Multiple curve interest rate modelling
With stochastic basis spreads
Multi kurve rentemodellering Med stokastiske basis spænd
Cand.Merc.(Mat) master thesis
Author: Katrine Holst Larsen (109891) Guiding Professor:
David Skovmand
September 15th 2020 Number of pages: 112
Form˚alet med denne specialeafhandling er at belyse fler-kurve rente teorien der udspringer fra et paradigme skifte affødt af finanskrisen i 2007 hvor det stod klart at LIBOR renten ikke er risiko fri. De stigende basis spænd resulterede i behovet for at kunne modellere diskontering- og forward kurverne separat. Dette projekt beskæftiger sig primært med den udvidede Libor Market Model formuleret af Mercurio & Xie (2012), hvor basis spændet modelleres som et additivt spænd ved brug af korrelationen med forward renten samt uafhængig stokastisk volatilitet, to faktorer der historisk har præget udviklingen i spændet og gør modellen relevant. Modellen fittes ovenp˚a en klassisk Vasicek en- faktor kort rente model, hvilken er kalibreret til markeds observerede OIS nul renter, med det form˚al at modellere 3 m˚aneders LIBOR renten. De negative renter i det aktuelle marked illustrerer hvordan modellen har positiv sandsynlighed for at give negative rente spænd, hvilket ikke er et realistisk markedsscenarie. Desuden viser en analyse af parameter effekter, blandt andet, at valget af volatilitet for den stokastiske basis faktor proces har stor indflydelse p˚a eksponeringsprofilen for b˚ade rente swaps og basis swaps under modellen. Til sidst konstrueres et stokastisk og deterministisk modelleret spænd til sammenligning med de historiske observerede spænd der i perioder udviser vidt forskellig grad af volatilitet og korrelation. Det vises til slut at den stokastiske model producerer mere realistiske eksponerings scenarier end den deterministiske.
1 Introduction 1
2 Problem statement and methodology 2
3 Introductory Theory 3
3.1 On OIS as the risk free rate . . . 3
3.2 On LIBOR as fixing rate . . . 4
4 Basic Interest Theory 4 4.1 Basic Interest Rate Notation . . . 4
4.2 Risk Neutral Valuation . . . 5
4.2.1 The PDE Approach . . . 8
4.2.2 The Martingale Approach . . . 11
4.2.3 Change of numeraire and forward measures . . . 15
4.3 OIS Risk measures and bond prices . . . 18
4.4 Affine Term Structures . . . 19
5 One-factor short rate models 22 5.1 The Vasicek Model . . . 22
5.2 The extended Vasicek model . . . 24
5.3 Calibrating the OIS term structure . . . 25
5.3.1 Inverting the yield curve . . . 25
5.3.2 Market data used for calibration . . . 26
5.3.3 Calibration using Vasicek type dynamics . . . 27
6 Multiple Curve Models 31 7 Modelling stochastic spreads 32 7.1 The Libor Market Model . . . 33
7.2 The stochastic spread model of Mercurio & Xie . . . 35
7.2.1 Model setup and assumptions . . . 35
7.2.2 Generic framework for basis spreads . . . 38
7.2.3 The choice of basis factor dynamicsdX(t) . . . 39
7.2.4 A note on calibration of spread parameters . . . 40
8.2 Interest Rate Swaps . . . 41
8.2.1 Pricing the interest rate swap . . . 43
8.3 Basis Swaps . . . 45
8.3.1 Pricing the basis swap . . . 47
9 Calibration and analysis 49 9.1 Simulating spreads and creating exposure profiles . . . 49
9.2 Dynamics under the current market . . . 50
9.3 Analysis of parameters for the stochastic basis spread . . . 55
9.3.1 Case 1: spreads mainly correlation driven . . . 55
9.3.2 Case 2: Low correlation . . . 59
9.3.3 Case 3: High volatility . . . 61
9.4 Basis Factor sensitivity analysis . . . 63
9.5 Deterministic vs stochastic spreads . . . 65
10 Conclusion 70 References 72 A Data 73 A.1 Observed OIS zero rates from the 15/07/2020 . . . 73
A.2 Parameters for spread: High correlation and low volatility . . . 75
A.3 Parameters for spread: decreasing vol, increasing correlation. . . 77
A.4 Spread levels for deterministic spread vs stochastic spread analysis . . . 80
B Source code 81
1 Introduction
As a consequence of the financial crisis in 2007 the approach for interest rate modelling in pricing inter- est rate derivatives changed drastically as the LIBOR reference rates on which many OTC derivatives contracts are linked, started to show clear signs of being affected by different risk factors, such as coun- terparty and liquidity risk. Driving it away from the Fed Funds rates resulting in a spread. Before the crisis the spreads between the LIBOR rates and the overnight rates Fed Funds rates was negligible, so LIBOR was considered to be the risk free rate on which both the discounting and forwarding of cashflows in many OTC interest derivatives contracts was based. This allowed for these contracts to be priced on a single curve setup by using the same short rate for producing the discounting curve and the forward curve. Then came the financial crisis and the acknowledgement that LIBOR is not a risk free rate, as otherwise perceived, led to a shift in paradigm in the way these interest rates was considered and modelled. As the OTC market for LIBOR based derivatives is one of the most traded markets, the need for modelling the LIBOR rates as risky rates for forwarding cahsflows on top of discounting using the overnight indexed rates (OIS). Since then a lot of work has been done in this area and many different modelling approaches has been proposed, many of which seek to extend the already known and established frameworks of single curve models. The historic basis spreads between the overnight indexed rates and LIBOR vary greatly in trends over time. A recent example of spreads widening drastically arose in the early stages of the still ongoing Covid-19 pandemic, which in March 2020 skyrocketed the basis spread. This is especially clear when looking at historic spreads between the USD OIS- and USD 3M-LIBOR on 3M maturity which rose to the level of around 140 bps from being at the level of 30 bps by the end of 2019. The uncertainty surrounding the pandemic created tension in the USD short term funding markets. A number of policy measures helped stabilise the markets, resulting in the spread returning to the levels as observed before the outbreak in a matter of months.
This again stresses why exploiting multiple curve frameworks for pricing derivatives is important.
One could argue that the outphasing of LIBOR, in 2022, makes the analysis on multiple curve mod- elling of LIBOR outdated. But it is still very important and relevant that we understand the risk factors and the dynamics of the multiple curve issue. This way we are better prepared for similar issues in the future. In this thesis we will explore how we move from single curve modelling to a multi curve framework in a way that allows us the specify the correlation between USD LIBOR and USD OIS, furthermore allowing LIBOR to move stochastically in a way that allow for fitting to extreme market situations like the one observed in March 2020. We will then have special focus on how the correlation and volatility in the chosen multi curve model affects the modelling of spreads under the current market reality of negative rates.
