L\'evy-Ito Models in Finance

We propose a class of financial models in which the prices of assets are L\'evy-Ito processes driven by Brownian motion and a dynamic Poisson random measure. Each such model consists of a pricing kernel, a money market account, and one or more risky assets. The Poisson random measure is associated with an $n$-dimensional L\'evy process. We show that the excess rate of return of a risky asset in a pure-jump model is given by an integral of the product of a term representing the riskiness of the asset and a term representing the level of market risk aversion. The integral is over the state space of the Poisson random measure and is taken with respect to the L\'evy measure associated with the $n$-dimensional L\'evy process. The resulting framework is applied to the theory of interest rates and foreign exchange, allowing one to construct new models as well as various generalizations of familiar models.


I. INTRODUCTION
Pricing models driven by Lévy processes have been considered by numerous authors; see references [1][2][3][4][5][6][7][8][9][10][11][12], for instance, alongside many other works. We shall be concerned here with a much more general family of models, namely, the so-called Lévy-Ito models. Such models are driven both by a Brownian motion and a Poisson random measure, where the Poisson random measure is taken to be associated with an underlying Lévy process. The Lévy-Ito class of models is general enough to include many familiar models as special cases, yet offers the opportunity for the creation of new models as well, while retaining a substantial overall level of analytic tractability. The need for a broad, systematic theory of Lévy-Ito models in finance is plain, for if an asset price is driven by a Lévy process, then the price process of an option or other derivative based on that asset cannot itself in general be represented by a Lévy model, but it can typically be represented by a Lévy-Ito model; and as we were taught many years ago by Black, Scholes and Merton [13,14], most securities, both corporate and sovereign, can be viewed as complex options based on the cash flows associated with one or more simpler underlying assets. Our intention, therefore, in that which follows, is to present the theory of Lévy-Ito models for asset pricing from a completely unified point of view, working exclusively in the real-world measure and emphasizing the role of the excess rate of return. In doing so we also present a number of specific examples of tractable Lévy-Ito models, ranging across a variety of different asset classes, thereby illustrating the flexibility and general utility of the resulting modelling framework.
The structure of the paper is as follows. In Section II we present a theory of risky assets driven by Lévy-Ito processes, and in Proposition 1 we deduce the general form that the price process for such an asset takes in an arbitrage-free economy. We comment, in particular, on the nature of the excess rate of return above the short rate of interest in a Lévy-Ito setting. In Section III we develop a theory of Lévy-Ito interest rate models, and in Proposition 2 we work out a general expression for the price of a discount bond in such a model. Then we show how the theory can be cast into HJM form under additional assumptions. As an example of a Lévy-Ito interest rate model, in Section IV we present a natural extension of the Vasicek model, summarized in Proposition 3, generalizing results of Norberg [15] and others. In Section V we show that the so-called "chaotic" interest rate models [16][17][18] upgrade naturally to the Lévy-Ito category. In Proposition 4, we prove that the pricing kernel in a Lévy-Ito model for interest rates can be written as the conditional variance of a random variable that admits a Ito-type chaos expansion. Then in Proposition 5 we give explicit formulae for a class of second-order chaos models, and we demonstrate how the model can be calibrated to an arbitrary initial term structure. Finally, in Section VI we consider Lévy-Ito models for foreign exchange, and in Proposition 6 we present a general expression of the exchange rate matrix in a Lévy-Ito setting. We conclude with a rather detailed analysis of Siegel's paradox in a multi-currency situation.
In the remainder of this section we present a brief overview of the Lévy-Ito calculus, which acts as the main workhorse of the theory, and give some examples of typical calculations. This will also give us the opportunity to establish our notation. In Lévy-Ito models, the price processes of financial assets are driven collectively by an n-dimensional Brownian motion and a "dynamic" Poisson random measure defined on R n × R + . We refer to the space-time dimension of the Poisson random measure as being (n, 1). Or if there is no ambiguity we speak of an n-dimensional Poisson random measure, where n refers to the dimensionality of the state space R n . For definiteness, we consider the class of models for which the Poisson random measure is associated with an n-dimensional Lévy process. That is to say, we assume the existence of an underlying Lévy process of dimension n, and we consider the Poisson random measure determined by this process via the Lévy-Ito decomposition. This restriction can be lifted for much of what we have to say, but in applications we find it useful to see the models we formulate as being generalizations of a corresponding class of Lévy models.
We proceed to introduce in more specific terms what we mean by a Lévy-Ito process. For simplicity, we discuss in detail first the situation where the Brownian motion and the Poisson random measure are each of dimension one; the higher dimensional situation can then be easily reconstructed by analogy with a slight adjustment of notation. Thus, when we model the dynamics of the price of a single risky asset, we find that for some purposes a one-dimensional model will suffice; but when we move on to consider collections of assets, as indeed we must for interest rates and foreign exchange, then the need for Lévy-Ito models with higher-dimensional state spaces becomes apparent.
