Arbitrage-Free Interpolation in Models of Market Observable Interest Rates
AARBITRAGE–FREE INTERPOLATION IN MODELS OF MARKET OBSERVABLEINTEREST RATES
ERIK SCHL ¨OGLA
BSTRACT . Models which postulate lognormal dynamics for interest rates which are compoundedaccording to market conventions, such as forward LIBOR or forward swap rates, can be con-structed initially in a discrete tenor framework. Interpolating interest rates between maturities inthe discrete tenor structure is equivalent to extending the model to continuous tenor. The presentpaper sets forth an alternative way of performing this extension; one which preserves the Mar-kovian properties of the discrete tenor models and guarantees the positivity of all interpolatedrates.
Practitioners have long priced caps, floors and other interest rate derivative contracts by Black/Scholes–like formulae, a practice usually attributed to the work of Black (1976). Initially, thisapproach was very much focused on the pricing of individual contracts, without regard to thearbitrage relationships between different fixed income instruments. The seminal paper of Hoand Lee (1986) instigated further research into models of the entire yield curve and fitting initialdata, where the yield curve was given either terms of zero coupon bond prices or continuouslycompounded short or forward rates. The latter approach was taken by Heath, Jarrow and Morton(1992) (HJM), who developed a general framework for models of the term structure of interestrates, deriving drift conditions for instantaneous forward rates which must be satisfied in any arbitrage–free model. Thus the fundamental objects of these models were mainly mathemati-cal constructs, not market observables. The first work to move away from this paradigm wasSandmann and Sondermann (1989, 1994), who construct a binomial term structure model withthe short rate compounded according to market conventions (e.g. three–month LIBOR) as thefundamental model variable. Taking the modelling of market observable interest rates to contin-uous time and bringing theoretical developments back into line with widespread market practice,Miltersen, Sandmann and Sondermann (1997) (MSS) embedded the pricing of caps and floorsby Black/Scholes–like formulae in a consistent HJM framework. The critical assumption for thisresult is that relative volatility of observable market forward rates such as LIBOR is determin-istic. Brace, Gatarek and Musiela (1997) (BGM) resolved key questions in the construction of sucha model, in particular concerning existence and measure relationships. Given the deterministicvolatility assumption, they explicitly identified forward LIBOR as lognormal martingales underthe forward measure to the end of the respective accrual periods. Their label
Market Models
Date : November 22, 2001. School of Finance and Economics, University of Technology, Sydney, PO Box 123,Broadway, NSW 2007, Australia. E-mail: [email protected] author would like to thank Marek Musiela for helpful discussions, but claims responsibility for any remainingerrors. This is an older, working paper version of the paper published as Schl¨ogl (2002a) (see references at the endof the paper). cf. Black and Scholes (1973). cf. such papers as Ho and Lee (1986) or El Karoui, Lepage, Myneni, Roseau and Viswanathan (1991). See also Sandmann and Sondermann (1994) and Sandmann, Sondermann and Miltersen (1995). The relative volatility of a diffusion process X ( t ) is σ if its quadratic variation is given by X ( t ) σ dt . For a succinct treatment of the significance of this assumption, see Rady (1997). a r X i v : . [ q -f i n . M F ] J un ERIK SCHL ¨OGL seems to have become the generally accepted descriptor for models which postulate lognormal-ity (under the appropriate probability measure) for market observable interest rates. The samemethodology can be applied to forward swap rates to derive a model which supports the marketpractice of pricing swaptions by a Black/Scholes–type formula, as demonstrated by Jamshidian(1997). Miltersen, Nielsen and Sandmann (2001) use a similar approach to construct a model oflognormal futures rates (implied by the prices of interest rate futures), which nests the lognormalforward LIBOR model.Starting point of our discussion is the work of Musiela and Rutkowski (1997), who bringimportant clarity into the construction of lognormal market models. Besides explicitly derivingthe relationships between forward measures of different maturities, they set up the model oflognormal forward LIBORs for a discrete set of maturities (the tenor structure ) first and thenextend it to a continuum of maturities by applying the assumptions of BGM, i.e. lognormalityof all forward rates with compounding period δ and zero volatility for all zero coupon bondswith time to maturity less than δ . This makes transparent the fact that extending the model tocontinuous tenor in effect stipulates how the model will interpolate between a discrete set ofmaturities. Note that in all cases considered in the present paper the stochastic dynamics ofinterest rates are modelled in continuous time. Interpolation occurs in the maturity dimension only. Furthermore, the rates modelled have a fixed maturity (as opposed to a fixed time–to–maturity). In simple terms, the question addressed is how to determine the continuous timedynamics of, say, the yield for an investment maturing next August given the continuous timedynamics of three–month forward rates for accrual periods starting in March, June, Septemberand December of every year.Specifying lognormal dynamics for all δ –compounded forward rates, as MSS and BGM do,has the obvious advantage that all caplets and floorlets on such rates are valued by Black/Scholes–like formulae. However, this elegance comes at a price. For one, the no–arbitrage with cashproperty cannot be guaranteed, i.e. although the forward LIBORs modelled as lognormal willalways be positive, other rates may not be. Second, the Markovian properties of the discretetenor model are lost; this is a fact which can cause considerable problems in numerical imple-mentations, including Monte Carlo simulation, as we will discuss below. Third, one may takethe pragmatic point of view that in reality one observes only a discrete number of rates on themarket, with the rest determined by some interpolation method of choice. Thus one may prefera version of the model which implies an interpolation method for future dates that is transparentand tractable. As the approach of Musiela and Rutkowski (1997) to the construction of a con-tinuous tenor model makes clear, once volatilities are specified for continuous tenor, one has thefreedom to arbitrarily interpolate initial observed market rates. Thereafter, interpolation is fixedby no arbitrage conditions and must be evaluated numerically.The alternative way of extending a lognormal market model from discrete to continuous tenorproposed here addresses each of these points. It is also worthy to note that while the practicalrelevance of the lognormal market models makes the deterministic volatility case a natural focusof attention, Musiela and Rutkowski’s forward measure approach to term structure modellingis valid for much more general volatility specifications, and this generality carries over to alarge part of the results presented here. In particular, this is true for other models constructed indiscrete tenor, such as the lognormal forward swap rate model of Jamshidian (1997).In the present paper, the extension to continuous tenor is initially performed by deterministi-cally interpolating zero coupon bond prices maturing before the earliest (future) date in the tenorstructure (the “short bonds”), not only for the initial term structure, but at any future time as Musiela and Rutkowski (1997) give an example of this.