2 Problem statement and methodology
This dissertation will be focused around the following main question:
How can we derive a multiple curve model for basis spread calibration that allows us to consider the correlation between the USD LIBOR rate of tenor 3M and the USD OIS rates while still displaying the market trend of LIBOR in periods moving away from OIS? And how does the choice of parameters affect the spread?
We will try to answer this question by a bottom-up approach. Firstly by taking root in classic interest theory on single curve modelling, especially within the family of short rate models.
How do we formulate and calibrate a single curve model to the current USD OIS market rates?
Given we then have a basis for the single curve framework, how do we then extend to the multiple curve framework? Here we will consider some different approaches but pursue the class of extended Libor Market Models. More specifically we will look at the work of Mercurio & Xie (2012).
How do we extend the single curve theory for multiple curve modelling?
How does the chosen model allow for specification of correlation and volatility in spreads?
We will then explore how to use the framework to price specific OTC derivatives on 3M-LIBOR, described in the context of adding a spread on top of the OIS rates, as proposed in the specific extended Libor market model considered in question two.
How do we price the common OTC interest rate swaps and basis swaps in the considered multiple curve framework?
Lastly we will do a practical calibration of the mulitple cuve model with a plug on top approach on the already calibrated single curve model for OIS in section one. With the developed model we will try to answer the following questions:
How does the current negative interest rates affect the spreads of the chosen multiple curve model.
How does the choice of parameters affect the spreads? And how does the correlation parameters affect the spreads and thereby the modelled LIBOR rates?
How does the chosen model fit into the current market conditions?
3 Introductory Theory
We will start out by introducing some of the basic concepts and terminology we will be using through- out the following sections.
3.1 On OIS as the risk free rate
A widely accepted risk free discounting rate, as post crisis alternative to LIBOR, are the market swap rates backed out of so called OIS swaps. This market swap follows a discrete payment structure like any other swap, but the floating reference rate is the discretely compounded overnight rates. By US market standards the overnight rates are those of theFederal Funds rates. So an OIS swap is a deriva- tive with the Federal Funds rates as underlying. Similar in the Euro market Eonia is the overnight reference rate used in OIS swaps.
When OIS swap rates currently is widely accepted to be risk free rates it is due to the collateralization of derivatives. OTC derivatives follow standard ISDA agreements that very often include CSAs, which areCredit Support Annex clauses that defines the terms of colleteral settlement on the contract. The purpose of these clauses is to ensure a certain coverage of losses due to possible counterparty failure to pay. Hence derivatives that are collateralized are less riskier than non-collateralized contracts. As the collateral should be discounted using the risk free rate and OIS is used for this purpose in the market, it is natural to consider this as the risk free rate. Figue 1 shows the current spread between the US OIS zero rate curve and US LIBOR 3M zero curve. Here we clearly see how LIBOR is affected by events triggering change in counterparty risk such as the Covid-19 outbreak in early 2020 where the spread raised drastically.
Figure 1: 3m US LIBOR-OIS historic spreads.
3.2 On LIBOR as fixing rate
London Interbank Offered Rate, or in short LIBOR, is used as reference rate in a wide variety of fixed income products. Especially many interest rate derivatives are tied up to LIBOR as the fixing rate.
Construction wise LIBOR is an interest rate based on the trimmed arithmetic mean1 of interbank lending rates which are reported daily by a large number international banks in London, also called panel bank, and published by the ICE Benchmark Administration (IBA).
The daily published LIBOR rates are quoted at different maturities ranging from the overnight rate to 12 months. In this project we will consider the most commonly used tenor for fixing, the 3M Libor rate. Furthermore we will limit our attention to the USD Libor rate, which can be seen as the interbank cost of borrowing funds in USD. This means we will limit our attention to one currency market only and furthermore we will only considerclean valuation of interest derivatives, meaning we will take into account the risk of counterparty default when valuing a contract.
As mentioned in the introduction is relevant to note that LIBOR as commonly used reference rate is being phased out and replaced by other reference rates such as the Secured Overnight Financing Rate, in short SOFR. SOFR also measures the overnight cost of interbank lending daily. LIBOR is being phased out to fix the security issues in market manipulation which has previously been seen. As many contracts are to this date still based on LIBOR we will proceed with studying the relationship between LIBOR and the OIS rates.
4 Basic Interest Theory
4.1 Basic Interest Rate Notation
We will give a brief overview of the basic notations and concepts of interest rate theory. The purpose is to clarify the fundamental relationsship of rates, which in the rest of the project is assumed to be known facts.
Define by P(t, T) the price of a zero coupon bond at time tpaying out one unit at maturity T. The time to maturity we define by τ =T −t.
The continously compounded spot interest ratewith maturityT for 0≤t≤T is defined as R(t, T) =−lnP(t, T)
τ
1Trimmed arithmetic mean refers to the mean interest rate where 25% of the highest and lowest submissions are removed from the calculation.
Now define for 0≤t≤S < T the continously compounded forward interest rateas R(t;S, T) =−lnP(t, T)−lnP(t, S)
τs
Whereτs=T−S.
For 0≤t≤T the simple compounded interest rateis Rsimple(t, T) = P(t, T)−1
τ P(t, T)
Then the simple compounded forward interest rate for 0≤t≤S < T is F(t;S, T) = 1
τs
P(t, S) P(t, T) −1
The instantaneous forward interest ratewith maturityT at time tis defined as fT(t) =−∂lnP(t, T)
∂T
From this theinstantaneous spot interest rate is defined as r(t) =ft(t)
4.2 Risk Neutral Valuation
As interest rates are stochastic of nature, we will in this section introduce some key results for pricing contracts in an arbitrage-free market with stochastic interest rates.
Consider again the timet zero coupon bond with maturityT. This is a fixed flow contract with only one coupon payment at maturity of one unit in the relevant currency. The graph over all zero coupon bond prices, we will denote as the term structure of coupon bonds at time t, namely P(t, T) for all maturitiesT.
In the following we will assume a frictionless market where there exists a liquid market ofT-bonds of all maturities. We will also assume there exists an infinite number of securities in this market, i.e. that there exists zero coupon bonds for all perceivable maturities. Furthermore we will assume perfectly divisible assets.
Assume now that P(t, T) is differentiable with respect toT for a given tand we will denote the first partial derivative as
PT(t, T) = ∂P(t, T)
∂T
We differ between three equivalent representations of the yield curve:
The discount curvewhich is the before mentioned graphical representation of the term struc- ture of zero coupon bonds at a given timet.
The spot rate curve,R(t, T), which is achieved by inverting the zero coupon bond yield curve.
