Long-term returns in stochastic interest rate models

Transcription

1 Long-term return in tochatic interet rate model G. Deeltra F. Delbaen Vrije Univeriteit Bruel Departement Wikunde Abtract In thi paper, we oberve the convergence of the long-term return, uing an extenion of the Cox-Ingeroll-Ro (985) tochatic model of the hort interet rate r. Uing the theory of Beel procee, we are able to prove the convergence almot everywhere of t Xd with X a t generalized Beelquare proce with drift with tochatic reverion level. Key word Long-term return, tochatic procee, generalized Beelquare procee, convergence almot everywhere. Abbreviated title: Long-term return G. Deeltra, Departement Wikunde, Vrije Univeriteit Bruel, Pleinlaan, B-5 Bruel, Belgium tel: fax: gdeeltr@vnet3.vub.ac.be

2 Introduction. In thi paper, we are intereted in the long-term return t t r udu where (r u ) u denote the intantaneou interet rate. Inurance companie promie a certain fixed percentage of interet on their inurance product uch a bond, lifeinurance, etcetera. We wonder how thi percentage hould be determined. It ha to be lower than the etimated random return in order to cover expene and to hold ome reerve o that the company can get through difficult period of economic crii without bankruptcy. On the other hand, the competitive truggle, trengthened by the open market of the European Community, force rival companie to take the wihe of the clientele into conideration. Naturally, the cotumer want a return a high a poible. In thi light, we think it i intereting to tudy and to model the long-term return in a mathematical way. We analye the convergence of the long-term return, uing an extenion of the Cox, Ingeroll and Ro (985) tochatic model of the hort interet rate r. Cox, Ingeroll and Ro expre the hort interet rate dynamic a dr t = κ(γ r t )dt + σ r t db t. with (B t ) t a Brownian motion and κ, γ and σ poitive contant. It i a wellknown fact that thi model ha ome empirically relevant propertie. In thi model, r never become negative and for κγ σ, r doe not reach zero. For κ > and γ, the randomly moving interet rate i elatically pulled toward the long-term contant value γ. However, it i reaonable to conjecture that the market will contantly change thi level γ and the volatility σ. In the foottep of Schaefer and Schwartz (984), Hull and White (99) and Longtaff and Schwartz (99), we extend the CIR model in order to reflect the time-dependence caued by the cyclical nature of the economy or by expectation concerning the future impact of monetary policie. We aume the reverion level to be tochatic and we alo generalize the volatility. In thi ituation, we examine the convergence of the long-term return t t r udu and we propoe ome application. We conider a family of tochatic procee X, which contain the Beelquare procee with drift. The many reult known about thee procee, e.g. Pitman-Yor (98), Revuz-Yor (99), convinced u that thee procee are very tractable. Uing the theory of Beel procee, we found the following theorem, which i very ueful for deducing the convergence almot everywhere of the long-term return in quite general ituation: Theorem Suppoe that a probability pace (Ω, (F t ) t, IP ) i given and that a Brownian

3 motion (B t ) t i defined on it. A tochatic proce X : Ω IR + IR + i aumed to atify the tochatic differential equation dx = (βx + δ )d + g(x )db IR + with β < and g : IR IR + a function, vanihing at zero and uch that there i a contant b with g(x) g(y) b x y. The meaurable and adapted proce δ : Ω IR + IR + i aumed to atify: δ u du a.e. δ with δ : Ω IR +. Under thee condition, the following convergence almot everywhere hold X u du a.e. δ β. We will give a proof of thi theorem in ection. In ection 3, we how an immediate application of theorem. We conider the long-term return in the two-factormodel: dr t = κ(γ t r t )dt + σ r t db t dγ t = κ(γ γ t )dt + σ γ t d B t with (B t ) t and ( B t ) t two Brownian Motion and with κ, κ, σ, σ and γ poitive contant. The hort interet rate proce ha a reverion level which i a tochatic proce itelf. We do not need any aumption about the correlation between the Brownian motion of the intantaneou interet rate and of the tochatic reverion level proce. We tre thi fact becaue it i not trivial. Mot author of two-factormodel require, for technical reaon, that the Wiener procee are uncorrelated or have a determinitic and fixed correlation. Without further notice we aume that the filtration (F t ) t atifie the uual aumption with repect to IP, a fixed probability on the igma-algebra F = t F t. Alo B i a continuou proce that i a Brownian motion with repect to (F t ) t. 3