RBITRAGE FREE TERM STRUCTURE INTERPOLATION 3 well (section 1). At any point in time, interpolation of longer–dated bonds is determined by noarbitrage conditions (section 2). Interpolating short bond prices from the two adjacent discretetenor LIBORs introduces volatility for the short bonds and the associated interest rates (section3). The dynamics of all interpolated rates remain Markovian in the state vector of discrete tenorforward LIBORs (section 4). The properties of the interpolated rates are discussed in section 5.Section 6 analyses the implications for the pricing of cap/floor contacts which do not fit into thediscrete tenor structure and section 7 concludes.1. I
NTERPOLATION OF S HORT – DATED B ONDS
Given a filtered probability space (Ω , {F t } t ∈ [0 ,T ∗ ] , IP T ∗ ) satisfying the usual conditions, let { W T ∗ ( t ) } t ∈ [0 ,T ∗ ] denote a d –dimensional standard Wiener process and assume that the filtration {F t } t ∈ [0 ,T ∗ ] is the usual IP T ∗ –augmentation of the filtration generated by { W T ∗ ( t ) } t ∈ [0 ,T ∗ ] .The model is set up on the basis of assumptions (BP.1) and (BP.2) of Musiela and Rutkowski(1997): (BP.1) For any date T ∈ [0 , T ∗ ] , the price process of a zero coupon bond B ( t, T ) , t ∈ [0 , T ] is astrictly positive special martingale under IP T ∗ . (BP.2) For any fixed T ∈ [0 , T ∗ ] , the forward process F B ( t, T, T ∗ ) = B ( t, T ) B ( t, T ∗ ) , ∀ t ∈ [0 , T ] follows a martingale under IP T ∗ .Note that assumption (BP.2) means that IP T ∗ can be interpreted as the time T ∗ forward measure and implies that the bond price dynamics are arbitrage–free.The objects to be modelled on the fixed income markets are the δ -compounded forward ratesdefined by(1) L ( t, T ) = δ − (cid:18) B ( t, T ) B ( t, T + δ ) − (cid:19) Since the compounding matches the market convention for rates such as the London InterbankOffer Rate, L ( t, T ) is also referred to as forward LIBOR.Consider a discrete tenor structure T = { T , . . . , T N } . For notational simplicity, let T = 0 also be the time origin and T i +1 − T i = δ ∀ i < N . The dynamics of the rates L ( · , T i ) , ≤ i ERIK SCHL ¨OGL as η ( t ) = max { i ∈ { , . . . , N }| T i − < t } Similarly to the extensions to continuous tenor proposed in Brace, Gatarek and Musiela (1997)and Musiela and Rutkowski (1997), we initially make the following1.1. A SSUMPTION . Let volatility be zero for all zero coupon bonds B ( t , t ) maturing by thenext (future) date in the tenor structure T , i.e. t ≤ t ≤ T η ( t ) . Consequently, at time T i the price of the zero coupon bond maturing in T i +1 is given by B ( T i , T i +1 ) = (1 + δL ( T i , T i )) − and its dynamics are deterministic thereafter. Since they are deterministic, interpolation of in-terest rates with time to maturity less than δ is the same as specifying the evolution of the bondprice, and arbitrary deterministic dynamics can be specified for B ( t, T i +1 ) , T i < t < T i +1 , with B ( T i +1 , T i +1 ) = 1 . For obvious reasons, these dynamics should be monotonically increasingand continuous, which means defining the continuously compounded short rate as some positivefunction r ( s, L ( T η ( s ) − , T η ( s ) − )) , such that (3) exp (cid:26)(cid:90) T i +1 T i r ( s, L ( T η ( s ) − , T η ( s ) − )) ds (cid:27) = 1 + δL ( T i , T i ) ∀ i < N Thus r ( t ) is an F η ( t ) − –measurable random variable. Consequently, B ( t, T η ( t ) ) = exp (cid:26) − (cid:90) T η ( t ) t r ( s, L ( T η ( s ) − , T η ( s ) − )) ds (cid:27) Note that while the dynamics of the abstract variable r are in general discontinuous at each T i ,any rate associated with a compounding period greater than length zero will have continuousdynamics.1.2. R EMARK . In a term structure model with zero volatility for zero coupon bonds B ( t, T η ( t ) ) for all T ≤ t ≤ T N , the risk neutral and rolling spot LIBOR measures are identical and theshort rate interpolation (3) is arbitrage free. P ROOF : The risk neutral equivalent martingale measure is conventionally defined as the measureassociated with taking the (continuously compounded) savings account as the numeraire asset,i.e. under this measure all assets discounted by the savings account are martingales. The savingsaccount β ( t ) defined by the interpolation is β ( t ) = exp (cid:26)(cid:90) t r ( s, L ( T η ( s ) − , T η ( s ) − )) ds (cid:27) = η ( t ) − (cid:89) i =0 (1 + δL ( T i , T i )) exp (cid:40)(cid:90) tT η ( t ) − r ( s, L ( T η ( s ) − , T η ( s ) − )) ds (cid:41) (4)For the rolling spot LIBOR measure as introduced by Jamshidian (1997), the numeraire is a roll–over strategy in which money is invested and subsequently reinvested at spot LIBOR. Investingone monetary unit in T i at the spot LIBOR L ( T i , T i ) is the same as buying k zero coupon bonds Note that this immediately implies that all interest rates generated by the model will be positive. Given Assumption 1.1, (3) is simply a restatement of the identity (23) in Miltersen, Sandmann and Sondermann(1997). RBITRAGE FREE TERM STRUCTURE INTERPOLATION 5 B ( T i , T i +1 ) , where k = B ( T i , T i +1 ) . Thus in continuous tenor the value at time t of the roll–overstrategy is(5) η ( t ) − (cid:89) i =0 (1 + δL ( T i , T i )) B ( t, T i +1 ) B ( T i , T i +1 ) Since r ( s, L ( T η ( s ) − , T η ( s ) − )) is F T i –measurable for T i ≤ s < T i +1 , we have by constructionthat B ( t, T i +1 ) B ( T i , T i +1 ) = exp (cid:40)(cid:90) tT η ( t ) − r ( s, L ( T η ( s ) − , T η ( s ) − )) ds (cid:41) for T i ≤ t ≤ T i +1 . Thus the numeraire processes are identical, which means that the associatedmartingale measures must be .The interpolation defined by (3) is consistent with no arbitrage if the savings account givenby (4) discounted by a numeraire asset is a martingale under the martingale measure associatedwith this numeraire. It is sufficient to verify this for one measure and by the above argument,this condition is trivially fulfilled since the value of the savings account is identically equal tothe value of the spot LIBOR roll–over strategy. (cid:50) In other words, since at any time t the dynamics of the shortest remaining bond to a discretetenor date B ( t, T η ( t ) ) are consistent with the deterministic interpolation for the continuouslycompounded short rate, all martingale properties (and thus no arbitrage) are preserved.The observation that the roll–over strategy in spot LIBOR corresponds to a roll-over strategyin the shortest remaining bond to a discrete tenor date also leads us to the following1.3. R EMARK . Conditional on the information at time T i , for all F T i +1 –measurable events, therolling spot LIBOR measure IP is identical to the forward measure of maturity T i +1 , i.e.IP { A |F T i } = IP T i +1 { A |F T i } ∀ A ∈ F T i +1 Thus IP can be interpreted as “pasting together” a sequence of conditional forward measures.This is valid for any arbitrage free term structure model. P ROOF : This is an immediate consequence of the fact that the time t value of the spot LIBORroll–over strategy (5) can be written, independently of how the discrete tenor model is extendedto continuous tenor, as