The forward rate curve, f(t, T), which is represented by the instantaneous forward rate at timet as defined in the previous chapter.
We can write the price of a zero coupon bond and the spot rate using the instantaneous forward rate in the following way
P(t, T) =e−RtTf(t,u)du R(t, T) = 1
T−t
− Z T
t
f(t, u)du
As the the interest rate evolves stochastically we cannot observe the forward rates in the market and will therefor have to introduce a way to calculate the expectation of the future rate. First we will introduce the stochastic process of interest rates.
Let (Ω,F,(Ft)t≥0, P) be the filtered probability space upon which we will consider a Wiener process (also called Brownian Motion) W which is defined as
Definition 4.1. A standard Brownian motionWtis a stochastic process taking values inR where
W0 = 0
the process increments Wt1−Ws1, ..., Wtn−Wsn are independent for s1 ≤t1 ≤s2 ≤t2 ≤...≤ sn≤tn.
for all s < t thenWt−Ws∼ N(0,(t−s)).
the functiont→Wt is continuous.
Assume thatWtis a standard Brownian motion in the sense from Definition 1. Using what we know about the process realization at times < t we can compute the conditional mean as
E(Wt|Fs) =E(Wt−Ws+Ws|Fs) =E(Ws|Fs) +E(Wt−Ws|Fs) =Ws
If we then look at the expectation conditional on only what the process was at time s, we get E(Wt|Ws) =E(Wt−Ws+Ws|Ws) =E(Wt−Ws|Ws) +E(Ws|Ws) =Ws
Which means that
E(Wt|Fs) =E(Wt|Ws) =Ws
The future is thus only dependent on the current state. From this we can conclude that the Brownian motion satisfies the Markov Property and also that it is a martingale, which makes it ideal for risk neutral pricing.
From now on we will refer to Wt as a standard Brownian Motion unless anything else is stated.
We will assume that there only exists one security
Bt=eR0tr(s)ds (1)
with the dynamics
dB(t) =r(t)B(t)dt B(0) = 1
(2)
Which corresponds to putting money in the bank at the stochastic short rate r(t). This is equivalent to investing in a self-financing roll-over trading strategy consisting only of bonds just maturing at t+dt, which corresponds to using the instantaneous forward rate, such that p(t, T) =e−RtTf(t,s)ds. It is then clear that these bonds then depend in some way on the future value of the short rate and therefor the bond prices are considered derivatives of the short rate.
Using the relationship between the short rates, the forward and the bond prices. If we specify the bond market in terms of one of these we will be able to utilize the relationship between the three to specify the others. This is one of the reasons for why we will see different modelling frameworks based on each of these as starting point.
As we will be using short rate models to determine the term structure of the discount curve later we will limit our attention to specifying the dynamics of the short rate.
We will therefor consider the following process dynamics under the real probability measure P
dr(t) =µ(t, r(t))dt+σ(t, r(t))dWt (3)
where the stochastic part of the process is driven by the Wiener process as defined in definition 1.
µ(t, r(t)) is a function of time and the short rate, and makes up the drift part of the short rate process.
Likewiseσ(t, r(t)) makes up the diffusion part of the process alongside with the stochastic realizations
of the Wiener process.
We will now seek to determine the short rate such that we obtain arbitrage free bond prices for all convievable times t.
The incomplete market problem
The only security we know and can build on is the bank account defined in equation (1). Due to the assumptions of the existence of T-bonds for all maturities T we have a market of only one security and infinetly many assets. As we have zero securities apart from the bank account and one random source the market is arbitrage free but incomplete according to the meta-theorem2. This means that we can not determine a unique price of a particular bond, but if we apply certain internal consistency relations on the bond prices and take the price of one bond as given, then we can price all the other bonds uniquely in terms of the given bond and our knowledge on the dynamics of the short rate r.
With one given bond price we have one security aside from the bank account and hence we have the same number of exogenously given securities and random sources implying that we now consider a complete market where we are able to determine prices uniquely.
4.2.1 The PDE Approach
We will now consider how to determine the short rate dynamics in an arbitrage free market.
In this market we let the price of a T-bond be determined by a smooth function F in the following manner
P(t, T) =FT(t, r(t))
Which denote the, at time tbond price with maturity T and short rate process r(t).
UsingItˆo’s lemma3 on FT we can derive the dynamics of the bond price process as
dFT =FTαTdt+FTσTdW (4)
WhereαT = ∂FT
∂t +µ∂F∂rT +12σ2∂∂r2F2T
/FT and σT =σ∂F∂rT/FT.
Now using a value process of a self-financing portfolio argument, consisting of two bonds of different maturity, we can show what must hold for the bond dynamics to stay arbitrage free. Let V be the
2Bj¨ork [1] Meta-theorem 8.3.1
3Bj¨ork[1] Proposition 4.11
value of a portfolio consistent of uT amount of T-bonds and uS amount of S-bonds. Then the value of the portfolio develops in accordance with the bond development as
dV =V
uT
dFT FT +uS
dFS FS
.
Inserting the bond price dynamics from equation (4)
dV =V (uT(αTdt+σTdW) +uS(αSdt+σSdW))
=V(uTαT +uSαS)dt+V(uTσT +uSσS)dW
Using the restriction on the relative portfolio uT +uS = 1 and uTσT +uSσS = 0 the risky term is eliminated resulting in a risk free representation of the stochastic differential equation (SDE) for the value portfolio
dV =V(uTαT +uSαS)dt Using the conditions and solving by parts we get that
uT = σS
σT −σS
uS = σT σT −σS
must hold.
Hence the drift of the value process of the relative portfolio is dV =V
αSσT −αTσS
σT −σS
dt.
According to Bj¨ork [1] Proposition 7.6 it must hold that αSσσT−αTσS
T−σS =r(t) ∀t to avoid arbitrage, ie.
the rate of return on the self-financing portfolio must equal that of the short rate of interest.
We can rewrite it as
αT(t)−r(t)
σT(t) = αS(t)−r(t) σS(t)
We see that the condition for arbitrage free prices depend in some way on the drift and volatility of the relative bond price process. But as we also see from the relation above, the maturity of the bonds does not affect the determination, therefore the dependence on bond drift and volatility stems from the dependence on the short rate. We then introduce the universal process λ as a function of time
and short rate as
λ(t, r) = αT(t)−r(t)
σT(t) , ∀T (5)
as the intuition behind αT is that it is the local rate of return of the T-bond and r(t) is the rate of return of the riskless bond, the difference between the two is what we call the market risk premium or simply market price of risk. Hence it measures the excess rate of return required by the market for it to be arbitrage free. This notion arises from the fact that agents on the market have different risk appetite, hence the supply and demand in combination with the risk appetite sets the amount of compensation by the risk premium needed for contracts to be traded.
