Affine LIBOR models driven by real-valued affine processes
AAffine LIBOR models driven by real-valued affineprocesses
Wolfgang M¨uller, Stefan Waldenberger
Keywords:
LIBOR rate models, forward price models, affine processes,volatility smile
Abstract:
The class of affine LIBOR models is appealing since it satisfiesthree central requirements of interest rate modeling. It is arbitrage-free,interest rates are nonnegative and caplet and swaption prices can be calcu-lated analytically. In order to guarantee nonnegative interest rates affineLIBOR models are driven by nonnegative affine processes, a restriction,which makes it hard to produce volatility smiles. We modify the affineLIBOR models in such a way that real-valued affine processes can be usedwithout destroying the nonnegativity of interest rates. Numerical examplesshow that in this class of models pronounced volatility smiles are possible.
1. Introduction
Market models, the most famous example being the LIBOR market model, are very popularin the area of interest rate modeling. If these models generate nonnegative interest ratesthey usually do not give semi-analytic formulas for both basic interest rate derivatives, capsand swaptions. One exception is the class of affine LIBOR models proposed by Keller-Resselet al. [14]. Using nonnegative affine processes as driving processes affine LIBOR modelsguarantee nonnegative forward interest rates and lead to semi-analytical formulas for capsand swaptions, so that calibration to interest rate market data is possible.This paper modifies the setup of Keller-Ressel et al. [14] to allow for not necessarily non-negative affine processes. This modification still leads to semi-analytical formulas for capsand swaptions and guarantees nonnegative forward interest rates, but allows for a wider classof driving affine processes and hence is more flexible in producing interest rate skews andsmiles. Fonseca et al. [8] also propose a modification of affine LIBOR models. There drivingprocesses are affine processes with values in the space of positive semidefinite matrices. Theapproach in this paper has the advantage that a flexible class of implied volatility surfacescan be produced with a much smaller number of parameters.The structure of this paper is as follows. In section 2 affine processes and their properties arereviewed. Section 3 introduces the necessary notation and market setup and reviews affineLIBOR models. It concludes with some comments on practical implementation. Section 4 isthe main section of this paper. The first part presents the modified affine LIBOR model andsemi-analytical pricing formulas for caps and swaptions are derived. The second part thengives some examples of usable affine processes with numerical calculations.1 a r X i v : . [ q -f i n . P R ] M a r ffine LIBOR models with real-valued processes
2. Affine processes
Let X = ( X t ) ≤ t ≤ T be a homogeneous Markov process with values in D = R m ≥ × R n realizedon a measurable space (Ω , A ) with filtration ( F t ) ≤ t ≤ T , with regards to which X is adapted.Denote by P x [ · ] and E x [ · ] the corresponding probability and expectation when X = x . X issaid to be an affine process, if its characteristic function has the form E x (cid:2) e u · X t (cid:3) = exp ( φ t ( u ) + ψ t ( u ) · x ) , u ∈ i R d , x ∈ D, (1)where φ : [0 , T ] × i R d → C and ψ : [0 , T ] × i R d → C d with i R d = { u ∈ C d : Re( u ) = 0 } and · denoting the scalar product in R d . By homogeneity and the Markov property the conditionalcharacteristic function satisfies E x (cid:2) e u · X t | F s (cid:3) = exp ( φ t − s ( u ) + ψ t − s ( u ) · X s ) . Accordingly affine processes can also be defined for inhomogeneous Markov processes (seeFilipovic [7]), in which case the above equality reads E x (cid:2) e u · X t | F s (cid:3) = exp ( φ s,t ( u ) + ψ s,t ( u ) · X s ) , u ∈ i R d , x ∈ D, with φ s,t : i R d → C and ψ s,t : i R d → C d for 0 ≤ s ≤ t . X is called an analytic affine process (see Keller-Ressel [11]), if X is stochastically continuousand the interior of the set V := (cid:40) u ∈ C d : sup ≤ s ≤ T E x (cid:104) e Re( u ) · X s (cid:105) < ∞ ∀ x ∈ D (cid:41) , (2)contains 0 . In this case the functions φ and ψ have continuous extensions to V , which areanalytic in the interior, such that (1) holds for all u ∈ V .The class of affine processes includes Brownian motion and more generally all L´evy processes.Since L´evy processes have stationary independent increments, in this case ψ t ( u ) = u and φ t ( u ) = tκ ( u ), where κ is the cumulant generating function of the L´evy process. Ornstein-Uhlenbeck processes are further important examples of affine processes. They are discussedin section 4.2.The standard reference for affine processes is Duffie et al. [4]. There they give a characteri-zation of affine processes, where φ and ψ are specified as solutions of a system of differentialequations . Of all the rich theory of affine processes the methods in this paper only use thespecific form (1) of their moment generating function and the following property. Lemma 1:
Let X be a one-dimensional analytic affine process and Re( u ) < Re( w ) , u, w ∈ V .Then Re( ψ t ( u )) < ψ t (Re( w )) , i.e. ψ t | V∩ R is strictly increasing. V can be described as the (convex) set, where the extended moment generating function of X t is defined forall times t ≤ T and all starting values x ∈ E . By Lemma 4.2 in Keller-Ressel and Mayerhofer [12] the set V is in fact equal to the seemingly smaller set (cid:110) u ∈ C d : ∃ x ∈ int( D ) : E x (cid:104) e Re( u ) · X T (cid:105) < ∞ (cid:111) . This also implies that X is conservative, i.e. P x ( X t ∈ D ) = 1 ∀ x ∈ D and 0 ≤ t ≤ T . The fact that this characterization holds for all stochastically continuous affine processes was first shown inKeller-Ressel et al. [13] and later for affine processes with more general state spaces in Keller-Ressel et al.[15] and Cuchiero and Teichmann [3]. ffine LIBOR models with real-valued processes Proof:
The case D = R + is already contained in Keller-Ressel et al. [14]. In case D = R thelemma follows from the fact that by Proposition 3.3 in Keller-Ressel et al. [13] ψ t ( u ) = e βt u for some constant β . Remark: If D = R + , it is known that both, ψ and φ , are monotonically increasing (Keller-Ressel et al. [14]). With D = R this stays true for ψ , but not φ , as the deterministic affineprocess X t = x − t shows.
