TRANSITION DENSITY FOR CIR PROCESS BY LIE SYMMETRIES ...

International Journal of Pure and Applied Mathematics Volume 88 No. 2 2013, 239-246 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v88i2.7

AP ijpam.eu

TRANSITION DENSITY FOR CIR PROCESS BY LIE SYMMETRIES AND APPLICATION TO ZCB PRICING Francesco Cordoni1 , Luca Di Persio2 § 1 Department

of Mathematics University of Trento Via Sommarive, 14 Trento, ITALY 2 Department of Computer Science University of Verona Via le Grazie, 14 Verona, ITALY

Abstract: Using a Lie algebraic approach we explicitly compute the transition density function for the solution of the stochastic differential equation defining the CIR process. Moreover we show how to use such a derivation to recover the well-known formula for the asscoiated ZCB fair price. AMS Subject Classification: 60H15, 35Q91 Key Words: CIR process, stochastic differential equations, Lie symmetry groups, diffusion processes, transition densities

1. Introduction and Preliminary Results In this section, following [3, 4], we will give the basic results needed to characterize the group of symmetries for the solutions of a particular class of PDE which will be later used to compute the transition density function of the CIR process. Received:

July 10, 2013

§ Correspondence

author

c 2013 Academic Publications, Ltd.

url: www.acadpubl.eu

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F. Cordoni, L. Di Persio

This process is widely used in finance to model short term interest rate, see [2] for details, and it is also used to model stochastic volatility in the Heston model. Theorem 1.1. Let us consider the PDE uτ =

σ2 γ x uxx + f (x)ux − g(x)u , γ 6= 2 2

(1)

and suppose that g and h := x1−γ f satisfy the second Riccati equation σ2 ′ σ2 1 A B 2−γ xh − h + h2 = x +C , x4−2γ + 2 2 2 2(2 − γ)2 2−γ

(2)

where A > 0 and B, C are arbitrary constants. Let u0 be a stationary, analytic solution of (1), then one can define the following family of symmetric solutions depending on a parameter ǫ which is linked to the Lie algebraic representaion of solutions of (1), see [3], Th. 2.5, for details: √ √ ¯ǫ (x, τ ) =(1 + ǫ2 (cosh( Aτ − 1) + 2ǫ sinh( A))−c U B √ √  cosh( Aτ ) + (1 + 2ǫ) sinh( Aτ ) σ2 √A(2−γ)  1 Bτ 2 F (x)− σ 2 (2−γ) 2 2 σ √ √ e × cosh( Aτ ) − (1 − 2ǫ) sinh( Aτ ) 2 2 ! √ √ √ −2 Aǫx2−γ (cosh( Aτ ) + ǫ sinh( Aτ )) √ √ × exp σ 2 (2 − γ)2 (1 + ǫ2 (cosh( Aτ ) − 1) + 2ǫ sinh( Aτ )) !! x 1 F × exp √ √ 1 σ2 (1 + ǫ2 (cosh( Aτ ) − 1) + 2ǫ sinh( Aτ )) 2−γ ! x × u0 , √ √ 1 (1 + ǫ2 (cosh( Aτ ) − 1) + 2ǫ sinh( Aτ )) 2−γ and c := 1−γ with F such that F ′ (x) = fx(x) γ 2−γ . Furthermore there exists a fundamental solution p(x, y, τ ) of (1) such that Z ∞ 2−γ e−λy u0 p(x, y, τ )dy = Uλ (x, τ ) , (3) 0

¯ σ2 (2−γ)2 λ (x, τ ), and u0 is the stationary solution of (1). where Uλ (x, τ ) = U √ 2 A

Remark 1.1. We would like to note that the time variable τ in equation (1), will be later defined as τ := T − t. The latter notation is motivated by standard financial arguments since τ actually refers to time left to maturity where T indicates, e.g., the expiration time of a certain financial contract.

TRANSITION DENSITY FOR CIR PROCESS BY...