We will later consider how to read this market price of risk from the quoted market prices as this factor is purely determined by the market and therefore considered exogenously given.
Rearranging equation (4) we can formulate the drift term of the bond price process as
αT(t)FT =FTλσT(t) +FTr (6)
Whereλ=λ(t, r).
Setting equal the formulation forαT from equation (4) and usingσT(t) =σ∂F∂rT/FT we get λσ∂FT
∂r +FTr= ∂FT
∂t +µ∂FT
∂r + 1
2σ2∂2FT
∂r2
⇔ ∂FT
∂r (µ−λσ) +∂FT
∂t +1
2σ2∂2FT
∂r2 =rFT
Which means we can describe the partial differential equation (PDE) under the risk neutral Q measure as
∂FT
∂t +{µ−λσ}∂F∂rT +12σ2∂∂r2F2T −rFT = 0 FT(T, r) = 1
(7)
Whereλ=λ(t, r). This result is reffered to asthe term structure equation.
Solving this partial differential equation will determine the term structure in an arbitrage free market.
But it is clear that in order to do this we will have to specifyµandσ as well asλwhich, as mentioned before, is not determined within the model but has to be exogenously given.
The result can be generalized to any contingent claim with the short rate process as underlying, simply by letting the boundary condition equal Φ(r), where Φ is some real valued function Φ.
An equivalent representation of the solution to the PDE in equation (7) is the Feynman-Kac formula.
If we fix the timetand the rate at time t,r(t) =r, corresponding to assuming we know the short rate realisation at current, and consider the, by the short rate, discounted T-bond price process
e−
Rs
t FT(s, r(s))
then by Feynman-Kac4 the risk neutral valuation of the bond prices are then given as FT(t, r) =Et,rQ
h e−
RT t r(s)dsi
whereQ is the risk neutral martingale measure related toλand the expectation shall be taken using the short rate dynamics
dr(s) =µ(s, r(s))dt+σ(s, r(s))dWt (8)
r(t) =r. (9)
As mentioned before this means that given we know one realisation of the short rate, we can use the dynamics of the short rate under the Q measure to derive the risk neutral bond prices as the expectation of the discounted value of one unit paid out at maturity.
We will later have a look at a couple of short rate models using the PDE approach for proposing a framework resulting in explicit representation of the term structure.
Now we will turn our attention to an alternative approach of deriving the risk neutral dynamics by using measure change, which later leads us to the results of forward measures through change of numeraire, which is an important result used in the theory of forward rate models, which we will consider later on.
4.2.2 The Martingale Approach
As mentioned another strategy for obtaining arbitrage free pricing is the one using martingale theory.
We will look at some of the main results from the martingale approach but will not be concerned with going into all details of the derivation5.
4Bj¨ork[1] Proposition 5.6
5for full details seeR¨oman[6, chapter 13.1]
First we state that a martingale is a stochastic process of the following form
dX(t) =g(t)dW(t) (10)
Whereg(t) is assumed to beL2 (integrable) and W is a Brownian motion.
X is then said to be a FtW-martingale if it satisfies the following conditions6
X is adapted on the filtration {FtW}t≥0.
For alltthe process must have finite first moments: E[|X(t)|]<∞
For all sand t with s≤t the expectation of future process realisations is the same as current, ie.
E[X(t)|FsW] =X(s) (11)
This means that the process has no drift as implied in equation (10). Which also implies that E[Rt
sg(u)dW(u)|FsW] = 0 must hold.
As previously we will assume there exists a local risk free security with the dynamics given in (2) and that the short rate dynamics can be described as in (3). We will furthermore assume that there exist a probability measure Q << P such that each process for the bond prices discounted by the bank account
ZT(t) = P(t, T) B(t) is aQ-martingale on [0, T].
Using that
B(t) =eR0tr(u)du
the bond prices for t≤s≤Tis therefore given be the expectation of the bond prices at a future time sdiscounted by the short rate of return of the bank account.
p(t, T) =EQ[P(s, T)e−
Rs
t r(u)du|Ft]
this is shown by using that ZT(t) are Q-martingales and stochastic processes under Q, which means the realisation of the process is the expectation under Q. It also follows that with s = T then we
6Refers to [2] definition 4.6.
arrive atp(t, T) =EQ[e−Rtsr(u)du|Ft] by definition ofP(T, T) = 1.
Using the processZT(t) we will now show that the bond price dynamics under Q is given by dP(t, T) =r(t)P(t, T)dt+ν(t, T)P(t, T)dV(t)
WhereV is a Brownian motion under Q.
First we will state that there exists adapted processes m(t, T) andν(t, T) such that the bond prices can be expressed in terms of the usual dynamics structure
dP(t, T) =m(t, T)P(t, T)dt+ν(t, T)P(t, T)dW(t) (12) As discussed previously we have only one source of randomness under the real measure P given by the Brownian motion W, meaning that the filtration Ft under the P measure is the one generated fromW. Using this along with the above assumptions, the converse of the Girsanov Theorem7 which tells us there exists an adapted processϕ, reffered to as theGirsanov kernel, such that the likelihood processL has the following dynamics
dL(t) =ϕ(t)L(t)dV(t) L(0) = 1
WhereV is a Brownian motion under Q.
From theGirsanov Theorem8 the relation betweenW and V can be expressed as
dW(t) =ϕ(t)dt+dV(t). (13)
Now using Itˆos lemma we can derive the dynamics of the discounted bond price process as dZT(t) = ∂ZT(t)
∂B(t) dB+ ∂Zt(t)
∂P(t, T)dP Using that ∂P∂Z(t,T)T(t) = B(t)1 and ∂Z∂B(t)T(t) =−P(t,T)
B(t)2 along with the dynamics from equation (2) and (12)
7Bj¨ork [1] Theorem 11.6.
8Bj¨ork[1]Theorem 11.3
we get
dZT(t) = 1
B(t)(m(t, T)P(t, T)dt+ν(t, T)P(t, T)dW(t))−P(t, T)
B(t)2 B(t)r(t)dt
=ZT(t) (m(t, T)−r(t))dt+ZT(t)ν(t, T)dW(t) Using the Girsanov kernel as in (13) we see that
dZT(t) =ZT(t) (m(t, T)−r(t))dt+ZT(t)ν(t, T){ϕ(t)dt+dV(t)}
=ZT(t) (m(t, T)−r(t) +ν(t, T)ϕ(t))dt+ZT(t)ν(t, T)dV(t) We now let
ϕ(t) = r(t)−m(t, T) ν(t, T)
Note that this relates to the market price of risk of previous section asϕ(t) =−λ(t).