3. Interest rate market models
Classical market models
Consider a tenor structure 0 < T < · · · < T N < T N +1 =: T and a market consisting of zerocoupon bonds with maturities T , . . . , T N +1 . Their price processes ( P ( t, T k )) ≤ t ≤ T k are as-sumed to be nonnegative semimartingales on a filtered probability space (Ω , A , ( F t ) ≤ t ≤ T , P ),which satisfy P ( T k , T k ) = 1 almost surely. If there exists an equivalent probability measure Q T such that the normalized bond price processes P ( · , T k ) /P ( · , T ) are martingales , the mar-ket is arbitrage-free. In this case we can define equivalent martingale measures Q T k for thenumeraires P ( t, T k ) instead of P ( t, T ) byd Q T k d Q T = 1 P ( T k , T ) P (0 , T ) P (0 , T k ) . (3)In particular under the measure Q T k the forward bond price process P ( · , T k − ) /P ( · , T k ) andthe forward interest rate process F k ( · ), F k ( t ) = 1∆ k (cid:18) P ( t, T k − ) P ( t, T k ) − (cid:19) , ∆ k = T k − T k − , (4)are martingales. This is the basic market setup used throughout the rest of the paper.In the classical LIBOR market models forward interest rate processes F k are modeled ascontinuous exponential martingales under their respective martingale measure Q T k . Henceforward interest rates are positive. Using driftless geometric Brownian motions as driving pro-cesses caplet prices are given by the Black formula (Black [1]) while swaption prices cannotbe calculated analytically. Alternatively one can start with modeling the forward bond priceprocesses P ( · , T k − ) /P ( · , T k ) instead of forward interest rate processes. Using again exponen-tial martingales like a driftless Brownian motion it is then possible to analytically calculatecaplet and swaption prices (see Eberlein and ¨Ozkan [5]). The drawback of this approach isthat forward interest rates will be negative with positive probability.Keller-Ressel et al. [14] proposed the affine LIBOR models, where forward interest ratesare nonnegative while swaption and caplet prices can still be calculated semi-analytically, i.e.up to a numerical integration. The above approaches model the individual forward interestrate processes (resp. forward bond price process) with respect to the individual measure One can extend bond price processes to [0 , T ] by setting P ( t, T k ) := P ( t,T ) P ( T k ,T ) for t > T k , so that P ( · , T k ) /P ( · , T ) is a martingale on [0 , T ] if and only if it is a martingale on [0 , T k ]. Economically thiscan be interpreted as immediately investing the payoff of a zero coupon bond into the longest-running zerocoupon bond. ffine LIBOR models with real-valued processes Q T k under which they are a martingale. Contrary Keller-Ressel et al. [14] model the priceprocesses P ( · , T k ) /P ( · , T ), which are all martingales under the same probability measure Q T . Remark:
Note that all models mentioned in this paper do not fully specify the whole termstructure, but only part of it. In order to price derivatives not contained within the specifiedtenor structure it is necessary to specify some kind of interpolation scheme. Arbitrary in-terpolations may lead to arbitrage, however one can always choose an interpolation method,such that the model stays arbitrage-free (Werpachowski [17]).
The affine LIBOR models
This section presents a summary of the affine LIBOR model introduced in Keller-Resselet al. [14]. On the filtered probability space (Ω , A , ( F t ) ≤ t ≤ T , Q T ) consider a nonnegativeanalytic affine process X with a fixed starting value x ∈ R d ≥ . For the tenor structure0 < T < · · · < T N < T N +1 =: T define for k = 1 , . . . , N and 0 ≤ t ≤ T k P ( t, T k ) P ( t, T ) := E Q T (cid:2) e u k · X T | F t (cid:3) = e φ T − t ( u k )+ ψ T − t ( u k ) · X t , u k ≥ , u k ∈ V , (5)where E Q [ · ] denotes the expectation with respect to a probability measure Q . These priceprocesses are martingales and the resulting model is arbitrage-free.Writing P ( t, T k − ) P ( t, T k ) = P ( t, T k − ) P ( t, T ) (cid:30) P ( t, T k ) P ( t, T ) (6)in (4) shows that forward interest rates being nonnegative is equivalent to normalized bondprices of (5) satisfying P ( t, T ) P ( t, T ) ≥ .. ≥ P ( t, T N ) P ( t, T ) ≥ . (7)Since for x ≥
0, e u T x is monotonically increasing in every component of u the monotonicityfor normalized bond prices in (7) is satisfied as long as u ≥ . . . u N ≥ u k in (5) should be determined, so that the starting values of normalizedbond prices P (0 , T k ) /P (0 , T ) = exp ( φ T ( u k ) + ψ T ( u k ) · x ) fit the initial term structure in-ferred from actual market data. For most affine processes every term structure can be fittedand for currently nonnegative forward interest rates this can be done using an decreasingsequence u ≥ · · · ≥ u N ≥ Remark:
