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Journal of Applied Mathematics and Stochastic Analysis
Volume 2009 (2009), Article ID 215817, 16 pages
http://dx.doi.org/10.1155/2009/215817
Research Article

On Modelling Long Term Stock Returns with Ergodic Diffusion Processes: Arbitrage and Arbitrage-Free Specifications

School of Actuarial Studies, Australian School of Business, University of New South Wales, NSW 2052, Australia

Received 13 March 2009; Accepted 29 July 2009

Academic Editor: Vo Anh

Copyright © 2009 Bernard Wong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the arbitrage-free property of stock price models where the local martingale component is based on an ergodic diffusion with a specified stationary distribution. These models are particularly useful for long horizon asset-liability management as they allow the modelling of long term stock returns with heavy tail ergodic diffusions, with tractable, time homogeneous dynamics, and which moreover admit a complete financial market, leading to unique pricing and hedging strategies. Unfortunately the standard specifications of these models in literature admit arbitrage opportunities. We investigate in detail the features of the existing model specifications which create these arbitrage opportunities and consequently construct a modification that is arbitrage free.

1. Introduction

Ever since the fundamental work of Black and Scholes, there has been extensive work in the literature on alternative stock price models. In a continuous time setting, these include, for example, jump diffusions, Levy processes, stochastic volatility models, and regime switching models. Extensive references can be found in Cont and Tankov [1] and Fouque et al. [2]. These models and tools have proven invaluable for long term asset liability management, in particular with applications to insurance and pensions. These include the modelling and pricing of long term embedded guarantees (see, e.g., Sherris [3], Bauer et al. [4], Milevsky and Salisbury [5], Zaglauer and Bauer [6]) and also with the study of optimal asset allocation problems (see, e.g., Cairns [7], Gerber and Shiu [8], Stamos [9]). Extensive references to the vast literature can also be found in the monographs of Hardy [10], Schmidli [11], Milevsky [12], and Møller and Steffensen [13].

In this paper we investigate the arbitrage-free property of the class of Brownian based stock price models where the local martingale component of the (log) stock returns is assumed to be an ergodic diffusion. This class of models was first investigated by Bibby and Sørensen [14] and Rydberg [15, 16] (henceforth “BSR”) who reported good fit of their models to financial data. The use of ergodic diffusions imply that the marginal distributions over a long horizon will be approximately equal to that of the specified stationary distribution, such as the Student-𝑡 (Bibby and Sørensen [14] provide additional discussion of this property). The dynamics of these models are time homogeneous and, as it is based on diffusions, tractable. The financial market under these models will be complete, and hence the valuation of options and guarantees can be performed without requiring extra assumptions regarding the market price of risk. In contrast, most alternative stock price models admit incomplete markets, with no unique pricing of options and guarantees available in general.

A significant drawback of the ergodic diffusion approach however was also noted by BSR (in particular, Bibby and Sørensen [14] and Rydberg [15]) who showed that no standard equivalent local martingale measure can exist for the ergodic diffusion model with a (generalized) hyperbolic ergodic distribution model they considered, and so the model they considered is only arbitrage free up to a stopping time. By the fundamental theorem of asset pricing (Harrison and Kreps [17], Harrison and Pliska [18], Delbaen and Schachermayer [19]) it follows that these models permit arbitrage. Furthermore the discussions of this issue in BSR for the (generalized) hyperbolic class of models further suggest that this feature may perhaps be present in many ergodic diffusion-based models in general.

Following on from the previous discussion in literature, in this paper we provide a detailed proof that any ergodic diffusion process used as a stock return model, and as specified in literature, will admit arbitrage. We further analyze in detail the cause for these arbitrage opportunities and consequently propose a modification that is arbitrage-free. This modification once again opens up the application of ergodic diffusion models to problems in insurance and finance.

The outline of this paper is as follows. In Section 2 we briefly review the construction of local martingales based on ergodic diffusion processes. Section 3 sets out the financial market we consider and defines the economic notions of portfolios and arbitrage. The standard model specification considered in the existing literature is investigated, and associated arbitrage opportunities identified, in Section 4. This analysis is consequently used to construct an alternative, arbitrage-free specification in Section 5. An extension of the arbitrage-free specification to stock markets with a stochastic term structure of interest rates can be found in Section 6. Section 7 concludes.