We shall assume that the reader is familiar with elements of the theory of Lévy processes and their applications, as represented in works such as [19][20][21][22][23][24][25][26][27]. We fix a probability space (Ω, F , P) and let {ξ t } t≥0 be a one-dimensional Lévy process. In our notation for stochastic processes the curly brackets {·} signify an indexed set of random variables. The index space is usually indicated explicitly when the process is defined, but can be dropped subsequently for brevity, unless we specifically wish to draw attention again to the index set. Thus, we can now refer to the process {ξ t }, since we have already mentioned the index set {t ∈ R : t ≥ 0}. A similar convention applies to filtrations.
It is well known that a Lévy process {ξ t } admits a so-called Lévy-Ito decomposition ( [26], theorem 19.2) of the form where α and β are constants and Here {W t } t≥0 is a standard Brownian motion and N(dx, dt) is an independent Poisson random measure defined on R × R + such that for every t ≥ 0 and A ∈ B(R) we have where {ν(A)} A ∈ B(R) is a Lévy measure. Here B(R) denotes the Borel sigma algebra, generated by the open sets of R. By a Lévy measure on R we mean a sigma-finite measure ν(dx) on (R, B(R)), not necessarily finite, such that ν({0}) = 0 and where a ∧ b = min(a, b). The two parts of the integral with respect toÑ(dx, ds) in the third term on the right side of (1) cannot in general be split into separate terms by use of (2); rather, the term as a whole is defined by a limiting procedure (see, e.g., [26], page 120). From the foregoing we have seen that once we specify a Lévy process on (Ω, F , P), we have a Brownian motion {W t } and an independent Poisson random measure N(dx, dt) with mean measure ν(dx) dt, where ν(dx) is a Lévy measure. By a Lévy-Ito process driven by {W t } and N(dx, dt) we then mean a process {Y t } of the form We require that the processes {α t } t≥0 , {β t } t≥0 , {γ t (x)} t≥0, x∈R and {δ t (x)} t≥0, x∈R should be predictable and that the following condition should hold for all t ≥ 0: We recall that an R-valued process {φ t } t≥0 on a probability space (Ω, F , P) with filtration {F t } t≥0 is said to be predictable if the map φ : R + ×Ω → R is measurable with respect to the predictable σ-algebra, which is the σ-algebra generated by all left-continuous {F t }-adapted processes on (Ω, F , P).
In the case of the map-valued processes {γ t (x)} and {δ t (x)} appearing in the final two terms in the stochastic integral (5), the predictable σ-algebra is defined to be the smallest σ-algebra P with respect to which every map of the form ψ : R × R + × Ω → R such that (a) for each t ∈ R + the map (x, ω) → ψ t (x, ω) is B(R) ⊗ F t -measurable, and (b) for each x ∈ R and ω ∈ Ω the map t → ψ t (x, ω) is left continuous, is P-measurable. Any process {ψ t (x)} t≥0, x∈R defined by a P-measurable map ψ : R×R + ×Ω → R is said to be predictable. We observe that if ψ is predictable, then the process t → ψ t (x) is adapted for each x ∈ R.
We define P 2 (R, R + ) to be the set of all mappings (modulo equivalence) of the form ψ : R × R + × Ω → R such that the process {ψ t (x)} is predictable and the condition holds for t ≥ 0. Two such processes are taken to be equivalent if they coincide almost surely with respect to ν × Leb × P on B (R) ⊗ B (R + ) ⊗ F . We refer to (7) as the P 2 condition. We note, in particular, by virtue of (6), that the process {γ t (x)} appearing in equation (5) is required to satisfy the P 2 condition. In calculations, one often finds it convenient to write (5) in differential form. Then the initial condition is implicit and we have As in the classical Ito calculus, the meaning of such a differential form comes from the corresponding integral expression. We proceed to consider a generalized version of Ito's lemma applicable in the case of Lévy-Ito processes. The formula is more complicated than the corresponding result for processes driven by Brownian motion, but just as useful. Let the map F : R → R admit a continuous second derivative and write F ′ (x) and F ′′ (x) for the first and second derivatives of F at x ∈ R. Let {Y t } t≥0 be a Lévy-Ito process given in the form (5) and assume that for t ≥ 0. Then for {F (Y t )} t≥0 we have the following generalized version of Ito's formula: (see, e.g., [19], theorem 4.4.7). Here, as usual, for any process {X t } t≥0 admitting left limits we write X t − = lim s ↑ t X s . The first three terms of (10) are analogous to the terms of the classical Ito formula, whereas the remaining three terms come from the Poisson random measure.
