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 N(t) be a standard Poisson process and

be i.i.d. random variables which are independent of N(t) 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 (B, S)-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 (Ω, ℬ, ℱ_t, ℙ) be a standard probability space with complete filtration ℱ_t and let

be a Markov chain in the phase space (X, 𝒳) with transition probability P(x, A), where x ∈ X, A ∈ 𝒳. Let

be a renewal process which is a sequence of independent and identically distributed (i.i.d.) random variables with a common distribution function F(x): = ℙ{w : θ_k(w) ≤ x}. 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 τ_k 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 X × ℝ₊ 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 X × ℝ₊ with the semi-Markov kernel Q(x, A, t) defined in (2.3), and let x(t): = x_v(t) be a semi-Markov process where the counting process v(t) is defined in (2.2). Let ρ(x) be a bounded continuous function on X such that ρ(x)>−1. We define the geometric Markov renewal process (GMRP)

as a stochastic functional S_t defined by

()

where S₀ > 0 is the initial value of S_t. We call this process

a geometric Markov renewal process by analogy with the geometric compound Poisson processes

()

where

, N(t) 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 C₀(ℝ₊) be the space of continuous functions on ℝ₊ vanishing at infinity, and let us define a family of bounded contracting operators D(x) on C₀(ℝ₊) as follows:

()

With these contraction operators D(x) we define the following jump semi-Markov random evolution (JSMRE) V(t) of the geometric Markov renewal processes

in (2.5):

()

Using (2.7) we obtain from (2.8)

()

where S_t is defined in (2.5) and S₀ = s. Let Q(x, A, t) be a semi-Markov kernel for Markov renewal process

, that is, Q(x, A, t) = P(x, A)G_x(t), where P(x, A) is the transition probability of the Markov chain

and G_x(t) is defined by G_x(t): = ℙ(θ_n+1 ≤ t∣x_n = x). Let

()

be the mean value of the semi-Markov random evolution V(t) in (2.9).

The following theorem is proved in [1, page 60] and [4, page 38].

Theorem 2.3. The mean value u(t, x) in (2.10) of the semi-Markov random evolution V(t) given by the solution of the following Markov renewal equation (MRE):

()

where

, g(x) is a bounded and continuous function on X.

2.2. Infinitesimal Operators of the GMRP

Let

()

A detailed information about ρ_T(x) 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 γ(t): = t − τ_v(t) and consider the process (x(t), γ(t)) on X × R₊. It is a Markov process with infinitesimal operator

()

where g_x(t): = dG_x(t)/dt,

, where f(x, t) ∈ C(X × R₊). The infinitesimal operator for the process ln S(t) has the form:

()

where z : = ln S₀. The process (ln S(t), x(t), γ(t)) is a Markov process on R₊ × X × R₊ 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 x(t): = x_v(t) is a Markov process with kernel

()

namely, G_x(t) = 1 − e^−λ(x)t, then g_x(t) = λ(x)e^−λ(x)t,

, and the operator

in (2.17) has the form:

()

The process (ln S(t), x(t)) on R₊ × X is a Markov process with infinitesimal operator

()

where

()

It follows that the process

()

is an ℱ_t-martingale, where ℱ_t : = σ(x(u); 0 ≤ u ≤ t).

2.3. Martingale Property of the GMRP

Consider the geometric Markov renewal processes

()

For t ∈ [0, T] let us define

()

where h(x) is a bounded continuous function such that

()

If EL_T = 1, then geometric Markov renewal process S_t in (2.25) is an (ℱ_t, P^*)-martingale, where measure P^* is defined as follows:

()

In the discrete case we have

()

Let

, EL₀ = 1, where h(x) is defined in (2.27). If EL_N = 1, then S_n is an (ℱ_t, P^*)-martingale, where dP^*/dP = L_N, and ℱ_n : = σ(x_k; 0 ≤ k ≤ n).