2. Local Martingales Based on Ergodic Diffusions

Let 𝑊() be a Brownian motion. Consider a local martingale 𝑋() of the form 𝑋𝑑𝑋(𝑡)=𝜎(𝑋(𝑡))𝑑𝑊(𝑡),(0)=0,(2.1) where 𝜎() is a continuous, strictly positive deterministic function. By Engelbert and Schmidt [20] it follows that a nonexploding solution to the stochastic differential equation (2.1) will always exist.

Consider an interval (𝑙,𝑟) with <𝑙<0<𝑟<. Let 𝑔() be a continuous, bounded probability density function which is strictly positive on (𝑙,𝑟) and zero outside (𝑙,𝑟). Typically in applications to stock returns we are interested in the case when (𝑙,𝑟) is (,), and we will make this assumption for the analysis in Sections 47.

Using standard diffusion theory (cf. Karlin and Taylor [21]), it was noted by Bibby and Sørensen [14] and Rydberg [15] that 𝑋() can be constructed to be an ergodic diffusion with stationary density 𝑔() by selecting 𝜎(𝑥)=𝑘𝑔(𝑥),(2.2) for some arbitrary constant 𝑘>0. In particular they considered 𝑔() from the (generalized) hyperbolic class, which includes as special cases the Student-𝑡 and Normal Inverse Gaussian distributions. As an example, a local martingale 𝑋() with coefficient 𝜎2(𝑥)=𝑘𝜋𝜐Γ(𝜐/2)Γ𝑥((𝜐+1)/2)1+2𝜐(𝜐+1)/2(2.3) will possess a Student-𝑡 (with 𝜐 degrees of freedom) stationary distribution, while a local martingale 𝑋() with 𝜎2(𝑥)=𝑘𝜋1+𝑥2(2.4) will possess a Cauchy stationary distribution. Other distributions can also be used, dependent on the characteristics of the process being modelled.

It is worth noting that the existence of a stationary distribution for (2.1) is closely related to the concept of volatility-induced stationarity in the interest rate modelling literature (Conley et al. [22], Nicolau [23]).

Note that the above is not the only method of constructing a Brownian-based model with a specified distribution. There are two alternative approaches. The first alternative (Bibby et al. [24], Borkovec and Klüppelberg [25]) constructs semimartingales with stationary density 𝑔() by considering diffusion processes with nonzero drift. We do not pursue this approach as our stock price construction considers local martingales, which necessarily have zero drift. The second alternative (Dupire [26], Madan and Yor [27]) uses the Fokker-Plank equation of a diffusion, with an additional time scaling assumption, to construct a diffusion with a specified marginal distribution. A significant drawback of this second alternative approach however is that the dynamics of the resulting local martingales will be time inhomogeneous in general. In contrast, the dynamics of the local martingale (2.1) is time homogeneous.

3. Financial Market

For the financial market we consider a probability space (Ω,,𝐏) and the time interval [0,𝑇], the filtration being generated by 1-dimensional Brownian Motion, augmented to satisfy the usual conditions. In cases where we consider two nonequivalent measures we will augment with respect to the null sets of both measures. Denote by 𝒫𝑃 the equivalence class of all progressively measurable processes 𝜑() satisfying 𝑃𝑇0𝜑2(𝑡)𝑑𝑡<=1.(3.1) For clarity of presentation, in Sections 35 we will assume that the only source of uncertainty in the financial market arises from the stock. Specifically, assume that there are two primary securities traded in the time interval [0,𝑇]. The first is a savings account 𝐵() with 𝐵(𝑡)=𝑒𝑟𝑡,(3.2) for some constant short rate 𝑟𝐑. The second is a strictly positive stock price 𝑆(), with 𝑆(0)=1. We consider alternative specifications for 𝑆() in Sections 4 and 5.

An extension of our framework to include stochastic interest rates can be found in the Section 6.

