Explosion in the quasi-Gaussian HJM model
EEXPLOSION IN THE QUASI-GAUSSIAN HJM MODEL
DAN PIRJOL AND LINGJIONG ZHU
Abstract.
We study the explosion of the solutions of the SDE in the quasi-Gaussian HJM modelwith a CEV-type volatility. The quasi-Gaussian HJM models are a popular approach for modelingthe dynamics of the yield curve. This is due to their low dimensional Markovian representationwhich simplifies their numerical implementation and simulation. We show rigorously that the shortrate in these models explodes in finite time with positive probability, under certain assumptions forthe model parameters, and that the explosion occurs in finite time with probability one under somestronger assumptions. We discuss the implications of these results for the pricing of the zero couponbonds and Eurodollar futures under this model. Introduction
The quasi-Gaussian Heath-Jarrow-Morton (HJM) models [3, 4, 5, 8, 31] are frequently used infinancial practice for modeling the dynamics of the yield curve [3]. They were introduced as asimpler alternative to the HJM model [16], which describe the dynamics of the yield curve f ( t, T )as the stochastic differential equation(1) df ( t, T ) = σ f ( t, T ) dW ( t ) + σ f ( t, T ) (cid:18)(cid:90) Tt σ f ( t, s ) ds (cid:19) dt , where W ( t ) is a vector Brownian motion under the risk-neutral measure Q , and ( σ f ( t, T )) t ≤ T is afamily of vector processes. The dynamical variable in the HJM models is the forward rate f ( t, T )for maturity T .The quasi-Gaussian HJM models assume a separable form for the volatility function σ f ( t, T ) = g ( T ) h t where g is a deterministic vector function and ( h t ) is a k × k matrix stochastic process. Suchmodels admit a Markov representation of the dynamics of the yield curve in terms of k + k ( k + 1)state variables. This simplification aids considerably with the simulation of these models, whichcan be performed using Monte Carlo or finite difference methods [6, 11].We consider in this paper the one-factor quasi-Gaussian HJM model with volatility specification σ f ( t, T ) = k ( t, T ) σ r ( r t ) where k ( t, T ) = e − β ( T − t ) , and σ ( r t ) is the volatility of the short rate r t = f ( t, t ). This model admits a two-state Markov representation.It has been noted in [26, 16] that in HJM models with log-normal volatility specification, that is forwhich σ f ( t, T ) = σf ( t, T ), the rates explode to infinity with probability one, and zero coupon bondprices approach zero. See also [34] for a general study of the conditions for the existence of strongsolutions to stochastic differential equations (SDEs) of HJM type. A similar explosion appears in atwo-dimensional model studied in [17]. It is natural to ask if such explosions are present also in thequasi-Gaussian HJM models. Models of this type with parametric volatility are used in financial Key words and phrases.
HJM model, stochastic modeling, multidimensional diffusions, explosion. a r X i v : . [ q -f i n . M F ] A ug DAN PIRJOL AND LINGJIONG ZHU practice for modeling the swaption volatility skew [9, 10, 6]. Non-parametric forms have been alsoconsidered recently in the literature [15, 7].We recall that singular behavior is also observed for certain derivatives prices in short rate log-normal interest rates models [2, 3]. It was observed by Hogan and Weintraub [18] that Eurodollarfutures prices are infinite in the Dothan and Black-Karasinski models. A milder singularity is alsopresent in finite tenor log-normal models, such as the Black-Derman-Toy model, manifested as arapid increase of the Eurodollar futures convexity adjustment as the volatility increases above athreshold value [29]. This singularity can be avoided by formulating the models by specifying thedistributional properties of rates with finite tenor [32]. This line of argument led to the formulationof the LIBOR Market Models which are free of singularities [3].In a recent work [30], we studied the small-noise limit of the log-normal quasi-Gaussian model, usinga deterministic approximation, and showed rigorously that the short rate may explode to infinityin a finite time. More precisely, it was shown in [30] that for sufficiently small mean-reversionparameter β , the small-noise approximation for the short rate r t has an explosion in finite time, andan upper bound is given on the explosion time, which is saturated in the flat forward rate limit.In this paper, we extend these results in two directions:(i) We consider a wider class of quasi-Gaussian HJM models with a constant elasticity of variance(CEV)-type volatility specification. This includes the log-normal model as a special case. We alsoconsider the case of the displaced log-normal model. These volatility specifications are widely usedby practitioners [9, 3].(ii) The Brownian noise is taken into account. This requires the study of the explosion of the solu-tions of a two-dimensional stochastic differential equation. Mathematically, it is well known that forone-dimensional diffusion processes there is the celebrated Feller criterion [14, 25] for explosion/non-explosion, see e.g. [22, 28] for overviews. This is a sufficient and necessary condition under whichthere is an explosion in finite time. We note that the distribution of the explosion time has beenalso studied recently [23].For d -dimensional stochastic differential equations with d >
1, to the best of our knowledge, there isno sufficient and necessary condition for explosion. Several sufficient conditions for explosions havebeen presented in the literature for multi-dimensional diffusions [33]. The Khasminskii criterion forexplosion is presented in [25, 28]. The method of the Lyapunov function was presented in [12, 24].This was extended to a non-linear Lyapunov method by [35]. The application of these conditionsis non-trivial, and checking that the conditions required hold is sometimes very challenging.We rely on the sufficient conditions for explosion with positive probability and explosion withprobability one given in [12, 24]. The main tool is the construction of some delicate Lyapunovfunctions that satisfy certain non-trivial conditions [12, 24].We show rigorously that under certain conditions, in the CEV-type model with exponent in acertain range ( , UASI-GAUSSIAN HJM MODEL 3 to that occurring in the log-normal HJM model [16], and an explosion of interest rate derivativeslinked to the LIBOR rate, in particular the Eurodollar futures prices. This introduces a limitationin the applicability of the model for pricing these products to maturities smaller than the explosiontime.The paper is organized as follows. In Section 2, we introduce the model, and discuss its use inthe literature. In Section 3, we present rigorous results giving sufficient conditions for explosionin finite time with positive probability in the quasi-Gaussian HJM model with CEV-like volatilityspecification. Furthermore, under stronger assumptions, we can show that the explosion occurs insome finite time with probability one. In Section 4, we discussion the implications of our results tothe pricing of the zero coupon bond and the Eurodollar futures. Finally, the proofs are collected inthe Appendix. 2.
