STABILITY RESULTS FOR LÉVY TERM STRUCTURE MODELS

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1 STABILITY RESULTS FOR LÉVY TERM STRUCTURE MODELS BARBARA RÜDIGER AND STEFAN TAPPE Astract. In this note, we study term structure models driven y Lévy processes and provide staility results for them. In reality, we can never e sure of the accuracy of a proposed model. With this motivation, we present sufficient conditions which ensure that the model has the tendency to recover from perturations. Our results include staility conditions for the forward rates, yield curves and option prices. 1. Introduction The value at time t of one monetary unit to e paid at time T t is expressed y a Zero Coupon Bond. A Zero Coupon Bond is a contract which guarantees the holder one monetary unit at the maturity date T. The corresponding ond prices till maturity can e written as the continuous discounting of one unit of cash T P t, T = exp ft, sds, where ft, T is the rate prevailing at time t for instantaneous orrowing at time T, the so-called the forward rate for date T. The classical continuous time framework for the evolution of the forward rates goes ack to Heath, Jarrow and Morton HJM 14]. They assume that, for every date T, the forward rates ft, T follow an Itô process of the form 1.1 ft, T = f, T + t α HJM s, T ds + t t σs, T dw s, t, T ] where W is a Wiener process. In this paper, we consider Lévy term structure models, which generalize the classical HJM framework y replacing the Wiener process W in 1.1 y a more general Lévy process X, also taking into account the occurrence of jumps. This extension has een proposed y Eerlein et al. 8, 7, 3, 4, 5, 6]. In the sequel, we therefore assume that, for every date T, the forward rates ft, T follow an Itô process ft, T = f, T + t α HJM s, T ds + t σs, T dx s, t, T ] with X eing a Lévy process. In reality, we can never e sure of the accuracy of a proposed model. Therefore, we are interested to know how much its corresponding quantities forward rates, option prices, etc. would change if we pertur the model i.e. the volatility σt, T and the initial forward curve f, T a it. In order to approach this staility prolem, we will switch to the Musiela parametrization of forward curves r t x = ft, t + x see Date: January 5, Mathematics Suject Classification. 91G8, 6H15. Key words and phrases. Lévy term structure model, staility result, stochastic partial differential equation, Heath-Jarrow-Morton-Musiela equation. We are grateful to Damir Filipović, Vidyadhar Mandrekar and Josef Teichmann for their helpful remarks and discussions. 1

2 2 BARBARA RÜDIGER AND STEFAN TAPPE 17], which allows us to consider the forward rates as the solution of a stochastic partial differential equation SPDE, the so-called HJMM Heath Jarrow Morton Musiela equation drt = d dx r t + α HJM r t dt + σr t dx t 1.2 r = h, and to apply staility results for Lévy driven SPDEs, which can, e.g., e found in 1, 12, 16]. Existence and uniqueness of the Lévy driven HJMM equation 1.2 has een investigated in 2, 11, 15, 18, 19]. In order to ensure that the implied ond market P t, T is free of aritrage opportunities, we assume the existence of an equivalent martingale measure. Under such a measure, the drift α HJM : H H in 1.2 is given y the HJM drift condition 1.3 α HJM h = d dx Ψ σhηdη = σhψ σhηdη, where Ψ denotes the cumulant generating function of the Lévy process, see 7, Sec. 2.1]. Therefore, the principal difficulty when applying staility results for SPDEs is to assure that not only the volatility σ, ut also the corresponding drift term α HJM which depends on σ, satisfy appropriate regularity conditions. The remainder of the note is organized as follows. In Section 2 we introduce the term structure model, and in Section 3 we present the announced staility results. 2. Presentation of the term structure model In this section, we introduce the Lévy term structure model. From now on, let Ω, F, F t t, P e a filtered proaility space satisfying the usual conditions, and let X = X t t e a real-valued Lévy process with drift R, Gaussian part c and Lévy measure ν, that is, the characteristic function of X 1 is given y ϕ X1 u = exp iu c 2 u2 + e iux 1 iux1 1,1] x νdx, u R. R In what follows, we assume the existence of constants N, ɛ > such that e zx νdx <, z 1 + ɛn, 1 + ɛn]. x >1} Then, the Lévy process X possesses moments of aritrary order. The cumulant generating function Ψz := ln Ee zx1 ] exists on 1 + ɛn, 1 + ɛn], and elongs to class C on the open interval 1 + ɛn, 1 + ɛn. We fix an aritrary constant β > and denote y H β the space of all asolutely continuous functions h : R + R such that 1/2 2.1 h β := h 2 + h x 2 e βx dx <. R + Spaces of this kind have een introduced in 9]. According to 13, Thm. 2.1], the space H β is a separale Hilert space, the shift semigroup S t t defined y S t h := ht + is a C -semigroup on H β, there are constants C 1, C 2 > such that h L R + C 1 h β, h H β, h h L 1 R + C 2 h β, h H β,