Then the discounted bond price dynamics reduces to diffusion only dZT(t) = ν(t, T)ZT(t)dV(t).
Which as the standard Brownian motion is martingale, and that dZT(t) has no drift further makes dZT(t) a martingale under Q, which we also assumed earlier.
Now using that P(t, T) =ZT(t)B(t) we can again use Itˆo’s lemma to derive the dynamics as dP(t, T) = ∂P(t, T)
∂ZT(t) dZ+ ∂P(t, T)
∂B(t) dB
=B(t)ν(t, T)ZT(t)dV(t) +ZT(t)r(t)B(t)dt
=r(t)P(t, T)dt+ν(t, T)P(t, T)dV(t)
We have now shown that under theQmeasure the drift of the bond process dynamics is given by the short rate r(t). Furthermore relating it to the real measureP by equation (13) the drift of the bond price process under P is given by r(t)−ϕ(t)ν(t, T). It is also clear from (13) that ϕ is the function determining the unique measureQ. Hence we will need to show thatϕ=−λ(t) is the one that ensures absence of arbitrage.
This is done by introducing a self-financing portfolio process using the above model and showing that the probability of this portfolio process presenting an arbitrage opportunity is 0. I will refer to refer- ence [6, p. 323 - 326] for the specific argument.
Key takeaways from the transformation to the equivalent martingale measure, as we have just consid- ered, is that the diffusion term of the bond dynamics is unaffected by the change of measure. Therefore
the volatility remains the same under Qas under P, but the drift term changes and we need to take this into consideration when we try to fit model parameters to observed data, which is observed under P.
To conclude the risk neutral valuation section we repeat the key conclusions into a general definition of pricing claims on an arbitrage free bond market.
Definition 4.2. LetX be a contingentT-claim of the formX = Φ(r(T)). In an arbitrage free market the price Π(t; Φ) will be given as
Π(t,Φ) =F(t, r(t))
WhereF(t, r(t)) solves the PDE, also called the boundary value problem,
∂F(t,r)
∂t + (µ−λσ)∂F∂r(t,r) +12σ2∂2F(t,r)∂r2 −rF = 0 F(t, r) = Φ(r).
Furthermore F has the stochastic representation F(t, r;T) =Et,rQ h
e−RtTr(s)dsΦ(r(T))i
Where the martingale measure Q and the subscripts t,r denote that the expectation shall be taken using the following dynamics
dr(s) = (µ−λσ)ds+σdW(s) r(t) =r
Which means we assume the realisation of the short rater at the initial time,tas given.
This means that the term structure as well as the prices of all interest rate derivatives are completely determined by the short rate dynamics, so determining this we can compute the rest and price all contracts on this market.
4.2.3 Change of numeraire and forward measures
In the above section 4.2.2 we have implicitly used the risk-free bank acount as a so-called numeraire by introducing the relative process for the bond prices
ZT(t) = P(t, T) B(t)
which is a martingale under Q.
Recalling the arbitrage free pricing of a contingent claim from definition 4.2, we can write the prices as F(t, r;T) = Et,rQ
h
e−RtTr(s)dsΦ(r(T)) i
in a computationally simpler manner by using the general pricing formula fromBj¨ork[1] proposition 26.8 as
Π(t; Φ(r(T))) =P(t, T)ET[Φ(r(T))|Ft]
where ET denote the expectation under the forward measure QT. Here again X denotes a T-claim.
As we see the bond prices P(t, T) does not have to be computed and could as such be observed in the market (if they are actually observable). Then we would only have to compute the the integral of the T-claim under the forward measure. The key to arriving at the forward measure is by change of numeraire.
By using the Radon-Nikodym deriavite as stated in Bj¨ork[1] Theorem A.52 result A.12 L10(T) =dQ1
dQ0
where Q0 is the martingale measure associated with a numeraire asset S0 on FT. It can be derived through using the general pricing formula for a T-claim in terms of the two measures as
Π(0;X) =S0(0)E0 X
S0(T)
Π(0;X) =S1(0)E1 X
S1(T)
WhereS is a process of underlying assets.
The likelihood process defining theQ1 martingale measure onFt, assuming thatS1is a possitve asset price, is then9
L10(t) = S0(0)
S1(0) ·S1(t)
S0(t), 0≤t≤T. (14)
Assuming now that the asset prices follow the price processes of a geometric brownian motion, the dynamics can be written as
dS1(t) =α1S1(t)dt+S1(t)σ1dW(t) dS2(t) =α2S2(t)dt+S2(t)σ2dW(t) whereW is a two dimensional standard wiener process.
9for full calculation seeBj¨ork[1] p. 401-403.
The dynamics of the likelihood process can through the multi dimensional Itˆo formula10 be written as dL1(t) =L10(t){σ1(t)−σ0(t)}dWQ(t) (15) Which means that the Girsanov kernel for transitioning between the two measures is the difference of volatilityϕ10(t) =σ1(t)−σ0(t).
For the bond market consisting of zero coupon bonds, as we have considered until now, we can choose the T-bond as our new numeraire S1 and as we know the bank account is the numeraire for the risk neutral martingale measure Q. Then by definition the T-forward measure QT is defined as the martingale measure for the numeraire process P(t, T) for a fixed maturityT.
We can arrive at the T-forward measure by the Radon Nikodym derivative defining the likelihood process forQT as
LT(t) = dQT
dQ , 0≤t≤T on Ft.
From (14) we can write the likelihood process onQT as LT(t) = B(0)
B(t)
P(t, T)
P(0, T) (16)
= P(t, T)
B(t)P(0, T) (17)
Where we have used that the bank account pr. definition is B(0) = 1.
Now using (16) and the bond dynamics we derived in the last section, namelig dP(t, T) =r(t)P(t, T)dt+ν(t, T)P(t, T)dV(t)
Where V(t) is the standard brownian motion under Q. Then the dynamics of the likelihood process forQT is
dLT(t) =LT(t)ν(t, T)dV(t)
As the dynamics of the bank account asset has no volatility the girsanov transformation is given by the volatility function of the T-bond price.
10Bj¨ork[1] Theorem 4.16
We then arrive at the key result of the price of a T-claim, which can be described as Π(t,Φ) =P(t, T)ET
Φ(r(T)) P(T, T)
=P(t, T)ET [Φ(r(T))|Ft] under the QT forward measure.
We will later look at how we model LIBOR rates directly under the respecitve forward measures. It is clear that for each maturity we will have a corresponding unique forward measure. We can change from one measure to another by using the Girsanov transformation that is the difference in volatilities.