Since X is nonnegative, the k-th normalized bond price is not only greater equalto one, but is bounded from below by the time-dependent constant exp ( φ T − t ( u k )) , which isstrictly greater than one. Accordingly in the affine LIBOR models forward interest rates arebounded from below by a strictly positive time-dependent constant.Affine LIBOR models lead to nonnegative forward interest rates. Additionally this specifica-tion is appealing because the density processes for changes of measures are again exponentially Since x is fixed, contrary to to section 2 any dependence of probability measures on the starting value ofthe Markov process X will be suppressed from now on. ffine LIBOR models with real-valued processes affine in X t , i.e. inserting (5) into (3) givesd Q T k d Q T = P (0 , T ) P (0 , T k ) e φ T − Tk ( u k )+ ψ T − Tk ( u k ) · X Tk . Moreover normalized bond prices and because of (6) also forward bond prices are of ex-ponential affine form. It follows that the moment generating function of the logartihm ofnormalized bond prices under Q T k is also of exponential affine form and that calculation ofcaplet prices is possible via a one-dimensional Fourier inversion. If the dimension of the driv-ing process is one, swaption prices can also be calculated via one-dimensional Fourier inversion(see Keller-Ressel et al. [14]). Hence this approach satisfies both, nonnegative interest ratesand analytical tractability of standard interest rate market instruments. If the dimension islarger than one, the exact price of swaptions can only be calculated via higher-dimensionalintegration, the dimension of which is the length of the underlying swap. Alternatively Grbacet al. [9] provide approximate formulas for swaptions. Practical application of the affine LIBOR model
Although this framework is elegant from a theoretical point of view, a practical implementa-tion faces several difficulties which shall be discussed here.First, calibration of interest rates and implied volatilities cannot be separated. The initialterm structure can be fitted using the u k , but the parameters u k also have a strong impacton implied volatilities. This can be seen by looking at the forward bond price1 P ( T k − , T k ) = exp (cid:0) φ T − t ( u k − ) − φ T − t ( u k ) + ( ψ T − t ( u k − ) − ψ T − t ( u k )) · X T k − (cid:1) , (8)which is the random variable responsible for the payoff of a caplet. The driving process X influences the distribution of this random variable through two different channels. First viathe parameters of the driving process itself and second via the parameters u k (depending on X and the initial interest rate term structure). Hence for changes in the yield curve differentparameters are required to reproduce the same implied volatility surface. If X is a L´evyprocess, then as mentioned in section 2 ψ t ( u ) = u and it follows that the distribution of (8)depends on the difference u k +1 − u k , which in turn is related to the steepness of the initial yieldcurve . Hence caplet implied volatilities are especially sensitive with regards the steepness ofthe initial yield curve.Second, interest rates and volatilities of this model depend on the final horizon T . Changingthe horizon T while using the same affine process X will lead to different results and thereis no general way of rescaling the parameters of X to negate such an effect. This is rathercounterintuitive, since extending the horizon of a model should not change the results forquantities already included with the shorter horizon.Third, the types of possible volatility surfaces is rather constrained in the fully analyticallytractable one-dimensional case. For example, we were only able to generate volatility skews . This is similar for most affine processes, but is best visible for L´evy processes. The smile example of Keller-Ressel et al. [14], figure 9.2, using an Ornstein-Uhlenbeck process seems to benumerically incorrect for strikes smaller than 0.4. With the mentioned initial yield curve the underlyinginterest rate is always larger than the strike, which corresponds to a zero implied volatility, destroying thedisplayed smile. ffine LIBOR models with real-valued processes This might be resolved by using higher-dimensional nonnegative processes. However, in multi-dimensional affine LIBOR models swaptions can no longer be calculated efficiently by Fouriermethods. On the other hand allowing arbitrary affine processes destroys the nonnegativityof forward interest rates, a central property of affine LIBOR models. We propose a modifi-cation, that preserves the nonnegativity of forward interest rates without the restriction tononnegative affine processes.
4. The modified affine LIBOR model
On the filtered probability space (Ω , A , ( F t ) ≤ t ≤ T , Q T ) consider an analytic one-dimensionalaffine process X with a fixed starting value x , i.e. the set V defined in (2) contains 0 in theinterior. For u ∈ V with − u ∈ V consider the martingales M u , M ut := E Q T [cosh( uX T ) | F t ] = 12 (cid:16) e φ T − t ( u )+ ψ T − t ( u ) X t + e φ T − t ( − u )+ ψ T − t ( − u ) X t (cid:17) . (9)By the symmetry of the cosinus hyperbolicus M u = M − u , hence one may restrict u to benonnegative. For the given tenor structure 0 < T < · · · < T N ≤ T N +1 = T and the marketsetup of section 3 define the normalized bond prices for k = 1 , . . . , N and t ≤ T k as P ( t, T k ) P ( t, T ) := M u k t , u k ∈ { v ∈ V : v ≥ , − v ∈ V} . With M u k t being a Q T -martingale the model is arbitrage-free. For every x ∈ R the function u (cid:55)→ cosh( ux ) is increasing in u ∈ R ≥ and satisfies cosh( ux ) ≥ u ≥ u ≥ · · · ≥ u N ≥ , equation (7) holds and forward interest rates F k ( t ) = 1∆ k (cid:18) M u k − t M u k t − (cid:19) , ≤ t ≤ T k − . are nonnegative for all t . To fit initial market data one has to choose the sequence ( u k ) sothat M u k = P (0 , T k ) /P (0 , T ) . The following lemma gives the condition for the affine process X under which a given initial term structure can be reproduced and shows that the u k areuniquely determined. Lemma 2: If P (0 , T ) /P (0 , T ) < sup u ∈V : − u ∈V E Q T [cosh( uX T ) | F ] , then the model can fit any term structure of nonnegative forward interest rates. Additionallythere exists a unique decreasing sequence u ≥ · · · ≥ u N , such that P (0 , T k ) /P (0 , T ) = E Q T [cosh( u k X T ) | F ] = M u k . If forward interest rates are strictly positive, the sequence is strictly decreasing.