3 STABILITY RESULTS FOR LÉVY TERM STRUCTURE MODELS 3 and there exist another separale Hilert space H β, a C -group U t t R on H β and continuous linear operators l LH β, H β, π LH β, H β such that πu t l = S t for all t R +. The latter result allows us to apply the staility results from 12], where SPDEs are understood as time-dependent transformations of SDEs. The particular representation of the Hilert space H β is not required in the sequel. Let Hβ e the suspace } 2.4 Hβ := h H β : lim hx =, x and let U Hβ e the set 2.5 U := h Hβ : } hηdη N. L R + For each h U we define the function Σh : R + R as Σh := h Ψ hηdη. Let C lip = C lip H β; Hβ e the linear space of all ounded Lipschitz functions σ : H β Hβ. The linear space Clip equipped with the norm σ lip = sup h H β σh β + sup h 1,h 2 H β h 1 h 2 is a Banach space. We define the suset F C lip F := σ C lip σh 1 σh 2 β h 1 h 2 β as : σh β U} Lemma. The following statements are valid: 1 We have ΣU Hβ, and Σ : U H β is locally Lipschitz continuous. 2 For each σ F we have Σ σ C lip, and for each n N there exists a constant M = Mn > such that for all σ F with σ lip n. Σ σ lip M Proof. By 11, Prop. 4.5] there exists a constant C 3 > such that for all h, g U we have Σh Σg β C h β + g β + g 2 β h g β, which provides oth assertions. Now let σ F e a volatility. Note that we can write the HJM drift term 1.3 which ensures the asence of aritrage as α HJM = Σ σ. Hence, y Lemma 2.1 we have α HJM C lip. Consequently, for each h H β there exists a unique mild solution for 1.2 with r = h on the state space H β of forward curves with càdlàg sample paths satisfying ] E sup r t 2 < for all T, t,t ] see, e.g., 11, Thm. C.1].

4 4 BARBARA RÜDIGER AND STEFAN TAPPE 3. Staility results In this section, we present the announced staility results for the Lévy term structure model presented in the previous section. Let volatilities σ F and σ n n N F e given. Furthermore, let initial conditions h H β and h n n N H β e given. In addition to the HJMM equation 1.2, we also consider the sequence of HJMM equations dr n t = d dx rn t + αhjm n rn t dt + σ n rt dx n t 3.1 r n = h n, where in order to ensure the asence of aritrage the corresponding drift terms are given y α n HJM = Σ σn. The following standing assumption assumptions prevail throughout this section: h n h in H β ; σ n h σh in H β for all h H β ; sup n N σ n lip <. Then, y Lemma 2.1 we have α n HJM h α HJMh in H β for all h H β ; sup n N α n HJM lip <. For what follows, we define the joint Lipschitz constant 3.2 L := sup max αhjm n lip, σ n lip } <. n N Denoting y r t t the mild solution for 1.2, for every T we have T ] ɛ n T, r := E α HJM r s αhjmr n s 2 βds 3.3 T 1/2 + E σr s σ n r s βds] 2, which follows from Leesgue s dominated convergence theorem. Now, we are ready to prove staility of the forward curves under the previous conditions Proposition. For all T there exists a constant K 1 = K 1 T, L > such that ] 1/2 3.4 E sup r t rt n 2 β K 1 h h n 2 β + ɛ2 n for n, t,t ] where ɛ n = ɛ n T, r was defined in 3.3. Proof. Since we have the joint Lipschitz constant 3.2, the assertion follows from 12, Prop. 9.1]. Besides the forward curve, the yield curve is another measurement of the ond market. Given a ond price P t, T, the yield Y t, T is the quantity ln P t, T Y t, T := = 1 ft, sds. T t T t t Switching to the Musiela parametrization, we thus define the yield curve operator y : H β CR + as the linear operator h, if x =, yhx := 1 x x T hηdη, if x > Lemma. We have yh β C R + and y LH β, C R +, that is, y is a continuous linear operator from H β to C R +.