One factor quasi-Gaussian HJM model
We will consider in this paper a class of one-factor quasi-Gaussian HJM models, defined by thevolatility specification(2) σ f ( t, T ) = σ r ( r t ) e − β ( T − t ) . Several parametric choices for the short rate volatility function σ r ( x ) have been considered in theliterature, including:(i) Log-normal model [6]: σ r ( x ) = σx ;(ii) Displaced log-normal model, also known in the literature as the linear Cheyette model [9, 3]: σ r ( x ) = σ ( x + a ) ;(iii) CEV-type model [6]: σ r ( x ) = σx γ , where γ ∈ (0 ,
1] .The simulation of the model with the volatility specification (2) can be reduced to simulating thestochastic differential equation for the two variables { x t , y t } t ≥ [31, 3] dx t = ( y t − βx t ) dt + σ r ( λ ( t ) + x t ) dW t , (3) dy t = ( σ r ( λ ( t ) + x t ) − βy t ) dt, with initial condition x = y = 0. Here λ ( t ) = f (0 , t ) is the forward short rate, giving the initialyield curve. The price of the zero coupon bond with maturity T is(4) P ( t, T ) = P (0 , T ) P (0 , t ) exp (cid:18) − G ( t, T ) x t − G ( t, T ) y t (cid:19) , with G ( t, T ) = β (1 − e − β ( T − t ) ) a non-negative deterministic function [3]. The short rate is r t := f ( t, t ) = λ ( t ) + x t .Under the CEV-type volatility σ r ( x ) = σx γ , the equations (3) can be expressed in terms of theshort rate as dr t = σr γt dW t + ( y t − βr t + βλ ( t ) + λ (cid:48) ( t )) dt , (5) dy t = ( σ r γt − βy t ) dt , with the initial condition r = λ := λ (0) > y = 0.One potential complication with the usual CEV volatility specification σ r ( x ) = σx γ is related tothe non-uniqueness of the solution of the SDE (5) for 0 < γ < DAN PIRJOL AND LINGJIONG ZHU model [13], given by the SDE with 0 < γ ≤ dx t = σx γt dW t + µx t dt , the origin x = 0 is a regular boundary for 0 < γ < , and an exit boundary for ≤ γ <
1. For thegeometric Brownian motion case γ = 1, the point zero is a natural boundary. For 0 < γ < , thesolution of the SDE is not unique, and an additional boundary condition must be imposed at x = 0in order to ensure uniqueness.In order to avoid singular behavior near the origin r = 0, practitioners use various modifications ofthe quasi-Gaussian model with CEV volatility specification near the r = 0 point, see e.g. Section4.3 of [19]. This work describes three possible modifications: (a) σ r ( x ) → σ | x | γ ; (b) σ r ( x ) → | x | ≤ ε ; (c) σ r ( x ) → σε γ for | x | ≤ ε with ε > σ r ( x ) = σx min( x γ − , ε γ − ) , with ε > < γ ≤
1. We call this the ε − CEV quasi-Gaussian HJM model.Note that as ε = 0, this reduces to the usual CEV volatility specification σ r ( x ) = σx γ , see e.g. [6].The modification ε > < r t < ε , where the process is identicalwith the log-normal model with the volatility σε γ − , and leaves unchanged the behavior of theprocess for large r t , which is relevant for the study of the explosions of r t . The modification is onlyrequired for 0 < γ <
1. When γ = 1, the equation (7) reduces to σ r ( x ) = σx , which coincides withthe log-normal model.With the volatility specification (7), we will study the 2-dimensional SDE with ε > dr t = ( y t − βr t + βλ ( t ) + λ (cid:48) ( t )) dt + σr t min( r γ − t , ε γ − ) dW t , (8) dy t = ( σ r t min( r γ − t , ε γ − ) − βy t ) dt, (9)with the initial condition r = λ (0) > ε and y = 0.In the special case γ = 1, (8),(9) reduces to the log-normal model dr t = σr t dW t + ( y t − βr t + βλ ( t ) + λ (cid:48) ( t )) dt , (10) dy t = ( σ r t − βy t ) dt , (11)with the initial condition r = λ := λ (0) > y = 0.Assume that λ (cid:48) ( t ) + βλ ( t ) ≥ r >
0. Then the solutions of (10), (11) are positive withprobability one(12) P ( r t >
0) = 1 , for all t ≥ . The result follows by noting that(13) y t = σ (cid:90) t r s e − β ( t − s ) ds > , almost surely for every t >
0, and then follows by an application of the comparison theorem(Theorem 1.1 in [36] and Theorem 5.2.18 in [22]). See also the Appendix D in [27] for a proof ofthis result.