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 v(t) in (2.5) in the new accelerated scale of time tT², that is, v ≡ v(tT²). 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 p(x) is ergodic distribution of Markov chain

. Then

, for all t ∈ R⁺. Consider

in the new scale of time tT²:

()

Due to more rapid jumps of v(tT²) the process S_T(t) will be fluctuated near the point S₀ as T → +∞. 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

, R₀ is a potential [3, page 68], of

, w(t) is a standard Wiener process. The last term in (3.3) goes to zero as T → +∞. Let

be the limiting process for S_T(t) in (3.3) as T → +∞. 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 k = 1,2, …, r where

is the supporting embedded Markov chain, p_k is the stationary density for the ergodic component X_k, m(k) is defined in [2], and conditions of reducibility of X 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 T → +∞. In this way, from (3.3) we obtain:

()

where

is the limit S_T(t) as T → +∞. 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

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 D_R[0, +∞) [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 A(x): = [ρ²(x)/2 + ρ(x)(R₀ − I)ρ(x)]d²/ds². From the balance condition (4.2) and equality

it follows that both equations in (4.3) simultaneously solvable and the solutions

are bounded functions, i = 1,2.

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 G₀(t) in DA

Let us introduce the family of functions:

()

where ϕ^T are defined in (4.7) and

is defined by

()

Functions ψ^T(s, t) are

-martingale by t. Taking into account the expression (4.6) and (4.7), we find the following expression:

()

where O(T⁻²) is the sum of terms with T⁻²nd order. Since ψ^T(0, t) is

-martingale with respect to measure Q_T, generated by process G_T(t) in (4.1), then for every scalar linear continuous functional

we have from (4.8)-(4.10):

()

where E^T is a mean value by measure Q_T. If the process

converges weakly to some process G₀(t) as T → +∞, then from (4.11) we obtain

()

that is, the process

()

is a continuous Q_T-martingale. Since

is the second order differential operator and coefficient

is positively defined, where

()

then the process G₀(t) is a Wiener process with variance

in (4.14): G₀(t) = σw(t). Taking into account the renewal theorem for v(t), namely, T⁻¹v(tT²)→_T→+∞t/m, and the following representation

()

we obtain, replacing [tT²] by v(tT²), that process G_T(t) converges weakly to the process

as T → +∞, which is the solution of such martingale problem:

()

is a continuous Q_T-martingale, where

, and

is defined in (4.5)-(4.5).

4.3. Weak Convergence of the Processes G_T(t) in DA

From the representation of the process G_T(t) 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 G_T(t) in DA.

4.4. Characterization of the Limiting Measure Q for Q_T as T → +∞ in DA

From Section 4.3 (see also Section 4.1.4 of [2]) it follows that there exists a sequence T_n such that measures

converge weakly to some measure Q on D_R[0, +∞) as T → +∞, where D_R[0, +∞) is the Skorokhod space [7, page 148]. This measure is the solution of such martingale problem: the following process

()

is a Q-martingale for all f(g) ∈ C²(R) and

()

for scalar continuous bounded functional

, E is a mean value by measure Q. 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 T → +∞. From equality (4.11) we find that

. Moreover, from the following expression

()

we obtain that there exists the measure Q on D_R[0, +∞) 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

, the sequence

()

is ℱ_n-martingale, where ℱ_n : = σ{x_k, θ_k; 0 ≤ k ≤ n}. 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 tT we obtain that

goes to zero as T → +∞ in probability, which follows from (4.27):

()

for all t ∈ R₊. In the diffusion approximation scheme for GMRP in scale of time tT² from (4.27) we obtain that characteristic

does not go to zero as T → +∞ 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 T → +∞ will be diffusion process

(see (3.10)). If m₀(t) is the limiting martingale for

in (4.22) as T → +∞, then from (4.31) and (3.10) we obtain

()

Since

, (because

and m₀(t) are zero-mean martingales) then from (4.32) we obtain:

()

Taking into account the balance condition ∫_Xπ(dx)ρ(x) = 0 and the central limit theorem for a Markov chain [4, page 98], we obtain

()

where C₁(t₀) is a constant depending on t₀,t ∈ [0, t₀]. From (4.33), (4.2), and (4.32) we obtain:

()

Thus, the rates of convergence in diffusion scheme has the order T⁻¹.