4.3 OIS Risk measures and bond prices
We have now seen how to price interest rate derivatives on a market of bonds such that the prices will be free of arbitrage. We will now iterate some of the main results using the terminology of OIS bonds as assets from which we can price any T-claimX under the risk-free measure. As mentioned in section 3.1, OIS is broadly considered to be the risk-free rate therefore we will denote the zero coupon bond prices as the ones given by the OIS short rate rt as underlying. We can in the same manner as before denote the bank account using the OIS short rate as
BOISt =e
Rt 0rsds
We will also refer to the OIS risk neutral martingale measure asQD, whereD denotes ”discounting”, which is in the same manner as in previous section equivalent to the real measure P. UnderQD all traded assets are discounted using the money market accountBtOIStherefor theQD specific numeraire isBtOIS yielding that we can write the OIS bond prices as
POIS(t, T) =EQD
BtOIS BTOIS|Ft
=EQD
e−RtTrsds|Ft
such that
POIS(t,T) BtOIS
t≤T areQD-martingales for each T.
As we saw in section 4.2.3 we can change from the QD measure to a so-called OIS forward measure by using the Radon-Nikodyn derivative
LT(t) = dQTD
dQD , 0≤t≤T
on the filtrationFt where
LT(t) = POIS(t, T)
BOIS(t) · BOIS(0)
POIS(0, T) = POIS(t, T)
BtOISPOIS(0, T) , 0≤t≤T Which means we can price any T-claim X as
Π(t,X) =POIS(t, T)EQTD[X |Ft]
In the same manner we can use the link between two OIS forward measures as given by LT2(t) = dQTD2
dQTD1 , 0≤t≤T1∧T2
for two OIST1- and T2 forward measures, whereT1, T2 ∈[0, T∗] and T∗ is the terminal maturity.
Especially the OIS forward measure will be of importance later where we will model the forward rates directly. We will again drop the superscript of the bond prices for sake of brief notation. So P(t, T) will from now implicitly be the OIS zero coupon bond price process.
4.4 Affine Term Structures
Now let the OIS short rate process be under theP measure with the same dynamics as used for the short rate previously
dr(t) =µ(t, r(t))dt+σ(t, r(t))dW(t)
Now we assume the existence of a risk neutral measure with the OIS compounded bank account as numeraire. Then the risk-free dynamics of the OIS short rate is
dr(t) =µ(t, r(t))dt+σ(t, r(t))dWQ(t)
Where the drift is now of the formµ−λσas given in definition 4.2. Then to price the OIS zero coupon bond using risk neutral valuation, we must solve the boundary value problem
∂FT
∂t +{µ(t)−λ(t)σ(t)}∂F∂rT +12σ2∂∂r2F2T −r(t)FT = 0 F(r, T, T) = 1
(18)
It turns out that by letting the term structure F be described by an affine function, then we can in some cases find closed form solutions. Furthermore this function class is also relatively easy to work with.
Definition 4.3. Affine term structure (ATS)
If the term structure {p(t, T); 0≤t≤T, T ≥0} has the form p(t, T) =F(r(t), t, T) Where
F(r, t, T) =eA(t,T)−B(t,T)r
and A,B are deterministic functions, then the model is said to be of affine term structure.
Assuming thatFT has the same structure as in definition 4.3 and using the short rate dynamics under Q we can rewrite the PDE through deriving the partial derivatives as
∂F(r, t, T)
∂t = ∂A(t, T)
∂t −∂B(t, T)
∂t reA(t,T)−B(t,T)r
∂F(r, t, T)
∂r =−B(t, T)eA(t,T)−B(t,T)r
∂2F(r, t, T)
∂r2 =B(t, T)2eA(t,T)−B(t,T)r We now insert these into the PDE
∂A(t, T)
∂t −∂B(t, T)
∂t reA(t,T)−B(t,T)r(t)−µ(t, r(t))B(t, T)eA(t,T)−B(t,T)r
+1
2σ(t, r(t))2B(t, T)2eA(t,T)−B(t,T)r−rF(r, t, T) = 0
and divide by F(r, t, T), giving us a PDE that depends on the drift, diffusion and deterministic functions of the affine term structure
∂A(t, T)
∂t −
1 +∂B(t, T)
∂t
r(t)−µ(t, r(t))B(t, T) +1
2σ(t, r(t))2B(t, T)2= 0.
Using the boundary condition F(r, T, T) = 1, it must hold that A(T, T) =B(T, T) = 0. We need to assume that µ(t, r(t)) and σ(t, r(t)) be affine functions in order for the PDE to become a seperable differential equation which we can solve. Hence we will assume that
µ(t, r) =a(t)r+b(t) (19)
σ(t, r(t))2=c(t)r+d(t) (20)
Inserting the affine structure of the drift function and diffusion function into the PDE we get the following seperabel differential equation.
∂A(t, T)
∂t −
1 +∂B(t, T)
∂t
r(t)−(a(t)r+b(t))B(t, T) +1
2(c(t)r+d(t))B(t, T)2= 0 Which can be separated into terms with and without the shortrater(t)
∂A(t,T)
∂t −b(t)B(t, T) +12d(t)B(t, T)2 = 0
∂B(t,T)
∂t +a(t)B(t, T)− 12c(t)B(t, T)2 =−1 A(T, T) =B(T, T) = 0
(21)
Solving the Ricatti equation (inB quadratic first order differential equation) and inserting to solve for A will yield a closed form solution ofA and B in which we can insert parameters and fit to market.
The general form of the SDE in these models is
dr(t) ={a(t)r(t) +b(t)}dt+p
c(t)r(t) +d(t)dWQ(t). (22)
5 One-factor short rate models
The family of one-factor short rate models contains a variety of different models using the general form of the SDE from (22). To mention just a few there is the Vasicek(1977) model with constant parameters, which we will explore, also there isthe extended Vasicek (Hull White, 1990) model which introduces time dependent parameters for a better fit. Furthermore the Cox-Ingersoll-Ross (1985) model that uses the squared short rate as volatility to prevent negative rates. There are of course many more short rate models as this field has been widely explored in the pre-financial crisis theory, where as mentioned the same curve was used for discounting and forwarding. The realisation that reference rates are not risk free have brought up the need for multi curve models. As we will later explore an extended LIBOR market model, which can be plugged on top of any short rate model for OIS, we will in this section introduce the basic notation of Vasicek as we will be generating the discount curve from this. We will also give a short mention to the Hull White model as using Vasicek comes with the price of a poor fit to the initial observed yield curve as we will also see later.
5.1 The Vasicek Model
Vasicek (1977) proposed the following short rate dynamics dr=κ(θ−r)dt+σdW Whereθ=b/a andκ=ain terms of the notation in (22).