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Corollary 1.1. Suppose that the conditions of Th. (1.1) hold and let g = 0, then, taking the stationary R ∞ solution u0 = 1, the resulting fundamental solution p(x, y, τ ) satisfies 0 p(x, y, τ )dy = 1. It follows that if ρ is also positive, then it is a probability density function. The following proposition, see [3], Prop. 2.11, for details, shows the link between fundamental solutions and the expectation of the associated SDE solutions. Proposition 1.1. Let X = {Xτ : τ ≥ 0} be an Itˆo process solution of p dXτ = f (Xτ )dτ + σ Xτ dWτ ,

where Wτ is a standard Brownian motion, and f is measurable. Suppose that there exist positive constants a, K such that |f (x)| ≤ Keax . Then there exists T > 0 such that u(x, τ, λ) := Ex [e−λXτ ] is the unique solution of u τ + λ2

σ2 uλ + λEx [f (Xτ )e−λXτ ] = 0 , 2

subject to u(x, 0, λ) = e−λx for 0 ≤ τ < T , λ > a.

Moreover, see [4] for details, the following holds Theorem 1.2. Let us suppose f is a solution of the equation 1 1 σf + f 2 + 2σµx2 = Ax2 + Bx + C, 2 2

with A > 0 .

(4)

Then there exists a fundamental solution of uτ = σxuxx + f (x)ux − µxu,

x≥0,

(5)

of the form p(x, y, τ ) = 

√

C1 (y)Iν √

2

! √ Axye−(F (x)−F (y)/(2σ)) A(x + y) Bτ √ √ − exp − 2σ 2σ sinh( Aτ /2) 2σ tanh( Aτ /2)

√ √    Axy Axy √ √ + (C2 (y)I−ν σ sinh( Aτ /2) σ sinh( Aτ /2) P (x/2)2n , is a first-order modified and Iν (x) := ( x2 )ν n≥0 n!Γ(ν+n+1)



where ν = σ σ+2C Bessel function.

(6)

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F. Cordoni, L. Di Persio 2. CIR Process

Let us now focus on the CIR process and on its symmetric group. We will study the stochastic process X = {Xτ : τ ≥ 0} solution of the CIR model, see [2] for details, described by the following SDE ( √ dXτ = k(θ − Xτ )dτ + σ Xτ dWτ . (7) X0 = x We want to recover the fundamental solution associated to the SDE (7) exploiting the method described in Section (1). In particular the PDE associated to equation (7) is of the form described by equation (1), i.e. it reads as follows uτ =

σ2 xuxx + k(θ − x)ux . 2

(8)

According to notation introduced in Th. (1.1), we have γ = 1 ; g(x) = 0 ; f (x) = k(θ − x) ; h(x) = f (x) = k(θ − x) , and in order h(x) to satisfy an equation of type (2) we obtain −

σ2 σ2 (kθ)2 k2 x2 A xk + xk − σkθ + + − k2 θx = x2 + Bx + C , 2 2 2 2 2

if and only if parameters A, B, C satisfy A = k2 ; B = −k2 θ , C =

kθ (kθ − σ 2 ) . 2

We can finally apply (1.1) with F (x) = kθ ln(x) − kx and u0 (x) = 1 to obtain the desired solution, namely kθ

U σ2 (2−γ) 2 λ (x, τ ) = √ 2 A

2k σ2 e 2

2k2 θτ σ2 2kθ

( σ2 λ(ekτ − 1) + kekτ ) σ2

exp

2

−λkx

2kθ

( σ2 λ(ekτ − 1) + kekτ ) σ2

in fact we have that −kθ cosh( kτ ) + (1 + 2ǫ) sinh( kτ ) σ2 2 2 Uǫ (x, τ ) = kτ cosh( kτ 2 ) − (1 − 2ǫ) sinh( 2 )   1 2k2 θτ × exp − 2 (kθ ln(x) − kx) + σ σ2

!

,

TRANSITION DENSITY FOR CIR PROCESS BY...

