Abstract
We consider the geometric Markov renewal processes as a model for a security market and study this processes in a diffusion approximation scheme. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes in diffusion scheme are presented. We present European call option pricing formulas in the case of ergodic, double-averaged, and merged diffusion geometric Markov renewal processes.
1. Introduction
Let be a standard Poisson process and be i.i.d. random variables which are independent of and . The geometric compound Poisson processes is a trading model in many financial applications with pure jumps [1, page 214]. Motivated by the geometric compound Poisson processes (1.1), Swishchuk and Islam [2] studied the Geometric Markov renewal processes (2.5) (see Section 2) for a security market in a series scheme. The geometric Markov renewal processes (2.5) are also known as a switched-switching process. Averaging and diffusion approximation methods are important approximation methods for a switched-switching system. Averaging schemes of the geometric Markov renewal processes (2.5) were studied in [2].
The singular perturbation technique of a reducible invertible-operator is one of the techniques for the construction of averaging and diffusion schemes for a switched-switching process. Strong ergodicity assumption for the switching process means that the singular perturbation problem has a solution with some additional nonrestrictive conditions. Averaging and diffusion approximation schemes for switched-switching processes in the form of random evolutions were studied in [3, page 157] and [1, page 41]. In this paper, we introduce diffusion approximation of the geometric Markov renewal processes. We study a discrete Markov-modulated -security market described by a geometric Markov renewal process (GMRP). Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes in diffusion scheme are presented. We present European call option pricing formulas in the case of ergodic, double-averaged, and merged diffusion geometric Markov renewal processes.
The paper is organized as follows. In Section 2 we review the definition of the geometric Markov renewal processes (GMRP) from [2]. Moreover we present notation and summarize results such as random evolution of GMRP, Markov renewal equation for GMRP, infinitesimal operator of GMRP, and martingale property of GMRP. In Section 3 we present diffusion approximation of GMRP in ergodic, merged, and double-averaging schemes. In Section 4 we present proofs of the above-mentioned results. Section 4 contains solution of martingale problem, weak convergence, rates of convergence for GMRP, and characterization of the limit measure. In Section 5 we present merged diffusion GMRP in the case of two ergodic classes. European call option pricing formula for ergodic, merged, and diffusion GMRP are presented in Section 6.
2. The Geometric Markov Renewal Processes (GMRP)
In this section we present the Geometric Markov renewal processes. We closely follow [2].
Let be a standard probability space with complete filtration and let be a Markov chain in the phase space with transition probability , where . Let be a renewal process which is a sequence of independent and identically distributed (i.i.d.) random variables with a common distribution function . The random variables can be interpreted as lifetimes (operating periods, holding times, renewal periods) of a certain system in a random environment. From the renewal process we can construct another renewal process defined by The random variables are called renewal times (or jump times). The process is called the counting process.
Definition 2.1 (see [1, 4]). A homogeneous two-dimensional Markov chain on the phase space is called a Markov renewal process (MRP) if its transition probabilities are given by the semi-Markov kernel
Definition 2.2. The process is called a semi-Markov process.
The ergodic theorem for a Markov renewal process and a semi-Markov process respectively can be found in [3, page 195], [1, page 66], and [4, page 113].
Let be a Markov renewal process on the phase space with the semi-Markov kernel defined in (2.3), and let be a semi-Markov process where the counting process is defined in (2.2). Let be a bounded continuous function on such that . We define the geometric Markov renewal process (GMRP) as a stochastic functional defined by where is the initial value of . We call this process a geometric Markov renewal process by analogy with the geometric compound Poisson processes where , is a standard Poisson process, are i.i.d. random variables. The geometric compound Poisson processes in (2.6) is a trading model in many financial applications as a pure jump model [5, 6]. The geometric Markov renewal processes in (2.5) will be our main trading model in further analysis.
Jump semi-Markov random evolutions, infinitesimal operators, and Martingale property of the GMRP were presented in [2]. For the convenience of readers we repeat them again in the following.
2.1. Jump Semi-Markov Random Evolutions
Let be the space of continuous functions on vanishing at infinity, and let us define a family of bounded contracting operators on as follows: With these contraction operators we define the following jump semi-Markov random evolution (JSMRE) of the geometric Markov renewal processes in (2.5): Using (2.7) we obtain from (2.8) where is defined in (2.5) and . Let be a semi-Markov kernel for Markov renewal process , that is, , where is the transition probability of the Markov chain and is defined by . Let be the mean value of the semi-Markov random evolution in (2.9).
The following theorem is proved in [1, page 60] and [4, page 38].
Theorem 2.3. The mean value in (2.10) of the semi-Markov random evolution given by the solution of the following Markov renewal equation (MRE): where , is a bounded and continuous function on .