a T η ( t ) − –measurable factor times B ( t, T η ( t ) ) , the numeraire of the time T η ( t ) forward measure. (cid:50) 2. I NTERPOLATION OF L ONGER – DATED B ONDS EMARK . Given a discrete tenor model and the interpolation of the short–dated zerocoupon bonds B ( t, T η ( t ) ) , the continuous tenor model is completely specified. In a continuous tenor model, the market is (dynamically) complete, therefore the martingale measure for agiven numeraire is unique. For an alternative proof, cf. Lemma 3.3 in Schl¨ogl (2002b). For a more formal statement of this proof, see Schl¨ogl (2002b). ERIK SCHL ¨OGL P ROOF : Consider a completely arbitrary pair of time points t , t , ≤ t < t ≤ T N . The time t price of a zero coupon bond maturing in t is given by(6) B ( t , t ) = B ( t , T η ( t ) ) η ( t ) − (cid:89) i = η ( t ) (1 + δL ( t , T i )) − B ( t , t ) B ( t , T η ( t ) ) where the short–dated bond B ( t , T η ( t ) ) is given by the interpolation and the bond price quotienton the far right by the no–arbitrage requirement that B ( t , t ) B ( t , T η ( t ) ) = E T η ( t [ B ( t , T η ( t ) ) − |F t ] Given a choice of interpolation (3), we can determine a function g such that(7) B ( t , t ) B ( t , T η ( t ) ) = E T η ( t [ g ( L ( T η ( t ) − , T η ( t ) − )) |F t ] In the lognormal case, the right hand side can easily be evaluated numerically and depends onlyon L ( t , T η ( t ) − ) and deterministic volatilities. (cid:50) When intermediate rates with LIBOR–type compounding are linearly interpolated by day–count fractions, (7) becomes very tractable for any model satisfying the assumptions (BP.1) and (BP.2) . Define r ( s, L ( T η ( s ) − , T η ( s ) − )) implicitly by setting(8) exp (cid:26)(cid:90) T i +1 t r ( s, L ( T η ( s ) − , T η ( s ) − )) ds (cid:27) = 1 + ( T i +1 − t ) L ( T i , T i ) ∀ T i ≤ t < T i +1 i.e. r ( s, L ( T i , T i )) = L ( T i , T i )1 + ( T i +1 − s ) L ( T i , T i ) Then B ( t , T η ( t ) ) = (1 + ( T η ( t ) − t ) L ( T η ( t ) − , T η ( t ) − )) − and B ( t , t ) B ( t , T η ( t ) ) = E T η ( t [ B ( t , T η ( t ) ) − |F t ]= 1 + ( T η ( t ) − t ) E T η ( t [ L ( T η ( t ) − , T η ( t ) − ) |F t ]= 1 + ( T η ( t ) − t ) L ( t , T η ( t ) − ) Note that by remark 2.1, if interpolation of short–dated bonds is specified for all times, there is nomodelling freedom to interpolate longer–dated bonds, including initial model inputs. In additionto being intuitively appealing, linear interpolation by day–count fractions has the added attractionof providing one consistent interpolation method for all times and all maturities. Other popularinterpolation methods, such as loglinear interpolation of discount factors or linear interpolationof continuously compounded yields, do not have this property. RBITRAGE FREE TERM STRUCTURE INTERPOLATION 7 3. I NTRODUCING V OLATILITY FOR THE S HORT B ONDS In some applications, setting short bond volatilities to zero may be unsatisfactory. Assumption1.1 can certainly be relaxed in any way which satisfies the arbitrage constraints. Remark 2.1remains valid, i.e. to extend the model to continuous tenor it is sufficient to specify the dynamicsof the short bonds. However, when the short bond dynamics are stochastic, the constraint givenby equation (3) is replaced by the more general formulation given in terms of instantaneousforward rates in equation (23) of Miltersen, Sandmann and Sondermann (1997):(9) exp (cid:26)(cid:90) T i +1 T i f ( T i , s ) ds (cid:27) = 1 + δL ( T i , T i ) ∀ i < N where f ( T i , s ) is the instantaneous forward rate at time T i for maturity s .One tractable and intuitively appealing way to introduce volatility for the interpolated shortbonds is to make the interpolated rates dependent on the closest remaining forward LIBOR L ( t , T η ( t ) ) . Set B ( t , T η ( t ) ) − = 1 + ( T η ( t ) − t )( α ( t ) L ( T η ( t ) − , T η ( t ) − ) + (1 − α ( t )) L ( t , T η ( t ) )) where lim ∆ (cid:38) α ( T i + ∆) = 1lim ∆ (cid:37) T i +1 − T i α ( T i + ∆) = 0 ∀ i = 0 , . . . , N − e.g. α ( t ) = T η ( t ) − tT η ( t ) − T η ( t ) − As in section 2, the prices of bonds with longer maturities must satisfy B ( t , t ) B ( t , T η ( t ) ) = E T η ( t (cid:20) B ( t , t ) B ( t , T η ( t ) ) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = 1 + ( T η ( t ) − t ) (cid:16) α ( t ) E T η ( t (cid:2) L ( T η ( t ) − , T η ( t ) − ) |F t (cid:3) +(1 − α ( t )) E T η ( t (cid:2) L ( t , T η ( t ) ) |F t (cid:3)(cid:17) Using the expected value of L ( t, T j ) under IP T j given in Rutkowski (1997): = 1 + ( T η ( t ) − t ) (cid:32) α ( t ) L ( t , T η ( t ) − ) + (1 − α ( t )) L ( t , T η ( t ) ) · T η ( t )+1 − T η ( t ) ) L ( t , T η ( t ) ) (cid:16) exp (cid:110)(cid:82) t t λ ( s, T η ( t ) ) ds (cid:111) − (cid:17) T η ( t )+1 − T η ( t ) ) L ( t , T η ( t ) ) (cid:124) (cid:123)(cid:122) (cid:125) “correction factor” (cid:33) (10)The “correction factor” will usually be quite close to one, unless volatility λ is very high or timeto maturity t − t is very long. Thus interpolation remains essentially linear. ERIK SCHL ¨OGL Note that since the short bonds are no longer deterministic, we also need to evaluate theappropriate expectation to derive B ( t, T ) for t < T < T η ( t ) : B ( t, T ) B ( t, T η ( t ) ) = E T η ( t ) (cid:20) B ( T, T ) B ( T, T η ( t ) ) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = 1 + ( T η ( t ) − T ) (cid:32) α ( T ) L ( t, T η ( t ) − ) + (1 − α ( T )) L ( t, T η ( t ) ) · T η ( t )+1 − T η ( t ) ) L ( t, T η ( t ) ) (cid:16) exp (cid:110)(cid:82) Tt λ ( s, T η ( t ) ) ds (cid:111) − (cid:17) T η ( t )+1 − T η ( t ) ) L ( t, T η ( t ) ) (cid:33) 4. F ORWARD M EASURES AND THE M ARKOV P ROPERTY Both Miltersen, Sandmann and Sondermann (1997) and Brace, Gatarek and Musiela (1997)set up the model of lognormal forward LIBORs in continuous tenor, i.e. equation (2) applies toall maturities T ∈ (0 , T ∗ − δ ] : dL ( t, T ) = L ( t, T ) λ ( t, T ) dW T + δ ( t ) with λ : IR × (0 , T ∗ ] → IR d a deterministic function of its arguments, i.e. each L ( t, T ) isa lognormal martingale under IP T + δ for a continuum of maturities up to the time horizon T ∗ .This has the advantage that all caps and floors on δ –compounded rates will be priced by theBlack/Scholes–type formulae favoured by market practitioners.The main disadvantage of this approach is revealed when one attempts to price instruments forwhich closed form solutions are unavailable. This is already the case if the cashflow underlyinga derivative does not fit neatly into a tenor structure of δ –compounded rates and even more so formany popular interest rate exotics. The problem is that the continuous tenor model is infinite–dimensional even if the driving Brownian motion is one–dimensional. That is, the Markovianstate variable for such a model is the entire yield curve. This causes considerable difficulties forall types of numerical methods. Even the method of last resort for high–dimensional problems,Monte Carlo simulation, cannot handle infinite–dimensional state variables.From the start, it has been part of the “folk wisdom” on these models that they do not per-mit a finite dimensional