The key point is that if {Y t } is a Lévy-Ito process subject to (9) and if F is continuously twice-differentiable, then {F (Y t )} is also a Lévy-Ito process. More explicitly, where we define It follows as a consequence of (6) and (9) that for t ≥ 0 we have which ensures that the integrals appearing in (11) are defined. One should note that in calculations, it is often convenient to write (10) in differential form, and we have Example 1. As a first step towards the construction of a pricing model we consider the problem of solving a stochastic differential equation of the form given the predictable processes {µ t } t≥0 , {Γ t (x)} t≥0, x∈R and {∆ t (x)} t≥0, x∈R as inputs, along with a strictly positive initial value Z 0 . We assume that {Γ t (x)} ∈ P 2 , and that for t ≥ 0 we have Γ t (x) > −1 for |x| < 1 and ∆ t (x) > −1 for |x| ≥ 1. These conditions ensure that {Z t } will not jump to zero or to a negative value. By use of Ito's formula we obtain after some rearrangement. Then the solution of (18) is given by We remark, finally, that in applications, it is sometimes convenient to write (20) in the alternative form where γ t (x) = log (1 + Γ t (x)) and δ t (x) = log (1 + ∆ t (x)).
Example 2. Next we consider the construction of exponential martingales in a Lévy-Ito framework. For this purpose, instead of (18) we look at the slightly modified equation the difference being that there is no drift term and we use the compensated random measure in both integrals. This opens up the possibility that we can make {Z t } a local martingale and even a martingale. In order for the compensator term to be defined in the |x| ≥ 1 integral we require for all t ≥ 0 that To obtain a solution we proceed as follows. We observe that as a consequence of (23) we can write (22) in the form But we see that (24) is exactly of the form (18) considered in Example 1, with It follows then by equation (20) in Example 1 that the solution takes the form Then {Z t } is a local martingale, and a sufficient condition to ensure that it is a martingale is that E[Z t ] = Z 0 for t ≥ 0. Next, we observe that if the process {∆ t (x)} also satisfies for t ≥ 0, then one can introduce a compensator term into the stochastic integral for |x| ≥ 1, and the expression for {Z t } can be put into the symmetrical form Note that if {∆ t (x)} is positive, then (23) implies (27). If (27) holds, we can abbreviate equation (28) by writing . Then the stochastic differential equation (22) satisfied by {Z t } takes the more compact form The solution (29) can also be written as where σ t (x) = log(1 + Σ t (x)). Thus, in contrast with the Brownian situation, in the case of a Lévy-Ito process, the volatility appears in two distinct forms, related on a one-to-one basis. We call {σ t (x)} t≥0 the exponential volatility and {Σ t (x)} t≥0 the dynamical volatility.

II. RISKY ASSETS
In our Lévy-Ito market model we introduce a money market account, a pricing kernel, and one or more risky assets driven by an n-dimensional Brownian motion {W t } t≥0 alongside an independent n-dimensional dynamic Poisson random measure Here for convenience we have written for the Poisson random measure associated with an n-dimensional Lévy process with Lévy measure {ν(A)} A ∈ B(R n ) . For ease of exposition we omit the Brownian component of the Lévy-Ito process in the discussion that follows; this can be easily restored.
In a general market model, the short rate of interest {r t } t≥0 is assumed to be an exogenously specified Lévy-Ito process and to satisfy The unit-initialized money market account is then defined by and the pricing kernel {π t } t≥0 is assumed to be given by an expression of the form Here λ : R n × R + → R is assumed to satisfy the P 2 condition, and to be such that and for t ≥ 0. We observe that the stochastic differential equation satisfied by {M t } is where Λ t (x) = 1 − e −λt(x) . It follows then that the pricing kernel takes the form and that the dynamical equation satisfied by the pricing kernel is given by Finally, we require that {λ t (x)} should satisfy conditions sufficient to ensure that E [M t ] = 1.
We consider now a risky financial asset with price {S t } t≥0 in a market endowed with the pricing kernel {π t }. Let us assume for simplicity that the asset is non-dividend paying and hence such that {π t S t } t≥0 is a martingale. We suppose, further, that this martingale takes the form and for t ≥ 0. It follows immediately that Gathering together the various terms, defining σ t (x) = β t (x) + λ t (x), and assuming that {σ t (x)} satisfies conditions analogous to those imposed on {β t (x)}, we obtain: The price of a non-dividend-paying risky asset in a Lévy-Ito market model takes the form where the excess rate of return above the interest rate is given by with Remarks. First, we observe that the risky asset satisfies the following stochastic differential equation: The dynamical volatility Σ t (x) represents the riskiness of the asset associated with the point x in the state space of the Poisson random measure at time t. Thus, Σ t (x) determines the multiplicative factor by which the price of the asset jumps if the jump in the underlying n-dimensional Lévy process is the vector x. The random variable Λ t (x) is the market price of risk associated with x at time t. The product Σ t (x)Λ t (x) is the excess rate of return per unit of jump intensity at x, and the Lévy measure ν(dx) determines the jump intensity. We note that a sufficient condition for the excess rate of return to be positive is that σ t (x) > 0 and λ t (x) > 0 for all t ≥ 0 and all x ∈ R. In that case, the excess rate of return is an increasing function of both the level of risk and level of risk aversion. Proposition 1 extends results known to hold in the case of models driven by Lévy processes [2,28].