Proof: m ( u ) = E Q T [cosh( uX T ) | F ] is a continuous function which is strictly increasing for u ≥
0. By the assumption of the theorem there exists u > m ( u ) > P (0 , T ) /P (0 , T ).6 ffine LIBOR models with real-valued processes Furthermore m (0) = 1, which proves the lemma. Remark:
Generalizing this approach to a d -dimensional driving process is possible by setting M ut = E (cid:34) d (cid:89) l =1 cosh (cid:16) u ( l ) X ( l ) T (cid:17) (cid:12)(cid:12)(cid:12) F t (cid:35) , u = ( u (1) , . . . , u ( d ) ) ≥ . In this case it is guaranteed that M ut ≥ M wt for u ≥ w , which guarantees the nonnegativityof forward interest rates. However, the option pricing formulas in the following sections donot generalize.As in the affine LIBOR model for a monotonically decreasing sequence ( u k ) forward interestrates are not only nonnegative, but bounded below by strictly positive time-dependent con-stants (the bounds can be calculated numerically). This is not a big issue if these bounds areclose to zero, but has to be checked during the calibration process.In the modified affine LIBOR model the change of measure to the T k -forward measure Q T k is given by d Q T k d Q T = P (0 , T ) P (0 , T k ) M u k T k = M u k T k M u k . (10)Here M u k t is a sum of exponentials of X t , while in the affine LIBOR model the correspondingterm is a single exponential. This means that contrary to the affine LIBOR model the process X is not an inhomogeneous affine process under Q T k and it is not possible to calculate themoment generating function of the logarithm of foward bond prices under Q T k . Neverthelessit is possible to get analytical formulas for the prices of caplets and swaptions. The derivation of the pricing formulas for caplets and swaptions is based on a method firstapplied in Jamshidian [10]. First caplets are dealt with, swaptions follow afterwards . Notethat if u k = u k − the corresponding forward interest rate F k always stays zero. To excludesuch pathological examples assume that the sequence ( u k ) is strictly decreasing. In thissection random variables are often viewed as functions of the value of the driving process X .Specifically consider the functions M ut : R → R , x (cid:55)→ M ut ( x ) := 12 (cid:16) e φ T − t ( u )+ ψ T − t ( u ) x + e φ T − t ( − u )+ ψ T − t ( − u ) x (cid:17) . (11)The time t value of martingale M u in (9) is then M ut = M ut ( X t ). In the rest of the paper M ut will denote both, the function and the value of the stochastic processes, where the correctinterpretation should be clear from context.The payoff of a caplet for the ( k + 1) th forward rate F k +1 ( T k ) with strike K is∆ k +1 (cid:16) F k +1 ( T k ) − K (cid:17) + = (cid:18) P ( T k , T k +1 ) − ˜ K (cid:19) + = (cid:32) M u k T k M u k +1 T k − ˜ K (cid:33) + , Actually caplet prices coincide with prices of swaptions with only one underlying period. The differencebetween those two derivatives is the payoff time. ffine LIBOR models with real-valued processes where ˜ K = 1 + ∆ k +1 K . Since this payoff has to be paid at time T k +1 the price of the capletand the corresponding floorlet isCpl( t, T k , T k +1 , K ) = P ( t, T k +1 ) E Q Tk +1 (cid:34)(cid:32) M u k T k M u k +1 T k − ˜ K (cid:33) + (cid:12)(cid:12)(cid:12) F t (cid:35) , Flt( t, T k , T k +1 , K ) = P ( t, T k +1 ) E Q Tk +1 (cid:34)(cid:32) ˜ K − M u k T k M u k +1 T k (cid:33) + (cid:12)(cid:12)(cid:12) F t (cid:35) . Since price processes are martingales, the put/call parity holds and prices of caplets followfrom floorlets and vice versa. Because Fourier analysis is easier for floorlets, where the payoffis bounded, formulas are derived for floorlets.Since the moment generating function of ln( M u k T k /M u k +1 T k ) is unknown, Fourier methods arenot directly applicable. However, the function x (cid:55)→ M u k T k ( x ) /M u k +1 T k ( x ) has a unique minimumand is monotonically increasing moving away from this minimum. Using this one can getrid of the positive part and use Fourier inversion to calculate the above expectations. Theabove mentioned monotonicity is very fortunate and follows from a close interplay betweenthe monotonicity of the sequence ( u k ) and the function ψ with properties of the cosinushyperbolicus. Details are laid out in the proof of the following lemma, which can be found inthe appendix. Lemma 3:
