2.3 Market Models
2.3.2 Markov-functional Model
The second class of models we will be interested in this thesis areMarkov-functional models
(MFMs). They were introduced by Hunt et al. (2000) and also by Balland and Hughston (2000) based on the earlier ideas of Hunt et al. (1998). The main idea of an MFM is that we express the prices of ZCBs as functions of the state of some Markov processx. This allows
us to implement an MFM by only keeping track of only processx. In this thesis we adopt
the definition of an MFM from Kennedy (2010).
Definition 2.20. A model of a T∗ < ∞ time horizon economy consisting of ZCBs with
maturities from a non-empty setT ⊂(0, T∗]4 is said to be Markov-functional if there exists
a numeraire pair (N,N)and a (d-dimensional) {Ft}t∈[0,T∗]-adapted process xsuch that:
1. xis a Markov process under the measureN;
2. for allT ∈ T andt≤T the timetprice of aT-maturity ZCBDt,T isσ(xt)-measurable.
Before we analyse Definition 2.20 in more detail let us make a trivial observation that the existence of a numeraire pair ensures that any MFM is arbitrage-free. Besides the numeraire pair, the central part of the definition of the MFM is a Markov processxwhich we will refer
to as thedriving process or thedriver. It is easy to see that for a given numeraire pair the
driving process is not unique. For example, letxbe a driving process anda >0 then the
processx0 defined byx0
t:=axtgives rise to the same model as the process x(we will discuss
this in greater detail in Section 4.4). Moreover, the driving processes need not be of the same dimension.
With this in mind we say that the dimension of an MFM isdif:
1. There exists ad-dimensional driving processx;
2. Dimension of any other driving process x0 (possibly corresponding to a different numeraire pair) is at leastd.
However, in practical application one typically singles out a specific numeraire pair and driving process when working with an MFM and refers to the dimension of the MFM as the dimension of the chosen driving process. We will see in Chapter 4 that the ‘true spirit’ of an MFM is in choosing the driving process and the EMM in advance and then determining the numeraire and the functional forms of the ZCBs by calibrating the model to prices of caplets or swaptions.
LIBOR Market Models as Markov-functional Models
Let us conclude this section with a comment how LMMs can be seen as MFMs. In particular, let us consider an LMM under the terminal measure consisting of forward LIBORsL1, . . . , Ln
given by the System of SDEs (2.54).
4In the context of the economyEwe setT ={T1, . . . , T
In order to identify, the LMM as the MFM we then have to find a driving process x
and show that the prices of ZCBs can be expressed in terms of the driving process. Recall that, in the LMM the prices of ZCBs were uniquely defined only on dates T1, . . . , Tn+1.
As a consequence, the embedding of an LMM into an MFM is not unique, however any two embedding (under the same measure) will share a common joint distribution of ZCB prices on the datesT1, . . . , Tn+1. In that sense, we will embed the LMM into the MFM
by specifying only the functional forms of ZCBs on datesT1, . . . , Tn+1 which is in practice
enough for pricing of the derivatives.
Recall, that the prices of ZCBs on the datesT1, . . . , Tn+1 can be expressed in terms of
forward LIBORs as in equation (2.58). We can then define the driving process to be the processx= (Li)n
i=1 (where we extend each Li to [0, Tn+1] in the sense of Remark 2.19).
Note, thatxis ann-dimensional Markov process under terminal measure and that the prices
of ZCBs on dates T1, . . . , Tn+1 can be expressed as functions of the state of the driving
process as in equation (2.58).
Note, that the resulting MFM isn-dimensional regardless of the dimension of the Brownian
motion driving the dynamics of the LMM. The connection between MFM and LMM will be further explored in 4 and 6.
Chapter 3
Classification of Two- and
Three-Factor Time-Homogeneous
Separable LIBOR Market Models
TheLIBOR market models (LMMs), introduced in Section 2.3 are one of the most popular
classes of term structure models. One of the reasons for their popularity can be attributed to the flexibility of their parameterisations. However, this flexibility comes with a major drawback, the dimension of an LMM is equal to the number of forward rates in the model. This makes them particularly cumbersome to use for pricing of derivatives with early exercise features.
To overcome the issue of high-dimensionality Pietersz et al. (2004) proposed theseparab- ilityconstraint on the volatility structure of the LMM and proved that a separable LMM
has an approximation with dimension equal to the number of Brownian motions driving the model dynamics. This process came with two drawbacks. Firstly, it greatly restricted the class of available parameterisations. In particular, it was noted in Joshi (2011) that the separability condition is too restrictive to use when the instantaneous volatilities are time-homogeneous. Secondly, the approximation obtained is not arbitrage-free and is only useful for time horizons up to 15 years.