III. LÉVY-ITO MODELS FOR INTEREST RATES
An interest-rate model consists of a pricing kernel {π t } t≥0 , a money market account {B t } t≥0 , and system of discount bonds {P tT } t≥0, T ≥0 . A discount bond with maturity T pays a single unit dividend at time T . Thus its value drops to zero at time T , and stays at that level for all t > T . In particular, we have lim s ↑ T P sT = 1 and P tT = 0 for t ≥ T . Occasionally, it is useful to refer to the associated discount function {P tT } 0≤t≤T <∞ , defined byP tT = P tT for 0 ≤ t < T < ∞ andP T T = 1 for T ≥ 0. The discount function is not defined for t > T . One regards P tT as being a price, whereasP tT is a discount factor. Interest rate models driven by Lévy processes and, more generally, by random measures, have been considered by various authors in the past; see, e.g., [29][30][31][32][33] and references cited therein. In what follows we present a general theory of interest rate models of the Lévy-Ito type. Before introducing our Lévy-Ito model, we make a few general remarks about interest rate modelling. There are several different ways of putting together interest rate models, depending on the purpose of the model and on which ingredients of the model one regards as primitives. This accounts for the various "types" of models and "approaches" that have been developed over the last few decades. But even in the case of a Brownian filtration the relationship of the various modelling frameworks is not easy to summarize in a few words (see, for example, Baxter [34], Hunt & Kennedy [35]), and once we add jumps the situation is even more complicated.
Generally speaking, there are three processes that play a key role in the formulation of an interest rate model with jump risk. These are the short rate {r t }, the market price of risk {Λ t (x)}, and the dynamical volatility {Ω tT (x)}. Alongside these processes we also make use of the risk aversion process In so-called short-rate models, the short rate of interest and the market price of risk are the "primitives" of the model. Once these have been specified, the remaining elements of the model can be worked out, such as the discount bond prices and volatilities. In so-called volatility models, which have been popular with practitioners, the discount bond volatilities and the market price of risk are the primitives, and from these we can work out the remaining elements of the model, such as the discount bond prices and the short rate.
Historically, in a Brownian context, short-rate models were the first to be developed, in the 1970s and 1980s; volatility models came later, in the late 1980s and on into the 1990s, in conjunction with the rise of interest rate derivatives markets. An influential variant on the volatility model idea, dating from the late 1980s, was to use the instantaneous forward rate volatilities as the primitives, along with the market price of risk [36]. A variant on the idea of the short-rate model dating from the early and mid 1990s was that of combining the short rate and the market price of risk together to form a so-called pricing kernel (or state-price density), and using that as the primitive ingredient [37][38][39].
From a broad perspective, and modulo the details of various technicalities, all of these approaches are more or less equivalent mathematically. Where they differ is in the naturalness and ease with which specific models of one sort or another can be developed, and in the facility with which parametric and functional degrees of freedom can be incorporated that can be used to calibrate the models to market data. When it comes to the formulation of Lévy-Ito models for interest rates, it will be convenient to begin with the volatility modelling approach. This is because the ideas that we have developed in the previous section concerning risky assets can be carried over directly. Therefore, we shall regard the bond volatility processes as being given, along with the risk aversion process. Hence, following the scheme outlined in the previous section, we treat each discount bond as a risky asset, and for a bond of maturity T we write where the excess rate of return is given by for t < T . We require that the exponential bond volatility system {ω tT (x)} 0≤t<T, x∈R n should satisfy and with lim s ↑ T ω sT = 0. It follows from the maturity condition on the discount bond that Substituting (53) into (49), we obtain the following: Proposition 2. Let the discount bond volatilities {ω tT (x)} and the risk aversion {λ t (x)} be given as elements of P 2 satisfying (36), (37), (51), (52). Then the price of a unit discount bond with maturity T takes the form where P 0tT = P 0T /P 0t denotes the forward price made at time 0 for purchase at time t of a T -maturity discount bond.
Thus we see that once the initial term structure, the risk aversion process, and the volatility processes have been specified, then the money market account and the discount bond prices for all maturities are determined.
Then to propose a specific interest rate model one needs to choose a form for the risk aversion function and the volatility function sufficiently general to allow one to calibrate the model to an appropriate range of market prices for discount bonds and interest rate derivative products.
For some purposes it is useful to assume that the interest rate model can be formulated in framework of the HJM type [36]. In that case one assumes the existence of a family of so-called instantaneous forward rates {f tT } 0≤t≤T <∞ such that The idea is that the instantaneous forward rates themselves should be Lévy-Ito processes with appropriate dynamics and initial conditions. The primitives of the model include (a) the initial discount function {P 0t }, (b) the risk aversion {λ t (x)} ∈ P 2 , and (c) a system of instantaneous forward rates {σ tT (x)} 0≤t≤T <∞, x∈R n ∈ P 2 . We ask that the initial discount function should admit a continuous first derivative with respect to T , and we set We require that and The instantaneous forward rates are then given by and the short rate of interest is given by r t = lim s ↑ t f st . Note that lim t ↑ T ω tT = 0 follows as a consequence of (56).