For i = 1 , . . . , n let u ≥ u i ≥ , where for at least one i u > u i . Let c i > bepositive constants. Define a function g : R → R by g ( x ) := n (cid:88) i =1 c i M u i t ( x ) M u t ( x ) . (12) Then g has a unique maximum at some point ξ ∈ R and and is strictly monotonically de-creasing to on the left and right side of ξ . For floorlet valuation this lemma is not directly applicable as u k > u k +1 , which is the wronginequality. However, there is only one summand and the lemma can be applied to the inverse M u k +1 T k ( x ) /M u k T k ( x ). It follows that M u k T k ( x ) /M u k +1 T k ( x ) has a unique minimum at some point ξ and is increasing to infinity to the left and right. Hence it is possible to write (cid:32) ˜ K − M u k T k ( x ) M u k +1 T k ( x ) (cid:33) + = (cid:32) ˜ K − M u k T k ( x ) M u k +1 T k ( x ) (cid:33) I { κ < x < κ } , (13)where κ and κ are two uniquely determined constants satisfying κ ≤ ξ ≤ κ . If κ = ξ = κ the payoff is zero, which corresponds to M u k T k /M u k +1 T k > ˜ K . This happens if the forward interestrate is bounded from below by K, which only happens for very low strikes K . Inserting (13)into the price of a floorlet it follows by a change of measure thatFlt( t, T k , T k +1 , K ) = P ( t, T k +1 ) E Q Tk +1 (cid:34)(cid:32) ˜ K − M u k T k M u k +1 T k (cid:33) I { κ < X T k < κ } (cid:12)(cid:12)(cid:12) F t (cid:35) = P ( t, T ) E Q T (cid:104)(cid:16) ˜ KM u k +1 T k − M u k T k (cid:17) I { κ < X T k < κ } (cid:12)(cid:12)(cid:12) F t (cid:105) . (14)8 ffine LIBOR models with real-valued processes ˜ KM u k +1 T k − M u k T k is the sum of exponentials of the random variable X T k . The expectation in(14) is calculated under the measure Q T , where the conditional moment generating function M X t | X s ( z ) := E Q T (cid:2) e zX t | F s (cid:3) = E Q T (cid:2) e zX t | X s (cid:3) = exp ( φ t − s ( z ) + ψ t − s ( z ) X s )is known for z ∈ V . Hence the expectation in (14) can be calculated via Fourier inversion.The Fourier inversion formula for terms of the above form is stated in Lemma 4, the proof ofwhich is given in the appendix. Lemma 4:
Assume that the function f : R → R has the representation f ( x ) = (cid:88) k C k e v k x I { κ < x < κ } , lim x ↓ κ f ( x ) = lim x ↑ κ f ( x ) = 0 , where the summation is over a finite index set and the C k and v k are real constants. Thenfor R ∈ V ∩ R the Fourier inversion formula E [ f ( X t ) | F s ] = 1 π (cid:90) ∞ Re (cid:16) M X t | X s ( i u + R ) ˆ f ( u − i R ) (cid:17) d u holds, where ˆ f is the analytic Fouier transform given by ˆ f ( z ) = 1 i z (cid:88) k C k v k v k − i z (cid:16) e ( v k − i z ) κ − e ( v k − i z ) κ (cid:17) , z (cid:54) = 0 , z (cid:54) = − i v k . (15)To calculate the price of a floorlet in (14) apply Lemma 4 to f Kk +1 ( X T k ) with f Kk +1 ( x ) := (cid:16) ˜ KM u k +1 T k ( x ) − M u k T k ( x ) (cid:17) I { κ < x < κ } . (16)Its Fourier transform isˆ f Kk +1 ( z ) = 1 i z (cid:0) (1 + ∆ k +1 K ) h T k κ ,κ ( − i z, u k +1 ) − h T k κ ,κ ( − i z, u k ) (cid:1) (17)with h tκ ,κ ( z, u ) := e φ T − t ( u ) ψ T − t ( u )2( z + ψ T − t ( u )) (cid:16) e ( z + ψ T − t ( u )) κ − e ( z + ψ T − t ( u )) κ (cid:17) + e φ T − t ( − u ) ψ T − t ( − u )2( z + ψ T − t ( − u )) (cid:16) e ( z + ψ T − t ( − u )) κ − e ( z + ψ T − t ( − u )) κ (cid:17) . (18)The case of swaptions is similar. Consider a swap which is part of the tenor structure. Thatis, consider 1 ≤ α < β ≤ N and the according interest rate swap with forward swap rate S α,β ( t ) = P ( t, T α ) − P ( t, T β ) (cid:80) βk = α +1 ∆ k P ( t, T k ) , ∆ k = T k − T k − . ffine LIBOR models with real-valued processes The payoff of a put swaption on the above swap with strike K is then β (cid:88) k = α +1 P ( T α , T k )∆ k ( K − S α,β ( T α )) + = (cid:32) P ( T α , T β ) + K β (cid:88) k = α +1 ∆ k P ( T α , T k ) − (cid:33) + = (cid:32) M u β T α M u α T α + β (cid:88) k = α +1 K ∆ k M u k T α M u α T α − (cid:33) + . Since the function M u β T α ( x ) /M u α T α ( x ) + (cid:80) βk = α +1 K ∆ k M u k T α ( x ) /M u α T α ( x ) is of the form of Lemma3, it has a unique maximum ξ and one can find constants κ ≤ ξ ≤ κ such that after achange of measure the value of a put swaption isPutSwaption( t, T α , T β , K ) = P ( t, T ) E Q T (cid:104) f Kα,β ( X T α ) (cid:12)(cid:12)(cid:12) F t (cid:105) , where f Kα,β ( x ) = (cid:32) M u β T α ( x ) − M u α T α ( x ) + β (cid:88) k = α +1 K ∆ k M u k T α ( x ) (cid:33) I { κ < x < κ } . (19)Again this of the form in Lemma 4 and in this caseˆ f Kα,β ( z ) = 1 i z (cid:16) h T α κ ,κ ( − i z, u β ) − h T α κ ,κ ( − i z, u α ) + K β (cid:88) k = α +1 ∆ k h T α κ ,κ ( − i z, u k ) (cid:17) , (20)where h tκ ,κ ( z, u ) is defined in (18). The pricing formulas are summarized in the followingtheorem. Theorem 5:
Let R ∈ V ∩ R . In the modified affine LIBOR model prices of a forward interestrate put and a put swaption are Flt( t, T k , T k +1 , K ) = P ( t, T ) π (cid:90) ∞ Re (cid:16) M X Tk | X t ( R + i u ) ˆ f Kk +1 ( u − i R ) (cid:17) d u, (21)PutSwaption( t, T α , T β , K ) = P ( t, T ) π (cid:90) ∞ Re (cid:16) M X Tα | X t ( R + i u ) ˆ f Kα,β ( u − i R ) (cid:17) d u, (22) The Fourier transforms ˆ f Kk +1 repectively ˆ f Kα,β are given in (17) respectively (20) for
R / ∈{ , u k , u k +1 } respectively R / ∈ { , u α , . . . , u β } . As in the floorlet case if κ = κ then the forward swap rate is always larger than the strike. Note that S α,β ( t ) can also be written as S α,β ( t ) = (cid:80) βk = α +1 w k ( t ) F k ( t ) with w k > ffine LIBOR models with real-valued processes In order to calculate ˆ f Ki respectively ˆ f Kα,β one has to find the roots κ , κ of the functions g Kk ( x ) := ˜ K − M u k T k ( x ) M u k +1 T k ( x ) , (23) g Kα,β ( x ) := M u β T α ( x ) M u α T α ( x ) + β (cid:88) k = α +1 K ∆ k M u k T α ( x ) M u α T α ( x ) − . (24)By Lemma 3 this amounts to finding the roots of a function which has a single optimumand is monotonic when moving away from this optimum. Numerical determination of theroots of such well-behaved one-dimensional functions poses no problem. Having determinedthose bounds valuation reduces to a one-dimensional integration of a function that is fallingat least like 1 /x (depending on the moment generating function of the affine process), soalso numerical integration is feasible. Note that besides caps, floors and swaptions, optionslike digital options or Asset-or-Nothing options can be calculated in a similar manner. The first part of this section looks at the benchmark case of a Brownian motion, whereeverything can also be calculated in closed form. Afterwards Ornstein-Uhlenbeck processesare discussed. The section concludes with examples of possible volatility surfaces.
Brownian motion
Choose X t = B t , a standard Brownian motion starting in 0. The conditional moment gener-ating function is M B T | B t ( u ) = E (cid:2) e uB T | F t (cid:3) = exp (cid:18) uB t + u T − t ) (cid:19) . Hence this is an affine process with φ t ( u ) = u t and ψ t ( u ) = u . Consider the time 0 priceof a floorlet as given in (14) with t = 0. Since M ut ( − x ) = M ut ( x ) one finds that in this case κ = κ and κ = − κ , where κ is the unique positive root of (23) if g Kk +1 (0) < κ = 0otherwise. By (14) the floorlet price Flt(0 , T k , T k +1 , K ) is P (0 , T ) E Q T (cid:20)(cid:18) ˜ K e u k +12 ( T − T k ) cosh( u k +1 B T k ) − e u k ( T − T k ) cosh( u k B T k ) (cid:19) I {| B T k | ≤ κ } (cid:21) . By the symmetry of a Brownian motion starting in 0 E [cosh( zB t ) I {| B t | ≤ κ } ] = E (cid:2) e zB t I {| B t | ≤ κ } (cid:3) = E (cid:2) e − zB t I {| B t | ≤ κ } (cid:3) = e tz (cid:18) Φ (cid:16) κ √ t − z √ t (cid:17) − Φ (cid:16) − κ √ t − z √ t (cid:17)(cid:19) , ffine LIBOR models with real-valued processes where Φ denotes the cumulative distribution function of a standard normal distributed randomvariable. HenceFlt(0 , T k , T k +1 , K ) = ˜ KP (0 , T )e u k +1 T (cid:18) Φ (cid:16) κ √ T k − u k +1 (cid:112) T k (cid:17) − Φ (cid:16) − κ √ T k − u k +1 (cid:112) T k (cid:17)(cid:19) − P (0 , T )e u k T (cid:18) Φ (cid:16) κ √ T k − u k (cid:112) T k (cid:17) − Φ (cid:16) − κ √ T k − u k (cid:112) T k (cid:17)(cid:19) . Slightly more complicated formulas exist when B is replaced with a Brownian motion withconstant drift and volatility and a starting value different from 0.Swaptions can be treated the same way. Let κ be the unique positive root of (24) if g Kα,β (0) > κ = 0 otherwise. ThenPutSwaption(0 , T α , T β , K ) = P (0 , T )e u β T (cid:18) Φ (cid:16) κ √ T α − u β (cid:112) T α (cid:17) − Φ (cid:16) − κ √ T α − u β (cid:112) T α (cid:17)(cid:19) − P (0 , T )e u α T (cid:18) Φ (cid:16) κ √ T α − u α (cid:112) T α (cid:17) − Φ (cid:16) − κ √ T α − u α (cid:112) T α (cid:17)(cid:19) + P (0 , T ) β (cid:88) k = α +1 K ∆ k e u k T (cid:18) Φ (cid:16) κ √ T α − u k (cid:112) T α (cid:17) − Φ (cid:16) − κ √ T α − u k (cid:112) T α (cid:17)(cid:19) . Ornstein-Uhlenbeck (OU) processes