4 Convergence a.e. of the long-term return. Uing the theory of Beel procee, ee Pitman-Yor (98), we found a theorem which i very ueful for deducing the convergence almot everywhere of the long-term return in quite general ituation. In thi ection, we give a proof of the convergence reult, which relie on the theory of tochatic differential equation and on Kronecker lemma. Firt, we recall Kronecker lemma which i a tandard lemma, even for tochatic integral [ee Revuz-Yor (99, p.75 exercie.6)]. For completene, we give a proof. Kronecker lemma Let u aume a continuou emimartingale Y and a trictly poitive increaing function f which tend to infinity. If dy u f(u) exit a.e., then Yt f(t) a.e. Proof If we denote dz t = dyt f(t), then Y t = Y + f(u)dz u. By partial integration, recall that the emimartingale Z i continuou, we find that: Conequently, Y t = Y + f(t)z t f()z = Y + f()z t + Z u df(u) (Z t Z u )df(u) f()z. Y t f(t) = Y + f()z t f()z + f(t) f(t) (Z t Z u )df(u). Since Z = dy u f(u) exit a.e., (Z t) t converge to Z a.e. and therefore up Z t < a.e.. Hence, the firt term converge to zero a.e.. Let u look at the econd term: f(t) (Z t Z u )df(u) = f(t) (Z t Z u )df(u) + f(t) (Z t Z u )df(u). Since (Z t ) t converge to Z a.e., we can chooe for a given ε >, a number large enough uch that Z t Z u < ε for all t, u. For thi fixed, we can chooe t uch that max u Z u f() f(t) < ε. q.e.d. 4

5 To tudy the convergence a.e. of the long-term return, we conider a family of tochatic procee X, which contain the Beelquare procee with drift. We define the (continuou) adapted proce X by the tochatic differential equation dx = (βx + δ )d + g(x )db IR + with β trictly negative and g a function, vanihing at zero and atifying a Hölder condition of order one half. In order to have a unique olution of thi tochatic differential equation, we uppoe that δ d < for all t. The unique olution X i non-negative and atifie ) X = e (X β + δ u e βu du + e βu g(x u ) db u. () In view of the application, we think that the problem of exitence and uniquene goe beyond the cope of thi paper. The intereted reader i referred to Deeltra-Delbaen (994). For thi family of tochatic procee, we firt prove a technical lemma needed in the proof of the main theorem. Lemma Suppoe that a probability pace (Ω, (F t ) t, IP ) i given and that a tochatic proce X : Ω IR + IR + i defined by the tochatic differential equation with β, dx = (βx + δ )d + g(x )db IR + g : IR IR + i a function, vanihing at zero and uch that there i a contant b with g(x) g(y) b x y, δ : Ω IR + IR + i an adapted and meaurable proce, δ udu < a.. for all t. Then for all t we have up ut X u i finite and if IE[δ u]du < we have for all t: IE[X ] = e β X + e β( u) IE[δ u ]du. 5

6 Proof Let u firt conider the cae E[δ u]du <. We define the equence (T n ) n of topping time by From (), we have that for all t e β( Tn) X Tn = X + T n = inf{u X u n}. Tn δ u e βu du + Tn g(x u )e βu db u. () Since X i bounded on the interval [, T n ] and ince g(x) i bounded by b ( x, Tn we obtain that g(x u )e βu db u i a martingale, bounded in L )t. Indeed, let u calculate the quare of the L norm: [ ] t Tn IE g (X u )e 4βu du [ ] t Tn e 4βt IE b X u du [ ] t Tn b e 4βt IE X u du b e 4βt nt <. Therefore, the expected value of T n e βu g(x u ) db u i zero and equation () reduce to [ ] ] Tn IE [e β( Tn) X Tn = X + IE δ u e βu du. Taking the limit for n going to infinity and applying Fatou lemma, we obtain for all t: [ ] IE[X ] e β X + e β IE δ u e βu du [ ] e β X + IE δ u du <. We now how that up t X i integrable. From the olution of the tochatic differential equation, it i known that ( )) up X up (e β X + δ u e βu du + g(x u )e βu db u t t X + δ u du + up g(x u )e βu db u. t 6