representation. Brace, Gatarek and Musiela (1997) derive the dynamicsof the continuously compounded short rate under a given set of assumptions on the volatilitystructure and show that they are highly path–dependent and thus not Markovian in any finiteset of state variables. Recently, Corr (2000) formalised this conjecture and showed that whenthe model is driven by a one–dimensional Brownian motion, a finite–dimensional representationcan only exist in the trivial case of zero volatility. He also derives similarly restrictive condi-tions on the finite–dimensional representability of continuous tenor LIBOR models driven bya multidimensional Brownian motion. These conditions, though stopping short of precludingfinite–dimensional cases, indicate that such cases are not likely to be practically useful if they doexist. The term “yield curve” generally denotes the term structure interest rates for a continuum of maturities up tothe time horizon. It can be represented in several equivalent ways, for example by the curve of all continuouslycompounded yields at time t , y t : ( t, T ∗ ] → IR with y t ( T ) defined by B ( t, T ) = exp {− y t ( T ) · T } RBITRAGE FREE TERM STRUCTURE INTERPOLATION 9 The extensions of the discrete tenor model to continuous tenor proposed in the previous sec-tions provide a way around this problem. Since the interpolated rates are specified as functionsof the discrete tenor rates, the Markovian structure of the discrete tenor model is preserved. Thisstructure can be characterised as follows:4.1. R EMARK . In the discrete tenor lognormal forward LIBOR model, consider the dynamicsof a rate L ( · , T i ) under some forward measure IP T j . These dynamics are Markovian in a statevariable vector consisting of n = | j − − i | + 1 rates L ( · , T k ) with min( i, j − ≤ k ≤ max( i, j − . P ROOF : Musiela and Rutkowski (1997) show that the relationship between forward measuresIP T k and IP T k +1 is given by the Radon/Nikodym derivative d IP T k d IP T k +1 (cid:12)(cid:12)(cid:12)(cid:12) F t = exp (cid:26)(cid:90) t γ ( u, T k , T k +1 ) dW T k +1 ( u ) − (cid:90) t γ ( u, T k , T k +1 ) du (cid:27) where W T k +1 is a Brownian motion under IP T k +1 and γ ( t, T k , T k +1 ) = δL ( t, T k )1 + δL ( t, T k ) λ ( t, T k ) with λ ( t, T k ) the (deterministic) volatility of L ( t, T k ) . Thus dW T k ( t ) = dW T k +1 ( t ) − δL ( t, T k )1 + δL ( t, T k ) λ ( t, T k ) dt Since the dynamics of L ( · , T k ) are given by dL ( t, T k ) = L ( t, T k ) λ ( t, T k ) dW T k +1 ( t ) the joint dynamics of all rates L ( · , T k ) with min( i, j − ≤ k ≤ max( i, j − under IP T max( i +1 ,j ) are(11) d L ( t ) = Λ( t, L ( t )) dW T max( i +1 ,j ) ( t ) − Ψ( L ( t )) (cid:96) ( L ( t )) dt where L ( t ) = L ( t, T min( i,j − ) L ( t, T min( i,j − ) ... L ( t, T max( i,j − ) If d is the dimension of the driving Brownian motion, Λ( t, L ( t )) is an n × d matrix with Λ hk ( t, L ( t )) = L ( t, T min( i,j − − h ) λ k ( t, T min( i,j − − h )Ψ( t, L ( t )) is an n × n matrix with Ψ hk ( t, L ( t )) = (cid:26) L ( t, T min( i,j − − h ) λ ( t, T min( i,j − − h ) λ ( t, T min( i,j − − k ) if k > h otherwiseFinally, (cid:96) ( L ( t )) is an n –dimensional vector with (cid:96) k ( L ( t )) = δL ( t, T min( i,j − − k )1 + δL ( t, T min( i,j − − k ) Note that the Markov property depends on the probability measure under consideration. ARBITRAGE FREE TERM STRUCTURE INTERPOLATION 10 . 05 4 . . 16 4 . 21 4 . 26 4 . 31 4 . 36 4 . 42 4 . 47 4 . 52 4 . 57 4 . 62 4 . 68 4 . 73 4 . 78 4 . 83 4 . 88 4 . 94 4 . 99 5 . 04 5 . 09 5 . 14 5 . . 25 5 . . 35 5 . . 46 5 . 51 5 . 56 5 . 61 5 . 66 5 . 72 5 . 77 5 . 82 5 . 87 5 . 92 5 . Maturity I n s t a n t a n e ou s f w d r a t es Figure 1 . 05 4 . . 16 4 . 21 4 . 26 4 . 31 4 . 36 4 . 42 4 . 47 4 . 52 4 . 57 4 . 62 4 . 68 4 . 73 4 . 78 4 . 83 4 . 88 4 . 94 4 . 99 5 . 04 5 . 09 5 . 14 5 . . 25 5 . . 35 5 . . 46 5 . 51 5 . 56 5 . 61 5 . 66 5 . 72 5 . 77 5 . 82 5 . 87 5 . 92 5 . Maturity I n s t a n t a n e ou s f w d r a t es Figure 2 Interpolation by daycount fractions vs. interpolation with short bond volatilityConversely, under IP T min( i +1 ,j ) we have(12) d L ( t ) = Λ( t, L ( t )) dW T min( i +1 ,j ) ( t ) + Ψ ( L ( t )) ` ( L ( t )) dt withΨ hk ( t, L ( t )) = ½ L ( t, T min( i,j − − h ) λ ( t, T min( i,j − − h ) λ ( t, T min( i,j − − k ) if 1 < k ≤ h L ( t ). This result does not rely on the assumption that the LIBOR volatilities λ ( t, T k ) are de-terministic functions of their arguments. Rather, if the λ ( t, T k ) are level dependent onsome or all rates in L ( t ), the Markov property still holds. Also, by remark 1.3, the aboveresult implies that under the rolling spot LIBOR measure, any rate L ( · , T i ) is Markov inthe state variable vector { L ( t, T k ) | ≤ k ≤ i } .5. A Look at the Interpolated Interest Rates This section illustrates some of the properties of the interpolated interest rates andcontrasts the two interpolation methods proposed in the previous sections. The methoddescribed in section 2 will be referred to as method 1 or interpolation by daycount fractions ;the one introduced in section 3 will be labelled method 2 or interpolation with short bondvolatility . For the inputs used to produce each of the plots, please refer to table 1 in theappendix. The algorithm from Brace, Musiela and Schl¨ogl (1998) was used to generatesample paths for the evolution of the term structure.5.1. Term Structures. Figures 1 and 2 show how the two methods interpolate instan-taneous forward rates. For reference, the stepwise constant rates resulting from loglinearinterpolation of zero coupon bond prices are included in each plot. The instantaneousforward rates jump when one moves from one δ –compounding period to the next. Fromequations (8) and (9), this is to be expected, as the maturities of the LIBORs underlyingthe interpolation are different on each interval. Instantaneous forward rates on f (0 , T ) aredetermined on each interval T ∈ ( T i , T i +1 ) by calculating − ∂ T ln B (0 , T ) B (0 , T η ( T ) ) F IGURE ARBITRAGE FREE TERM STRUCTURE INTERPOLATION 10 . 05 4 . . 16 4 . 21 4 . 26 4 . 31 4 . 36 4 . 42 4 . 47 4 . 52 4 . 57 4 . 62 4 . 68 4 . 73 4 . 78 4 . 83 4 . 88 4 . 94 4 . 99 5 . 04 5 . 09 5 . 14 5 . . 25 5 . . 35 5 . . 46 5 . 51 5 . 56 5 . 61 5 . 66 5 . 72 5 . 77 5 . 82 5 . 87 5 . 92 5 . Maturity I n s t a n t a n e ou s f w d r a t es Figure 1 . 05 4 . . 16 4 . 21 4 . 26 4 . 31 4 . 36 4 . 42 4 . 47 4 . 52 4 . 57 4 . 62 4 . 68 4 . 73 4 . 78 4 . 83 4 . 88 4 . 94 4 . 99 5 . 04 5 . 09 5 . 14 5 . . 25 5 . . 35 5 . . 46 5 . 51 5 . 56 5 . 61 5 . 66 5 . 72 5 . 77 5 . 82 5 . 87 5 . 