2.2. Infinitesimal Operators of the GMRP
Let A detailed information about and can be found in Section 4 of [2]. It can be easily shown that To describe martingale properties of the GMRP in (2.5) we need to find an infinitesimal operator of the process Let and consider the process on . It is a Markov process with infinitesimal operator where , , where . The infinitesimal operator for the process has the form: where . The process is a Markov process on with the infinitesimal operator where the operators and are defined in (2.17) and (2.18), respectively. Thus we obtain that the process is an -martingale, where . If is a Markov process with kernel namely, , then , , , and the operator in (2.17) has the form: The process on is a Markov process with infinitesimal operator where It follows that the process is an -martingale, where .
2.3. Martingale Property of the GMRP
Consider the geometric Markov renewal processes For let us define where is a bounded continuous function such that If , then geometric Markov renewal process in (2.25) is an -martingale, where measure is defined as follows: In the discrete case we have Let ,, where is defined in (2.27). If , then is an -martingale, where , and .
3. Diffusion Approximation of the Geometric Markov Renewal Process (GMRP)
Under an additional balance condition, averaging effect leads to diffusion approximation of the geometric Markov renewal process (GMRP). In fact, we consider the counting process in (2.5) in the new accelerated scale of time , that is, . Due to more rapid changes of states of the system under the balance condition, the fluctuations are described by a diffusion processes.
3.1. Ergodic Diffusion Approximation
Let us suppose that balance condition is fulfilled for functional : where is ergodic distribution of Markov chain . Then , for all . Consider in the new scale of time : Due to more rapid jumps of the process will be fluctuated near the point as . By similar arguments similar to (4.3)–(4.5) in [2], we obtain the following expression: Algorithms of ergodic averaging give the limit result for the second term in (3.3) (see [1, page 43] and [4, page 88]): where . Using algorithms of diffusion approximation with respect to the first term in (3.3) we obtain [4, page 88]: where , is a potential [3, page 68], of , is a standard Wiener process. The last term in (3.3) goes to zero as . Let be the limiting process for in (3.3) as . Taking limit on both sides of (3.3) we obtain where and are defined in (3.4) and (3.5), respectively. From (3.6) we obtain Thus, satisfies the following stochastic differential equation (SDE):
In this way we have the following corollary.
Corollary 3.1. The ergodic diffusion GMRP has the form and it satisfies the following SDE:
3.2. Merged Diffusion Approximation
Let us suppose that the balance condition satisfies the following: for all where is the supporting embedded Markov chain, is the stationary density for the ergodic component , is defined in [2], and conditions of reducibility of are fulfilled. Using the algorithms of merged averaging [1, 3, 4] we obtain from the second part of the right hand side in (3.3): where using the algorithm of merged diffusion approximation that [1, 3, 4] obtain from the first part of the right hand side in (3.3): where The third term in (3.3) goes to 0 as . In this way, from (3.3) we obtain: where is the limit as . From (3.16) we obtain Stochastic differential equation (SDE) for has the following form: where is a merged Markov process.
In this way we have the following corollary.
Corollary 3.2. Merged diffusion GMRP has the form (3.17) and satisfies the SDE (3.18).
3.3. Diffusion Approximation under Double Averaging
Let us suppose that the phase space of the merged Markov process consists of one ergodic class with stationary distributions . Let us also suppose that the balance condition is fulfilled: Then using the algorithms of diffusion approximation under double averaging (see [3, page 188], [1, page 49] and [4, page 93]) we obtain: where and and are defined in (3.13) and (3.15), respectively. Thus, we obtain from (3.20):
Corollary 3.3. The diffusion GMRP under double averaging has the form and satisfies the SDE
4. Proofs
In this section we present proofs of results in Section 3. All the above-mentioned results are obtained from the general results for semi-Markov random evolutions [3, 4] in series scheme. The main steps of proof are (1) weak convergence of in Skorokhod space [7, page 148]; (2) solution of martingale problem for the limit process ; (3) characterization of the limit measure for the limit process ; (4) uniqueness of solution of martingale problem. We also give here the rate of convergence in the diffusion approximation scheme.
4.1. Diffusion Approximation (DA)
Let and the balance condition is satisfied: Let us define the functions where and are defined as follows: where and . From the balance condition (4.2) and equality it follows that both equations in (4.3) simultaneously solvable and the solutions are bounded functions, .
We note that and define where and are defined in (4.4) and (4.5), respectively. We note, that .
4.2. Martingale Problem for the Limiting Problem in DA
Let us introduce the family of functions: where are defined in (4.7) and is defined by Functions are -martingale by . Taking into account the expression (4.6) and (4.7), we find the following expression: where is the sum of terms with nd order. Since is -martingale with respect to measure , generated by process in (4.1), then for every scalar linear continuous functional we have from (4.8)-(4.10): where is a mean value by measure . If the process converges weakly to some process as , then from (4.11) we obtain that is, the process is a continuous -martingale. Since is the second order differential operator and coefficient is positively defined, where then the process is a Wiener process with variance in (4.14): . Taking into account the renewal theorem for , namely, , and the following representation we obtain, replacing by , that process converges weakly to the process as , which is the solution of such martingale problem: is a continuous -martingale, where , and is defined in (4.5)-(4.5).