The OU process X generated by a L´evy process L is defined as the unique strong solution of(see Sato [16], section 17) d X t = − λX t d t + d L t , X = x . (25)Then Y t := e λt X t = x + (cid:82) t e λs L s d s. Using the key formula of Eberlein and Raible [6] itfollows that E (cid:2) e uX t (cid:3) = E (cid:104) exp (cid:16) e − λt uY t (cid:17)(cid:105) = exp (cid:18) e − λt ux + (cid:90) t κ (e − λs u ) d s (cid:19) , where κ ( u ) = ln ( E [ L ]) is the cumulant generating function of the L´evy process L . Hencethis process is affine with ψ t ( u ) = e − λt u and φ t ( u ) = (cid:90) t κ (e − λs u ) d s = 1 λ (cid:90) − λt κ ( vu ) v d v. (26)By Corollary 2.10 in Duffie et al. [4] every affine process with state space R is in fact an OUprocess. Hence in the context of affine processes defined on the real line OU processes are theright class to consider. For application it should be possible to calculate the integral in (26)analytically. Two examples where this is possible are presented below. Remark: If L is a martingale, the process in (25) is mean-reverting to zero, however shiftingthe mean to θ is easily done by using Z t = θ + X t . Then d Z t = λ ( θ − Z t ) d t + d L t and E (cid:2) e uZ t (cid:3) = exp (cid:16) ( φ t ( u ) + θu (1 − e − λt )) + ψ t ( u ) Z (cid:17) . ffine LIBOR models with real-valued processes Hence Z is again affine with ψ θt ( u ) = ψ t ( u ) and φ θt ( u ) = φ t ( u ) + θu (1 − e − λt ). Note that thisprocess is then generated by the L´evy process ˜ L t = L t + θλt , i.e. the original L´evy processplus an additional drift of θλ .The first example is the classical OU process generated by a Brownian motion σB , where κ ( u ) = σ u . This process is described byd X t = − λX t d t + σ d B t , X = x. The integral in (26) is φ t ( u ) = 1 λ (cid:90) − λt κ ( vu ) v d v = σ u λ (1 − e − λt ) . (27)With Brownian motion describing the continuous part of L´evy processes., for the secondexample we consider a pure jump process, namely a Double Γ-OU process. Γ-OU processesare generated by a compound Poisson process with jump intensity λβ ( λ being the same as in(25)) and exponentially distributed jumps with expectation value α . The limit distributionof this process is a Γ-distribution, which gives the process its name. As the generatingcompound Poisson process is strictly increasing, the generated Γ-OU process is a subordinatorand stays above 0. In order to find an OU process with values in R consider the difference oftwo independent compound Γ-OU processes L + , L − with parameters α + , β + , α − , β − and set λ + = λβ + , λ − = λβ − . Then L = L + − L − is a compound Poisson process, where positivejumps with expected jump size α + are arriving at rate λ + , while negative jumps with expectedjump size α − are arriving at rate λ − .The cumulant generating function of a compound Poisson process with exponential jumpsis λβuα − u , which is defined for u < α . Hence for u ∈ ( − α − , α + ) the moment generating functionof the combined process L is E (cid:2) e uL (cid:3) = E (cid:104) e uL +1 (cid:105) E (cid:104) e − uL − (cid:105) = exp (cid:18) λ ( β + + β − ) u + ( β + α − − β − α + ) u ( α + − u )( α − + u ) (cid:19) . Inserting this into (26) straightforward calculations show that the function φ for the resultingOU process is given by φ t ( u ) = β + + β − (cid:18) ( α + − e − λt u )( α − + e − λt u )( α + − u )( α − + u ) (cid:19) + β + − β − (cid:18) ( α + − e − λt u )( α − + u )( α + − u )( α − + e − λt u ) (cid:19) . (28)It is also possible to combine the two approaches by considering an OU process generatedby a L´evy process which is the difference of two compound Poisson processes plus a Brownianmotion, all of which are independent. The resulting φ then follows by adding up the twofunctions (27) and (28) and for this process V = { u ∈ C : − α − < Re( u ) < α + } . By theprevious remark it is also possible to shift this process by θ . Such OU processes are used inthe following numerical examples. 13 ffine LIBOR models with real-valued processes M a t u r i t y Figure 1: Implied volatility skew of caplets generated by an OU process withparameters λ = 0 . , α + = 12 , α − = 10 , β + = 50 , β − = 5 , σ = 0 . , θ =0 . , x = 0 . T = 10. Volatility surfaces
With the OU process of the previous section it is possible to generate volatility smiles as wellas volatility skews. For illustration we consider a term structure with constant interest ratesof 3 . .
02 to 0 .