7 Conequently, [ ] IE up X X + t IE[δ u ] du + up t g(x u )e βu db u. Uing Doob inequality, we remark that the lat term i bounded. up g(x u )e βu db u t 4 g(x u )e βu db u [ ] 4e 4βt IE g(x u ) du <. [ 4e 4βt IE ] b X u du 4e 4βt b IE[X u ] du Thi how that (X Tn ) t,n i a uniformly integrable family and that we are allowed to interchange limit and expectation in the expreion ( [ [ ] ]) Tn lim IE e β( Tn) X Tn = lim X + IE e βu δ u. n n We conclude that in the cae of IE[δ u]du <, the reult i obtained: IE[X ] = e β X + e β( u) IE[δ u ]du. Let u now look at the general cae with the local aumption δ udu < a.e. for all t. We define the equence (σ n ) n by σ n = inf{t δ u n} and we denote δ u [,σn ] by δ u (n). The tochatic differential equation dx (n) = ( βx (n) + δ (n) ) d + g(x (n) )db ha a unique olution and by the definition of σ n, we have IE[δ(n) u ]du n. Applying the firt part of the proof, we obtain up t X (n) < a.e.. On [, σ n ], all X (k), k n are equal by the uniquene of the olution of the tochatic differential equation. Since [, σ n ] [, t], the reult hold under the local aumption δ udu < a.e. for all t. q.e.d. 7

8 Now, we are ready for the convergence theorem itelf. Theorem Suppoe that a probability pace (Ω, (F t ) t, IP ) i given and that a Brownian motion (B t ) t i defined on it. A tochatic proce X : Ω IR + IR + i aumed to atify the tochatic differential equation dx = (βx + δ )d + g(x )db IR + with β < and g : IR IR + a function, vanihing at zero and uch that there i a contant b with g(x) g(y) b x y. The meaurable and adapted proce δ : Ω IR + IR + i aumed to atify: δ u du a.e. δ with δ : Ω IR +. Under thee condition, the following convergence almot everywhere hold X u du a.e. δ β. Proof Integrating the tochatic differential equation dx = (βx + δ )d + g(x )db IR + over the time-interval [,t] and dividing thi integral by β(t + ), give u the equality: t + (X + δ β )d = g(x ) β(t + ) db + X t X β(t + ). (3) It remain to prove that both term on the right hand ide converge to zero almot everywhere. In order to how that g(x ) β(t+) db converge to zero a.e., we ue Kronecker lemma and check the exitence a.e. of g(x u) u+ db u. Let u introduce the equence (T n ) n of topping time: { } δ u T n = inf t (u + ) du n. Since by hypothei + δ udu a.e. δ, we obtain that u δ d K(u + ) a.e. for ome contant K, depending on ω. Straightforward calculation how that 8

9 δ u (u+) du < a.e.: δ u (u + ) du = lim u <. u δ d (u + ) u δ d (u + ) + + ( u K(u + ) (u + ) 3 du ) du δ d (u + ) 3 Hence, {T n = } Ω and conequently, we only need to prove the exitence a.e. of g(x Tn u ) u+ db u on {T n = }. Moreover, ince g(x Tn u ) u+ db u i a local martingale, it uffice to remark that g(x Tn u ) u+ db u i a L -bounded martingale: = g(xu Tn ) u + db u IE [ g (X Tn u ) ] (u + ) du IE [ ] b Xu Tn (u + ) du. In order to evaluate thi lat integral, we remark that IE [ ] [ Xu Tn = IE Xu (utn)] e βu IE [ e βu ] X u (utn) ] e βu IE [e β(u Tn) X u Tn. In lemma, we obtained the equality [ ] Tn IE[e β( Tn) X Tn ] = X + IE e βu δ u du. Conequently: IE [ ] Xu Tn [ ]) u Tn e (X βu + IE e β δ d Uing thi reult, we obtain u e βu X + e βu e β IE [ δ (Tn)] d. IE [ ] Xu Tn (u + ) du 9

10 X + e βu (u + ) du e βu (u + ) du u e β IE [ δ (Tn)] d. Obviouly, the firt term i uniformly bounded in t. It remain to look at the econd term. We apply Fubini theorem to find a bound which i not depending on t: = e βu (u + ) du u e β IE [ δ (Tn)] d e β IE [ ] t e βu δ (Tn) d (u + ) du IE [ ( ) ] δ (Tn) ( + ) d β [ ] Tn β IE n β. δ ( + ) d In order to how that the econd term of (3), namely Xt X β(t+), converge to zero a.e., we divide the olution of the tochatic differential equation () by t + : X t t + = eβt X t + + e β(t u) t + δ udu + e β(t u) t + g(x u)db u. (4) Under the hypothei + δ udu a.e. δ, the econd term in (4) can be made arbitrarily mall, ince for all ω Ω and for all ε > : t + t + e β(t u) δ(ω, u)du t Given ε >, the firt term can be rewritten t + t e β t δ(ω, u)du + t + t δ(ω, u)du. t t + t e β t δ(ω, u)du t δ( + ε)e β t. e β t δ(ω, u)du