UASI-GAUSSIAN HJM MODEL 5
This implies that the origin r = 0 is a natural boundary for this diffusion. For the time-homogeneouscase λ ( t ) = λ we use a similar argument to prove the same result for the ε − CEV model with general γ ∈ ( , σ r ( x ) = σ ( x + a ) reduces to that for the log-normalcase by the substitutions r t + a → r t , λ ( t ) + a → λ ( t ). Expressed in terms of r t , y t , this is dr t = σ ( r t + a ) dW t + ( y t − βr t + βλ ( t ) + λ (cid:48) ( t )) dt , (14) dy t = ( σ ( r t + a ) − βy t ) dt , (15)with initial conditions r = λ (0) , y = 0. Defining ˜ r t = r t + a the shifted short rate, we have d ˜ r t = σ ˜ r t dW t + ( y t − β ˜ r t + β ( λ ( t ) + a ) + λ (cid:48) ( t )) dt , (16) dy t = ( σ ˜ r t − βy t ) dt , (17)started at ˜ r t = λ (0) + a, y = 0. Redefining λ ( t ) + a → λ ( t ), the shift parameter a disappears, andthe resulting SDE is identical to that for the log-normal γ = 1 model.Under this model negative values for r t can also be accommodated, with a floor on the short rate r t > − a . We will assume that r + a >
0, and then r t > − a for any t >
0. All the results for γ = 1apply also to the displaced log-normal model with minimal substitutions.In [30], we studied the small-noise deterministic limit of the SDE (8),(9) in the log-normal case γ = 1 r (cid:48) ( t ) = y ( t ) − βr ( t ) + βλ ( t ) + λ (cid:48) ( t ) , (18) y (cid:48) ( t ) = σ ( r ( t )) − βy ( t ) , with r (0) = λ (0) = λ > y (0) = 0. In the small-noise limit, it is proved rigorously in [30] thatfor sufficiently large β or sufficiently small σ , the short rate r ( t ) is uniformly bounded, and hencethere is no explosion. When β = 0, the short rate explodes in a finite time, and an upper bound isgiven for the explosion time (Proposition 4 in [30]). Under the further assumption that λ ( t ) ≡ λ ,the upper bound for the explosion time given in Proposition 4 of [30] is sharp. The β > λ ( t ) ≡ λ . For this caseit is shown in [30] that when β < β C := σ √ λ , the explosion occurs at a finite time and when β ≥ β C , we have lim t →∞ r ( t ) = β σ (1 − (cid:113) − σ λ β ), and there is no explosion.In this paper we would like to study directly the original stochastic system (8),(9) in the presenceof random noise. We will show rigorously that the solutions of the stochastic system (8), (9) mayexplode with non-zero probability for γ ∈ ( , Numerical example for γ = 1 . Such explosions are indeed observed in numerical simulationsof the stochastic system (8),(9). We illustrate this phenomenon for the log-normal model γ = 1in Figure 1, which shows sample paths for { r t } t ≥ for several choices of the model parameters σ , λ (:= λ ) , β . These results were obtained by numerical simulation of the stochastic differentialequations (8),(9) by Euler discretization with time step τ = 0 . σ = 0 . , λ = 0 . β . The left plot shows sample pathsfor r t with β = 0. The paths explode at various times, which is expected in the presence of theBrownian noise. In the small-noise limit studied in [30], the explosion time is deterministic. The DAN PIRJOL AND LINGJIONG ZHU b=0.0 t 0 10 20 30 40 50 60 70246810 b=0.05 t Figure 1.
Sample paths for { r t } t ≥ for σ = 0 . λ = 0 . β = 0. The red vertical line isat t exp = 47 . β = 0 . λ ( t ) = λ , and is given in Proposition4 of [30]. The prediction is shown in Figure 1 (left) as the red vertical line.The right plot in Figure 1 shows sample paths for β = 0 .
05. There is still explosion, but theexplosion tends to occur at longer maturities. This is in qualitative agreement with the behaviorexpected in the small-noise limit [30] where it was shown that increasing β delays the explosiontime, and suppresses it completely for β ≥ β C = σ √ λ . For the parameters considered in Figure 1the small-noise critical value is β C = 0 . β = 0 . Explosion of the CEV-type quasi-Gaussian HJM model
Assume that the forward rate λ ( t ) satisfies the inequality,(19) λ (cid:48) ( t ) + βλ ( t ) ≥ βλ (0) . By a comparison argument, the solutions of (8),(9) are bounded from below by the solutions ofthe time-homogeneous SDE obtained by replacing λ ( t ) → λ = λ (0). Thus, for the purpose ofstudying the explosions of the solutions of the SDE (8),(9) it is sufficient to study the correspondingtime-homogeneous SDE with constant λ ( t ) = λ dr t = ( y t − βr t + βλ ) dt + σr t min( r γ − t , ε γ − ) dW t , (20) dy t = ( σ r t min( r γ − t , ε γ − ) − βy t ) dt, (21)with the initial condition r = λ > ε and y = 0.The coefficients of this SDE satisfy a local Lipschitz condition. For 0 < γ ≤ they also satisfy asublinear growth condition and global Lipschitz condition. Thus we can apply the standard result,see for example Theorem 5.2.9 in [22], to conclude that the SDE has a unique strong solution,which is furthermore square integrable and thus non-explosive. On the other hand we show thatfor < γ ≤ UASI-GAUSSIAN HJM MODEL 7 Γ R 2RR2R ry D Figure 2.
Regions D and Γ for the application of Proposition 1.The infinitesimal generator of this diffusion is L ε V ( r, y ) = ( σ r min( r γ − , ε γ − ) − βy ) ∂ y V (22) +( y − βr + βr ) ∂ r V + 12 σ r min( r γ − , ε γ − ) ∂ r V .
We would like to study the explosion time of this diffusion, defined as(23) τ := sup { t > y t < ∞ , r t < ∞} . We present a few preliminary results which will be used in our proof. The following theorem wasproved in [12], see Theorem 1.
