Modelling the term structure of interest rates \'{a} la Heath-Jarrow-Morton but with non Gaussian fluctuations

We consider a generalization of the Heath Jarrow Morton model for the term structure of interest rates where the forward rate is driven by Paretian fluctuations. We derive a generalization of It\^{o}'s lemma for the calculation of a differential of a Paretian stochastic variable and use it to derive a Stochastic Differential Equation for the discounted bond price. We show that it is not possible to choose the parameters of the model to ensure absence of drift of the discounted bond price. Then we consider a Continuous Time Random Walk with jumps driven by Paretian random variables and we derive the large time scaling limit of the jump probability distribution function (pdf). We show that under certain conditions defined in text the large time scaling limit of the jump pdf in the Fourier domain is \tilde{omega}_t(k,t) \sim \exp{-K/(\ln(k t))^2} and is different from the case of a random walk with Gaussian fluctuations. We also derive the master equation for the jump pdf and discuss the relation of the master equation to Distributed Order Fractional Diffusion Equations.


Introduction
The Arbitrage Pricing Theory (APT) is a generalization of the Capital Asset Pricing Model (CAPM), and it states that capital assets are priced according to a set of factors. It does not say what these factors are, just that only idiosyncratic (firm specific) risk is priced. So, two different stocks could have two different APT formulations.
However, the law of one price states that two identical assets should have the same value on the financial market and deviations from rational behavior of traders for example, significantly larger returns on Fridays the 13 th, Lucey (2000) have been evidenced. Yet, market efficiency is an assumed prerequisite for CAPM to hold and contemporary finance still uses APT as a quantitative tool. Heath-Jarrow-Morton (1992) (see also Musiela-Rutkowski (1991)) have formulated a general theory (to be referred to as the HJM model) and a unifying framework for the existing APTs.
Presuming the forward interest rate and the bond price to be a continuous time stochastic process described by a Stochastic Differential Equation (SDE) they derive the generic form of the SDE that ensures that the bond price, related to the zero coupon bond (risk-less security issued by the central bank at a fixed interest rate called the spot rate), has a mean zero as a function of time.
The HJM model has been generalized by Sornette (1998), who, drawing on some ideas from string theory, derives a general condition for a class of second order SDEs that must be satisfied to ensure absence of arbitrage opportunities. He obtains a whole class of, nonlinear second order SDEs that, in some cases, can be solved exactly and that also provide extensions to the theory of pricing options Musiela-Rutkowski (1991).
These models have a rich structure but are based on Gaussian fluctuations. In this paper we develop a corresponding theory that allows the fluctuations to have an arbitrary distribution, in particular they may have power-law behavior at the high end.

Theoretical analysis
The purpose of this section is to provide overall understanding of stochastic processes that involve asymptotically large, power-law distributed, fluctuations. The analysis lacks mathematical rigor and draws more on physical ideas of Langévin processes (even though they are not explicitly mentioned in the text) than on the abstract theory of stochastic processes. The derivation of strict mathematical foundations for the analysis (e.g. foundations for the Lemma and the Statement in the next section) is left for future work.

Definitions and heuristic reasoning
The forward interest rate f (t, T ) is an interest rate that is agreed at time t and that gives the trader a right to trade in an asset at time T (to be termed as maturity). The underlying assumption in the HJM model is to describe f (t; T ) through a following SDE: where ǫ > 0. Here the functions α and σ, called drift and volatility are in general dependent on the process itself, and the fluctuations δη =.
η (θ)dθ have a zero mean, are uncorrelated with one another for different times and conform to a generic, non-Gaussian distribution with a density ρ(δη). An asset which at maturity has a value one Euro has at some earlier time t a smaller value that is expressed through a discount factor involving the forward rate (1) as follows: The bond price is related to the price of a zero-coupon bond M(t) whose stochastic dynamics is given by a SDE that involves the spot rate of interest f (θ, θ) and some fluctuations: and conditions are derived for the discounted bond price M(t)B (t; T ) to have a zero drift in order to ensure absence of arbitrage opportunities. The zerodrift condition is also referred to as the "martingale property" and states that the conditional mean of the bond price at time s > t discounted back to time zero equals the discounted bond price at time t. Here E [M(s)B(s; T ) |F t ] = M(t)B(t; T ) and the conditioning is on the so called "filtration" F t of information meaning a set of information about the bond price up to time t (see Musiela-Rutkowski (1991)).
The main problem is now to derive a SDE for the bond price (2). For computing a differential of a function of Gaussian fluctuations one can use the lemma by Ito (1997). This lemma states that the differential is obtained by expanding the function into a Taylor series to the first order in time and to the second order in the fluctuations and by replacing differentials of the fluctuations via their averages. If the function of the stochastic process f has a form given by equation (2) we may use a theorem by Girsanov [see Musiela-Rutkowski (1991)] that simplifies the computation.
If, however, the fluctuations are not Gaussian, in particular if the likelihood for them to be large is much larger than in the Gaussian case then the higher moments and auto-covariances of the process may be infinite (as in the case of a Lévy or a Pareto distribution) and the procedure for computing the differential may fail. It may also happen that the probability distribution of the squared fluctuation is of the same kind as that of the fluctuation itself which will certainly change the structure of the SDE that describes the bond price (2). In the following we examine the mathematical difficulties that arise. Since we are not aware of any generalization of Itô's lemma that can be used in a straightforward manner for the calculation of the differential of the bond price, we must perform the computations from scratch. The procedure consists in first working out a small change of the bond price, secondly in expressing it via the forward rate and finally in performing the limit ǫ → 0. Here the change reads : where δ ǫ f (t; ξ) := f (t + ǫ; ξ) − f (t; ξ). Now we expand the right hand side in a Taylor series and retain terms proportional to ǫ neglecting higher powers of ǫ. The expression in square brackets on the right hand side of (4) reads: Now the whole problem consists in a proper treatment of the products of fluctuations n i=1 δη(θ i ) in the last term on the right hand side in (5). This is the point where the analysis differs essentially from the Gaussian case. We assume the fluctuations to conform to a Lévy (or Paretian) distribution. We set δη = Lévy µ (0, 1, 0)dt which means that. η = δη/dt is (parameters in brackets from the left to the right) a Paretian distribution with a characteristic exponent µ, skewness zero, dispersion one and mean zero (to be termed standard Lévy variable). The Paretian distribution has a pdf whose asymptotic expansion reads: From this expansion follows an important property for the product of many Paretian variables. Lemma: (A) If δη = Lévy µ (0, 1, 0)dt then (δη) n = Lévy µ/n (0, 1, 0)(dt/n) (B) For independent Lévy variables δη 1 and δη 2 we have Z = δη 1 δη 2 ∼ Lévy µ (0, 1, 0)(dt) 2.
Both statements regard the high end of the distribution. Proof: (A) From the conservation of elementary probabilities f Y (y)dy = f X (x)dx it follows that the pdf of the n-th power Y = X n of a Paretian variable X = Lévy µ (0, 1, 0)dt reads: where a m denote coefficients in the expansion (6) for the pdf of a Paretian variable. (B) If X 1, X 2 = Lévy µ (0, 1, 0)dt are independent Paretian variables then the pdf of their product Z = X 1 X 2 reads: where (8) holds only for large values of z since we have replaced the pdfs by the first term of the asymptotic expansion (6). From the last expression in (8) it follows that,up to a logarithmic correction, Z is a Paretian with a squared dispersion q.e.d.
The Lemma implies that the only factors proportional to the first power of ǫ in the last sum on the right hand side of equation (5) are those which contain "diagonal" terms of the