11 Conequently, the firt term converge to zero for t going to infinity. The econd term alo tend to zero ince lim t t + = lim t t δ(ω, u)du t ( t ) t t δ(ω, u)du t + t + δ(ω, u)du t ) t + + t t t + δ(ω, u)du t t t t t + t + δ(ω, u)du t ( + lim t = δ δ + lim t =. In order to check the convergence a.e. of eβt t+ e βu g(x u )db u to zero, we again ue Kronecker lemma and we look at the exitence of e βt g(x t) db (t+)e βt t. However, ince thi integral i equal to g(x t) t+ db t, the reult follow from the calculation above. q.e.d. Thi theorem can be generalized to tochatic procee X with a time-dependent trictly negative drift rate β. Let u define the proce X by the tochatic differential equation dx = (β X + δ )d + g(x )db for all IR + where the function g and the proce δ atify the hypothei of theorem ; and where up β < and dvie[δ v ]e v βd v following convergence almot everywhere hold: e u βd (u+) ( X u + δ ) u du a.e.. β u du <. Then, the

12 3 A two-factor CIR model. In thi ection, we give an example of theorem. We tudy the two-factormodel dr t = κ(γ t r t )dt + σ r t db t dγ t = κ(γ γ t )dt + σ γ t d B t with κ, κ > ; γ, σ and σ poitive contant and (B t ) t and ( B t ) t two Brownian motion. The hort interet rate r follow an extended Cox, Ingeroll and Ro quare root proce with reverion level (γ t ) t, which follow a quare root proce itelf. We know that the time-dependent reverion level (γ t ) t i itelf elatically pulled toward the long-term contant value γ. We are intereted in the convergence of the long term return t t r udu. Remark that we do not make any aumption about the way the two Wiener procee are correlated. In contrat with mot author, we do not demand the Brownian motion to be independent, they may have an arbitrary random correlation. Before looking at the convergence almot everywhere of the long-term return r udu itelf, we firt ue theorem to check that indeed γ u du a.e. γ. If we define Y u = 4 σ γ u, then Y u atifie the tochatic differential equation : ( ( ) ) 4 κγ κ dy u = σ + Y u du + Y u d B u. Thu, (Y u ) u i a Beelquare proce with drift, namely in the notation of Pitman-Yor : Y κ/ 4 κγ σ IQ 4γ. σ In general, dy u = (δ u + βy ) u du + g (Y u ) d B u with β = κ < δ u = 4 κγ σ u IR + g (Y u ) = Y u. Trivially, the condition of theorem are atified and it follow that Y u du a.e. 4γ σ

13 and conequently: γ u du a.e. γ. Analogouly, we conider the intantaneou interet rate r u itelf. The tranformation X u = 4 σ r u atifie the tochatic differential equation dx u = In term of theorem : with β = κ < δ u = 4κγu σ u IR + g (X u ) = X u. ( 4κγu σ + ( κ ) X u ) du + X u db u. dx u = (δ u + βx u ) du + g (X u ) db u Since δ i a tranformation of the continuou, adapted proce γ, δ itelf i meaurable and adapted. Becaue γ udu a.e. γ, we know that δ u du = ( ) 4κ γ u du σ a.e. γ 4κ σ. Therefore, the condition of theorem are fulfilled and we find that and finally that X u du a.e. 4γ σ r u du a.e. γ. We conclude that the long-term return converge a.e. to γ, the long-term contant value toward which the drift rate i pulled in thi two-factormodel. 3

14 REFERENCES. Cox, J.C., J.E.Ingeroll and S.A. Ro (985). A theory of the term tructure of interet rate. Econometrica 53, Deeltra G. and F. Delbaen (994). Exitence of Solution of Stochatic Differential Equation related to the Beel proce. Submitted Hull J. and A. White (99). Pricing Interet-Rate-Derivative Securitie. The Review of Financial Studie, Vol. 3, nr 4, Longtaff, F.A. and E.S. Schwarz (99). Interet rate volatility and the term tructure: a two-factor general equilibrium model. The Journal of Finance 47, nr 4, Pitman J. and M. Yor (98). A decompoition of Beel Bridge. Zeitchrift für Wahrcheinlichkeittheorie und Verwandte Gebiete, 59, Revuz D. and M. Yor (99). Continuou Martingale and Brownian motion. Springer-Verlag, Berlin-Heidelberg, New-York. Schaefer, S.M. and E.S. Schwarz (984). A two-factor model of the term tructure: An approximate analytical olution. Journal of Financial and Quantitive Analyi, 9,