Proposition 1 (Theorem 1, [12]) . Let
D ⊂ R d be a bounded open set with regular boundary ∂ D andlet D c be the complement of D . Consider the d -dimensional diffusion dX t = σ ( X t ) dW t + b ( X t ) dt where the coefficients σ ( · ) , b ( · ) are Lipschitz continuous on any compact subset of R d for any t ≥ t .Moreover, there exists a positive function V ( t, x ) ∈ C , ([ t , ∞ ) × D c ) and positive constants K , K < K and C such that(A.1) sup t ≥ t ,x ∈D c V ( t, x ) = K < ∞ .(A.2) sup t ≥ t ,x ∈ ∂ D V ( t, x ) = K < inf t ≥ t ,x ∈ Γ V ( t, x ) = K , for some set Γ ⊂ D c .(A.3) L V ( t, x ) ≥ CV ( t, x ) for every t ≥ t , x ∈ D c , where L is the infinitesimal generator of X t .Then, the explosion eventually occurs with positive probability if the process starts at a future time t ≥ t at a point x ∈ Γ . Proposition 2 (Theorem 2 in [12]) . Assume the conditions in Proposition 1 are satisfied. Then,we have the almost sure explosion provided the additional assumptions hold:(A.4) inf t ≥ t ,x ∈D c V ( t, x ) = K > ;(A.5) For any t ≥ t , x ∈ ∂ D , P t,x ( τ Γ < ∞ ) = 1 , where τ Γ is the first hitting time of the set Γ . DAN PIRJOL AND LINGJIONG ZHU
Theorem 1 in [12] is a generalization of Theorem 3.6 in [24]. Intuitively, it relates the explosion of thesolution of a stochastic differential equation to the behavior of an appropriately defined Lyapunovfunction for large values in the space domain.Unlike the case of one-dimensional diffusion processes, where a sufficient and necessary condition forexplosion is given by Feller’s criterion [25], for multidimensional diffusions, there are many differenttheoretical results giving sufficient conditions for explosions [25, 12, 24]. For our purpose, Theorem 1in [12] suffices. The strategy of the proof will be to construct an appropriate Lyapunov function, andshow that for the two-dimensional SDE model (20), (21), explosion occurs with positive probability.The main result of this paper follows.
Theorem 1.
Assume λ ( t ) ≡ r > ε > and β ≥ .(a) For γ ∈ (0 , ] the solution of the SDE (20) , (21) { r t , y t } t ≥ is non-explosive.(b) For γ ∈ ( , the solution of the SDE (20) , (21) { r t , y t } t ≥ explodes with non-zero probability P ( τ < ∞ ) > provided that any one of the following conditions is satisfied for at least one set of ( δ , δ ) , where δ , δ > are positive constants satisfying (1 + δ )(1 + δ ) = 2 γ .(i) sup R ≥ ε F ( R ; β, σ ) > where the function F ( R ; β, σ ) is defined by (24) F ( R ; β, σ ) := R γ − (cid:18) β + 12 σ δ ( δ + 1) (cid:19) (cid:18) δ σ (1 + R ) δ +1 + 1 δ R γ − (1 + R ) δ +1 (cid:19) . (ii) sup R ≥ ε ( G ( R ) − (2 β + σ δ ( δ + 1))) ≥ where the function G ( R ) is defined by (25) G ( R ) := δ R (1 + R ) δ +1 . Remark 1.
Under the assumptions in Theorem 1, V , K , K , K and C from the conditions inProposition 1 are given by V ( r, y ) = C − C (1 + y ) δ − C (1 + r ) δ , (26) K = C , (27) K = sup ( r,y ) ∈ ∂ D V ( r, y ) = C − C (1 + R ) δ − C (1 + R ) δ , (28) K = inf ( r,y ) ∈ Γ V ( r, y ) = C − C (1 + 2 R ) δ − C (1 + 2 R ) δ , (29) C = 2 β + 12 σ δ ( δ + 1) , (30) C = C + C , (31) where R ≥ ε , ε > is sufficiently small, δ , δ > are positive constants satisfying (1 + δ )(1 + δ ) =2 γ , and C , C > are determined separately for each case as follows (this is a restatement of theinequalities for ( a, b ) following from Lemma 2).(i) For case (i) of Theorem 1, C , C > satisfy the inequalities (32) R δ ( δ +2) ≤ δ C δ C σ (cid:18) R R (cid:19) δ − δ ≤ R γ − δ − − κ κ , where κ , κ are defined in (73), (74), and R is in the range allowed by condition (i) of Theorem 1. UASI-GAUSSIAN HJM MODEL 9 g=0.6g=3/4g=1 sb Figure 3.
Region in the ( σ, β ) plane allowed by the condition (ii) in Theorem 1.For given γ ∈ ( , β below the curves shown. (ii) For case (ii) of Theorem 1, C , C > satisfy the inequality δ C δ C σ (cid:18) R R (cid:19) δ − δ ≤ min (cid:26) R δ ( δ +2) , κ R − δ − κ (cid:27) , (33) where κ , κ are defined in (73), (74), and R = δ . Remark 2.
The constraints of Theorem 1 can be made stronger by replacing (34) 2 β + 12 σ δ ( δ + 1) → max { δ , δ } β + 12 σ δ ( δ + 1) . See the discussion around Eq. (67) about the choice of the constant C . This gives a wider regionfor β . Numerical study.
We study here the regions for ( σ, β ) allowed by Theorem 1. We discussonly the condition (ii) which is more amenable to an analytical treatment. The resulting regionfor ( β, σ ) includes all the typical values of these parameters which are relevant for applications0 < σ < . < β < .
1, see for example [6]. The constraint on β can be weakened further,see Remark 2.The condition (ii) of Theorem 1 is satisfied in the region below the curves shown in Figure 3. Foreach < γ ≤ δ taking values in 0 < δ < γ − G ( R ) = δ R (1+ R ) δ with δ > G ( R ) vanishes for R → R → ∞ . The function G ( R ) increases for R < R ( δ ) and decreasesfor R > R ( δ ), with R ( δ ) = δ .(ii) G ( R ) has a maximum at R ( δ ). At this point the value of the function is(35) G ( R ) = (cid:18) δ δ (cid:19) δ +1 . Fix the values of γ and σ . By scanning over δ ∈ [0 , γ − δ ∗ := arg max δ ∈ [0 , γ − (cid:26) G ( R ( δ )) − σ δ ( δ + 1) (cid:27) . Then the values of β allowed by the condition (ii) of Theorem 1 (for given σ, γ ) are(37) 0 ≤ β ≤ G ( R ( δ ∗ )) − σ δ ∗ ( δ ∗ + 1) , where δ ∗ is given by (36). This region for β is shown in Fig. 3 for several values of γ . The regionbecomes smaller as γ approaches and disappears at this point.The value δ ∗ decreases with σ , at fixed γ . (Recall that this determines also the range of allowedvalues for R , which includes the point R = 1 /δ ∗ .) In a range of sufficiently small σ , the maximumin (36) is realized at the maximally allowed value δ ∗ = 2 γ −
1. In this region the curves formaximally allowed β with different values of γ are distinct, as seen in Figure 3. For σ above acertain value, which depends on γ , the value of δ ∗ decreases from 2 γ − β curves are overlapping, since δ ∗ is independent of γ .There is a maximum value of σ for which positive values of β are allowed. At this maximum value,which depends on β , δ ∗ reaches zero. For β = 0, this maximum value is σ max = √
2. This followsfrom the small- δ expansion(38) G ( R ( δ )) = δ + δ (log δ + 1) + O ( δ ) . Substituting into (37) gives β ≤ δ ∗ (cid:18) − σ (cid:19) + O ( δ ∗ ) . (39)Requiring the cancellation of the O ( δ ∗ ) term gives the maximal value σ max = √ Almost sure explosion.