5 STABILITY RESULTS FOR LÉVY TERM STRUCTURE MODELS 5 Proof. For h H β we estimate, y using 2.2, yh L R + = sup 1 x hηdη x, x sup 1 x, x finishing the proof. h L R + C 1 h β, x hη dη We define y t t as the C R + -valued process y t x := yr t x, where r t t denotes the mild solution for the HJMM equation 1.2. Then, the time t yield curve is given y Y t, T = y t T t, T t Proposition. For all T there exists a constant K 2 = K 2 T, L > such that ] 1/2 3.5 E sup y t yt n 2 L R + K 2 h h n 2 β + ɛ2 n for n, t,t ] where ɛ n = ɛ n T, r was defined in 3.3. Proof. This is a consequence of Proposition 3.1 and Lemma 3.2. Next, we analyze the staility of option prices under perturations of the interest rate model 1.2. For this purpose, we will assume that the HJMM equation 1.2 only produces positive forward curves. This is a reasonale condition, as negative forward rates are rarely oserved at the market. More precisely, from now on we suppose that the HJMM equation 1.2 is positivity preserving, that is, for all h P we have Pr t P = 1, t where r t t denotes the mild solution for 1.2 with r = h, and where P H β denotes the suset P = h H β : h } consisting of all nonnegative forward curves. We note that the conditions σhξ =, ξ, and h H β with hξ =, h + σhx P, h P and ν almost all x R. are necessary and sufficient for the positivity preserving property of the HJMM equation 1.2, see 13, Cor. 4.23]. Now, let us fix a future date T > and a payoff profile φ : P R depending on the forward curve r T. Since we model the HJMM equation 1.2 under a risk-neutral proaility measure, the time t price of φ is given y π t φ = E e T t rsds φr T F t ], t, T ] where r t denotes the short rate at time t Examples. Let us consider the following examples: 1 φ 1 is the payoff profile of a T -ond. 2 We fix another future date S with T < S and a strike rate K, and set 3.6 φh = exp S T + hηdη K, h P. Then we have S T + φr T = exp r T ηdη K = P T, S K +,