Portfolios are formed by holding an amount of 𝜋0() of the savings account and 𝜋() of the stock, with 𝜋0(),𝜋() being progressively measurable. The value 𝑉() of a self-financing portfolio should satisfy for all 𝑡[0,𝑇], 𝜋𝑑𝑉(𝑡)=0(𝑡)𝐵(𝑡)𝑑𝐵(𝑡)+𝜋(𝑡)𝑆(𝑡)𝑑𝑆(𝑡),𝑉(𝑡)=𝜋0(𝑡)+𝜋(𝑡).(3.3) To ensure that there are no pathological doubling strategies we also require that the discounted value of a portfolio 𝑉()/𝐵() is bounded below by 1. Portfolios satisfying this constraint are referred to as tame (Dybvig and Huang [28]). Finally, an arbitrage opportunity is defined as a tame portfolio with value process 𝑉() satisfying 𝑉(0)=0,𝑉(𝑡)𝑉𝐵(𝑡)1,(𝑇)>0.(3.4)

4. Arbitrage Opportunities under the Standard Specification

Ergodic diffusion-based stock price models were first considered by Bibby and Sørensen [14] and Rydberg [15, 16]. Their models for the stock price process 𝑆() are of the form 𝑆(𝑡)=exp{𝜇𝑡+𝑋(𝑡)},(4.1) with 𝑆(0)=1, constant 𝜇𝐑, and 𝑋() being a driftless ergodic diffusion with stationary density 𝑔() and associated state space (,). In the following we will call ergodic diffusion based models of the form (4.1) as being of the standard specification.

As 𝑋() is nonexplosive by construction, it follows that 𝑃(𝑆(𝑇)>0)=1.(4.2) The model can equivalently be represented by the following stochastic differential equation by an application of Ito's formula: 𝑘𝑑𝑆(𝑡)=𝜇+𝑆2𝑔(ln𝑆(𝑡)𝜇𝑡)(𝑡)𝑑𝑡+𝑘𝑆𝑔(ln𝑆(𝑡)𝜇𝑡)(𝑡)𝑑𝑊(𝑡).(4.3) Representation (4.3) is sometimes more convenient for calculation purposes.

Bibby and Sørensen [14] and Rydberg [15, 16] report a good fit of the generalized hyperbolic ergodic diffusion to financial data. Unfortunately they also noted that no standard equivalent martingale measure can exist for the models they considered. By the fundamental theorem of asset pricing (Harrison and Kreps [17], Harrison and Pliska [18], Delbaen and Schachermayer [19]) it follows that arbitrage opportunities exist in these models. Furthermore, their discussion and proof suggests that the same problem may also apply to any ergodic diffusion-based model of the form (4.1). In the following we provide a detailed proof of the arbitrage opportunity for general ergodic diffusion models and perform further analysis on the technical features of this model specification which created these opportunities. Consequently in Section 5 we construct an alternative model specification that is arbitrage free.

Theorem 4.1. The financial market with stock price modelled by (4.1) admits arbitrage opportunities.