92 5 . Maturity I n s t a n t a n e ou s f w d r a t es Figure 2 Interpolation by daycount fractions vs. interpolation with short bond volatilityConversely, under IP T min( i +1 ,j ) we have(12) d L ( t ) = Λ( t, L ( t )) dW T min( i +1 ,j ) ( t ) + Ψ ( L ( t )) ` ( L ( t )) dt withΨ hk ( t, L ( t )) = ½ L ( t, T min( i,j − − h ) λ ( t, T min( i,j − − h ) λ ( t, T min( i,j − − k ) if 1 < k ≤ h L ( t ). This result does not rely on the assumption that the LIBOR volatilities λ ( t, T k ) are de-terministic functions of their arguments. Rather, if the λ ( t, T k ) are level dependent onsome or all rates in L ( t ), the Markov property still holds. Also, by remark 1.3, the aboveresult implies that under the rolling spot LIBOR measure, any rate L ( · , T i ) is Markov inthe state variable vector { L ( t, T k ) | ≤ k ≤ i } .5. A Look at the Interpolated Interest Rates This section illustrates some of the properties of the interpolated interest rates andcontrasts the two interpolation methods proposed in the previous sections. The methoddescribed in section 2 will be referred to as method 1 or interpolation by daycount fractions ;the one introduced in section 3 will be labelled method 2 or interpolation with short bondvolatility . For the inputs used to produce each of the plots, please refer to table 1 in theappendix. The algorithm from Brace, Musiela and Schl¨ogl (1998) was used to generatesample paths for the evolution of the term structure.5.1. Term Structures. Figures 1 and 2 show how the two methods interpolate instan-taneous forward rates. For reference, the stepwise constant rates resulting from loglinearinterpolation of zero coupon bond prices are included in each plot. The instantaneousforward rates jump when one moves from one δ –compounding period to the next. Fromequations (8) and (9), this is to be expected, as the maturities of the LIBORs underlyingthe interpolation are different on each interval. Instantaneous forward rates on f (0 , T ) aredetermined on each interval T ∈ ( T i , T i +1 ) by calculating − ∂ T ln B (0 , T ) B (0 , T η ( T ) ) F IGURE T min( i +1 ,j ) we have(12) d L ( t ) = Λ( t, L ( t )) dW T min( i +1 ,j ) ( t ) + Ψ (cid:48) ( L ( t )) (cid:96) ( L ( t )) dt with Ψ (cid:48) hk ( t, L ( t )) = (cid:26) L ( t, T min( i,j − − h ) λ ( t, T min( i,j − − h ) λ ( t, T min( i,j − − k ) if < k ≤ h otherwiseBy the Markov property of Ito diffusions, the dynamics given by (11) resp. (12) are Markov in L ( t ) . (cid:50) This result does not rely on the assumption that the LIBOR volatilities λ ( t, T k ) are deterministicfunctions of their arguments. Rather, if the λ ( t, T k ) are level dependent on some or all ratesin L ( t ) , the Markov property still holds. Also, by remark 1.3, the above result implies thatunder the rolling spot LIBOR measure, any rate L ( · , T i ) is Markov in the state variable vector { L ( t, T k ) | ≤ k ≤ i } .5. A L OOK AT THE I NTERPOLATED I NTEREST R ATES This section illustrates some of the properties of the interpolated interest rates and contrasts thetwo interpolation methods proposed in the previous sections. The method described in section2 will be referred to as method 1 or interpolation by daycount fractions ; the one introduced insection 3 will be labelled method 2 or interpolation with short bond volatility . For the inputsused to produce each of the plots, please refer to table 1 in the appendix. The algorithm fromBrace, Musiela and Schl¨ogl (1998b) was used to generate sample paths for the evolution of theterm structure.5.1. Term Structures. Figures 1 and 2 show how the two methods interpolate instantaneousforward rates. For reference, the stepwise constant rates resulting from loglinear interpolation ofzero coupon bond prices are included in each plot. The instantaneous forward rates jump whenone moves from one δ –compounding period to the next. From equations (8) and (9), this is tobe expected, as the maturities of the LIBORs underlying the interpolation are different on eachinterval. Instantaneous forward rates on f (0 , T ) are determined on each interval T ∈ ( T i , T i +1 ) by calculating − ∂ T ln B (0 , T ) B (0 , T η ( T ) ) RBITRAGE FREE TERM STRUCTURE INTERPOLATION 11 ARBITRAGE FREE TERM STRUCTURE INTERPOLATION 11 . 05 4 . . 16 4 . 21 4 . 26 4 . 31 4 . 36 4 . 42 4 . 47 4 . 52 4 . 57 4 . 62 4 . 68 4 . 73 4 . 78 4 . 83 4 . 88 4 . 94 4 . 99 5 . 04 5 . 09 5 . 14 5 . . 25 5 . . 35 5 . . 46 5 . 51 5 . 56 5 . 61 5 . 66 5 . 72 5 . 77 5 . 82 5 . 87 5 . 92 5 . Figure 3. Interpolated forward LIBORsusing (9) for method 1 and (10) for method 2.Instantaneous forward rates give the most disaggregate representation of the term struc-ture and therefore effectively demonstrate the implications of the choice of interpolationmethod. However, one should not judge the usefulness of a particular method from thisperspective alone. At first glance the “sawtooth” patterns of the interpolated rates forfalling term structures in method 1 and rising term structures in method 2 seem to rep-resent serious drawbacks to using either method, but one should keep in mind that theinstantaneous forward rates are also purely mathematical artifacts, while the objects beingmodelled are market observable rates such as LIBOR. Forward LIBORs are interpolatedby L ( t, T ) = 1 δ µ B ( t, T ) B ( t, T + δ ) − ¶ = 1 δ µ B ( t, T ) B ( t, T η ( T ) ) B ( t, T η ( T ) ) B ( t, T η ( T )+1 ) B ( t, T η ( T )+1 ) B ( t, T + δ ) − ¶ = 1 δ µ B ( t, T ) B ( t, T η ( T ) ) (1 + δL ( t, T η ( T ) )) B ( t, T η ( T )+1 ) B ( t, T + δ ) − ¶ (13)For method 1 this becomes L ( t, T ) = 1 δ ¡ (1 + ( T η ( T ) − T ) L ( t, T η ( T ) − ))(1 + δL ( t, T η ( T ) )) · (1 + ( T η ( T )+1 − ( T + δ )) L ( t, T η ( T ) )) − − ¢ Figure 3 shows interpolated forward three–month LIBORs for a continuum of start datesbetween four and six years forward. The piecewise linear graph is actually two plots so close F IGURE 3. Interpolated forward LIBORsusing (9) for method 1 and (10) for method 2.Instantaneous forward rates give the most disaggregate representation of the term structureand therefore effectively demonstrate the implications of the choice of interpolation method.However, one should not judge the usefulness of a particular method from this perspective alone.At first glance the “sawtooth” patterns of the interpolated rates for falling term structures inmethod 1 and rising term structures in method 2 seem to represent serious drawbacks to usingeither method, but one should keep in mind that the instantaneous forward rates are also purelymathematical artifacts, while the objects being modelled are market observable rates such asLIBOR. Forward LIBORs are interpolated by L ( t, T ) = 1 δ (cid:18) B ( t, T ) B ( t, T + δ ) − (cid:19) = 1 δ (cid:18) B ( t, T ) B ( t, T η ( T ) ) B ( t, T η ( T ) ) B ( t, T η ( T )+1 ) B ( t, T η ( T )+1 ) B ( t, T + δ ) − (cid:19) = 1 δ (cid:18) B ( t, T ) B ( t, T η ( T ) ) (1 + δL ( t, T η ( T ) )) B ( t, T η ( T )+1 ) B ( t, T + δ ) − (cid:19) (13)For method 1 this becomes L ( t, T ) = 1 δ (cid:0) (1 + ( T η ( T ) − T ) L ( t, T η ( T ) − ))(1 + δL ( t, T η ( T ) )) · (1 + ( T η ( T )+1 − ( T + δ )) L ( t, T η ( T ) )) − − (cid:1) Figure 3 shows interpolated forward three–month LIBORs for a continuum of start dates betweenfour and six years forward. The piecewise linear graph is actually two plots so close as to be ARBITRAGE FREE TERM STRUCTURE INTERPOLATION 12 Short Rate Dynamics . 06 0 . 11 0 . 17 0 . 22 0 . 28 0 . 33 0 . 39 0 . 44 0 . . 55 0 . . 66 0 . 71 0 . 77 0 . 82 0 . 88 0 . 93 0 . 99 1 . 04 1 . . 15 1 . 21 1 . 26 1 . 32 1 . 37 1 . 43 1 . 48 1 . 54 1 . 59 1 . 65 1 . . 76 1 . 81 1 . 87 1 . 