In Theorem 1, we showed that under certain conditions, the explo-sion occurs with positive probability. Under some additional assumptions, one can further provethe almost sure explosion, that is, that explosion occurs with probability one.
Theorem 2.
Suppose the assumptions of Theorem 1 are satisfied. Assume β > . For sufficientlylarge r so that (40) r > max (cid:26) eβ (4 βR + β + σ ) , σ β e e Rσ (4 βR + β + σ ) − R − (cid:27) , where R is determined such that either of the conditions (i) or (ii) of Theorem 1 holds, we have thealmost sure explosion, that is, P ( τ < ∞ ) = 1 . Remark 3.
Under the assumptions in Theorem 2, K from the conditions in Proposition 2 is givenby (41) K = min (cid:26) C − C (1 + R ) δ − C , C − C − C (1 + R ) δ (cid:27) , where C , C , C , R, δ , δ are defined in Remark 1. UASI-GAUSSIAN HJM MODEL 11 Implications for zero coupon bond prices and Eurodollar futures
The explosion of ( r t , y t ) is equivalent to the explosion of r t due to the explicit form of y t in termsof ( r s ) ≤ s ≤ t . The explosion of r t thus implies that the prices of zero coupon bonds P ( t, T ) becomezero almost surely for all t > τ , with τ the explosion time of r t .This follows from Eq. (4) for the zero coupon bond price, which gives(42) P ( T, T + δ ) = P (0 , T + δ ) P (0 , T ) exp (cid:18) − G ( T, T + δ ) x T − G ( T, T + δ ) y T (cid:19) , where we recall that x T = r T − λ ( T ). Suppose the assumptions in Theorem 1 are satisfied, then P ( τ < ∞ ) >
0, which implies that for sufficiently large T , P ( τ < T ) >
0. As a result, with positiveprobability, and sufficiently large T , the zero coupon bond price P ( T, T + δ ) collapses to zero.This implies that interest rates L ( T , T ) explode for all T > τ . Recall that the rate L ( T , T ) isrelated to P ( T , T ) as L ( T , T ) = T − T ( P − ( T , T ) − L ( T , T ) such as interest rate caps, swaptions, CMSswaps, and Eurodollar futures also become infinite. We will show this explicitly for the prices ofEurodollar futures contracts. Using Eq. (42) for the zero coupon bond price P ( T, T + δ ) we get (43) E Q [ P − ( T, T + δ )] = P (0 , T ) P (0 , T + δ ) E Q (cid:20) exp (cid:18) G ( T, T + δ ) x T + 12 G ( T, T + δ ) y T (cid:19)(cid:21) . Suppose the assumptions in Theorem 1 are satisfied, then x T = y T = ∞ with positive probabilityfor sufficiently large T . It follows that the Eurodollar futures price explodes to infinity, that is, E Q [ P − ( T, T + δ )] = ∞ , for sufficiently large T .In practical applications the explosions of the short rate r t could be avoided by capping the shortrate volatility to a finite value c , possibly using a prescription of the same type as that proposedin [16], σ r ( r t ) → min { max { , σ r ( r t ) } , c } . With this change, the diffusion coefficients satisfy thesub-linear growth condition of Theorem 5.2.9 in [22], which ensures that the solution r t exists andis non-explosive. 5. Appendix: Proofs
Proof of Theorem 1. (a) For 0 < γ ≤ the coefficients of the 2-d diffusion (20), (21) satisfy theconditions of Theorem 5.2.9 in [22], which we recall here briefly for convenience.Consider the SDE for the d − dimensional vector X t ∈ R d (44) X t = σ ( t, x ) dW t + b ( t, x ) dt where x ∈ R d and W t is a d -dimensional Brownian motion. Assume that the coefficients satisfy theglobal Lipschitz and linear growth conditions || b ( t, x ) − b ( t, y ) || + || σ ( t, x ) − σ ( t, y ) || ≤ K || x − y || , (45) || b ( t, x ) || + || σ ( t, x ) || ≤ K (1 + || x || ) , for every 0 ≤ t < ∞ , x ∈ R d , y ∈ R d and K is a positive constant. Under these conditions, thereexists a continuous, adapted process X = { X t ; 0 ≤ t < ∞} which is a strong solution of the SDE(44) with initial condition X , and is furthermore square-integrable. Note that in Eq. (43) in [30], there is a typo: the factor on the right-hand side of this equation should be P (0 ,T ) P (0 ,T + δ ) . The SDE (20), (21) with 0 < γ ≤ satisfies the conditions (45), and thus { r t , y t } t ≥ does notexplode. We study next the case < γ ≤
1, where the linear growth condition does not hold.(b) The boundary r t = 0, y t = 0 is unattainable. Indeed, for r t < ε , we have dr t = σr t dW t + ( y t − βr t + βr ) dt with(46) y t = σ (cid:90) t r s min( r γ − s , ε γ − ) e β ( s − t ) ds > , and since the term βr > y t > r t , by comparing r t with a geometricBrownian motion, we have r t >
0. Therefore, ( y t , r t ) ∈ R + × R + . This generalizes to γ ∈ ( ,
1] theresult of (12) and (13). Although this is proved here for the time-homogeneous case λ ( t ) = λ , theresult is easily seen to hold also under the weaker assumption λ (cid:48) ( t ) + βλ ( t ) ≥ D := (0 , R ) × (0 , R ), with R ≥ ε , where we define D c = R + × R + \D . It is clear that D is a bounded open set. The boundary ∂ D is regular since βr > r t . It isalso easy to see that, for < γ ≤
1, on any compact subset of R + × R + , the coefficients of the SDE(20),(21), are continuous and Lipschitz.Assume the following form for the Lyapunov function(47) V ( r, y ) = C − C (1 + y ) δ − C (1 + r ) δ , with C , C , C > δ , δ > δ )(1 + δ ) = 2 γ . We would like to test the conditions (A.1),(A.2) and (A.3) of Proposition 1.(1)
Condition (A.1).