Proof. By Ito's lemma the process 𝑆()=𝑆()/𝐵() satisfies the stochastic differential equation 𝑑𝑘𝑆(𝑡)=𝜇𝑟+2𝑔ln𝑆(𝑡)(𝜇𝑟)𝑡𝑆(𝑡)𝑑𝑡+𝑘𝑔ln𝑆(𝑡)(𝜇𝑟)𝑡𝑆(𝑡)𝑑𝑊(𝑡).(4.4) In a Brownian setting, the fundamental theorem of asset pricing states that there is no arbitrage if and only if there exists an equivalent measure such that 𝑆() is a local martingale under the equivalent measure. As there is only one source of uncertainty, the market is complete. It follows that an equivalent local martingale measure exists if and only if there exists a strictly positive martingale 𝑍() with 𝑍(0)=1, and 𝑍(𝑡)=exp𝑡01𝛾(𝑢)𝑑𝑊(𝑢)2𝑡0𝛾2(,𝑢)𝑑𝑢(4.5) where 𝛾() is the market price of risk process, with 𝛾(𝑡)=𝜇𝑟𝑘𝑔(𝑋(𝑡))+𝑘21𝑔(𝑋(𝑡)).(4.6) Notice that as 𝑍() is a supermartingale as it is a local martingale that is bounded below. Consequently 0 will be absorbing if reached. As the processes 𝑔(𝑋()) and 1/𝑔(𝑋()) are continuous and non-explosive by construction, 𝛾()𝒫𝑃, and hence 𝑃(𝑍(𝑇)>0)=1(4.7) (cf. Kazamaki [29], Liptser and Shiryaev [30, 6.1.1]).
The martingale property of 𝑍() can be investigated by considering a candidate measure 𝑄 (cf. Kadota and Shepp [31], Delbaen and Shirakawa [32], Wong and Heyde [33], Rogers and Veraart [34] for an application of this technique in different settings), which is not assumed to be equivalent to 𝑃 a priori, with Brownian motion 𝑊𝑄(), and a process 𝑆𝑄() satisfying 𝑑𝑆𝑄(𝑡)=𝑘𝑔𝑆ln𝑄𝑆(𝑡)(𝜇𝑟)𝑡𝑄(𝑡)𝑑𝑊𝑄(𝑡),(4.8) with 𝑆𝑄(0)=1. Under 𝑄 we assume that 𝑆𝑄() will be stopped if explosion (to 0 or ) occurs. Denote this stopping time as 𝜏𝑆𝑄. If 𝑃 and 𝑄 are equivalent measures, then 𝑆() and 𝑆𝑄() will be equivalent in law under 𝑄.
Note that 𝑆𝑄() can also be represented in terms of a process 𝑋𝑄(), with 𝑑𝑆𝑄(𝑡)=𝑘𝑔𝑋𝑄𝑆(𝑡)𝑄(𝑡)𝑑𝑊𝑄𝑋(𝑡),(4.9)𝑄(𝑡)=𝑡0𝑘𝑔𝑋𝑄(𝑢)𝑑𝑊𝑄(𝑢)𝑡01(𝜇𝑟)+2𝑘𝑔𝑋𝑄(𝑢)𝑑𝑢,(4.10) with 𝑋𝑄() also being stopped at the explosion time 𝜏𝑆𝑄. Note also that if 𝑄𝜏𝑆𝑄𝑇>0,(4.11) then the stochastic integral in (4.10) should be interpreted in the Liptser-Shiryaev [30, 4.2.9] sense.
For calculation purposes it is convenient to consider, under the measure 𝑃, the related process 𝑆() defined by 𝑆(𝑡)=𝑒𝜇𝑡𝑆(𝑡)=𝑒𝑋(𝑡),(4.12) which, by the definition of 𝑋(), satisfies 𝑃𝑆(𝑇)>0=1.(4.13) Correspondingly, consider under the measure 𝑄 a process 𝑆𝑄(), defined by 𝑆𝑄(𝑡)=𝑒(𝜇𝑟)𝑡𝑆𝑄(𝑡)=𝑒𝑋𝑄(𝑡).(4.14) By Ito's lemma 𝑆𝑄() satisfies 𝑑𝑆𝑄𝑆(𝑡)=(𝑟𝜇)𝑄(𝑡)𝑑𝑡+𝑘𝑔𝑆ln𝑄𝑆(𝑡)𝑄(𝑡)𝑑𝑊𝑄(𝑡).(4.15) Define the 𝑄 local martingale 𝜂() by 𝜂(𝑡)=exp𝑡0𝜇𝑟𝑘𝑔𝑋𝑄(𝑢)𝑑𝑊𝑄1(𝑢)2𝑡0𝜇𝑟𝑘2𝑔𝑋𝑄.(𝑢)𝑑𝑢(4.16) Observe that, by the boundedness of 𝑔() and Novikov's condition, 𝜂() is a strictly positive, true martingale. Hence we can define a measure 𝑄𝜂 equivalent to 𝑄 by 𝑄𝜂(𝐴)=𝐸𝑄𝜂(𝑇)1𝐴,(4.17) for some 𝑇 measurable event 𝐴, and where 1𝐴 represents the indicator function for 𝐴. Girsanov's theorem shows that under 𝑄𝜂 there exists a Brownian motion 𝑊𝑄𝜂(), and where 𝑆𝑄() satisfies 𝑑𝑆𝑄(𝑡)=𝑘𝑔𝑆ln𝑄𝑆(𝑡)𝑄(𝑡)𝑑𝑊𝑄𝜂(𝑡).(4.18) Consider the probability of explosion to 0 in finite time. This can be checked with a criterion of Delbaen and Shirakawa [32]. Denote 𝐺() as the cumulative distribution function corresponding to 𝑔(). By the assumptions on 𝑔() in Section 2 we have, for all 𝜖,𝑀>0, with >𝑀𝜖, >sup𝜖𝑢𝑀𝑘𝑢𝑔(ln(𝑢))inf𝜖𝑢𝑀𝑘𝑢𝑔(ln(𝑢))>0(4.19) as required by Delbaen and Shirakawa [32, Theorem  1.4]. Consequently the probability of hitting 0 is decided by the convergence or divergence of the integral 10𝑔(ln(𝑦))𝑘𝑦𝑑𝑦=𝐺(0)𝐺()𝑘,(4.20) with divergence being equivalent to a 0 probability of the process reaching 0. As 0𝐺()1 it follows that (4.20) is finite, and hence 𝑆𝑄() can hit 0 in finite time. As 𝑆𝑄() is a 𝑄𝜂 local martingale bounded below by 0, it is also a supermartingale. Hence 0 is absorbing, and we have 𝑄𝜂𝑆𝑄(𝑇)=0>0,(4.21) and by equivalence 𝑄𝑆𝑄(𝑇)=0>0,(4.22) which by comparison to (4.13) implies that 𝑃 and 𝑄 cannot be equivalent measures.

