eu SMALL NOISE EXPANSION FOR THE LÉVY PERTURBED VASICEK MODEL

In the present paper we shall provide rigorous small noise expansion results for the Lévy perturbed Vasicek model. Our analysis is based on [3], in the setting proposed in [9, Sec.6.2]. Let us underline that during recent years a wide range of small noise expansion techniques have been developed, particularly with respect to the so called Loval Volatility Models (LVMs), see, e.g., [4, 8, 11, 13]. LVMs are commonly used to analyse options markets where the underlying volatility strongly depends on the level of the underlying itself, therefore LVMs are also widely accepted as tools to model interest-rate derivatives as is the case for the Vasicek model. The paper is organized as follows: in Sect. 2 the we


Introduction
In the present paper we shall provide rigorous small noise expansion results for the Lévy perturbed Vasicek model. Our analysis is based on [3], in the setting proposed in [9,Sec.6.2]. Let us underline that during recent years a wide range of small noise expansion techniques have been developed, particularly with respect to the so called Loval Volatility Models (LVMs), see, e.g., [4,8,11,13]. LVMs are commonly used to analyse options markets where the underlying volatility strongly depends on the level of the underlying itself, therefore LVMs are also widely accepted as tools to model interest-rate derivatives as is the case for the Vasicek model. The paper is organized as follows: in Sect. 2 the we present the approach developed in [3] is presented and then applied, in Sect. 3, to provide order corrections to both the Vasicek model and its zero coupon bond price.

The General Setting
Let us consider the following stochastic differential equation (SDE), indexed by a parameter ǫ > 0 where L t , t ∈ [0, ∞), is a Lévy noise of jump diffusion type and µ ǫ : R d → R, σ ǫ : R d → R d×d are Borel measurable functions for any ǫ ≥ 0. In order to guarantee existence and uniqueness of strong solutions of (1) we assume µ ǫ and σ ǫ to be locally Lipschitz and with sublinear growth at infinity, see, e.g., [10]. We note that, if the Lévy process L t has a jump component, then X ǫ t in eq. (1) has to be understood as X ǫ t− := lim s↑t X ǫ s , see, e.g., [12] for details. In what follows we denote by stlim n→∞ X n := X, the limit in probability, namely, lim n→∞ P (|X n − X| > δ) = 0 .
Let us state the following Hypothesis 2.1. Let us assume that: (i) µ ǫ , σ ǫ ∈ C k+1 (R) in the space variable, for any fixed value ǫ > 0 and for all k ∈ N + (ii) the maps ǫ → α ǫ (x), where α = µ, σ, are of C M (I) in ǫ, for some M ∈ N, for every fixed x ∈ R and I : It has been shown in [1,3] that under hypothesis 2.1 on µ ǫ and σ ǫ , a solution X ǫ t of equation (1) can be represented as a power series with respect to the parameter ǫ, namely where X i : [0, ∞) → R, i = 0, . . . , N , are continuous functions, while |R N (t, ǫ)| ≤ C N (t)ǫ N +1 , ∀N ∈ N and ǫ ≥ 0, for some C N (t) independent of ǫ, but in general dependent of randomness, through X 0 t , X 1 t , . . . , X N t . The functions X i t are determined recursively as solutions of random differential equations in terms of the X j t , j ≤ i − 1, ∀i ∈ {1, . . . , N }. If we assume that x = x(ǫ), with ǫ → x(ǫ) in C N +1 , 0 ≤ ǫ < ǫ 0 , 0 < ǫ 0 < 1 and x(0) = x 0 independent from ǫ, then the following Taylor type expansion result holds Proposition 2.2. Let assume x(ǫ) can be written as a power series and let f ǫ ∈ C k+1 , k ∈ N + , be of the following form The general case reads as where B f k is a real function depending on (x 0 , x 1 , . . . , x k−1 ) only.
The following theorem establish our main expansion result with respect to the solutions to the SDE in eq. (1) Theorem 2.3. Let us assume that the coefficients α ǫ , α = µ, σ, of the stochastic differential equation (1) are in C Kα (I) as function of ǫ, ǫ ∈ [0, ǫ 0 ], and in C Mα (R) as function of x. Let us also assume that α ǫ are such that there exists a solution X ǫ t in the probabilistic strong, resp. weak sense of (1). Let us also assume that the recursive system of random differential equations has a unique solution.
Then there exists a decreasing sequence {ǫ n } n∈N , ǫ n ∈ (0, ǫ 0 ], with ǫ 0 > 0 as in hyp. 2.1 (ii), and ǫ n n→+∞ −→ 0, such that X ǫn t has an asymptotic expansion in powers of ǫ n , up to order N , in the following sense for some deterministic C N > 0.
Remark 2.4. It can be seen that in general the k−th equation for X k t in Th. 2.3 is a nonhomogeneous linear equation in X k t dL t (6) for some function f k , g k ,g h and h k .
Example 2.1. Let µ ǫ = ax + ǫbx and σ ǫ = σ 0 x + ǫσ 1 x with a, b, σ 0 and σ 1 some real constants. Applying Proposition 2.2 we get In particular, if we consider the special case of µ ǫ (x) = ax + b independent of ǫ, σ ǫ (x) = cx + ǫdx, for some constants a, b, c andd, and where the Lévy process is taken to be a standard Brownian motion, L t = W t , then by eq. (5) we have that X k t satisfies a linear equation with constant coefficients for any k ∈ N, thus applying standard results, see, e.g., [6,9], an explicit solution for X k t can be retrieved. Moreover, if L t = W t is a Brownian motion and we consider a set of K coupled linear stochastic equations with random coefficients, we have that the k − th equation is of the form where for any k with all the functions A k , B k , f k and g are assumed to be Lipschitz and with linear growth. A solution of equation (8) is given by where Φ t is the fundamental K × K matrix solution of the corresponding homogeneous equation, i.e. it is the solution of the problem Remark 2.5. In the trivial case of K = 1 then Φ reduces to a scalar and we have In the more general case where L t is a Lévy process composed by a Brownian motion plus a jump component, i.e.
whereÑ (dt, dx) is a Poisson compensated measure to be understood in the following sense. We haveÑ (t, A) := N (t, A) − tν(A) for all A ∈ B(R, 0), 0 ∈Ā, N being a Poisson random measure on R + × R 0 and ν(A) = E (N (1, A), while R 0 := R \ {0} and R 0 (|x| 2 ∧ 1)ν(dx) < ∞. Then we have that a solution to eq. (11) is explicitly given by where ∆η J (s) := X s − X s − being the jump at time s ∈ (0, t). This object is called Doóleans-Dade exponential (or stochastic exponential) and it is denoted by Φ(t) = E(X t ). The Doóleans-Dade exponential has a wide use in finance since it is the natural extension to the Lévy case of the standard geometric Brownian motion, see, e.g., [6,9] for a more extensive treatment on fundamental solution of homogeneous equation and [5] for more details on the Doóleans-Dade exponential.