It is apparent that this is a special case of the general form in equation (22) wherea(t) =−a,b(t) =b, c(t) = 0 and d(t) =σ2 which reduces the system of equations in (21) to
∂A(t,T)
∂t −bB(t, T) +σ22B(t, T)2= 0
∂B(t,T)
∂t −aB(t, T) =−1 A(T, T) =B(T, T) = 0
.
Using Vasicek term-structure the Ricatti reduces to a first order linear ordinary differential equation (ODE) in B which can easily be solved by using the general solution for x· +a(t)x =b(t) where we have a(t) =aand b(t) =−1 as constants, so the solution is of the form
B(t, T) =Ce−a(T−t)+e−a(T−t) Z
−ea(T−t)dt
=Ce−a(T−t)+e−a(T−t)1 aea(T−t)
=Ce−a(T−t)+ 1 a
Solving the initial value problem withB(T, T) = 0 we getC=−1a, so we arrive at the following closed form solution for B
B(t, T) =−1
ae−a(T−t)+1 a
= 1 a
1−e−a(T−t) .
We now insert B(t, T) into the equation forA A(t, T) =−b
a Z T
t
1−e−a(T−s)ds+ σ2 2a2
Z T t
1−e−a(T−s)2
ds
=−b a
T −t−1 a
1−e−a(T−t)
+ σ2 2a2
T−t+ 1 2a
1−e−2a(T−t)
−2 a
1−e−a(T−t)
The Vasicek term structureis then defined as
P(t, T) =eA(t,T)−B(t,T)r(t)
(23) WhereA(t, T) and B(t, T) is as derived above.
The Vasicek model is a Gaussian shortrate model with mean and variance given by E[r(t)] =r(0)e−at+ b
a(1−e−at) (24)
V ar[r(t)] = σ2
2a(1−e−2at) (25)
Furthermore the short rate dynamics is a mean-reverting stochastic process under Q. If we let r = θ = b/a, the shortrate has no drift, which means θ is the long-term mean and κ = a is the rate at which the process converges toθ. If we consider the above mean fort→ ∞ thene−at→0 and E(r(t))→b/a=θ.
What we can also gather from the above, is that the variance is increasing in time and will tend to σ2a2 by the same argumentation as for the mean, so the largerathe lower the asymptotic variance, which means the higherathe faster the process reaches its asymptotic mean.
As we will see in the modelling section, the shape of the yield curve can be divided into a mean reversion level according toθ and initial curvature byκ.
This leads us to the drawbacks of the Vasicek models. There are several points to consider when modelling the shortrate using this type of dynamics:
Given that the Vasicek is a Gaussian model, there is positive probability of generating negative shortrates. Currently negative rates are present in the market, hence this is not an unrealistic
scenario at present.
The model parameters κ, θ and σ are constants, which means the only information from the market yield curve used in modelling the short rate is the spot rate r(0). This sets some serious limitations to the shape of the generated yield curve which in some cases will not fit the actual observed curve. Other shortrate models such as the Hull White (extended Vasicek) uses information of the whole yield curve and hence will create a better fit in many cases.
Below I will give a short mention of the extended Vasicek model, which could just as well be considered as base model of the mutli curve model in consideration. Though we will stick to modelling using Vasicek due to its tractability and intuitiveness.
5.2 The extended Vasicek model
The Hull White model (1990) uses Vasicek type dynamics but introduces time dependent parameters to accommodate the drawback of poor model fit to observed curves in the Vasicek model. Therefore the Hull White dynamics can be written as
dr= (θ(t)−a(t)r)dt+σ(t)dW(t)
Where σ(t) is a deterministic function and W is a standard Brownian motion under the risk neutral measure Q. The simplified Hull White model uses constantaand σ but still use time dependent θ.
Still using the affine term structure of the bond prices p(t, T) =eA(t,T)−B(t,T)r as our solution guess to the PDE in definition 4.2, the derivation of closed form solutions for A(t, T) and B(t, T) is a bit more involved in the Hull White setup due to the time dependency of the parameters.
The general idea is to fit the theretical prices to the observed prices using the forward rates. As we recall it is possible to derive the bond prices directly from the forward rates.
The Hull White term structure forα and σ fixed can ultimately be shown as P(t, T) = P∗(0, T)
P∗(0, t) exp
B(t, T)f∗(0, t)−σ2
4aB2(t, T)(1−e−2at)−B(t, T)r(t)
(26) WhereB(t, T) is the same as for the Vasicek term structure andP∗ andf∗ denotes the observed bond prices and forward rates.
5.3 Calibrating the OIS term structure
We have introduced the theory behind risk neutral valuation of term structures using the short rate models. First we will give a brief introduction to how model parameters can be fit to market data using the inverted yield curve. Then we will calibrate the parameters in the Vasicek model to recent market data. But before that we give an introduction to market data retrieved for the calibration, both for this section but also for the multiple curve calibration later on.
5.3.1 Inverting the yield curve
We need a method for estimating the parameters of the martingale models above in order to get a term structure representation that reflect the current market.
In particular we will consider how to fit the parameters of the Vasicek model, namelya,band σ. As we determine the short rate dynamics straight under theQdynamics we can not observe these param- eters from historical prices as these are observed under P. Though we recall that we in chapter 4.3 saw how the change of measure fromP toQonly affect the drift term of a diffusion, hence in principal σ could be obtained fromP observed prices. But we need to find another method for estimating the parameters for theQ-drift.
As we discussed in chapter 4.2.1 the interpretation of the market price of risk is that it is the excess return of a risky bond relative to the risk free security and therefore it depends on the demand and supply as well as the risk apetite of the different agents on the market and λ works as risk compensation. Therefore it is the market that determines how λ is denfined and it is hence also the market that chooses the Q martingale measure. So in order to obtain information about the Q-drift parameters we need to collect price information from the market andinvert the yield curve.
Theoretically what we do is:
1. Choose model involving the parameters and denoteαas the full parameter vector. Then we can write theQshort rate dynamics as a function of α
dr(t) =µ(t, r(t);α)dtσ(t, r(t);α)dW(t)
2. Solving the term structure equation (23) for every maturity we obtain the theoretical term structure as a function of the parameter vectorP(t, T;a).
3. In the actual bond market we may, often indirectly, observe the empirical term structure as a series of bonds at time 0 for every maturity T: {P∗(0, T);T ≥0}.
4. Now a parameter vectorα∗is chosen such that{P(0, T);α∗;T ≥}fits the observed term structure as well as possible.
Inserting α∗ into µ(t, r(t);α) and σ(t, r(t);α) we have now determined the current martingale measure Q.