As this is a complete market we can identify a specific arbitrage portfolio and strategy using the techniques of Levental and Skorohod [35]. Firstly note that as 𝑍() is not a true martingale, we have 𝐸𝑃[]𝑍(𝑇)=𝑐,(4.23) for some constant 𝑐<1. Define a 𝑃-martingale 𝑍() by 𝑍(𝑡)=𝐸𝑃𝑍(𝑇)𝑡,(4.24) which is a strictly positive martingale. It follows by Ito's lemma that there exist ̃𝜃()𝒫𝑃 such that we can represent 𝑍() in stochastic exponential form 𝑍(𝑡)=𝑐exp𝑡0̃1𝜃(𝑢)𝑑𝑊(𝑢)2𝑡0̃𝜃2(.𝑢)𝑑𝑢(4.25) Consider the process 𝑉(), with 𝑉𝑍(𝑡)=𝐵(𝑡)(𝑡)𝑍(𝑡)+1𝑐.𝑍(𝑡)(4.26) Applying Ito's lemma shows that 𝑉() is the value process of a self financing portfolio, with 𝜋(𝑡)=(𝜇𝑟)𝑔(𝑋(𝑡))𝑘+12𝑉(𝑡)+(𝜇𝑟)𝑔(𝑋(𝑡))𝑘+12̃𝜃(𝑡)𝑔(𝑋(𝑡))𝑘𝑍(𝑡)𝑍(𝑡)𝐵(𝑡).(4.27) As 𝑍0(),𝑍0()>0, we have 𝑉(0)=0,𝑉(𝑡)=𝐵(𝑡)𝑍(𝑡)+𝑍(𝑡)1𝑐𝑍(𝑡)11,𝑉(𝑇)=(1𝑐)𝐵(𝑇)𝑍(𝑇)>0,(4.28) implying that 𝑉() is an arbitrage opportunity.

5. Arbitrage-Free Specification

In the previous section we have shown that the model specification (4.1) will always admit arbitrage opportunities. In this section we further investigate the technical features of this specification that create the arbitrage and consequently construct a modification that is arbitrage free.

As 𝑄 is not an equivalent measure to 𝑃, the 𝑄-local martingale 𝑍𝑄(), where 𝑍𝑄(0)=1, and 𝑍𝑄(𝑡)=exp𝑡0𝛾𝑄(𝑢)𝑑𝑊𝑄(𝑢)𝑡012𝛾2𝑄(,𝑢)𝑑𝑢(5.1) is not strictly positive, with 𝛾𝑄(𝑡)=𝜇𝑟𝑘𝑔𝑋𝑄(𝑡)𝑘21𝑔𝑋𝑄,(𝑡)(5.2) and 𝑋𝑄() defined by (4.10). By Kazamaki [29] and Liptser and Shiryaev [30, 6.1.1] we have 𝑍𝑄(𝑇)=0𝑇0𝛾2𝑄,(𝑡)𝑑𝑢=(5.3) where all processes are stopped at the explosion time of 0𝛾2𝑄(𝑡)𝑑𝑢, which we will denote as 𝜏𝛾𝑄. Note that, by comparing (4.9) and (4.10) with (5.2), and as 𝑔() is bounded, we have 𝜏𝛾𝑄𝑆=𝜏𝑄 by construction.

