters are set to the values that
were obtained by the fit to the excited market of 05.11.2001. Then the parameters
kn and kx as well as dn and dx are varied by the fit to the excited market 05.11.2001.
This basically means that for both curves, normal and excited, the fit can change
the overall level and in addition the level at the short end in connection with the
height of the hump. The latter is due to the fact that the k factors change a and b
at the same rate, where a is mainly responsible for the level at the short end and b is
mainly responsible for the height of the hump (see Sec. 2.3.1). It turns out that this
choice of fit parameters yields a more than 20% improved sum of squares compared
to the four-parameter fit without regime switches. This result is quite remarkable
since the number of fitted parameters is the same and the initial normal parameters
were the best fit results from a different day. The shapes of both forward-volatility
term structures is shown in Fig. 4.2.
Similar to the case for the normal day that is displayed in Fig. 4.1 the curves have
some of the expected features. The weak point is again at the long end of the curves
where the fit result does not agree with the behavior observed in the market. The
excited shape crosses the normal one at about τ = 11. As pointed out in the case of
the fit to the normal day, the fit with an independent variation of normal and excited
parameters does not yield the observed volatility term structure.
With the result of this section in mind I present a different approach in the next
section. Obviously, the normal and excited parameters should not be considered
independently but the features observed in the market should be implemented into
the model right from the start.
34
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0 2 4 6 8 10 12 14 16
σ i
τ
Figure 4.1: The forward-volatility term structure (in %) as a result of the 5-parameterfit for an, bn, cn, dn, dx to the data form 05.01.1998. The upper line corresponds tothe excited state and the lower one to the normal state. The values of the parametersare given in the text.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 2 4 6 8 10 12 14 16
σ i
τ
Figure 4.2: Same as in Fig. 4.1 with data from a four-parameter fit for kn, dn, kx, dx
to the data form 05.11.2001.
35
4.2 Fits for Many Trading Days
The parameterization that is used in this section has only four fit parameters and
attempts to incorporate all the findings of the preceding sections. The goal is to
choose a parsimonious parameterization that comprises the main features observed
in market data. The model should yield the expected shapes for the normal and
excited instantaneous volatility term structures, i.e. a humped shape for the normal
and a decaying shape for the excited curve where the excited curve is well above the
normal at the short end and gets to about the same level as the normal at the long
end, as shown in Fig. 4.3. In order to obtain a decaying shape I choose bx = 0. As a
somewhat arbitrary choice I set ax = 10 × an to ensure that the excited curve starts
out well above the normal at the short end. For the long end I make the simple but
again somewhat arbitrary choice dx = dn. As fit parameters I vary an, bn, cn and
dn (which therefore uniquely determine ax and dx). Since bx was set to zero only cx
remains at the initial choice of the best fit value from the four-parameter fit without
regime switches. In the previous studies the variation of cx did not have a large effect.
This parameterization is the result of the effects that have been seen in the variation
of the various parameters in Sec. 2.3.1, the analysis of market data (see Sec. 3.1 and
[19]) and the fits with various parameterizations in the previous section. A typical
forward volatility term structure that can be obtained by the model for normal and
excited curves is given in Fig. 4.3.
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0 2 4 6 8 10 12 14 16
σ i
τ
Figure 4.3: These are typical shapes for the normal and excited curves in the param-eterization used for the fits in this section. For this plot I used the best-fit resultsfrom 01.04.1998.
36
To summarize the setup:
• Fit parameters: an, bn, cn and dn
• Constraints: ax = 10 × an, bx = 0, dx = dn
• All other parameters are set to “reasonable” values
In the following table I summarize all the parameters that were kept at a fixed value
during the fits.
cx 0.075363546 β 0λxn 20.0 NSim 1000λxn 2.0 NSteps 50α 1.18013071052 Initial State 0 or 1
With this setup I perform fits for the first trading day of each month in the years
1998, 1999, 2000 and 2001 and January 2002, a total of 49 months. For all these days
I perform the four-parameter fit in the described parameterization
• with a normal state as the initial state,
• with an excited state as the initial state.
It might seem strange to run both setups for the whole data set of trading days. Of
course, the model that has the normal state initially should fit better for a market
that is in a normal state. But many times it is not easy to decide whether the market
is normal or excited by just looking at the data. So the idea is to run both setups
for all days and then decide ex post which of the days was normal or excited. If the
fit that started with the excited state yields the lower sum of squared deviations the
market is in an excited state, otherwise the market is normal. In order to compare
these results to the model without regime switches I also fit the model without regime
switches as it was discussed in Sec. 3.2.1 to the data of each trading day.