As for the above point (1) we will choose the short rate dynamics to be that of the Vasicek model in order to estimate the Vasicek term structure corresponding to the OIS bond prices. Though as these prices are not directly observable in the market, the bond prices has to be calculated using the OIS zero curve backed out of OIS based traded market instruments, such as Fed Fund Futures prices for the short end of the curve and OIS swaps for the longer maturities. We will not be concerned with the specifics of the zero curve bootstrapped from prices of liquid market products in this project as the already bootstrapped curves are readily availabe in the Reuters repository. This leads us to describe the data we will use in this project.
5.3.2 Market data used for calibration
As mentioned bootstrapping the zero curve from derivatives prices of liquid market instruments, with the relevant rate as underlying, is not done within the scope of this project. Instead we utilize the large database of financial market data through access to the Reuters Eikon terminal and the Thomson Reuters Excel add-in from which we have pulled historic time series data.
First we need to identify the data we need. As the ultimate goal is to calibrate a multi curve model to calculate present value of US plain vanilla swaps and basis swaps we will need the US 3M-LIBOR zero rate and the OIS zero rate for all quoted tenors. Furthermore we would like to relate the spreads to the historic development, therefore the 10 year history of the US 3M-LIBOR- and OIS zero rates are retrieved using the following Reuters RIC codes
USDOISXXZ =R USDSBQLXXZ =R
where XX denote the maturity. Note that the RIC for LIBOR is the one of a USD swap on semi-bond fixed payments and quarterly LIBOR. But the implied LIBOR zero rate is singled out in the time series curve constituents in the functions of the Thomson Reuters Excel add-in.
Data is pulled for all business days from 12/04/2010 to 15/07/2020 marking by the last date the data for which we will calibrate the parameters of the Vasicek model in the next section.
Both zero curves were available for quarterly maturities up till 10Y after that for maturities XX ={12Y,15Y,20Y,25Y,30Y,40Y,50Y}.
Data used for OIS Vasicek term structure calibration can be found in appendix A.1.
To fill in the missing maturities, for the sake of fitting a term structure model on 3M tenor, Hermite interpolation is used as it ensures continuous forward rates.
Figure 2 shows a graph over the historic observations for the 3M USD OIS zero rate.
Figure 2: Historic USD OIS zero rates for 3M maturity.
5.3.3 Calibration using Vasicek type dynamics
We have now chosen which model to use for our short rate dynamics. Furthermore we have retrieved the underP observed OIS bond term structure, which we will denote{POIS(0, Tk);k= 0.25,0.5, ...,30}
corresponding to quarterly maturities up until 30 years, which we will denote as T∗. We now need to chooseα = (a, b, σ)| according to step 4 in the process of yield invertion. This can be done either by Maksimum Likelihood estimation as we know the distribution of the short rate to be normal with mean and variance as given in equation (24) and (25).
Another method is by using Least Square Estimation where we aim to minimize the sum of squared residuals, ie. as the solution α∗ that minimizes the following objective function
minα T∗
X
k=1
POIS(0, Tk)−P(0, Tk;α)2
The calibration has been performed on data from the, at the time, latest market observations from the 15th of July 2020 using the last of the two techniques, namely Least Square estimation. The Microsoft Office Excel Solver tool has been used for calibrating α such that it minimizes the sum of squared residuals. Specifically I have used the GRG Nonlinear Solving Method which uses the Generalized Reduced Gradient algorithm for nonlinear optimization.11
11Documentation for solver algorithms found at Microsoft support website [11] .
The estimates for the Vasicek parameters fit to the market on the 15th of July 2020 are Table 1: Vasicek parameter estimates
ˆ
a ˆb σˆ r(0)
0.24727 0.00178 0.00196 -0.00177 Recall that the Vasicek short rate diffusion is given by
dr(t) =κ(θ−r(t))dt+σdW(t)
where κ = a and θ = b/a which means that the Q-drift on the 15th of July 2020 is determined as κ(θ−r(t)) = 0.24727(0.00720−r(t)).
The interpretation of the parameters for the drift remains the same. The convergence speed towards the mean reversion level,b/a= 0.00720, is given bya= 0.24727. Note that the parameters have been calibrated on interest rates in terms of percentages, which means the Vasicek short rate realisations will be of a factor 1/100, therefore when we state the mean reversion level to be 0.007 it refers 0.7%.
Figure 3: Fitted Vasicek yield curve (blue) plottet against market implied OIS zero curve (black).
It is obvious from Figure 3 that the Vasicek yield curve does not offer a very good fit to the observed rates. As mentioned earlier this is one of the drawbacks of using Vasicek dynamics. Due to the con- stant parameters, it is simply not able to mimick the downward curve in the short end of the observed yield. Using the Hull White dynamics instead with time dependent mean reversion levelθ(t) we would get a better fit, but also at the expence of the simplicity of the Vasicek model. It becomes a trade off
between simplicity and fit.
Another usually highlighted drawback of the Vasicek short rate model is the possibility of negative yields. But as we see in the market implied zero curve, negative yields are occurring at current, therefor it is not unrealistic, actually for this market, it is preferred that the process can also produce negative rates.
(a) Vasicek bond prices for maturity up to 30Y. (b) Vasicek bond prices for maturity up to 2Y.
Figure 4: a) shows the modelled Vasicek term structures up until 30Y and b) shows a zoom in on the short end of the curve displaying discount factors above 1 due to the negative short rate.
Having commented on the issues with the model fit, I will continue with the Vasicek curve as the discounting curve upon which I will later put on top both a deterministic and stochastic spread for multicurve modelling. This is chosen in order to keep it simple. It is important to note that by assum- ing the above Vasicek yield curve to be the true yield curve, we are no longer able to relate market instrument prices to the ones produced on discount factors of this model, which means calculating prices of interest rate driven instruments on this curve would yield wrong prices.
We will now take a quick look at how the Hull White dynamics may offer a better fit than the one of Vasicek. We can give an indication of the estimate of θ(t) using just some fixed value fora andσ.
For this example we are mostly interested in the shape ofθ(t), not the level.
Here we are using that the deterministic functionθ(t) in the Hull White model according toBj¨ork[2, p. 385 equation 24.47] is expressed in terms of
θ(t) =f∗(0, t) + 1 a
∂f∗(0, t)
∂t + σ2
2k2(1−e−2kt)
Where f∗(0, T) = −∂lnP∂T(0,T) is the instantaneous forward rate which we will derive from the zero curve using finite difference approximation of the derivatives of the ZCB prices.
Here we see how time dependent theta mimicks some of the same structures as in the market zero curve, we just have to remember that this will be the structure of the forward curve in terms of Hull
Figure 5: Example of theta structure for market implied OIS zero curve from the 15/07/2020.
White modelling the term structure using calibration of the theoretical forward rates to the observed ones. The bumps are a result of the interpolation.