As 𝑍𝑄() is not strictly positive, we have 𝑄𝜏𝛾𝑄𝑇>0,(5.4) and as 𝑔() is bounded it follows that the term in (5.2) causing (5.4) is of the form 1/𝑔(𝑋𝑄(𝑡)). In comparison to the standard model specification (4.1) it is apparent that this term arose from a drift adjustment for the quadratic variation of 𝑋(). This suggests that an alternative stock price model of the form 𝑑𝑆(𝑡)=𝜇𝑆(𝑡)𝑑𝑡+𝑆(𝑡)𝑑𝑋(𝑡),(5.5) or, equivalently, 1𝑆(𝑡)=exp𝜇𝑡2,𝑋(𝑡)+𝑋(𝑡)(5.6) where 𝑋() is the quadratic variation process of 𝑋(), defined by 𝑋(𝑡)=𝑡0𝑘𝑔(𝑋(𝑢))𝑑𝑢,(5.7) will be arbitrage free.

Theorem 5.1. The financial market with stock model (5.6)-(5.7) does not admit arbitrage opportunities.

Proof. Note firstly that we have, by the assumptions on 𝑔() in Section 2, 𝑃(𝑋(𝑇)<)=1.(5.8) Hence by Kazamaki [29] and Liptser and Shiryaev [30, 6.1.1] it follows that 𝑆() is strictly positive over [0,𝑇].
From Ito's lemma the discounted stock price process 𝑆()=𝑆()/𝐵() satisfies 𝑑𝑆𝑆(𝑡)=(𝜇𝑟)(𝑡)𝑑𝑡+𝑘𝑆𝑔(𝑋(𝑡))(𝑡)𝑑𝑊(𝑡).(5.9) An equivalent local martingale measure exists if and only if there exists a strictly positive martingale 𝑍(), with 𝑍(0)=1, and 𝑍(𝑡)=exp𝑡01𝜃(𝑢)𝑑𝑊(𝑢)2𝑡0𝜃2(,𝑢)𝑑𝑢(5.10) where 𝜃() is the market price of risk process, with 𝜃(𝑡)=𝜇𝑟𝑘𝑔(𝑋(𝑡)).(5.11) By construction the process 𝑔(𝑋()) is continuous and bounded. Hence by Novikov's condition it follows that 𝑍() is a strictly positive martingale and an (unique) equivalent local martingale measure 𝑄 exists. By the fundamental theorem of asset pricing it follows that the model is arbitrage free.

Under the unique equivalent local martingale measure 𝑄 the discounted stock price 𝑆() satisfies 𝑑𝑆(𝑡)=𝑘𝑆𝑔(𝑋(𝑡))(𝑡)𝑑𝑊𝑄(𝑡),(5.12) with 𝑋(𝑡)=𝑡0𝑘𝑔(𝑋(𝑢))𝑑𝑊𝑄(𝑢)𝑡0(𝜇𝑟)𝑑𝑢.(5.13)

6. Stock Markets with Stochastic Term Structure of Interest Rates

In Sections 3 to 5 we considered a financial market with a constant interest rate for clarity of presentation. In practical applications however the long term nature of many problems (e.g., in insurance and pensions) imply that such an assumption may be inappropriate. In this section we extend Theorem 5.1 to stock markets with a stochastic term structure of interest rates.

To allow for imperfect correlation between stock and interest rates we now consider a probability space (Ω,𝒢,𝐏) and the time interval [0,𝑇], the filtration being generated by 2 dimensional Brownian motion (𝑊(),𝑊𝑟()), augmented to satisfy the usual conditions. Following Section 5, we consider a stock price model of the form 𝑑𝑆(𝑡)=𝜇𝑆(𝑡)𝑑𝑡+𝑆(𝑡)𝑑𝑋(𝑡),(6.1) where the local martingale 𝑋() is assumed to be 𝑑𝑋(𝑡)=𝑘𝑔(𝑥)𝜌𝑑𝑊(𝑡)+1𝜌2𝑑𝑊𝑟,(𝑡)𝑋(0)=0,(6.2) for constants 𝑘>0,𝜌(1,1), and stationary density 𝑔().