In this chapter we mainly address the first issue. We show that the separability condition can be relaxed by allowing the components of the driving Brownian motions to be correlated. Under the relaxed separability condition we characterise two- and three-factor separable LMMs with time-homogeneous instantaneous volatilities and show that they are of practical interest.
We briefly comment on the second issue, namely that the approximation considered admits arbitrage, by pointing out the ideas presented in Bennett and Kennedy (2005). In particular, we note that by defining a suitable Markov-functional model one can retain the benefits of low-dimensionality while avoiding problems with arbitrage.
driven by correlated Brownian motions and introduce the single time step approximation. The separability condition is discussed and generalised in Section 3.2. In Section 3.3 we characterise the two- and three-factor separable LMM with time-homogeneous instantaneous volatilities. In Section 3.4 we discuss the models obtained from a practical point of view. Section 3.5 concludes the chapter.
3.1
LIBOR Market Model
The LIBOR market model was briefly introduced in Section 2.3. In particular, we have specified the LMM under the terminal measure in equation (2.54) and introduced the instantaneous volatility functionsσinst,i, i= 1, . . . , n, and instantaneous correlation function
ρinst
i,j , i, j= 1, . . . , n in equations (2.59) and (2.60) respectively. Moreover, we noted that
the instantaneous volatility functions and correlation functions uniquely define the model dynamics.
In particular, by specifying the instantaneous volatility functions one implicitly specifies the evolution of the term structure of volatilities over time via equation (2.62). In practice one often does not have a particular view on the dynamics of the term structure of volatilties and is faced with two natural choices. Either one chooses the implied volatilities to be constant functions of time (i.e. they only depend on the maturityTi of the caplet) or so
that the implied volatilities are a function of the time to maturity (i.e. they depend on the differenceTi−t) (see Section 6.2 in Rebonato (2002)). In this chapter we will focus on the
latter choice. It is easy to see that the implied volatilities of caplets will depend on the time to maturity if the instantaneous volatility functions satisfy thetime-homogeneity condition
σinst,i(t) =σinst(Ti−t), t≤Ti, i= 1, . . . , n, (3.1)
whereσinst : [0, T
n]→R+ is some bounded measurable function. In particular,σinst is often taken to be of the form
σinst(τ) = a+bτexp
−cτ+
d, (3.2)
whereτ is the time to maturity. To ensure that equation (3.2) represents a valid paramet-
erisation the parameters must be chosen such thatσinst is non-negative and bounded on [0, Tn]. This parameterisation was proposed by Rebonato (1999) and remains a popular
choice amongst practitioners.
Let us now turn our attention back to the specification of the LMM. Recall that we assumed that the Brownian motion driving the model dynamics has independent components. While this assumption is in general non-restrictive, it turns out to be beneficial to relax it when there are additional constraints associated with functionsσi, i= 1, . . . , n, in equation (2.54).
Letρ: [0, Tn]→[−1,1]d×d be a measurable function, such thatρtis a correlation matrix
fort∈[0, Tn]. We will construct an LMM driven by ad-dimensional Brownian motionW,
satisfying
and bounded measurable volatility functionsσi : [0, T
i]→Rd, i= 1, , . . . , n.