6 6 BARBARA RÜDIGER AND STEFAN TAPPE and hence φ is the payoff profile of a call option on a S-ond. Note that the cash flow of a floor is up to a constant equivalent to the cash flow of a call option on a ond, see 1, Chap. 2]. 3 Accordingly, the function φh = K exp S T + hηdη, h P is the payoff profile of a put option on an S-ond, and its cash flow is up to a constant equivalent to the cash flow of a cap Proposition. For all T and all Lipschitz continuous payoff profiles φ : P R there exists a constant K 3 = K 3 T, φ, L, r > such that 3.7 sup E π t φ πt n φ ] K 3 h h n 2 β + ɛ2 n for n, t,t ] where ɛ n = ɛ n T, r was defined in 3.3. Proof. By the Lipschitz continuity of φ, there exist constants L φ, K φ > such that φh φg L φ h g β, h, g H β φh K φ 1 + h β, h H β. Note that ev : H β R, ev h = h is a continuous linear operator y 13, Thm. 2.1]. Hence, using the Cauchy-Schwarz inequality we calculate e sup E π t φ πt n φ ] sup E T t rsds φr T e ] T t rn s ds φr n T t,t ] t,t ] ] sup E t,t ] + sup E t,t ] K φ sup t,t ] e T t e T rsds e T ds t rn s φr T ] t rn s ds φr T φrt n T ] E 1 + r T β r s rs n ds + L φ E r T rt n β ] t T K φ E1 + r T β 2 ] 1/2 E 2T K φ 1 + E rt 2 β] 1/2 E T 2T Kφ 1 + E rt 2 β] 1/2 ev + L φ E Now, applying Proposition 3.1 completes the proof. ] 2 1/2 r s rs n ds + L φ E r T rt n 2 β] 1/2 ] 1/2 r t rt n 2 dt + L φ E r T rt n 2 β] 1/2 1/2 sup r t rt n β] 2. t,t ] 3.6. Remark. Note that Proposition 3.5 applies to all payoff profiles presented in Examples 3.4. For instance, the payoff profile 3.6 of a call option on an S-ond can e written as with ψ : P R eing defined as ψh = exp φh = ψh K +, h P S T hηdη, h P.

7 STABILITY RESULTS FOR LÉVY TERM STRUCTURE MODELS 7 Hence, it suffices to prove Lipschitz continuity of ψ. For aritrary h, g P this is estalished, using estimate 2.2, y the calculation S T S T ψh ψg = exp hηdη exp gηdη S T S T S T hηdη gηdη hη gη dη S T h g L R + S T C 1 h g β. Consequently, the prices of caps and floors are stale under perturations of the term structure model. A Zero Coupon Bond P t, T has the payoff profile φ 1. Applying Proposition 3.5 yields for every T the estimate sup E P t, T P n t, T ] K 3 h h n 2 β + ɛ2 n for n t,t ] with a constant K 3 = K 3 T, L, r >. Now, we shall improve this result y considering the ond curve T P t, T at time t. Note that we can express the ond prices as T P t, T = exp ft, sds. t Switching to the Musiela parametrization, we thus introduce the ond curve operator p : H β CR + y ph := exp hηdη Lemma. The following statements are valid: 1 We have pp C R +. 2 There exists a constant L 1 > such that ph 1 ph 2 L R + L 1 h 1 h 2 β, h 1, h 2 P with h 1 = h 2. 3 For every x R + there exists a constant L 2 = L 2 x > such that ph 1 ph 2 L,x ] L 2 h 1 h 2 β, h 1, h 2 P. Proof. It is clear that ph C R + for all h P. For h 1, h 2 P with h 1 = h 2 we otain, y using estimate 2.3, e x ph 1 ph 2 L R + = sup h1ηdη e x h2ηdη x R + x sup h 1 η h 2 ηdη h 1 h 2 L 1 R + C 2 h 1 h 2 β. x R + Similarly, for x R + and h 1, h 2 P, y 2.2 we get ph 1 ph 2 L,x ] = sup e x h1ηdη e x h2ηdη sup x,x ] x x,x ] x h 1 η h 2 ηdη h 1 η h 2 η dη x h 1 h 2 L R + x C 1 h 1 h 2 β. This completes the proof.