IV. VASICEK MODEL OF THE LÉVY-ITO TYPE
As an example of an interest rate model derived via the short-rate method, we construct a Vasicek model of the Lévy-Ito type. In the Lévy-Ito Vasicek model, the short rate {r t } t≥0 is taken to be a mean-reverting process of the Ornstein-Uhlenbeck (OU) type, satisfying The constants k and θ denote respectively the mean reversion rate and the mean reversion level. We assume that k and θ are strictly positive. The deterministic function σ : R n → R determines the volatility of the short rate. We shall assume that σ is non-negative and that The initial value of the short rate is r 0 and the initial value of the money market account is B 0 = 1. The risk aversion process in the Lévy-Ito Vasicek model is taken to be constant in time but not in space. Thus we have a non-negative function λ : R n → R + chosen so that and such that the process {m t } t≥0 defined by is a martingale. The dynamical equation (60) can be solved to give We observe that the mean of r t is θ + (r 0 − θ) e −kt , and that for the variance we have To obtain explicit formulae for the money market account and the pricing kernel, we require an expression for the integrated short rate, Substituting (64) into (66), we obtain Now, by the product rule, we have Integrating each side of this equation and rearranging the result, we obtain Substituting (69) back to (67), we see that Note that we can replace the u in equation (70) with an s. Using (70), we thus deduce that the pricing kernel takes the form A useful alternative expression for the integrated short rate can be obtained by combining (64) and (70). We get It follows that the money market account is given by and that the pricing kernel can be expressed in the form We proceed to derive an expression for the price of a discount bond, using the standard valuation formula P tT = 1(t < T ) E t [π T ]/π t . The conditional expectation of π T is given for t < T . Now, for any deterministic left-continuous process {α(x, t)} t≥0, x∈R n satisfying t 0 |x|<1 and t 0 |x|≥1 for t ≥ 0, we can make use of the so-called exponential formula As a consequence, if we define then by use of (61) and (62) we obtain Finally, using (64), (75), (79) and (80), we arrive at the following : Proposition 3. In a Lévy-Ito interest rate model for which the short rate of interest {r t } satisfies an Ornstein-Uhlenbeck equation of the form (60) and the risk aversion function is stationary and deterministic, the discount bond system is given for 0 ≤ t < T by Thus, by use of a pricing kernel technique we have obtained an expression for the price of a unit discount bond of maturity T in the Lévy-Ito Vasicek model, generalizing results of Vasicek [40], Cairns [41], Norberg [15], and Brody, Hughston & Meier [42]. The extra freedom provided by the functions {λ(x)} and {σ(x)} gives the model flexibility when it comes to fitting it to market data. Indeed, one of the novel features of our approach is that by allowing risk aversion to vary as a function of jump size one can let agents be, for example, more risk-averse to negative jumps than to positive jumps. It is reasonable to conjecture that the model can be generalized further, in the spirit of [43], by incorporating an element of deterministic time dependence in the mean reversion rate, the mean reversion level, the risk aversion function, and the volatility function.

V. CHAOS MODELS
The rather general class of Lévy-Ito interest rate models that we shall investigate in this section can be regarded as an example of the use of the pricing kernel method and has the property that the pricing kernel can be expressed as the conditional variance of an F ∞ -measurable square-integrable random variable.
We assume that interest rates are positive and that the model supports the existence of a perpetual floating rate note paying the short rate {r t } t≥0 on a unit principal on a continuous basis. The value of such a note is unity. Thus, by the standard valuation formula we have where F t denotes the σ-algebra generated by the Poisson random measure over the interval [0, t]. The intuition behind the pricing formula is that if interest is paid on a unit principal on a continuous basis, then the account will accumulate in value on an exponential basis-this leads to the standard expression for a continuous money market account; but if the interest is paid out on a continuous basis as a dividend, then the account itself must remain constant in value, and we are led to (82). It follows from the foregoing considerations that the pricing kernel can be expressed as a conditional expectation of the form where the integrand is positive. Consider now the random variable X defined by where It should be evident by construction that X is F ∞ -measurable, and that the existence of the stochastic integral appearing on the right-hand side of (84) is guaranteed since We proceed to calculate the conditional variance of X, defined by To work out (87) we use the conditional Ito isometry for Poisson random measure, given by which holds under the square-integrability condition A short calculation making use of (84), (87) and (88) then shows that Thus, we have established the following surprising fact: Proposition 4. In any arbitrage-free positive interest rate model driven by the Poisson random measure associated with an n-dimensional Lévy process and supporting the existence of a continuous floating rate note, the pricing kernel can be expressed as the conditional variance of a square-integrable F ∞ -measurable random variable.