The expectation is, to find fits of the same quality or slightly better when regime
switches are used with a normal initial state in normal periods as compared to the
model without regime switches. In the aftermath of excited periods, where the model
without regime switches does not yield satisfying fits I expect to find an improved
description by the model with regime switches and the excited initial state. It is quite
clear that this parameterization with a couple of arbitrarily chosen parameters still
leaves room for improvement. But as I show in the remainder of this section despite
37
these shortcomings the model with regime switches yields a noticeable improvement
in the description of market data.
The results of the three different fits are presented in Figs. 4.4–4.8. Each of these
figures contains two panels. The upper panel displays the results from the fit without
regime switches as a solid line, the results from the fit with regime switches that
started in a normal state as a dashed line and the results for the fit that started in
the excited state as a dotted line. In all these figures the lower panel gives the results
for the combined model with regime switches, i.e. at each day I decide whether the
market is normal or excited by comparing the sums of squared deviations of the two
fits with regime switches. I use the model that has a normal state as the initial state
when this model yields the better fit and the model that starts with an excited state
otherwise.
In the upper panel of Fig. 4.4 I show the sum of squared deviations for all three
runs. The quality of the fit without regime switches and the ones with regime switches
that has a normal initial state are of similar quality. On average the model without
regime switches is slightly better but the effect is very small. For both of these
models the sum of squared deviations is a lot higher in periods after market turmoil.
The times after the Russia default and LTCM crisis in 1998 and the rate cuts in
early 2001 and the terror attacks in September 2001 are clearly visible in the top
panel of Fig. 4.4 in the peaks of sums of squared deviations of the described models.
However, the situation changes for the model with regime switches that has an excited
state as the initial state. This model is slightly worse than the other two models in
normal market situations but is a lot better for the periods following excited market
situations. Obviously, the choice whether the initial state is normal or excited really
corresponds to the current market situation.
This result shows that the model does behave in the expected way and that
the model with regime switches is able to describe a broader range of real market
situations than the model without regime switches. For these results it is important
to keep in mind that many of the parameter choices in the models with regime switches
were rather arbitrary and therefore one can even expect a lot of room for improvement.
As described above, in the lower panel of Fig. 4.4 I use the quality of the fits for the
two parameterizations with regime switches as an indicator whether the market is
excited or normal, i.e. the market is in a normal state if the model that starts with
the normal state yields the lower sum of squared deviations and the market is excited
if the model that starts in the excited state yields the better fit. In other words for
each month the figure gives the lower of the two possible values, which is the result
38
of the combined model with regime switches that always starts with the “correct”
state. The average deviation of this combined model is 63 basis points in implied
volatility. For the fit without regime switches the average deviation is 76 basis points
(in [19] where all the trading days were taken into account instead of one per month,
the average deviation is 79 basis points). This needs to be compared to the usual
bid-offer spread of 50 to 100 basis points. So the improvement that stems form
regime switches is really remarkable and was obtained with the same number of fit
parameters in both models.
Now, I turn to the fit parameters that were obtained by the different setups.
In Figs. 4.5–4.8 I use the same formatting as in Fig. 4.4. The top panels display
the results from the three setups and the bottom panel displays the result from the
“combined” model. The very remarkable result that can be read off these plots is
that in case of the combined model the fit parameters are a lot more stable than in
the model without regime switches. Especially in the cases of an and dn the effect
is clearly visible. The smallest improvement is found for bn. As a measure for the
variation of the fit parameters over time I give the standard deviations for the four
parameters in the combined model and the model without regime switches:
Model an bn cn dn
No regime switches 0.01175 0.00197 0.28206 0.01065Combined model 0.00116 0.00174 0.11346 0.00245
The stability of the model parameters is a very crucial measure for the quality of a
model. Strong day-to-day variations in the model parameters cause high (re-)hedging
costs. If the parameters change much then the prediction of the future volatility
changes which also changes the future re-hedging costs. This model is therefore a
very interesting improvement to the widely used four-parameter setup without regime
switches. The next section briefly describes possible further steps in order to make
this model usable in practical applications. As a final plot in this section I show the
swaption matrix for 02.11.1998 where the market was in an excited state (Fig. 4.9).
This is the day with the greatest improvement in the sum of squared deviations.
39
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
01.01.1998 01.01.1999 01.01.2000 31.12.2000 31.12.2001
Sum
of S
quar
ed D
evia
tions
Date
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
01.01.1998 01.01.1999 01.01.2000 31.12.2000 31.12.2001
Sum
of S
quar
ed D
evia
tions
Date
Figure 4.4: In the top panel the sum of squared deviations is shown for all threeparameterizations. The model without regime switches is represented by the solidline, the model with regime switches and the normal state as the initial state is givenby the dashed line and the model with regime switches and the excited state as theinitial state is represented by the dotted line. By using the “correct” initial statethe sum of squared deviations should be the minimum of both models with regimeswitches shown in the lower panel.