5. Merged Diffusion Geometric Markov Renewal Process in the Case of Two Ergodic Classes

5.1. Two Ergodic Classes

Let P(x, A): = ℙ{x_n+1 ∈ A∣x_n = x} be the transition probabilities of supporting embedded reducible Markov chain

in the phase space X. Let us have two ergodic classes X₀ and X₁ of the phase space such that:

()

Let

be the measurable merged phase space. A stochastic kernel P₀(x, A) is consistent with the splitting (5.1) in the following way:

()

Let the supporting embedded Markov chain

with the transition probabilities P₀(x, A) be uniformly ergodic in each class X_k, k = 0,1 and have a stationary distribution π_k(dx) in the classes X_k, k = 0,1:

()

Let the stationary escape probabilities of the embedded Markov chain

with transition probabilities P(x, A): = ℙ{x_n+1 ∈ A∣x_n = x} 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

()

, k = 0,1, then

has virtual transitions. Intensities

of sojourn times

, k = 0,1, 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 P^* for the process in (6.1) is:

()

where

()

Under P^*, the process

is a martingale and the process w^*(t) = w(t) + θt 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

()

Φ(x) is a normal distribution and σ_ρ 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

()

Φ(x) 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.

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Citing Literature

All articles

Diffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas

Abstract

1. Introduction

2. The Geometric Markov Renewal Processes (GMRP)

2.1. Jump Semi-Markov Random Evolutions

2.2. Infinitesimal Operators of the GMRP

2.3. Martingale Property of the GMRP

3. Diffusion Approximation of the Geometric Markov Renewal Process (GMRP)

3.1. Ergodic Diffusion Approximation

3.2. Merged Diffusion Approximation

3.3. Diffusion Approximation under Double Averaging

4. Proofs

4.1. Diffusion Approximation (DA)

4.2. Martingale Problem for the Limiting Problem G₀(t) in DA

4.3. Weak Convergence of the Processes G_T(t) in DA

4.4. Characterization of the Limiting Measure Q for Q_T as T → +∞ in DA

4.5. Calculation of the Quadratic Variation for GMRP

4.6. Rates of Convergence for GMRP

5. Merged Diffusion Geometric Markov Renewal Process in the Case of Two Ergodic Classes

5.1. Two Ergodic Classes

5.2. Algorithms of Phase Averaging with Two Ergodic Classes

5.3. Merged Diffusion Approximation in the Case of Two Ergodic Classes

6. European Call Option Pricing Formulas for Diffusion GMRP

6.1. Ergodic Geometric Markov Renewal Process

6.2. Double Averaged Diffusion GMRP

6.3. European Call Option Pricing Formula for Merged Diffusion GMRP

Acknowledgment

References

Citing Literature

References

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Diffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas

Abstract

1. Introduction

2. The Geometric Markov Renewal Processes (GMRP)

2.1. Jump Semi-Markov Random Evolutions

2.2. Infinitesimal Operators of the GMRP

2.3. Martingale Property of the GMRP

3. Diffusion Approximation of the Geometric Markov Renewal Process (GMRP)

3.1. Ergodic Diffusion Approximation

3.2. Merged Diffusion Approximation

3.3. Diffusion Approximation under Double Averaging

4. Proofs

4.1. Diffusion Approximation (DA)

4.2. Martingale Problem for the Limiting Problem G0(t) in DA

4.3. Weak Convergence of the Processes GT(t) in DA

4.4. Characterization of the Limiting Measure Q for QT as T → +∞ in DA

4.5. Calculation of the Quadratic Variation for GMRP

4.6. Rates of Convergence for GMRP

5. Merged Diffusion Geometric Markov Renewal Process in the Case of Two Ergodic Classes

5.1. Two Ergodic Classes

5.2. Algorithms of Phase Averaging with Two Ergodic Classes

5.3. Merged Diffusion Approximation in the Case of Two Ergodic Classes

6. European Call Option Pricing Formulas for Diffusion GMRP

6.1. Ergodic Geometric Markov Renewal Process

6.2. Double Averaged Diffusion GMRP

6.3. European Call Option Pricing Formula for Merged Diffusion GMRP

Acknowledgment

References

Citing Literature

References

Related

Information

4.2. Martingale Problem for the Limiting Problem G₀(t) in DA

4.3. Weak Convergence of the Processes G_T(t) in DA

4.4. Characterization of the Limiting Measure Q for Q_T as T → +∞ in DA