This leads us to extensions of results obtained in the Brownian case by Hughston & Rafailidis [16], Brody & Hughston [17], Rafailidis [18], Grasselli & Hurd [44], Tsujimoto [45], and Grasselli & Tsujimoto [46], which we shall now discuss. It is well known that in the case of a probability space equipped with the filtration generated by a standard Brownian motion in one or more dimensions any square-integrable F ∞ -measurable random variable admits a so-called Wiener chaos expansion [47,48]. The chaos expansion expresses the random variable in the form of a uniquely-determined convergent sum of multiple stochastic integrals, where the k-th term involves an integrand given by a function of k time variables defined on a triangular domain, satisfying a square-integrability condition. This property extends to the case when the filtration is generated by a Poisson random measure in n dimensions [49][50][51], in which case the k-th term of the chaos expansion involves an integrand given by a function of k time variables and k space variables, each such space integration being over a copy of R n . As a consequence, the random variable X associated with the pricing kernel in any interest rate model of the Lévy-Ito type driven by the Poisson random measure associated with an n-dimensional Lévy process admits a chaos expansion. If the chaos expansion admits terms only up to order j, then we say that we have a general j-th order chaos model. If the expansion consists exclusively of the term of order j, they we say that we have a pure j-th order chaos model.
We shall present the form of the discount bonds in a general second-order chaos model driven by Poisson random measure. In this case we are given a pair of deterministic functions and ∞ s=0 x∈R n s These two functions are used to define an F ∞ -measurable random variable given by where for the integration range we have x ∈ R n and x 1 ∈ R n . The first step in the determination of the associated interest rate model is to calculate the conditional variance of the random variable X. Thus, if we set we find that the pricing kernel takes the following form: This formula for the pricing kernel allows one to work out expressions for the discount bond prices. Now, the price at time t of a bond with maturity T is given by A calculation making use of the conditional Ito isometry (88) shows that Then by inserting (96) and (98) into (97), we are able to determine the bond price in the general second-order chaos model. As a special case of the second-order chaos model one can consider what we shall call factorizable models, corresponding to the situation where the second-order chaos coefficient factorizes into a product of the form Under this simplifying assumption we find that the pricing kernel is linear combination of a pair of martingales. More precisely, if we define the process {M t } t≥0 by we find that {M t } is a square-integrable martingale for which the associated so-called predictable quadratic variation process {Q t } t≥0 is given by Then one can check that the process {M 2 t − Q t } t≥0 is also a martingale, and that the pricing kernel takes the form where the deterministic coefficients A t , B t and C t are defined as follows : Taking the conditional expectation of π T , and using the martingale condition, we obtain Equations (102) and (106) then show that the bond price is a rational function of M t . More specifically, we see that P tT takes the form of a ratio of a pair of quadratic polynomials in M t with deterministic coefficients : Alternatively, one can view the bond price as being given by a linear rational function of a pair of martingales. It is interesting to note that the structure of the bond price system is similar to that arising in the factorizable second-order Brownian chaos model [16][17][18], which also exhibits a linear rational structure. We proceed to consider the calibration of the factorizable second-order chaos model to market data. The first requirement that one can impose on any interest rate model with freely specifiable time-dependent degrees of freedom is that we should be able to calibrate the model to an arbitrarily specified initial yield curve. Thus, in the present context we assume that the initial discount function {P 0t } t≥0 is given in the form of a strictly decreasing function admitting a continuous first derivative. The problem is to choose the deterministic functions {φ t (x)}, {β t (x)}, {γ t (x)} in such a way that for t ≥ 0 we havē First, we notice that we can rescale {φ t (x)} and {β t (x)} by a common constant factor, without changing the resulting bond prices, in such a way that A 0 = 1. Once this is done, we must choose the renormalized functions The next step is to differentiate each side of this equation with respect to t and define the instantaneous forward rate where {p t (x)}, {q t (x)} ∈ P 2 are non-negative and satisfy (115) .
The remaining degrees of freedom can be then used to calibrate the model to other market instruments. How well such a calibration will perform remains to be seen, but it is worth taking note of the results obtained in the Brownian case by Grasselli & Tsujimoto [45,46]. One can also use the Lévy measure itself as a functional degree of freedom for the purpose of calibration, as discussed by Bouzianis & Hughston [52].

VI. LÉVY-ITO MODELS FOR FOREIGN EXCHANGE
We consider a system of exchange rates {F ij t } t≥0 for N currencies (i, j = 1, ..., N). Here F ij t denotes the price at time t of one unit of currency i expressed in units of currency j. As in our earlier considerations, we let N(dx, dt) denote the Poisson random measure associated with an n-dimensional underlying Lévy process with intensity ν(dx). Typically, we require that n ≥ N − 1 in order to ensure that the model has sufficient freedom. The idea is that we fix one of the currencies as a base currency (or "domestic" currency) and we consider the dynamics of the prices of the N − 1 remaining currencies when these prices are expressed in units of the base currency. Therefore, we would like the state space of the Lévy-Ito process to be at least of dimension N −1. For instance, in the case of three currencies, an underlying two-dimensional Lévy process is the necessary minimal structure.