For any ( r, y ) ∈ D c , we have(49) V ( r, y ) ≥ min (cid:26) C − C (1 + R ) δ − C , C − C − C (1 + R ) δ (cid:27) > , provided that we take(50) C ≥ C + C . Thus V ( r, y ) defined on D c is a positive function. Since r, y >
0, it is clear that V ( r, y ) ≤ C . Thusthe condition (A.1) is satisfied.(2) Condition (A.2).
Note that ∂ D = { ( R, y ) : 0 ≤ y ≤ R } ∪ { ( r, R ) : 0 ≤ r ≤ R } . Thus, we have(51) K = sup ( r,y ) ∈ ∂ D V ( r, y ) = C − C (1 + R ) δ − C (1 + R ) δ . On the other hand, taking Γ = [2 R, ∞ ) × [2 R, ∞ ) ⊂ D c , we obtain(52) K = inf ( r,y ) ∈ Γ V ( r, y ) = C − C (1 + 2 R ) δ − C (1 + 2 R ) δ > K . Hence, the condition (A.2) holds.
UASI-GAUSSIAN HJM MODEL 13 (3)
Condition (A.3).
Finally, let us check the condition (A.3). Note that L ε V ( r, y ) = δ C σ min( r γ , r ε γ − )(1 + y ) δ +1 − δ C β y (1 + y ) δ +1 + δ C y (1 + r ) δ +1 (53) − δ C βr (1 + r ) δ +1 + δ C βr r ) δ +1 − δ ( δ + 1) C σ min( r γ , r ε γ − )(1 + r ) δ +2 . Therefore, L ε V − CV = δ C σ min( r γ , r ε γ − )(1 + y ) δ +1 + δ C y (1 + r ) δ +1 + C (1 + y ) δ (cid:20) C − βδ y y (cid:21) + C (1 + r ) δ (cid:20) C − δ βr r − δ ( δ + 1) σ min( r γ , r ε γ − )(1 + r ) (cid:21) + δ C βr (1 + r ) δ +1 − CC ≥ δ C σ min( r γ , r ε γ − )(1 + y ) δ +1 + δ C y (1 + r ) δ +1 + C (1 + y ) δ [ C − βδ ] + C (1 + r ) δ (cid:20) C − δ β − σ δ ( δ + 1) min( r γ , r ε γ − )(1 + r ) (cid:21) + δ C βr (1 + r ) δ +1 − CC . Furthermore, for 1 < γ ≤
2, we have0 < r γ (1 + r ) ≤ , r ≥ , (54) 0 < r ε γ − (1 + r ) ≤ , ≤ r ≤ ε , (55)( ε is small, say less than 1) such that we have L ε V − CV ≥ δ C σ min( r γ , r ε γ − )(1 + y ) δ +1 + δ C y (1 + r ) δ +1 + C (1 + y ) δ [ C − βδ ] + C (1 + r ) δ (cid:20) C − δ β − σ δ ( δ + 1) (cid:21) + δ C βr (1 + r ) δ +1 − CC . Let us choose C to be a fixed constant so that(56) C ≥ max (cid:26) δ β, δ β + 12 σ δ ( δ + 1) (cid:27) . Then we have L ε V − CV ≥ δ C σ min( r γ , r ε γ − )(1 + y ) δ +1 + δ C y (1 + r ) δ +1 − CC . Recall that for ( r, y ) ∈ D c , we have either y ≥ R or r ≥ R , and we chose ε < R . (I) If y ≥ R and r < R , then we have for both 0 ≤ r ≤ ε and ε < r < R , by positivity of the firstterm, L ε V − CV ≥ δ C σ min( r γ , r ε γ − )(1 + y ) δ +1 + δ C y (1 + r ) δ +1 − CC ≥ δ C R (1 + R ) δ +1 − CC . (II) If r ≥ R and y < R , then we have (since r ≥ R > ε ) L ε V − CV ≥ δ C σ r γ (1 + y ) δ +1 + δ C y (1 + r ) δ +1 − CC ≥ δ C σ R γ (1 + R ) δ +1 − CC . (III) If y ≥ R and r ≥ R , then we have (again by r ≥ R > ε ) L ε V − CV ≥ δ C σ r γ (1 + y ) δ +1 + δ C y (1 + r ) δ +1 − CC ≥ δ C σ (cid:18) R R (cid:19) δ +1 r γ y δ +1 + δ C (cid:18) R R (cid:19) δ +1 yr δ +1 − CC = δ C σ (cid:18) R R (cid:19) δ +1 x δ +1 + δ C (cid:18) R R (cid:19) δ +1 x − CC , (57)where we denoted(58) x = r δ +1 y . The condition 2 γ = (1 + δ )(1 + δ ) was used to reduce the dependence on ( r, y ) to a function of x in the last step.The sum of the first two terms is bounded from below by the following Lemma. Lemma 1.