07. Figure 1 shows a skewed volatility surface whilefigure 2 shows a very pronounced smile, both of which are generated by an OU process of thejust introduced type. As mentioned in the previous chapters forward interest rates in thistype of model will be bounded from below. The bounds in these examples are at 1% for theforward interest rate expiring after half a year and decrease to basically 0% for the forwardinterest rate which expires in 5 years. Hence they are well within reasonable boundaries. Forcompleteness an example of at-the-money implied volatilities for swaptions with maturitiesand underlying swap rates ranging from 2 to 7 years is displayed in figure 3. Conclusion
Classical interest rate market models are not capable of simultaneously allowing for semi-analytical pricing formulas for caplets and swaptions and guaranteeing nonnegative forwardinterest rates. One exception are the affine LIBOR models presented in Keller-Ressel et al.[14]. This paper modifies their approach to also allow for driving processes which are notnecessarily nonnegative. Caplet and swaption valuation is possible via one-dimensional nu-merical integration. This allows for a fast calculation of implied volatilities for these typesof interest rate derivatives. With the additional flexibility of real-valued affine processes thistype of model is capable of producing skewed implied volatility surfaces as well as impliedvolatility surfaces with pronounced smiles. 14 ffine LIBOR models with real-valued processes M a t u r i t y Figure 2: Implied volatility smile of caplets generated by an OU processwith parameters λ = 0 . , α + = 50 , α − = 5 , β + = 50 , β − = 10 , σ = 0 , θ =0 , x = 1 and T = 10. S w ap t i on E x p i r y S w ap Leng t h Figure 3: Swaption implied volatilites generated by an OU process withparameters λ = 0 . , α + = 12 , α − = 10 , β + = 50 , β − = 5 , σ = 0 . , θ =0 . , x = 0 . T = 10. 15 ffine LIBOR models with real-valued processes A. Proofs
Proof of Lemma 3:
For a function f ( x ) denote its even and odd part by f e ( x ) = 12 ( f ( x ) + f ( − x )) , f o ( x ) = 12 ( f ( x ) − f ( − x )) . Note that if f is monotonically increasing, the same is true for f o . Then (11) can be writtenas M ut ( x ) = 12 (cid:16) e φ T − t ( u )+ ψ T − t ( u ) x + e φ T − t ( − u )+ ψ T − t ( − u ) x (cid:17) = e φ eT − t ( u )+ ψ eT − t ( u ) x cosh( φ oT − t ( u ) + ψ oT − t ( u ) x )and M u i t ( x ) M u t ( x ) = e ( φ eT − t ( u i ) − φ eT − t ( u ))+( ψ eT − t ( u i ) − ψ eT − t ( u )) x cosh( φ oT − t ( u i ) + ψ oT − t ( u i ) x )cosh( φ oT − t ( u ) + ψ oT − t ( u ) x ) . (29)If u i = u , then (29) is constant and has no influence regarding monotonicity or maxima.Hence from now on assume u > u i for all i . The function g of equation (12) can be writtenas g ( x ) = n (cid:88) i =1 c i e A i e a i x cosh( B i + b i x )cosh( B + b x ) , where for i = 0 , . . . , nA i = ( φ eT − t ( u i ) − φ eT − t ( u )) , B i = φ oT − t ( u i ) ,a i = ( ψ eT − t ( u i ) − ψ eT − t ( u )) , b i = ψ oT − t ( u i ) . Since ψ is monotonically increasing (see Lemma 1), also ψ o is monotonically increasing. With ψ o (0) = 0 it follows that b i ≥ i . Furthermore note that a i < b − b i is equivalentto ψ T − t ( u i ) < ψ T − t ( u ) and − a i < b − b i is equivalent to ψ T − t ( − u ) < ψ T − t ( − u i ). Since u > u i ≥ ψ yields | a i | < b − b i . (30)An elementary calculation gives g (cid:48) ( x ) = 1cosh( B + b x ) n (cid:88) i =1 c i e A i e a i x f i ( x ) , where f i ( x ) = a i cosh( B i + b i x ) cosh( B + b x ) + b i sinh( B i + b i x ) cosh( B + b x ) − b cosh( B i + b i x ) sinh( B + b x ) . ffine LIBOR models with real-valued processes The derivative of f i is f (cid:48) i ( x ) = b i cosh( B + b x ) (cid:16) a i sinh( B i + b i x ) + ( b i − b ) cosh( B i + b i x ) (cid:17) + b cosh( B i + b i x ) (cid:16) a i sinh( B + b x ) + ( b i − b ) cosh( B + b x ) (cid:17) . (31)Using (30) the terms inside the brackets of each row in (31) are strictly less than | a i | ( sgn ( a i ) sinh( B j + b j x ) − cosh( B j + b j x )) ≤ , ( j = i, . The last inequality is true since cosh( x ) ± sinh( x ) ≥
0. The terms outside of the brackets in(31) are all positive. Hence f (cid:48) ≥ f i are monotonically decreasing. Using (30) asimple calculation shows that lim x →−∞ f i ( x ) = ∞ and lim x →∞ f i ( x ) = −∞ . Since c i > i the same is true for (cid:80) ni =1 c i e a i x f i ( x ). Hence g has a single maximum and is decreasingto the left and right of it. Furthermore g ( x ) ≥ x →∞ g ( x ) =lim x →−∞ g ( x ) = 0. Proof of Lemma 4: f is continuous with compact support. Hence the extended Fouriertransform ˆ f ( z ) = (cid:82) R f ( x )e − i zx d x exists for all z ∈ C and is analytic. For z (cid:54) = 0 , z (cid:54) = − i v k , itis given by ˆ f ( z ) = (cid:90) κ κ e − i zx f ( x ) d x = 1 i z (cid:90) κ κ e − i zx f (cid:48) ( x ) d x = 1 i z (cid:88) k C k v k v k − i z (cid:16) e ( v k − i z ) κ − e ( v k − i z ) κ (cid:17) . Since ˆ f ( u − i R ) = O ( u − ) for fixed R , it is absolutely integrable. By Fourier inversion f ( x ) = 12 π (cid:90) Im( z )= − R e i zx ˆ f ( z ) d z = 12 π (cid:90) ∞ Re (cid:16) e ( i u + R ) x ˆ f ( u − i R ) (cid:17) d u, where the last equation follows from the fact that f is real valued and the symmetry ˆ f ( z ) =ˆ f ( − z ). Since (cid:82) E (cid:2) | e ( i z + R ) X t || F s (cid:3) | ˆ f ( z ) | d z = M X t | X s ( R ) (cid:82) | ˆ f ( z ) | d z is bounded if R ∈ V ∩ R ,conditional expectation and integration can be interchanged. References [1] Fischer Black. The pricing of commodity contracts.
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