To construct the general form of the exchange rate matrix we model a system of N pricing kernels {π i t } t≥0 , one for each currency, by setting Here again we have suppressed the n-dimensional Brownian component of the Lévy-Ito process, with the assumption that the model is driven by a pure-jump process; the general case including the Brownian component can be easily reconstructed. Note that when we consider foreign exchange it is convenient to give each pricing kernel a distinct initial value. Then the fundamental property of the exchange rate matrix is that for each currency pair the relevant component of the matrix is given by the ratio of the pricing kernels associated with the two currencies [39,53,54]. More precisely, we have If we combine (118) and (119), a straightforward calculation then leads to the following: Proposition 6. In a general Lévy-Ito setting, the exchange matrix takes the form where the excess rate of return is given by and for the exchange rate volatility we have It is interesting to observe that for each pair of currencies the exchange rate volatility decomposes into a pair of distinct terms, one for each of the two currencies. The significance of this fact is that one cannot model exchange rate volatility "directly" by simply positing an ad hoc form for {σ ij t (x)}. In particular, the cyclic identity leads to a set of conditions that have to be satisfied by the volatilities, namely for x ∈ R n and t ≥ 0, and one is immediately led back to an expression of the form (122) for the exchange rate volatility for some choice of the processes {λ i t (x)}. These relations apply to any exchange rate system in the absence of trading frictions.
There is, of course, a substantial literature devoted to attempts at modelling exchange rate volatility, and it has to be said that much of this is carried out without taking into account the risk aversion functions associated with each currency and the decomposition given by equation (122). We claim therefore that such investigations are misguided. It is clearly more natural if the modelling is pursued at the level of the individual risk aversion functions for the various currencies. One sees from (120) that once the short rates and the risk aversion processes have been specified for each of the currencies, along with the initial exchange rates, then the exchange rate dynamics are completely determined.
We turn now to consider the excess rate of return, which in a pure-jump Lévy-Ito model for foreign exchange takes the form (121). It is interesting to ask if it is possible for R ij t to be positive for all currency pairs. If a model has this property, we say that it satisfies Siegel's conditions. Siegel [55] seems to have been the first to identify the seemingly paradoxical fact that in a stochastic model it is consistent, for example, for the EUR-USD exchange rate and the USD-EUR exchange rate to exhibit positive excess rates of return simultaneously, even though the exchange rates are inverses of one another. The problem of determining whether it is possible for R ij t to be positive for all currency pairs is especially challenging in a setting with N currencies, where we need to show that N (N − 1) different exchange rates have positive excess rates of return. The intuition is that if any of these rates were to show a negative excess rate of return, then investors would sell off the overpriced currency, and would keep selling until a new price level was reached with the property that the excess rate of return was no longer negative, at which point normal trading would resume. We shall prove the existence of arbitrage-free N-currency models of the Lévy type in which all N (N −1) excess rates of return are strictly positive. The argument is non-trivial even in the Brownian case, so we consider that first. Then we look at a class of N-currency Merton-type models, i.e. compound Poisson with Gaussian jumps. Finally, we consider an N-currency model driven by an n-dimensional generalization of the variance gamma process. On the basis of these examples we are led to conjecture that Seigel's conditions can be satisfied in a broad class of Lévy-Ito models for foreign exchange. Geometric Brownian motion model. In the Brownian case we let {F ij t } denote a set of exchange rates between N currencies (i = 1, . . . , N) driven by a family of n independent Brownian motions. The pricing kernel for currency i is taken to be a geometric Brownian motion of the form where r i is the interest rate for currency i, λ i is a vector in R n for each value of i, and {W t } is a Brownian motion taking values in R n . The dot denotes the usual inner product between vectors in R n . It follows from (119) that where σ ij = λ j − λ i and R ij = σ ij · λ j . Thus, the question is whether we can choose the λ i vectors (i = 1, . . . , N) in such a way that for all i, j (i = j). The answer turns out to be yes, as the following construction shows. Let λ i (i = 1, . . . , N) be a set of N distinct vectors, each of the same length, so we have λ i · λ i = L 2 for some fixed length L > 0, for all i. Then for each pair i, j (i = j) we have where θ ij is the angle between the two vectors. We have assumed that the N equal-length vectors are distinct, so it must hold that θ ij = 0 for each pair i, j (i = j). As a consequence we see that cos θ ij < 1 for each such pair, and this leads to the desired result (127). Thus we have demonstrated the existence of N-currency geometric Brownian motion models in which Siegel's conditions hold for each currency pair.