Approximation of an Option Price
The Vasicek model (together with the CIR model) is one of the most used short rate modes. It assumes that the interest rate under the the risk neutral measure Q evolves according to a mean reverting Ornstein-Uhlenbeck process with constant coefficients, see, e.g. [7] for details. In particular the interest rate r t is the solution of the following linear stochastic equation with κ, θ, σ and r 0 some positive constants. The price of a pure discounted bond, better known as zero-coupon bond (ZCB), in the Vasicek model can be explicitly computed, see, e.g. [7] as The price of an option with payoff Φ(r T ) written on the interest rate r t is given by In the particular case of an European call/put option, as the one introduced in the previous BS model, the formula can be explicitly computed, see, e.g. [7] Sec. 3.2.1.
From Theorem 2.3 we deduce that Φ(r ǫ t ) has an asymptotic expansion in powers of ǫ of the form with sup and the coefficients can be computed from the expansions coefficients of r ǫ t , as discussed in section 2, where also the Taylor coefficients of Φ are treated.

The Vasicek Model: A First Order Correction
Applying the results in Sec. 2, let us then consider the following perturbed Vasicek model with σ 0 and σ 1 some positive constants, f a smooth real valued function, 0 ≤ ǫ ≤ ǫ 0 . Let us now consider the particular case f (r) = e αr , for some α ∈ R, then we get the following proposition.
Proposition 3.1. For the particular case where f (r) = e αr , for some α ∈ R 0 , we have that r ǫ t can be written a power series, namely where the expansion coefficients read as The particular choice of f (r) = e αr can easily be extended to any real function which can be written as a Fourier transform, resp. Laplace transform, f (r) = R 0 e irα λ(dα), resp. f (r) = R 0 e αr λ(dα) of some positive measure λ on R 0 (e.g. a probability measure) resp. which has finite Laplace transform. Formulae (20) holds with K α e αr 0 τ replaced by R 0 K α e iαr 0 τ λ(dα), resp. R 0 K α e αr 0 τ λ(dα), which are finite if, e.g. R 0 |K α |λ(dα) < ∞, resp. λ has compact support. In fact eq. (18) gets replaced by By repeating the steps used before we get the statements in Prop. 3.3 extended to these more general cases.

The Vasicek Model: A First Order Correction with Jumps
In the present section we will deal with the previous model with an addition of a small perturbed Poisson compensated measure N . In particular we will assumeÑ (t, A) := N (t, A)−tν(A) for all A ∈ B(R, 0), 0 ∈Ā, N being a Poisson random measure on R + × R 0 and ν(A) = E (N (1, A), while R 0 := R \ {0} and R 0 (|x| 2 ∧ 1)ν(dx) < ∞. Eventually the Poisson random measure is assumed to be independent of the Brownian motion W t . We refer to [5] for details on Levy processes. Under previous conditions let us assume we are given an interest rate r ǫ t evolving according to the SDE with the notation as previously introduced. Let us again consider the particular case f (r) = e αr , for some α ∈ R, then we get the following proposition. Proposition 3.3. For the particular case where f (r) = e αr , for some α ∈ R 0 , we have that r ǫ t can be written a power series, namely where the expansion coefficients read as with constants as in Prop. 3.4.
Proof. The proof is completely analogous to the one of Prop. 3.4 just taking into account the Poisson random measure.

Application to Pricing
Expanding the payoff function Φ, assumed to be smooth, according to eq. (14) we have that the first order correction to the fair price of an option written on the underlying r ǫ is given by If we consider the particular case of pricing a zero coupon bond (ZCB), namely we consider a terminal payoff Φ = 1, we have the following.
where ZCB is the price given in eq. (13).