92 1 . Time in years Fixed Maturity Instantaneous Fwd Rate . 05 0 . . 14 0 . 19 0 . 24 0 . 29 0 . 33 0 . 38 0 . 43 0 . 48 0 . 52 0 . 57 0 . 62 0 . 66 0 . 71 0 . 76 0 . 81 0 . 85 0 . . 95 1 1 . 04 1 . 09 1 . 14 1 . 19 1 . 23 1 . 28 1 . 33 1 . 38 1 . 42 1 . 47 1 . 52 1 . 57 1 . 61 1 . 66 1 . Time in years Fixed Time to Maturity Instantaneous Fwd Rates . 05 0 . 11 0 . 16 0 . 21 0 . 26 0 . 32 0 . 37 0 . 42 0 . 47 0 . 53 0 . 58 0 . 63 0 . 68 0 . 73 0 . 79 0 . 84 0 . 89 0 . 94 1 1 . 05 1 . . 15 1 . 21 1 . 26 1 . 31 1 . 36 1 . 42 1 . 47 1 . 52 1 . 57 1 . 63 1 . 68 1 . 73 1 . 78 1 . 84 1 . Time in years Short Rate vs. Fixed TTM Fwd Rate . 06 0 . 13 0 . 19 0 . 25 0 . 31 0 . 38 0 . 44 0 . . 56 0 . 62 0 . 69 0 . 75 0 . 81 0 . 87 0 . 94 1 1 . 06 1 . 12 1 . 19 1 . 25 1 . 31 1 . 37 1 . 44 1 . . 56 1 . 62 1 . 69 1 . 75 1 . 81 1 . 87 1 . Time in years Figure 4. Rate dynamics under interpolation by daycount fractionsas to be indistinguishable: Interpolation by daycount fractions and loglinear interpolationof zero coupon bond prices results in nearly identical forward three–month LIBORs, eventhough the resulting instantaneous forward rates are very different. This also implies thatif one is only concerned with rates such as forward three–month LIBORs, using loglinearinterpolation of zero coupon bond prices is for all intents and purposes arbitrage free.Introducing short bond volatility as per method 2 makes a difference for the interpolatedLIBORs only when the interest rate term structure changes slope. The vertical gridlinesin figure 3 represent boundaries between the accrual periods in the original discrete tenor.Interpolated forward LIBORs with start date in an accrual period immediately precedinga change in the slope of the term structure depart from the linearly interpolated plots andcurve “outward”. This is due to the fact that a forward LIBOR for a given start date T is interpolated using the forward LIBOR for the preceding and the two following discretetenor start dates T η ( T ) − , T η ( T ) and T η ( T )+1 , as can easily be seen by appropriately inserting(10) into (13).5.2. Rate Dynamics. The dynamics which the two methods imply for interpolated ratesare illustrated in Figures 4 and 5. Not only, as in the previous section, do equations (8)and (9) imply discontinuities in the instantaneous rates in the maturity dimension, butalso in the time dimension. This becomes particularly clear in the dynamics of the continu-ously compounded short rate under interpolation by daycount fractions: This rate evolvesdeterministically within each accrual period of the discrete tenor structure and jumps atthe accrual period boundaries. For instantaneous forward rates f ( t, T ) the dynamics arestochastic as long as the time to maturity T − t is greater than the accrual period length δ . The maturity T of the forward rate determines which forward LIBORs are used in the F IGURE 4. Rate dynamics under interpolation by daycount fractionsindistinguishable: Interpolation by daycount fractions and loglinear interpolation of zero couponbond prices results in nearly identical forward three–month LIBORs, even though the resultinginstantaneous forward rates are very different. This also implies that if one is only concernedwith rates such as forward three–month LIBORs, using loglinear interpolation of zero couponbond prices is for all intents and purposes arbitrage free.Introducing short bond volatility as per method 2 makes a difference for the interpolated LI-BORs only when the interest rate term structure changes slope. The vertical gridlines in figure 3represent boundaries between the accrual periods in the original discrete tenor. Interpolated for-ward LIBORs with start date in an accrual period immediately preceding a change in the slope ofthe term structure depart from the linearly interpolated plots and curve “outward”. This is due tothe fact that a forward LIBOR for a given start date T is interpolated using the forward LIBORfor the preceding and the two following discrete tenor start dates T η ( T ) − , T η ( T ) and T η ( T )+1 , ascan easily be seen by appropriately inserting (10) into (13).5.2. Rate Dynamics. The dynamics which the two methods imply for interpolated rates are il-lustrated in Figures 4 and 5. Not only, as in the previous section, do equations (8) and (9) implydiscontinuities in the instantaneous rates in the maturity dimension, but also in the time dimen-sion. This becomes particularly clear in the dynamics of the continuously compounded shortrate under interpolation by daycount fractions: This rate evolves deterministically within eachaccrual period of the discrete tenor structure and jumps at the accrual period boundaries. Forinstantaneous forward rates f ( t, T ) the dynamics are stochastic as long as the time to maturity T − t is greater than the accrual period length δ . The maturity T of the forward rate determines RBITRAGE FREE TERM STRUCTURE INTERPOLATION 13 ARBITRAGE FREE TERM STRUCTURE INTERPOLATION 13 Short Rate Dynamics . 06 0 . 11 0 . 17 0 . 22 0 . 28 0 . 33 0 . 39 0 . 44 0 . . 55 0 . . 66 0 . 71 0 . 77 0 . 82 0 . 88 0 . 93 0 . 99 1 . 04 1 . . 15 1 . 21 1 . 26 1 . 32 1 . 37 1 . 43 1 . 48 1 . 54 1 . 59 1 . 65 1 . . 76 1 . 81 1 . 87 1 . 92 1 . Time in years Fixed Maturity Instantaneous Fwd Rate . 05 0 . . 14 0 . 19 0 . 24 0 . 29 0 . 33 0 . 38 0 . 43 0 . 48 0 . 52 0 . 57 0 . 62 0 . 66 0 . 71 0 . 76 0 . 81 0 . 85 0 . . 95 1 1 . 04 1 . 09 1 . 14 1 . 19 1 . 23 1 . 28 1 . 33 1 . 38 1 . 42 1 . 47 1 . 52 1 . 57 1 . 61 1 . 66 1 . Time in years Fixed Time to Maturity Instantaneous Fwd Rates . 05 0 . 09 0 . 14 0 . 18 0 . 23 0 . 27 0 . 32 0 . 36 0 . 41 0 . 45 0 . . 54 0 . 58 0 . 63 0 . 67 0 . 72 0 . 76 0 . 81 0 . 85 0 . . 94 0 . 99 1 . 03 1 . 08 1 . 12 1 . 17 1 . 21 1 . 26 1 . . 35 1 . 39 1 . 44 1 . 48 1 . 53 1 . 57 1 . 62 1 . Time in years Short Rate vs. Fixed TTM Fwd Rate . 06 0 . 13 0 . 19 0 . 25 0 . 31 0 . 38 0 . 44 0 . . 56 0 . 62 0 . 69 0 . 75 0 . 81 0 . 87 0 . 94 1 1 . 06 1 . 12 1 . 19 1 . 25 1 . 31 1 . 37 1 . 44 1 . . 56 1 . 62 1 . 69 1 . 75 1 . 81 1 . 87 1 . Time in years Figure 5. Rate dynamics under interpolation with short bond volatilityinterpolation, so if one considers an instantaneous forward rate for a fixed maturity, thisrate will be interpolated from the same forward LIBORs at all times. Since the LIBORdynamics are modelled to be continuous, the interpolated rate will not jump (cf. plotin upper right of Figure 4). On the other hand, holding time to maturity (TTM) fixedmeans that there will be jumps when the maturity T moves across a boundary betweenaccrual periods in the original discrete tenor structure. The bottom two plots in Figure4) illustrate this. The time to maturity of the instantaneous forward rate shown here is1 . δ , resulting in a jump at three–quarters of every δ –interval of the tenor structure; thesepoints are marked by vertical gridlines in the plot. Thus different points on the forwardrate curve jump at different times.The same observations can be made for the dynamics of instantaneous rates under inter-polation with short bond volatility, as shown in Figure 5. The only qualitative difference