40
0
0.01
0.02
0.03
0.04
0.05
0.06
01.01.1998 01.01.1999 01.01.2000 31.12.2000 31.12.2001
a n
Date
0
0.01
0.02
0.03
0.04
0.05
0.06
01.01.1998 01.01.1999 01.01.2000 31.12.2000 31.12.2001
a n
Date
Figure 4.5: Best-fit values for the parameter an. The line types in the top panel areas given in Fig. 4.4. The bottom panel shows the combined model.
41
-0.005
0
0.005
0.01
0.015
01.01.1998 01.01.1999 01.01.2000 31.12.2000 31.12.2001
b n
Date
-0.005
0
0.005
0.01
0.015
01.01.1998 01.01.1999 01.01.2000 31.12.2000 31.12.2001
b n
Date
Figure 4.6: Same as Fig. 4.5 for fit parameter bn
42
0
0.2
0.4
0.6
0.8
1
1.2
1.4
01.01.1998 01.01.1999 01.01.2000 31.12.2000 31.12.2001
c n
Date
0
0.2
0.4
0.6
0.8
1
1.2
1.4
01.01.1998 01.01.1999 01.01.2000 31.12.2000 31.12.2001
c n
Date
Figure 4.7: Same as Fig. 4.5 for fit parameter cn
43
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
01.01.1998 01.01.1999 01.01.2000 31.12.2000 31.12.2001
d n
Date
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
01.01.1998 01.01.1999 01.01.2000 31.12.2000 31.12.2001
d n
Date
Figure 4.8: Same as Fig. 4.5 for fit parameter dn
44
8
10
12
14
16
18
20
22
0 5 10 15 20 25 30
σ im
p
Swaption
Figure 4.9: The swaption matrix on 02.11.1998 as given by market data (dashed), thefit without regime switches (dotted) and the fit for the model with regime switchesand the excited initial state (solid).
45
4.3 Necessary and Possible Future Steps
After the previous section showed the vast improvements that can be obtained by
the introduction of regime switches even though the model definition contains some
arbitrariness it is now time to mention possible future steps. An obvious improvement
to the model presented in the last section would be a systematic study on how to
choose the parameters that were set to “reasonable” values as well as how to possibly
choose the constraints ax = 10×an and dx = dn in a better way. A very large number
of fits would be needed for such a study. With the current implementation it is very
time consuming to analyze a large variety of parameterizations and a large range of
trading days. Even with only four parameters one fit can take up to 10 minutes on a
standard pc. For a further analysis it would be necessary to improve the performance
of the code and then systematically study reparameterizations and fits for many more
trading days.
By switching from Python as a programming language that was used for this
prototype to, e.g., C++, would certainly improve the performance a bit. However,
it would be more important to further optimize the algorithm where this is possible
and maybe find approximations for the most time-consuming steps.
The goal of a possible re-parameterization should be to further reduce the day-
to-day variations in the best-fit parameters and, of course, to further improve the
quality of the fits. The results presented in the previous section indicate that the
idea of linking an and ax as well as dn and dx works very well. Maybe these relations
can be optimized and similar constraints can be found for bn and cn. When the other
model parameters, i.e. transition probabilities λnx and λxn, the displacement α and
the correlation parameter β or even the whole correlation function, are investigated it
is important to examine the correlation among these parameters. If two parameters
show a high correlation it does not make a lot of sense to vary both parameters at
the same.
The ideal model can be reasonably fast calibrated, with stable best-fit parameters
and has a maximum deviation that is for each swaption in the range of the bit-offer
spread and not only on average, as it was obtained by the combined model in the
previous section.
46
Chapter 5
Conclusions
In this work I presented the implementation of a model for the instantaneous forward
volatility term structure of LIBOR rates. Together with the correlations among for-
ward rates the model contains all degrees of freedom that a LIBOR Market Model
has. Therefore the calibration of this model yields the specification of a LIBOR Mar-
ket Model . After the introduction, where I deduced the fact that all drift terms
in a LIBOR Market Model are given by the no-arbitrage condition, I explained the
definitions of the model for the LIBOR forward volatility term structure in detail and
showed how this model can be calibrated to either caplets and floorlets or swaptions.
I carried out a careful analysis of the possible shapes that can be obtained by varying
the different model parameters. From the large set of market data I presented two
samples that clearly show the swaption matrix evolves over time in a normal and an
excited market situation.
Based on this observation of more or less sudden switches between normal market
situations and excited markets the model incorporates transition probabilities from
a normal market situation where the instantaneous LIBOR forward volatility has a
“humped” shape to an excited market with an exponentially decaying shape and vice
versa. By using a Levenberg-Marquardt algorithm for adjusting the model parameters
to the swaption matrix for two sample trading days I showed that the implementation
works very well by comparing the limiting case of no regime switches with the results
of an earlier work by [19].