8 8 BARBARA RÜDIGER AND STEFAN TAPPE We define p t t as the C R + -valued process p t x := pr t x, where r t t denotes the mild solution for the HJMM equation 1.2. Then, the time t ond curve is given y P t, T = p t T t, T t Proposition. For all T, x there exists a constant K 4 = K 4 T, x, L such that ] 1/2 3.8 E sup p t p n t 2 L,x ] K 4 h h n 2 β + ɛ2 n as n, t,t ] where ɛ n = ɛ n T, r was defined in 3.3. Proof. This is a consequence of Proposition 3.1 and Lemma 3.7. For the rest of this section, we suppose that the following stronger conditions are satisfied: h n h in H β ; σ n σ in C lip. Then, for all h H β we have σh σ n h β σ σ n lip for n, and, y Lemma 2.1, there exists a constant L 1 > depending on σ such that for all h H β we have α HJM h α n HJMh β = Σσh Σσ n h β L 1 σh σ n h β L 1 σ σ n lip. Consequently, for any T there is a constant L 2 = L 2 T such that we can estimate ɛ n = ɛ n T, r defined in 3.3 y ɛ n L 2 σ σ n lip. Therefore, we can improve the estimates y replacing the right-hand sides for i = 1, 2, 3, 4 y 3.9 K i h h n 2 β + σ σn 2 lip as n, showing that the dependence of the considered quantities on the initial curves and on the volatilities is locally Lipschitz. References 1] Aleverio, S., Mandrekar, V., Rüdiger, B. 29: Existence of mild solutions for stochastic differential equations and semilinear equations with non Gaussian Lévy noise. Stochastic Processes and Their Applications 1193, ] Barski, M., Zaczyk, J. 212: Heath-Jarrow-Morton-Musiela equation with Lévy perturation. Journal of Differential Equations 2539, ] Eerlein, E., Jacod, J., Raile, S. 25: Lévy term structure models: no-aritrage and completeness. Finance and Stochastics 91, ] Eerlein, E., Kluge, W. 26: Exact pricing formulae for caps and swaptions in a Lévy term structure model. Journal of Computational Finance 92, ] Eerlein, E., Kluge, W. 26: Valuation of floating range notes in Lévy term structure models. Mathematical Finance 162, ] Eerlein, E., Kluge, W. 27: Caliration of Lévy term structure models. Advances in Mathematical Finance, pp , Birkhäuser Boston, Boston, MA. 7] Eerlein, E., Özkan, F. 23: The defaultale Lévy term structure: ratings and restructuring. Mathematical Finance 132, ] Eerlein, E., Raile, S. 1999: Term structure models driven y general Lévy processes. Mathematical Finance 91,

9 STABILITY RESULTS FOR LÉVY TERM STRUCTURE MODELS 9 9] Filipović, D. 21: Consistency prolems for Heath Jarrow Morton interest rate models. Springer, Berlin. 1] Filipović, D. 21: Term-structure models: A graduate course. Springer, Berlin. 11] Filipović, D., Tappe, S. 28: Existence of Lévy term structure models. Finance and Stochastics 121, ] Filipović, D., Tappe, S., Teichmann, J. 21: Jump-diffusions in Hilert spaces: Existence, staility and numerics. Stochastics 825, ] Filipović, D., Tappe, S., Teichmann, J. 21: Term structure models driven y Wiener processes and Poisson measures: Existence and positivity. SIAM Journal on Financial Mathematics 1, ] Heath, D., Jarrow, R., Morton, A. 1992: Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica 61, ] Marinelli, C. 21: Local well-posedness of Musiela s SPDE with Lévy noise. Mathematical Finance 23, ] Marinelli, C., Prévôt, C., Röckner, M. 21: Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise. Journal of Functional Analysis 2582, ] Musiela, M. 1993: Stochastic PDEs and term structure models. Journées Internationales de Finance, IGR-AFFI, La Baule. 18] Peszat, S., Zaczyk, J. 27: Heath-Jarrow-Morton-Musiela equation of ond market. Preprint IMPAN 677, Warsaw. 19] Tappe, S. 212: Existence of affine realizations for Lévy term structure models. Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences , Bergische Universität Wuppertal, Fachereich C Mathematik und Informatik, Gaußstraße 2, D-4297 Wuppertal, Germany address: ruediger@uni-wuppertal.de Leiniz Universität Hannover, Institut für Mathematische Stochastik, Welfengarten 1, 3167 Hannover, Germany address: tappe@stochastik.uni-hannover.de