Interest rate variability will be introduced via the second Brownian motion 𝑊𝑟() and its augmented filtration 𝑟(), with 𝑟𝒢. Following Heath et al. [36] (cf. Musiela and Rutkowski [37, Chapter  11]), assume that for every 𝑈𝑇, the instantaneous forward rate 𝑓(𝑡,𝑈) follows 𝑓(𝑡,𝑈)=𝑓(0,𝑈)+𝑡0𝛼𝑓(𝑢,𝑈)𝑑𝑢+𝑡0𝜎𝑓(𝑢,𝑈)𝑑𝑊𝑟(𝑢),(6.3) for a Borel measurable function 𝑓(0,), and 𝑟 progressively measurable processes 𝛼𝑓(,𝑈) and 𝜎𝑓(,𝑈) satisfying 𝑈0||𝛼𝑓||(𝑢,𝑈)𝑑𝑢+𝑈0||𝜎𝑓||(𝑢,𝑈)𝑑𝑢2<.(6.4) Assume 𝛼𝑓(𝑡,𝑈), 𝜎𝑓(𝑡,𝑈), and 𝑓(0,𝑈) are differentiable with respect to 𝑈, with bounded first derivatives 𝛼𝑓(𝑡,𝑈), 𝜎𝑓(𝑡,𝑈), and 𝑓(0,𝑈). It is known (cf. Musiela and Rutkowski [37, Proposition  11.1.1]) that the short rate process 𝑟() is a continuous semimartingale, with 𝑟(𝑡)=𝑟(0)+𝑡0𝜁(𝑢)𝑑𝑢+𝑡0𝜎𝑓(𝑢,𝑢)𝑑𝑊𝑟(𝑢),(6.5) where 𝜁(𝑡)=𝛼𝑓(𝑡,𝑡)+𝑓(0,𝑡)+𝑡0𝛼𝑓(𝑢,𝑡)𝑑𝑢+𝑡0𝜎𝑓(𝑢,𝑡)𝑑𝑊𝑟(𝑢).(6.6) Finally, assume that there exists a 𝑟 progressively measurable process 𝜆()𝒫𝑃 such that, for any 𝑈𝑇, we have 𝑈𝑡𝛼𝑓1(𝑡,𝑣)𝑑𝑣=2𝑈𝑡𝜎𝑓(𝑡,𝑣)𝑑𝑣2𝑈𝑡𝜎𝑓(𝑡,𝑣)𝑑𝑣𝜆(𝑡),(6.7) and such that the process 𝑍𝑟(), with 𝑍𝑟(𝑡)=exp𝑡0𝜆(𝑢)𝑑𝑊𝑟(1𝑢)2𝑡0𝜆2(,𝑢)𝑑𝑢(6.8) is a strictly positive martingale. This assumption is standard in literature and can intuitively be interpreted as assuming that the interest rate market is internally arbitrage free.

Under the above setup the savings account 𝐵() satisfies 𝐵(𝑡)=𝑒𝑡0𝑟(𝑢)𝑑𝑢,(6.9) and the price 𝐵(,𝑈) of a 𝑈-maturity Zero Coupon Bond process, with initial value 𝐵(0,𝑈), satisfies 𝑑𝐵(𝑡,𝑈)=𝑎(𝑡,𝑈)𝐵(𝑡,𝑈)𝑑𝑡+𝑏(𝑡,𝑈)𝐵(𝑡,𝑈)𝑑𝑊𝑟(𝑡),(6.10) where 𝑎(𝑡,𝑈)=𝑓(𝑡,𝑡)𝑈𝑡𝛼𝑓1(𝑡,𝑣)𝑑𝑣+2𝑈𝑡𝜎𝑓(𝑡,𝑣)𝑑𝑣2,𝑏(𝑡,𝑈)=𝑈𝑡𝜎𝑓(𝑡,𝑣)𝑑𝑣.(6.11)

Theorem 6.1. The financial market with stock model (6.1)-(6.2) and interest rate term structure modelled by (6.3)–(6.8) does not admit arbitrage opportunities.