In the next step I used fits where the parameters for the normal and excited
shapes were varied independently. Again, these fits were carried out for one trading
day where the market was in a normal state and another trading day where the mar-
ket was in an exciting trading day shortly after the LTCM crisis. These fits showed
that the obtained shapes from varying excited and normal parameters independently
led to shapes for the instantaneous volatility term structure that was inconsistent
47
with the observations in the market. Therefore I suggested a more constrained pa-
rameterization with a close connection between normal and excited parameters.
As a final step I analyzed the quality of fits using this model with regime switches
and four fit parameters. For a total of 49 trading days that span the period from
January 1998 to January 2002, I performed fits of the reparameterized model with
regime switches and for the standard four-parameter model without regime switches.
Crucial for the quality of the model with regime switches is that the simulation starts
in the state that corresponds to the current market situation. This model then yields
improved fits to the marked data as compared to the model without regime switches.
While the model without regime switches yields an absolute deviation of 76 basis
points in implied volatility the model with regime switches reduces this value to 63
basis points. In addition the best-fit parameters that are obtained by the model with
regime switches show a lot less day-to-day variations compared to the model without
regime switches. This is very important for estimating re-hedging costs since stable
model parameters yield reliable hedging costs over a long time.
Even though some parameter choices in the model were just obtained by an ed-
ucated guess the results are very good. This leaves some room for improvement.
In further studies it would be worthwhile to improve the performance of the algo-
rithm and implementation of the model in order to systematically study the effects
of the parameters that were kept at a fixed value during the fit. Also some other
re-parameterizations can be studied and with a faster calibration more trading days
can be taken into account.
48
Appendix A
Bootstrapping
In order to obtain all the discount factors that I need, I use a series of swap rates
(6m, 1y, 2y, 5y, 10y, 20y, 30y) and linearly interpolate the rates in between these
market rates. After that I have a grid of rates with a step size of half a year. From
the obtained swap rates I bootstrap the discount factors for the same time grid. The
first and second discount factor are
D0 = 1 (A.1)
D1 =1
1 + τr0
(A.2)
where τ is the tenor, i.e. half a year, and r0 is the first rate in our array of rates.
All the other discount factors can be obtained from the following consideration. If I
have a notional of 1 that I invest and I obtain semi annual coupons that I properly
discount and at maturity I get the notional back, then the present value is 1.
1 = D1r0τ + D2r1τ + · · · + Dnrn−1τ + Dn (A.3)
The last term corresponds to the payment of the notional. I can therefore use this
equation for a range of different n and obtain the discount factors by the following
formula:
Dn =1 −∑n−1
i=1 τri−1Di
1 − τrn−1
(A.4)
49
Appendix B
The Levenberg-Marquardt Method
The original idea for this algorithm dates back to 1944 when it was first published by
[9] and later by [10] (which we will follow here). The implementation that is used in
the code is a function of the MINPACK software, which is documented in [13]. It is
a method to iteratively solve a nonlinear optimization problem.
For the optimization the model is given by the function y = f(x, β) where x is
a vector of length m representing the independent variables of the model, like swap
maturity and swaption expiry. The vector β contains the k model parameters, i.e. αn,
βn, and so on. In addition there is a set of (market) parameters Y which is a vector of
length n corresponding to the Black volatilities in this case, with a set of n vectors Xi
of the values of independent variables at which the data where taken. The problem
now consists of computing the set of parameters β that minimizes the expression
Φ =n∑
i=1
(f(Xi, β) − Yi)2 (B.1)
where Yi refers to the i-th element of Y . The standard approach, also in other
optimization algorithms, is to use a taylor series
f(Xi, β + δt) ≃ f(Xi, β) +
k∑
j=1
∂fi
∂βj
(δt)j ≡ f0 + Jδt (B.2)
with the vector δj being small and fi ≡ f(Xi, β), f0 is the unperturbed value and J
is the Jacobian. Plugging this into Eq. (B.1) and using ∂Φ∂βj
= 0 for all j leads to the
minimization condition for δt
JT Jδt = JT (Y − f0) . (B.3)
Note that J is a matrix of size n×k, δt is a vector of size k, and Y and f0 are vectors
of size n. In the gradient methods the perturbation δ is taken to be in the direction
50
of the gradient
δg = −(
∂Φ
∂β1,∂Φ
∂β2, ...,
∂Φ
∂βk
)
. (B.4)
For a valid problem the direction of the negative gradient and the true correction
that minimizes the problem δt are orthogonal.
Levenberg and Marquardt improved these methods by using a strategy that in-
terpolates between deltat and deltag. By modifying Eq. (B.3) to
(
JT J + λ)
δt = JT (Y − f0) (B.5)
where J is the matrix with elements ∂fi
∂βj/
√
∑n
l=0
(
∂fl
∂βj
)2
. By iteratively varying λ
such that the estimates of Φ decrease it is possible to obtain rapid convergence.
51
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