First observe, that since ρ(t), t ≤Tn, is a correlation matrix, in particular a positive
semidefinite matrix, there exists a unique positive semidefinite matrixRt, called the principal
square root, such thatRtRt=ρ(t). Then we can define the matrix valued functionR:t7→Rt,
moreover one can show that taking the principal square root is a continuous operation and thereforeRtis a measurable function. Now let ˜W be a d-dimensional standard Brownian
motion. Then we can define a processW = (Wt)t∈[0,Tn] by
Wt:=
Z t
0 Rtd ˜
Wt (3.4)
note thatW satisfies
dWtdWtT = (RtdW˜t)(RtdW˜t)T =Rt(dW˜tdW˜tT)Rt=R2tdt=ρ(t)dt. (3.5)
Therefore, the process W as defined in equation (3.4) is a Brownian motion with the
correlation structure we desired. Next we define functions ˜σi: [0, T
i]→Rd, i= 1, . . . , n, by
˜
σi(t) :=Rtσi(t). (3.6)
It is easy to see that ˜σi, i= 1, . . . , nare bounded measurable functions. Then we can define
an LMM driven by ˜W and volatility functions ˜σi, i= 1, . . . , nas in equation (2.54), i.e. for
i∈ {1, . . . , n} dLit=Lithσ˜i(t), dW˜ti −Lit n X j=i+1 αjLjth˜σi(t),σ˜j(t)i 1 +αjLjt dt, t≤Ti. (3.7) Note that hσ˜i(t), dW˜ti=hRtσi(t), dW˜ti=hσi(t), RtdW˜ti=hσi(t), dWti (3.8) and hσ˜i(t),˜σj(t)i=hRtσi(t), Rtσj(t)i=hσi(t), R2tσj(t)i=hσi(t), ρ(t)σj(t)i, (3.9)
then we can rewrite (3.7) fori= 1, . . . , n, as dLi t=Lithσi(t), dWti −Lit n X j=i+1 αjLjthσi(t), ρ(t)σj(t)i 1 +αjLjt dt, t≤Ti. (3.10)
We will refer to the collection of functions{σi
}n
i=1 as the volatility structure and will say
that an LMM is parametrised by the pair ({σi
}n
i=1, ρ). We can express the instantaneous
volatility and correlation functions in terms of functionsσ1, . . . , σn andρas
σinst,i(t) =p
hσi(t), ρ(t)σi(t)i, t≤T
and
ρinsti,j (t) = h
σi(t), ρ(t)σj(t)i
σinst,i(t)σinst,j(t), t≤Ti∧Tj, i, j= 1, . . . , n. (3.12)
Remark 3.1. Note that we allowedρto be of any rank. In particular, ifρ(t)is of rankd0< d
fort≤Tn, we get ad-factor parameterisation of ad0factor LMM. This may seem suboptimal
for implementation purposes, however as we will later observe this is not necessarily the case.
Let us conclude this section by briefly discussing the implementation of the LMM. We have noted in Section 2.3 that one of the biggest challenges when implementing the LMM comes from the state dependent drifts occurring in the SDEs for the forward LIBORs (see equations (2.54) and (3.10)). In particular this ensures that the LMM is Markovian in dimension n regardless of the dimension of the Brownian motion driving the dynamics.
Furthermore, there are no closed form solutions for the joint distribution of the LIBORs at any datet >0. Therefore, in order to implement the LMM it is necessary to use a suitable
approximation. This is usually done in the log-space since
dlogLit=hσi(t), dWti − 1 2σinst,i(t)2+ n X j=i+1 αjLjthσi(t), ρ(t)σj(t)i 1 +αjLjt dt (3.13)
and the distribution ofRt2
t1hσ
i(t), dW
tiis known explicitly.
In this chapter we will focus on an approximation in which the forward LIBORs are evolved from time 0 to timetin a single time-step. An early description of this method can
be found in Hunter et al. (2001), however we will closely follow the approach and notation in Pietersz et al. (2004). Let us denote byZ a vector valued process, where theith component, i= 1, . . . , n,Zi is given by Zi(t) := Z t 0 hσ i(t), dW ti, t≤Ti. (3.14)
We say that (LA,i)n
i=1 is a single time-step approximationof (Li)ni=1 if
logLA,it = logLi0+Zi(t) +µi(t, Z(t)), t≤Ti, i= 1, . . . , n, (3.15)
whereµi is defined by the drift approximation used (e.g. Euler, Brownian bridge, see Joshi
and Stacey (2008)). Note that the drift approximation implicitly depends on the initial term structure. Furthermore, observe that the processZ is in general ann-dimensional Markov
process.
Remark 3.2. Observe that the process Zi is only well defined for t≤Ti, hence the drift
approximation µj at timet
≤Tj may only depend on theith component of vectorZ ift≤Ti.
However, this does not cause problems since the drift part oflogLj only depends on the state
of the Lj+1, . . . , Ln.
Remark 3.3. Instead of approximating the LMM under the terminal measure, we could
The single-time step approximation is a powerful computational tool, however it does come with one major drawback. Like most approximations of the LMM it is not arbitrage-free. In particular, the quality of approximation decreases with time and care must be taken when using it over long time horizons. This is typically less of a problem for the schemes that use many time steps to evolve the forward LIBORs in time. Nevertheless, the single time-step approximation is a useful method for short- and medium-term time horizons and its true power will be demonstrated in Section 3.2.
Remark 3.4. In this thesis we do not explore the accuracy of the single time-step approxim- ation. For separable LMMs (see the next section) this has been studied Pietersz et al. (2004) and Ng (2009). A comparison of drift approximations can be found in Joshi and Stacey (2008).