Proof. The discounted stock price process 𝑆()=𝑆()/𝐵() satisfies 𝑑𝑆𝑆(𝑡)=(𝜇𝑟(𝑡))(𝑡)𝑑𝑡+𝑘𝑆𝑔(𝑋(𝑡))(𝑡)𝜌𝑑𝑊(𝑡)+1𝜌2𝑑𝑊𝑟(𝑡)(6.12) under the measure 𝑃. As 𝑍𝑟() is a strictly positive 𝑃 martingale by assumption, we can define a measure 𝑃𝑍𝑟 equivalent to 𝑃 by 𝑃𝑍𝑟(𝐴)=𝐸𝑃𝑍𝑟(𝑇)1𝐴,(6.13) for some 𝒢𝑇 measurable event 𝐴, and where 1𝐴 represents the indicator function for 𝐴. By Girsanov's theorem, under 𝑃𝑍𝑟 we have a 2-dimensional Brownian motion (𝑊(),𝑊𝑟,𝑃𝑍𝑟()), with 𝑊𝑟,𝑃𝑍𝑟(𝑡)=𝑊𝑟(𝑡)𝑡0𝜆(𝑢)𝑑𝑢,(6.14) with 𝜆() defined by (6.7). Under 𝑃𝑍𝑟 the short rate process 𝑟() satisfies 𝑟(𝑡)=𝑟(0)+𝑡0𝜁(𝑢)+𝜎𝑓(𝑢,𝑢)𝜆(𝑢)𝑑𝑢+𝑡0𝜎𝑓(𝑢,𝑢)𝑑𝑊𝑟,𝑃𝑍𝑟(𝑢),(6.15) which is continuous and is independent of 𝑊() by the assumptions on the coefficients of (6.3).
Consider now the nonnegative 𝑃𝑍𝑟 local martingale 𝑍(), with 𝑍(𝑡)=exp𝑡01𝜃(𝑢)𝑑𝑊(𝑢)2𝑡0𝜃2(,𝑢)𝑑𝑢(6.16) and where 𝜃() is the market price of risk process corresponding to 𝑊(), with 𝜃(𝑡)=𝜇𝑟(𝑡)𝜌𝑘𝑔(𝑋(𝑡))+1𝜌2𝜌𝜆(𝑡).(6.17) Notice that 𝑟()𝒫𝑃 by the continuity of 𝑟(). As 𝜆()𝒫𝑃 we also have (𝑟()+𝜆())𝒫𝑃, and hence by equivalence 𝑃𝑍𝑟𝑇0(𝑟(𝑢)+𝜆(𝑢))2𝑑𝑢<=1.(6.18) Consequently, by the boundedness of 𝑔(), we have 𝑃𝑍𝑟𝑇0𝜃2(𝑢)𝑑𝑢<=1,(6.19) and, by noting that 𝑟() and 𝜆() are independent of 𝑊(), we have (cf. Liptser and Shiryaev, [30, Example  6.2.4]) 𝐸𝑃𝑍𝑟𝑒(1/2)𝑇0𝜃2(𝑡)𝑑𝑡𝑟𝑇<.(6.20) It follows by Novikov's condition that 𝐸𝑃𝑍𝑟𝑍(𝑇)𝑟𝑇=1(6.21) and in particular, 𝐸𝑃𝑍𝑟[]𝑍(𝑇)=𝐸𝑃𝑍𝑟𝐸𝑃𝑍𝑟𝑍(𝑇)𝑟𝑇=1.(6.22) Equations (6.19) and (6.22) imply that 𝑍() is a strictly positive 𝑃𝑍𝑟 martingale, and hence we can define a measure 𝑄 equivalent to 𝑃𝑍𝑟 by 𝑄(𝐴)=𝐸𝑄𝑍(𝑇)1𝐴,(6.23) for some 𝒢𝑇 measurable event 𝐴, and where 1𝐴 represents the indicator function for 𝐴. By Girsanov's Theorem it follows that discounted asset prices are local martingales under 𝑄. As 𝑄 is equivalent to 𝑃𝑍𝑟 which is in turn equivalent to 𝑃, 𝑄 is an equivalent local martingale measure. By the fundamental theorem of asset pricing it follows that the model is arbitrage free.

7. Conclusions

In this paper we investigated the arbitrage-free property of the class of stock price models where the local martingale component is based on an ergodic diffusion with a specified stationary distribution. The dynamics of these models are time homogeneous and, as it is based on Brownian motion, tractable. The financial market under these models will be complete, and hence the valuation of options and guarantees can be performed without requiring extra assumptions regarding the market price of risk. In this paper we provided a detailed proof that any ergodic diffusion process used as a stock return model, and as specified in the existing literature, will admit arbitrage in general. We further analyzed the technical cause for these arbitrage opportunities and consequently constructed a modification that is arbitrage-free. This arbitrage free property is shown to be true in financial markets both with and without stochastic interest rates. Our modification once again opens up the application of ergodic diffusion models to problems in insurance and finance.

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