isthat now rates with times to maturity less than δ , including the continuously compoundedshort rate, also have stochastic dynamics (cf. the upper left plot of Figure 5).As in the previous section, the observed discontinuities apply only to instantaneousrates. Of more practical relevance is the fact that interpolated rates for accrual periods ofany arbitrary length, for example a δ –compounded forward LIBOR L ( t, T ) for a “brokendate” maturity T = T η ( T ) , will have continuous dynamics: The instantaneous rates arecontinuous almost everywhere under the Lebesgue measure on the time line, thus anyjumps are integrated out once one considers accrual periods of greater than infinitesimallength. F IGURE 5. Rate dynamics under interpolation with short bond volatilitywhich forward LIBORs are used in the interpolation, so if one considers an instantaneous for-ward rate for a fixed maturity, this rate will be interpolated from the same forward LIBORs at alltimes. Since the LIBOR dynamics are modelled to be continuous, the interpolated rate will notjump (cf. plot in upper right of Figure 4). On the other hand, holding time to maturity (TTM)fixed means that there will be jumps when the maturity T moves across a boundary between ac-crual periods in the original discrete tenor structure. The bottom two plots in Figure 4) illustratethis. The time to maturity of the instantaneous forward rate shown here is . δ , resulting ina jump at three–quarters of every δ –interval of the tenor structure; these points are marked byvertical gridlines in the plot. Thus different points on the forward rate curve jump at differenttimes.The same observations can be made for the dynamics of instantaneous rates under interpola-tion with short bond volatility, as shown in Figure 5. The only qualitative difference is that nowrates with times to maturity less than δ , including the continuously compounded short rate, alsohave stochastic dynamics (cf. the upper left plot of Figure 5).As in the previous section, the observed discontinuities apply only to instantaneous rates.Of more practical relevance is the fact that interpolated rates for accrual periods of any arbi-trary length, for example a δ –compounded forward LIBOR L ( t, T ) for a “broken date” maturity T (cid:54) = T η ( T ) , will have continuous dynamics: The instantaneous rates are continuous almost ev-erywhere under the Lebesgue measure on the time line, thus any jumps are integrated out onceone considers accrual periods of greater than infinitesimal length. ARBITRAGE FREE TERM STRUCTURE INTERPOLATION 14 . 01 3 . 02 3 . 03 3 . 04 3 . 05 3 . 06 3 . 07 3 . 08 3 . 09 3 . . 11 3 . 12 3 . 13 3 . 14 3 . 15 3 . 16 3 . 17 3 . 18 3 . 19 3 . . 21 3 . 22 3 . 23 3 . 24 3 . Maturity Figure 6 . 01 3 . 02 3 . 03 3 . 04 3 . 05 3 . 06 3 . 07 3 . 08 3 . 09 3 . . 11 3 . 12 3 . 13 3 . 14 3 . 15 3 . 16 3 . 17 3 . 18 3 . 19 3 . . 21 3 . 22 3 . 23 3 . 24 3 . Maturity Figure 7 Implied volatilities resulting frominterpolation by daycount fractions vs. interpolation with short bond volatility6. Broken Date Caplets: A Comparison Another point of interest is how the choice of interpolation method affects the pricing ofderivatives. In this context, the volatility of interpolated rates is of key importance. Whenextending the discrete tenor model to continuous tenor, one is faced with a tradeoff ofcompleting the model via volatilities or interest rate interpolation. By fixing the volatilitiesof rates for all maturities, say all δ –compounded forward LIBORs L ( · , T ), T ∈ [0 , T ∗ − δ ]as in MSS and BGM, the model is completely specified and there is no need to interpolateinterest rates. On the other hand, when the interpolated rates are given as functions ofthe discrete tenor forward LIBORs, their volatilities can be calculated by an applicationof Ito’s Lemma. The former approach has the disadvantage of rather intractable dynamicsfor rates other the δ –compounded forward LIBORs, while latter implies that the “brokendate” forward LIBORs L ( t, T ), T = T η ( T ) are not lognormal. Applying Ito’s Lemma to(13), the relative volatility of L ( t, T ) isvol[ L ( t, T )] = L ( t, T ) − · N − X i =0 ∂ L ( t,T i ) µ δ µ B ( t, T ) B ( t, T η ( T ) ) (1 + δL ( t, T η ( T ) )) B ( t, T η ( T )+1 ) B ( t, T + δ ) − ¶¶ L ( t, T i ) λ ( t, T i )For method 1 this becomesvol[ L ( t, T )] = 1 δ L ( t, T ) − µ ( T η ( T ) − T ) 1 + δL ( t, T η ( T ) )1 + ( T η ( T ) − T ) L ( t, T η ( T ) ) λ ( t, T η ( T ) − ) L ( t, T η ( T ) − )+ (1 + ( T η ( T ) − T ) L ( t, T η ( T ) − )) · δ (1 + ( T η ( T ) − T ) L ( t, T η ( T ) )) − ( T η ( T ) − T )(1 + δL ( t, T η ( T ) ))(1 + ( T η ( T ) − T ) L ( t, T η ( T ) )) λ ( t, T η ( T ) ) L ( t, T η ( T ) ) ¶ Figures 6 and 7 illustrate how the two interpolation methods presented in the previoussections affect the prices for “broken date” caplets. In each plot, we vary the maturity T of the forward LIBOR L ( t, T ) underlying the caplet within one δ –accrual period of thediscrete tenor structure (i.e. the caplets on the two endpoints of each plot are based on F IGURE ARBITRAGE FREE TERM STRUCTURE INTERPOLATION 14 . 01 3 . 02 3 . 03 3 . 04 3 . 05 3 . 06 3 . 07 3 . 08 3 . 09 3 . . 11 3 . 12 3 . 13 3 . 14 3 . 15 3 . 16 3 . 17 3 . 18 3 . 19 3 . . 21 3 . 22 3 . 23 3 . 24 3 . Maturity Figure 6 . 01 3 . 02 3 . 03 3 . 04 3 . 05 3 . 06 3 . 07 3 . 08 3 . 09 3 . . 11 3 . 12 3 . 13 3 . 14 3 . 15 3 . 16 3 . 17 3 . 18 3 . 19 3 . . 21 3 . 22 3 . 23 3 . 24 3 . Maturity Figure 7 Implied volatilities resulting frominterpolation by daycount fractions vs. interpolation with short bond volatility6. Broken Date Caplets: A Comparison Another point of interest is how the choice of interpolation method affects the pricing ofderivatives. In this context, the volatility of interpolated rates is of key importance. Whenextending the discrete tenor model to continuous tenor, one is faced with a tradeoff ofcompleting the model via volatilities or interest rate interpolation. By fixing the volatilitiesof rates for all maturities, say all δ –compounded forward LIBORs L ( · , T ), T ∈ [0 , T ∗ − δ ]as in MSS and BGM, the model is completely specified and there is no need to interpolateinterest rates. On the other hand, when the interpolated rates are given as functions ofthe discrete tenor forward LIBORs, their volatilities can be calculated by an applicationof Ito’s Lemma. The former approach has the disadvantage of rather intractable dynamicsfor rates other the δ –compounded forward LIBORs, while latter implies that the “brokendate” forward LIBORs L ( t, T ), T = T η ( T ) are not lognormal. Applying Ito’s Lemma to(13), the relative volatility of L ( t, T ) isvol[ L ( t, T )] = L ( t, T ) − · N − X i =0 ∂ L ( t,T i ) µ δ µ B ( t, T ) B ( t, T η ( T ) ) (1 + δL ( t, T η ( T ) )) B ( t, T η ( T )+1 ) B ( t, T + δ ) − ¶¶ L ( t, T i ) λ ( t, T i )For method 1 this becomesvol[ L ( t, T )] = 1 δ L ( t, T ) − µ ( T η ( T ) − T ) 1 + δL ( t, T η ( T ) )1 + ( T η ( T ) − T ) L ( t, T η ( T ) ) λ ( t, T η ( T ) − ) L ( t, T η ( T ) − )+ (1 + ( T η ( T ) − T ) L ( t, T η ( T ) − )) · δ (1 + ( T η ( T ) − T ) L ( t, T η ( T ) )) − ( T η ( T ) − T )(1 + δL ( t, T η ( T ) ))(1 + ( T η ( T ) − T ) L ( t, T η ( T ) )) λ ( t, T η ( T ) ) L ( t, T η ( T ) ) ¶ Figures 6 and 7 illustrate how the two interpolation methods presented in the previoussections affect the prices for “broken date” caplets. In each plot, we vary the maturity T of the forward LIBOR L ( t, T ) underlying the caplet within one δ –accrual period of thediscrete tenor structure (i.e. the caplets on the two endpoints of each plot are based on F IGURE ROKEN D ATE C APLETS : A C OMPARISON Another point of interest is how the choice of interpolation method affects the pricing ofderivatives. In this context, the volatility of interpolated rates is of key importance. When ex-tending the discrete tenor model to continuous tenor, one is faced with a tradeoff of completingthe model via volatilities or interest rate interpolation. By fixing the volatilities of rates for allmaturities, say all δ –compounded forward LIBORs L ( · , T ) , T ∈ [0 , T ∗ − δ ] as in MSS and BGM,the model is completely specified and there is no need to interpolate interest rates. On the otherhand, when the interpolated rates are given as functions of the discrete tenor forward LIBORs,their volatilities can be calculated by an application of Ito’s Lemma. The former approach hasthe disadvantage of rather intractable dynamics for rates other the δ –compounded forward LI-BORs, while latter implies that the “broken date” forward LIBORs L ( t, T ) , T (cid:54) = T η ( T ) are notlognormal. Applying Ito’s Lemma to (13), the relative volatility of L ( t, T ) isvol [ L ( t, T )] = L ( t, T ) − · N − (cid:88) i =0 ∂ L ( t,T i ) (cid:18) δ (cid:18) B ( t, T ) B ( t, T η ( T ) ) (1 + δL ( t, T η ( T ) )) B ( t, T η ( T )+1 ) B ( t, T + δ ) − (cid:19)(cid:19) L ( t, T i ) λ ( t, T i ) For method 1 this becomesvol [ L ( t, T )] = 1 δ L ( t, T ) − (cid:18) ( T η ( T ) − T ) 1 + δL ( t, T η ( T ) )1 + ( T η ( T ) − T ) L ( t, T η ( T ) ) λ ( t, T η ( T ) − ) L ( t, T η ( T ) − )+ (1 + ( T η ( T ) − T ) L ( t, T η ( T ) − )) · δ (1 + ( T η ( T ) − T ) L ( t, T η ( T ) )) − ( T η ( T ) − T )(1 + δL ( t, T η ( T ) ))(1 + ( T η ( T ) − T ) L ( t, T η ( T ) )) λ ( t, T η ( T ) ) L ( t, T η ( T ) ) (cid:19) Figures 6 and 7 illustrate how the two interpolation methods presented in the previous sectionsaffect the prices for “broken date” caplets. In each plot, we vary the maturity T of the forwardLIBOR L ( t, T ) underlying the caplet within one δ –accrual period of the discrete tenor structure(i.e. the caplets on the two endpoints of each plot are based on non-interpolated rates). The strike RBITRAGE FREE TERM STRUCTURE INTERPOLATION 15 is chosen at 1.25 times the at–the–money level and the volatility function is two–dimensional ex-ponentially decaying. This choice should provide a representative example well away from thetrivial at–the–money, one–dimensional constant volatility case. The caplet prices were generatedby Monte Carlo simulation; the middle line in each plot give the Monte Carlo estimate, whilethe outside lines are the confidence interval boundaries two standard deviations to either side ofthe Monte Carlo estimate. One million MC runs make these confidence bounds reasonably tight.We represent the caplet prices in terms of their Black/Scholes implied volatility, i.e. in terms ofthe one–dimensional constant relative volatility of the underlying rate which would result in thesame caplet price. The dip in the implied volatilities is due to the fact that the interpolation of L ( T, T ) at maturity of the caplet is in part based on L ( T η ( T ) − , T η ( T ) − ) , i.e. a rate which evolvesstochastically only until T η ( T ) − < T . This effect is less pronounced under interpolation withshort bond volatility, as in this case L ( T η ( T ) − , T η ( T ) − ) plays a lesser role.The disadvantage that the “broken date” forward LIBORs are not lognormal is greatly al-leviated by the fact that their distribution is actually very close to lognormal. One can applythe argument which was first used to derive an approximate swaption formula in a lognormalforward LIBOR model. Noting that the level dependence of the volatility of the interpolatedforward LIBORs varies slowly compared to the rates themselves, one can derive an approxi-mate closed–form solution for the “broken date” caplet by calculating the level dependence withrespect to the initial rates, e.g. for method 1 by settingvol [ L ( t, T )] = L (0 , T ) − · (cid:18) ∂L (0 , T ) ∂L (0 , T η ( T ) − ) L (0 , T η ( T ) − ) λ ( t, T η ( T ) − ) + ∂L (0 , T ) ∂L (0 , T η ( T ) ) L (0 , T η ( T ) ) λ ( t, T η ( T ) ) (cid:19) = 1 δ L (0 , T ) − (cid:32) ( T η ( T ) − T ) 1 + δL (0 , T η ( T ) )1 + ( T η ( T ) − T ) L (0 , T η ( T ) ) λ ( t, T η ( T ) − ) L (0 , T η ( T ) − )+ (1 + ( T η ( T ) − T ) L (0 , T η ( T ) − )) · δ (1 + ( T η ( T ) − T ) L (0 , T η ( T ) )) − ( T η ( T ) − T )(1 + δL (0 , T η ( T ) ))(1 + ( T η ( T ) − T ) L (0 , T η ( T ) )) · λ ( t, T η ( T ) ) L (0 , T η ( T ) ) (cid:33) The Black/Scholes implied volatility is then given by √ T (cid:115)(cid:90) T ( vol [ L ( s, T )]) ds For details see appendix. This argument first appears in Brace, Gatarek and Musiela (1997). It was developed further in Brace, Dunand Barton (1998a) and formalized by Brace and Womersley (2000). This approximation works well not only forpricing, but also for hedging, as demonstrated in Dun, Barton and Schl¨ogl (2001). = 1 √ T (cid:32) ∂L (0 , T ) ∂L (0 , T η ( T ) − ) L (0 , T η ( T ) − ) ∂L (0 , T ) ∂L (0 , T η ( T ) ) L (0 , T η ( T ) ) cov ( T η ( T ) − , T η ( T ) )+ (cid:18) ∂L (0 , T ) ∂L (0 , T η ( T ) − ) L (0 , T η ( T ) − ) λ (0 , T η ( T ) − ) (cid:19) + (cid:18) ∂L (0 , T ) ∂L (0 , T η ( T ) ) L (0 , T η ( T ) ) λ (0 , T η ( T ) ) (cid:19) (cid:33) with cov ( T η ( T ) − , T η ( T ) ) = (cid:90) T η ( T ) − λ ( s, T η ( T ) − ) λ ( s, T η ( T ) ) dsλ (0 , T η ( T ) − ) = (cid:115)(cid:90) T η ( T ) − λ ( s, T η ( T ) − ) dsλ (0 , T η ( T ) ) = (cid:115)(cid:90) T η ( T ) λ ( s, T η ( T ) ) ds As illustrated by Figure 6, the resulting approximation of Black/Scholes implied volatilities isvery accurate. 7. C ONCLUSION Extending the models of market observable interest rates from discrete to continuous tenor byinterpolation is particularly useful in implementations where there are some financial productswhich must be priced numerically, as the Markovian structure of the discrete tenor model ispreserved. It is important to note that the interpolation method cannot be chosen arbitrarily forall maturities. Rather, it must take into account the relevant no–arbitrage conditions.Shifting the focus from instantaneous forward rates to market observables such as forwardLIBOR, the rate dynamics implied by the proposed interpolation methods are reasonable. Thesemethods supply alternatives to the continuous tenor versions of the lognormal forward rate Mar-ket Models proposed in the literature and in those cases where moving to continuous tenor byinterpolation entails a loss of tractability, for example for “broken date” caplets, very accurateapproximations exist. 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