The infimum of the function ˆ F ( x ) : (0 , ∞ ) → (0 , ∞ ) defined as (59) ˆ F ( x ) := ax δ +1 + bx , a, b > is given by (60) inf x> ˆ F ( x ) = κ δ a δ b δ δ , where (61) κ δ = ( δ + 2)( δ + 1) − δ δ > . Proof of Lemma 1.
The minimum of ˆ F ( x ) is achieved at ˆ F (cid:48) ( x ) = 0, which gives(62) ˆ F (cid:48) ( x ) = a ( δ + 1) x δ − bx = 0 , which implies that(63) min x> ˆ F ( x ) = ( δ + 2)( δ + 1) − δ δ a δ b δ δ . UASI-GAUSSIAN HJM MODEL 15
To see that κ δ is larger than 1 for δ ∈ [0 , G ( δ ) := log( δ + 2) − δ +1 δ +2 log( δ + 1).We will show that ˆ G is decreasing in δ ∈ [0 , G (cid:48) ( δ ) = − log( δ + 1)( δ + 2) < κ δ = e ˆ G ( δ ) is decreasing in δ ∈ [0 , δ = 1, we have κ δ =(1 + 2)(1 + 1) − = >
1. Hence, κ δ is larger than 1. (cid:3) Therefore, following (57), we have L ε V − CV ≥ κ δ (cid:32) δ C σ (cid:18) R R (cid:19) δ +1 (cid:33) δ (cid:32) δ C (cid:18) R R (cid:19) δ +1 (cid:33) δ δ − CC . Hence, from (I), (II) and (III), we conclude that L ε V ≥ CV for any ( r, y ) ∈ D c if we have CC ≤ min (cid:40) δ C R (1 + R ) δ +1 , δ C σ R γ (1 + R ) δ +1 , (65) κ δ (cid:32) δ C σ (cid:18) R R (cid:19) δ +1 (cid:33) δ (cid:32) δ C (cid:18) R R (cid:19) δ +1 (cid:33) δ δ (cid:41) . To summarize, in order to have the Lyapunov function V ( r, y ) to be bounded, positive and satisfy(A.1), (A.2), (A.3), we need the conditions (50), (56) and (65) to hold simultaneously. Taking C = C + C , (66) C = max { δ , δ } · β + 12 σ δ ( δ + 1) , (67)then (50) and (56) are satisfied. This can be simplified by replacing max { δ , δ } → < γ ≤ δ , δ ≤ (cid:18) β + 12 σ δ ( δ + 1) (cid:19) ( C + C )(68) ≤ min (cid:40) δ C R (1 + R ) δ +1 , δ C σ R γ (1 + R ) δ +1 ,κ δ (cid:32) δ C σ (cid:18) R R (cid:19) δ +1 (cid:33) δ (cid:32) δ C (cid:18) R R (cid:19) δ +1 (cid:33) δ δ (cid:41) . The study of the inequality (68).
We study next the conditions for ( β, σ ) for which the inequality(68) is satisfied in a region of ( C , C ), at least for one value of R . We start by writing it in anequivalent way as(69) κ a + κ b ≤ min (cid:110) κ δ a ε b ε , aR γ − δ − , bR − δ (cid:111) , where(70) ε = 1 δ + 2 , ε = δ + 1 δ + 2satisfying ε + ε = 1, and we defined the new variables(71) a = δ C σ (cid:18) R R (cid:19) δ +1 , b = δ C (cid:18) R R (cid:19) δ +1 , and denoted the constants κ δ := ( δ + 2)( δ + 1) − δ δ , (72) κ := 1 δ σ (cid:18) β + 12 σ δ ( δ + 1) (cid:19) (cid:18) RR (cid:19) δ +1 , (73) κ := 1 δ (cid:18) β + 12 σ δ ( δ + 1) (cid:19) (cid:18) RR (cid:19) δ +1 . (74)We would like to obtain the region in the ( a, b ) ∈ R plane where the inequality (69) holds, andfind conditions on κ , κ (or equivalently β, σ )) for which this region is non-empty, at least for onevalue of R . These regions are given by the following Lemma. Lemma 2.
The inequality (75) κ a + κ b ≤ min (cid:110) κ δ a ε b ε , aR γ − δ − , bR − δ (cid:111) , with κ , κ , ε , ε > , κ δ > , ε ( δ + 2) = 1 and ε + ε = 1 , holds in two regions of the ( a, b ) plane:(i) A wedge-like region of the positive quadrant of the ( a, b ) plane, contained between the two straightlines passing through origin (76) aR δ ( δ +2) ≤ b ≤ a R γ − δ − − κ κ . (ii) A wedge-like region of the positive quadrant of the ( a, b ) plane, below a straight line passingthrough origin given by (77) b ≤ a min (cid:26) R δ ( δ +2) , κ R − δ − κ (cid:27) . Proof of Lemma 2.