Merton model. We proceed to establish an analogous result for a class of pure-jump Lévy models. In particular, we consider an N-currency model driven by an (N − 1)-dimensional pure-jump process of the Merton type [56]. It will suffice to show the details of a threecurrency model driven by a two-dimensional Merton process; the reader will be able to supply the straightforward generalization to the N-currency situation. Thus, we consider a two-dimensional compound Poisson process given by a pair of processes of the form where the (X κ ) κ∈N constitute an independency of identically distributed random variables, the (Y κ ) κ∈N constitute another such independency, and {N t } t≥0 is an independent Poisson process. For fixed κ, the random variables X κ and Y κ are not necessarily independent, and for a typical such pair X, Y we write under the assumption that the moment generating function is finite for a non-trivial range of values of α and β. The associated Lévy exponent is then defined by and a calculation shows that where m is the intensity of the underlying Poisson process. Thus, in this example the jump times of the two processes coincide, but the jump sizes are random and generally distinct.
In the case of a Merton-type model, we have X, Y ∼ N(µ 1 , µ 2 , σ 1 , σ 2 , ρ), and hence ψ(α, β) = m exp α µ 1 + β µ 2 + 1 2 We introduce the vectors and For the pricing kernels associated with the three currencies we set for i = 1, 2, 3. The exchange rate matrix is then given by where It follows by (133) and (138) that to establish the existence of a three-currency pure-jump model satisfying Siegel's conditions it suffices to show that one can choose the parameters of the bivariate normal distribution along with the three vectors {λ i } i=1,2,3 so that e (λ j −λ i ) µ T + 1 2 (λ j −λ i ) C (λ j −λ i ) T + e −λ j µ T + 1 2 (λ j ) C (λ j ) T > e −λ i µ T + 1 2 (λ i ) C (λ i ) where µ = (µ 1 , µ 2 ), ( · ) T denotes the transpose operation, and C is the covariance matrix of the N(µ 1 , µ 2 , σ 1 , σ 2 , ρ) distribution. To construct an explicit example, let us assume that µ 1 = 0, µ 2 = 0, σ 1 = 1, σ 2 = 1, and ρ = 0. Then condition (139) takes the form where a i = (a 1 , a 2 , a 3 ) and b i = (b 1 , b 2 , b 3 ). The inequality (140) is manifestly satisfied if we choose the vectors {λ i } i=1,2,3 so that they are distinct and of equal length; that is to say, and For then we have for each currency pair, but also and hence (140). Thus we have demonstrated the existence of a non-trivial three-currency finite-activity pure-jump Lévy model satisfying Siegel's conditions for all six exchange rates. The corresponding construction for any number of currencies is straightforward.
Variance-gamma model. An interesting example of an infinite activity Lévy process leading to a foreign exchange model satisfying Siegel's conditions for any number of currencies can be obtained as follows. We present the three-currency case in full. First, let us recall a few details of the theory of the variance-gamma process [11,57,58]. Let {Γ t } t≥0 be a gamma process for which the parameters are chosen such that E [Γ t ] = t, and Var [Γ t ] = t/m. We shall refer to such a process as a standard gamma subordinator with intensity m, following [2,59]. For further aspects of the gamma process see [60][61][62]. Then by a variance-gamma process with intensity m, we mean a process {ξ t } t≥0 of the form ξ t = W Γt , where {W t } t≥0 is a standard Brownian motion and {Γ t } t≥0 is an independent standard gamma subordinator with intensity m. It is a straightforward exercise to check that for α such that − √ 2 m < α < √ 2 m .
In what follows we consider a three-currency exchange-rate system driven by a generalization of the variance-gamma process. Let {X t } t≥0 , and {Y t } t≥0 be independent Brownian motions, let {Γ t } t≥0 be an independent standard gamma subordinator with intensity m, and set Then the vector {ξ 1 t , ξ 2 t } t≥0 is a two-dimensional Lévy process, and the associated Lévy exponent is given by for α, β such that 0 ≤ α 2 + β 2 < 2m .
Let us define the vector ξ t as in equation (134), the vectors {λ i } i=1,2,3 as in equation (135), and {π i t } i=1,2,3 as in equation (136). Then the exchange rate matrix is given by (137), and the excess rate of return is given by (138). It should be evident by virtue of (149) that in order for the pricing kernels to be well defined the risk aversion vectors must be such that for i = 1, 2, 3. To construct a class of models satisfying Siegel's conditions, we proceed thusly. Fix m, and let the vectors {λ i } i=1,2,3 be distinct and of equal length. It follows immediately that for each currency pair we have Then the excess rate of return for each currency pair is well defined and strictly positive if and only if ψ λ j − λ i > 0 , for all i, j such that i = j, or equivalently Since the risk aversion vectors have been assumed to be distinct, it follows that R ij > 0 for any currency pair if and only if Now, writing L for the common length of the risk aversion vectors, we have where θ ij denotes the angle between λ i and λ j . Hence, R ij > 0 if and only if On the other hand, since L < √ 2 m by (150), a sufficient condition to ensure that the excess rate of return is positive for each currency pair is that is to say, that the angle between each of the risk aversion vectors is less than sixty degrees. With this choice, we have thus shown the existence of a three-currency infinite activity Lévy model satisfying Siegel's conditions for all six exchange rates. In fact, if then the risk aversion vectors can be at any angle relative to one another and Siegel's conditions will hold. The extension of the argument to four or more currencies is straightforward.