× exp

σ2 2 (1

243

−kǫx(cosh(kτ ) + ǫ sinh(kτ ))

+ 2ǫ2 (cosh(kτ ) − 1) + 2ǫ sinh(kτ ))  kθ2 !  σ x + ln 2 1 + 2ǫ (cosh(kτ ) − 1) + 2ǫ sinh(kτ )    kx 1 × exp − 2 σ 1 + 2ǫ2 (cosh(kτ ) − 1) + 2ǫ sinh(kτ ) −kθ cosh( kτ ) + (1 + 2ǫ) sinh( kτ ) σ2 k2 θτ 2 2 e σ2 kτ kτ cosh( 2 ) − (1 − 2ǫ) sinh( 2 ) = "  − kθ2 # σ x(1 + ǫ2 (cosh(kτ ) − 1) + 2ǫ sinh(kτ )) × exp ln x | {z } ≡P art1   kx 1 − 2ǫ(cosh(kτ ) − ǫ sinh(kτ ) × exp . (9) σ 2 1 + 2ǫ2 (cosh(kτ ) − 1) + 2ǫ sinh(kτ ) | {z } ≡P art2

Let us focus our attention separately on P art1 and P art2 of the equation (9). For what concerns P art1 we have −kθ cosh( kτ ) + (1 + 2ǫ) sinh( kτ ) σ2 k2 θτ 2 2 e σ2 P art1 = kτ cosh( kτ ) − (1 − 2ǫ) sinh( ) 2 2

× [1 + 2ǫ2 (cosh(kτ ) − 1) + 2ǫ sinh(kτ )] σ2  kτ  −kθ kτ −kτ e 2 + e −kτ σ2 2 2 − e 2 + (1 + 2ǫ) e k2 θτ σ2   = kτ e −kτ −kτ kτ e 2 + e 2 − (1 − 2ǫ) e 2 − e 2 −kθ

× [1 + ǫ2 (ekτ + e−kτ − 2) + ǫ(ekτ − e−kτ )] σ2 kτ kθ2 e 2 2ǫ + e −kτ σ 2 (2 − 2ǫ) = kτ e 2 (2 + 2ǫ) − e −kτ 2 2ǫ   kθ2 2 σ k θτ 1 e σ2 × kτ 2 −kτ 2 2 e (ǫ + ǫ) + e (ǫ − ǫ) − 2ǫ + 1 ! kθ2 σ k2 θτ ekτ σ2 , = e 2 (ǫ − ekτ (1 + ǫ)) −kθ

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F. Cordoni, L. Di Persio

 2kθ2  kτ 2 σ . hence, taking ǫ = σ2kλ we get P art1 = σ2 λ(ekτ2ke −1)+2kekτ Concerning the second part of equation (9), we have  1 + 2ǫ2 (cosh(kτ ) − 1) + 2ǫ sinh(kτ ) − 1 − 2ǫ(cosh(kτ ) − 2ǫ2 sinh(kτ )) 1 + 2ǫ2 (cosh(kτ ) − 1) + 2ǫ sinh(kτ )  2   kx 2ǫ (cosh(kτ ) − 1) + 2ǫ sinh(kτ ) − 2ǫ cosh(kτ ) − 2ǫ2 sinh(kτ ) = exp σ2 1 + 2ǫ2 (cosh(kτ ) − 1) + 2ǫ sinh(kτ )  kτ −kτ   kτ −kτ     kτ −kτ     2 ekτ +e−kτ − 2ǫ e +e − 2ǫ2 e −e − 1 + 2ǫ e −e 2ǫ 2 2 2 2 kx  = exp  2  σ 1 + 2ǫ2 (cosh(kτ ) − 1) + 2ǫ sinh(kτ )    kτ 2 2 −kτ 2 2 e (ǫ − ǫ + ǫ − ǫ) + e (2ǫ − 2ǫ) − 2ǫ kx      = exp  2  kτ −kτ kτ −kτ σ 1 + 2ǫ2 e +e − 1 + 2ǫ e −e

P art2 = exp



= exp



kx σ2



kx σ2



2

2

2

2

e (2ǫ − 2ǫ) − 2ǫ ekτ (ǫ2 + ǫ) + e−kτ (ǫ2 − ǫ) − 2ǫ2 + 1 −kτ



= exp



−2kxǫ −σ 2 ǫ + ekτ σ 2 (1 + ǫ)