We prove that the inequality (69) is satisfied in the regions (76) and (77).The line(78) b = aR γ − δ + δ − = aR δ ( δ +2) divides the first quadrant of the ( a, b ) ∈ R plane into two regions:(i) Region 1 with b > aR δ ( δ +2) ;(ii) Region 2 with b < aR δ ( δ +2) .We show that the inequality (69) simplifies in each of these regions as follows.(i) Region 1 with b > aR δ ( δ +2) . UASI-GAUSSIAN HJM MODEL 17
In this region we have clearly(79) min (cid:110) bR − δ , aR γ − δ − (cid:111) = aR γ − δ − . Furthermore, we have κ δ a ε b ε > κ δ aR ε δ ( δ +2) = κ δ aR δ ( δ +1) (80) = κ δ aR δ (2 γ − δ − ≥ aR δ (2 γ − δ − , since κ δ > κ a + κ b ≤ aR γ − δ − . This gives the upper bound on b in (76).(ii) Region 2 with b < aR δ ( δ +2) .In this region we have clearly(82) min (cid:110) bR − δ , aR γ − δ − (cid:111) = bR − δ . Furthermore, we have the lower bound(83) κ δ a ε b ε > κ δ ( bR − δ ( δ +2) ) ε b ε = κ δ bR − ε δ ( δ +2) > bR − δ since κ δ >
1. Thus the inequality (69) reduces in this region to the linear inequality(84) κ a + κ b ≤ bR − δ . This gives an upper bound on b (85) b ≤ κ R − δ − κ a which is useful only if R − δ > κ , or equivalently if(86) (cid:18) β + 12 σ δ ( δ + 1) (cid:19) (1 + R ) δ +1 < Rδ . This is obtained using the expression (74) for κ .If this condition is satisfied, then we get that (69) is satisfied in the subset of region 2(87) b ≤ min (cid:26) R δ ( δ +2) , κ R − δ − κ (cid:27) a . This is either the entire region 2, or a subset, bounded by the real axis and the line b = κ R − δ − κ a . (cid:3) Finally, let us get back to the proof of Theorem 1. In order for the region (76) to be non-empty,the following inequality must hold(88) κ R δ ( δ +2) ≤ R γ − δ − − κ . Substituting here the expressions (73), (74) for κ , κ , this becomes(89) R γ ≥ (cid:18) β + 12 σ δ ( δ + 1) (cid:19) (cid:18) δ σ (1 + R ) δ +1 + 1 δ R δ δ + δ + δ (1 + R ) δ +1 (cid:19) . In order for the region (77) to be non-empty one requires R − δ > κ which gives the inequality(90) R ≥ δ (cid:18) β + 12 σ δ ( δ + 1) (cid:19) (1 + R ) δ +1 . The inequality (88) yields the statement (i) of Theorem 1, and the inequality (90) the statement(ii). This completes the proof of Theorem 1. (cid:3)
Proof of Theorem 2.
We would like to test the conditions (A.4) and (A.5) of Proposition 2.(1)
Condition (A.4).
Let V ( r, y ) be the Lyapunov function (47) defined in our Theorem 1. We haveto check that its infimum on D c is positive. We can compute that K := inf ( r,y ) ∈D c V ( r, y ) = C − sup ( r,y ) ∈D c (cid:18) C (1 + y ) δ + C (1 + r ) δ (cid:19) (91) = min (cid:26) C − C (1 + R ) δ − C , C − C − C (1 + R ) δ (cid:27) . By (50) we have C ≥ C + C which gives K >
0. Thus, (A.4) holds.(2)
Condition (A.5) . According to Theorem 3.9. and the discussion at the beginning of Chapter3.7. in [24], it suffices to show that there exists a non-negative function V ( r, y ) for ( r, y ) ∈ Γ c thatis twice differentiable in ( r, y ) such that(92) L ε V ( r, y ) ≤ − α , for any ( r, y ) ∈ Γ c , where α > V to distinguish it from the Lyapunov function V defined in Theorem 1.Let us recall from the proof of Theorem 1 that Γ = [2 R, ∞ ) × [2 R, ∞ ). Therefore, Γ c = { ( r, y ) :0 < y < R or 0 < r < R } . Let us define(93) V ( r, y ) = e − r + e − y . Then V is non-negative and twice differentiable. We can compute that(94) L ε V ( r, y ) = ( − σ min( r γ , r ε γ − )+2 βy ) e − y + (cid:18) βr − y − βr + 12 σ min( r γ , r ε γ − ) (cid:19) e − r . For ( r, y ) ∈ Γ c , either one of the inequalities 0 < y < R or 0 < r < R holds.(a) If 0 < r < R , we distinguish between 0 < r ≤ ε and ε < r < R . In the latter case we have L ε V ( r, y ) ≤ − βr e − R + sup
1. Moreover, for any r ≥ e r ≥ r + r so that(98) (cid:18) βr + 12 σ ( r + r ) (cid:19) e − r ≤ ( β + σ ) r + σ r r + r ≤ β + σ . Hence, by plugging (98) into (97), we get(99) L ε V ( r, y ) ≤ − σ r γ e − R + 4 βR + β + σ − βr e − r . Denote H ( r ) := σ r γ e − R + βr e − r , r ≥
0. Let us give a lower bound of H ( r ) over r ≥
0. For0 ≤ r ≤
1, we have H ( r ) ≥ βe r , and for r >
1, we have H ( r ) ≥ σ re − R + βr e − r since γ ∈ ( , H ( r ) := σ re − R + βr e − r , and we can compute that(100) ˜ H (cid:48) ( r ) = σ e − R − βr e − r , which is negative for r < log( βr σ ) + 2 R and positive for r > log( βr σ ) + 2 R . Thus(101) ˜ H ( r ) ≥ σ e − R (cid:20) log (cid:18) βr σ (cid:19) + 2 R (cid:21) + βr e − log( βr σ ) − R = σ e − R (cid:20) log (cid:18) βr σ (cid:19) + 2 R + 1 (cid:21) . Hence,(102) H ( r ) = σ r γ e − R + βr e − r ≥ min (cid:26) βe r , σ e − R (cid:20) log (cid:18) βr σ (cid:19) + 2 R + 1 (cid:21)(cid:27) . Hence, we conclude that(103) max r,y ≥ L ε V ( r, y ) < , if r is sufficiently large so that(104) min (cid:26) βe r , σ e − R (cid:20) log (cid:18) βr σ (cid:19) + 2 R + 1 (cid:21)(cid:27) > βR + β + σ , which holds if(105) r > max (cid:26) eβ (4 βR + β + σ ) , σ β e e Rσ (4 βR + β + σ ) − R − (cid:27) . For both cases (a) and (b), for sufficiently large r the inequality max r,y ≥ L ε V ( r, y ) < (cid:3) Acknowledgements
We would like to thank Camelia Pop for discussions about boundary conditions of SDEs. Theauthors are also grateful to the REU students Ruby Oates, Alex Pollack and Kelsey Paetschow fortheir help with Figure 1. Lingjiong Zhu acknowledges the support from NSF Grant DMS-1613164.
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