4.3. Weak Convergence of the Processes in DA
From the representation of the process it follows that This representation gives the following estimation: Taking into account the same reasonings as in [2] we obtain the weak convergence of the processes in DA.
4.4. Characterization of the Limiting Measure for as in DA
From Section 4.3 (see also Section of [2]) it follows that there exists a sequence such that measures converge weakly to some measure on as , where is the Skorokhod space [7, page 148]. This measure is the solution of such martingale problem: the following process is a -martingale for all and for scalar continuous bounded functional , is a mean value by measure . From (4.19) it follows that , and it is necessary to show that the limiting passing in (4.1) goes to the process in (3.12) as . From equality (4.11) we find that . Moreover, from the following expression we obtain that there exists the measure on which solves the martingale problem for the operator (or, equivalently, for the process in the form (4.12)). Uniqueness of the solution of the martingale problem follows from the fact that operator generates the unique semigroup with respects to the Wiener process with variance in (4.14). As long as the semigroup is unique then the limit process is unique. See [3, Chapter 1].
4.5. Calculation of the Quadratic Variation for GMRP
If , the sequence is -martingale, where . From the definition it follows that the characteristic of the martingale has the form To calculate let us represent in (4.22) in the form of martingale-difference: From representation it follows that , that is why Since from (4.22) it follows that then substituting (4.27) in (4.23) we obtain In an averaging scheme (see [2]) for GMRP in the scale of time we obtain that goes to zero as in probability, which follows from (4.27): for all . In the diffusion approximation scheme for GMRP in scale of time from (4.27) we obtain that characteristic does not go to zero as since where .
4.6. Rates of Convergence for GMRP
Consider the representation (4.22) for martingale . It follows that In diffusion approximation scheme for GMRP the limit for the process as will be diffusion process (see (3.10)). If is the limiting martingale for in (4.22) as , then from (4.31) and (3.10) we obtain Since , (because and are zero-mean martingales) then from (4.32) we obtain: Taking into account the balance condition and the central limit theorem for a Markov chain [4, page 98], we obtain where is a constant depending on ,. From (4.33), (4.2), and (4.32) we obtain: Thus, the rates of convergence in diffusion scheme has the order .
5. Merged Diffusion Geometric Markov Renewal Process in the Case of Two Ergodic Classes
5.1. Two Ergodic Classes
Let be the transition probabilities of supporting embedded reducible Markov chain in the phase space . Let us have two ergodic classes and of the phase space such that: Let be the measurable merged phase space. A stochastic kernel is consistent with the splitting (5.1) in the following way: Let the supporting embedded Markov chain with the transition probabilities be uniformly ergodic in each class and have a stationary distribution in the classes , : Let the stationary escape probabilities of the embedded Markov chain with transition probabilities be positive and sufficiently small, that is, Let the stationary sojourn time in the classes of states be uniformly bounded, namely, where
5.2. Algorithms of Phase Averaging with Two Ergodic Classes
The merged Markov chain in merged phase space is given by matrix of transition probabilities As , , then has virtual transitions. Intensities of sojourn times , , of the merged MRP are calculated as follows: And, finally, the merged MRP in the merged phase space is given by the stochastic matrix Hence, the initial semi-Markov system is merged to a Markov system with two classes.
5.3. Merged Diffusion Approximation in the Case of Two Ergodic Classes
The merged diffusion GMRP in the case of two ergodic classes has the form: which satisfies the stochastic differential equation (SDE): where is a merged Markov process in with stochastic matrix in (5.9).
6. European Call Option Pricing Formulas for Diffusion GMRP
6.1. Ergodic Geometric Markov Renewal Process
As we have seen in Section 3, an ergodic diffusion GMRP satisfies the following SDE (see (3.10)): where The risk-neutral measure for the process in (6.1) is: where Under , the process is a martingale and the process is a Brownian motion. In this way, in the risk-neutral world, the process has the following form Using Black-Scholes formula (see [8]) we obtain the European call option pricing formula for our model (6.6): where is a normal distribution and is defined in (6.3).
6.2. Double Averaged Diffusion GMRP
Using the similar arguments as in (6.1)–(6.7), we can get European call option pricing formula for a double averaged diffusion GMRP in (3.24): where and are defined in (3.21), (see also (3.13)), and (3.15). Namely, the European call option pricing formula for a double averaged diffusion GMRP is: where is a normal distribution and is defined in (3.21).
6.3. European Call Option Pricing Formula for Merged Diffusion GMRP
From Section 3.2, the merged diffusion GMRP has the following form: where and are defined in Section 3.2 (see (3.18). Taking into account the result on European call option pricing formula for regime-switching geometric Brownian motion (see [4, page 224, corollary]), we obtain the option pricing formula for the merged diffusion GMRP: where is a Black-Scholes value and is a distribution of the random variable where is a merged Markov process.
Acknowledgment
This research is partially supported by the University of Prince Edward Island major research grants (MRG) of M. S. Islam and NSERC grant of A. Swishchuk.