,

  2 . thus, taking ǫ = σ2kλ , we obtain P art2 = exp σ2 λ(ekτ−2kxλ −1)+2kekτ Combining P art1 and P art2, we finally have Z ∞ U σ2 (2−γ) e−λy p(x, y, τ )dy 2 λ (x, τ ) = √ 0

2 A

2kθ

=

k σ2 e 2

2k2 θτ σ2

2kθ

( σ2 λ(ekτ − 1) + kekτ ) σ2

exp



−2λkx 2 kτ σ λ(e − 1) + 2kekτ



,

which is nothing but the Laplace transform of p(x, y, τ ), moreover it can be inverted, see [3, 4] for details, to have that the fundamental solution of (8) reads as follows pCIR (x, y, τ ) 2kθ   −2k(x + ekτ y) 2kek( σ2 +1)τ  y  ν2 = 2 kτ Iν exp σ (e − 1) x σ 2 (ekτ − 1)

√ 2k xy σ 2 sinh( kτ 2 )

!

, (10)

where ν := 2kθ − 1. σ2 Then by proposition (1.1) we have that the fundamental solution represented in (10) is in fact the transition density function of the CIR process. Moreover, exploiting Th. (1.2), we can compute the ZCB fair price, see, e.g., [1] for a description of such type of bond. To this end let us recall that given a differential operator L of elliptic type L such that uτ − Lu = 0

(11)

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245

with initial datum u(0, x) = f (x), where (x, τ ) ∈ Ω × [0, T ], and Ω is a measurable subset of R representing the domain of definition of f which is assumed to be sufficiently smooth on it, a fundamental solution for (11) is a kernel p(x, y, τ ) such that the following conditions hold: (i) for fixed y R∈ Ω, p(x, y, τ ) is a solution of the PDE (11) on Ω × (0, T ]; (ii) u(x, τ ) = Ω f (x)p(x, y, τ )dy is a solution of the Cauchy problem for a given initial datum f . We anyway refer to [5] for a complete treatment of fundamental solutions for partial differential equations of parabolic type. Then it follows that if the ZCB associated interested rate is driven by the CIR process defined by equation (7), then its fair price is given by theorem (1.2) setting µ = 1 ; A = k2 + 2σ 2 ; B = −k2 θ , C =

θk (θk − σ 2 ) ; 2

F (x) = θk ln(x) − kx ; C1 = 1 ; C2 = 0 . Then denoting with Pt the fair price of our ZCB with underlying interest rate process given by (7) we can recover a classical solution from the fundamental one exploiting point (ii) above. In particular for every t ∈ [0, T ) the price at time t of the ZCB reads as follows 2 −2θk)/2σ 2

2

2

k

y (σ +2θk)/2σ e σ2 (x−y) Pt :=u(x, t) = σ 2 sinh(γ(T − t)/2)   20 k θ(T − t) γ(x + y) − 2 × exp σ2 σ tanh(γ(T − t)/2)   √ 2γ xy × I σ2 −2kθ dy σ 2 sinh(γ(T − t)/2) σ2 " #
 2kθ −2x eγ(T −t) − 1 2γ exp [(γ + k)(T − t)/2] σ2 × exp , (γ + k)(eγ(T −t) − 1) + 2γ (γ + k)(eγ(T −t) − 1) + 2γ (12) Z

where γ :=

√

∞

γx(σ

k2 + 2σ 2 .

References [1] D. Brigo, F. Mercurio, Interest Rates Model: Springer-Verlag, Berlin (2006).

Theory and Practice,

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[2] J.C. Cox, J.E. Ingersoll, S.A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica, 53, No. 2 (1985), 385407, doi: 10.2307/1911242 [3] M. Craddock, Fundamental solutions, transition densities and the integration of Lie symmetries, J. Differential Equations, 246, No. 6 (2009), 2538-2560, doi: 10.1016/j.jde.2008.10.017. [4] M. Craddock, K.A. Lennox, The calculation of expectations for classes of diffusion processes by Lie symmetry methods, The Annals of Applied Probability, 19, No. 1 (2009), 127-157, doi: 10.1214/08-AAP534. [5] A. Friedman, Partial Differential Equations of Parabolic Type, PrenticeHall Inc., Englewood Cliffs, N.J. (1964).