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1. Introduction

The Markov regime switching markets contain dramatic change in macroeconomic by incorporating a continuous-time Markov chain. In fact the rare events information reflect on stock price in those frame work. As known the regime switching markets are incomplete. So the pricing of regime switching risk gets an important issue. Option pricing is one of the most important concept in modern finance. Black and Scholes developed the methodology of option valuation. A major challenge in the Black- Scholes model is that interest rate and the volatility rate are assumed to be constants which are not consistent with reality [3].

To get more realistic models, many extensions to the Black-Scholes model have been presented. Among those the regime-switching models provide more realistic description for asset price dynamics. In these models the parameters are functions of a finite-state Markov chain [5, 6, 9, 13].

Because of several previous studies and the display of the dates, we added two stochastic volatility with three jumps. An excellent contribution of the proposed model is developing the model of stochastic volatility. In fact, in this study, we model the stock price process by the Markov-modulated jump diffusion model with double stochastic volatility with three jumps. So our model better corresponds with reality than the another one.

A unique equivalent martingale measure by minimizing the quadratic utility of the losses is identified by Follmer and Sondermann. Then the minimal martingale measure and risk-minimizing hedging were further developed by several researchers [1, 4, 8, 10, 11, 12, 15, 16, 17].

As it's well known, equivalent martingale measure is not unique in the incomplete market [14]. In this paper, Firstly, we investigate the minimal martingale measure. Then we address risk minimizing option pricing under our proposed model.

The rest of the paper is organized as follows. In Section 2, we present the notation, assumptions, and model for the underlying market. In Section 3, we investigate an explicit representation of the density process of the minimal martingale measure. In Section 4, a PDE of the option pricing is driven. The locally risk-minimizing strategy is studied in Section 5.

2. Preliminaries

Let ([OMEGA], F, {[F.sub.t]}, P) be the complete probability space. Suppose the states of an economy are modeled by a finite state continuous-time Markov chain {[X.sub.t] : t [greater than or equal to] 0}. Without loss of generality, we can identify the state space of {[X.sub.t] : t [greater than or equal to] 0} with a finite set of unit vectors x := {[e.sub.1], [e.sub.2], ..., [e.sub.N]}, where [e.sub.i] = (0, ..., 1, ..., 0) [member of] [R.sup.N],

whose transition probabilities satisfy

[mathematical expression not reproducible],

when [delta] [right arrow] 0, where [q.sub.ij] [greater than or equal to] 0, i [not equal to] j; [q.sub.ii] = [[summation].sup.N.sub.j=1] [q.sub.ij]. Let Q = [[q.sub.ij]] denote the generating Q-matrix of the Markov chain. The financial market itself is consisting of a riskless asset [([B.sub.t]).sub.t[member of][0; T]] and a risky asset [([S.sub.t]).sub.t[member of][0,T] which [S.sub.t] is square integrable and [S.sub.0] > 0 is a constant, dynamics of [([B.sub.t]).sub.t[member of][0; T]] and [([S.sub.t]).sub.t[member of][0; T]] are as follows:

[mathematical expression not reproducible],

where [W.sup.1.sub.t], [W.sup.2.sub.t], [W.sup.3.sub.t], and [W.sup.4.sub.t] are standard Brownian motions, that

d[W.sup.1.sub.t] x d[W.sup.2.sub.t] = [[rho].sub.1]dt, d[W.sup.3.sub.t] x d[W.sup.4.sub.t] = [[rho].sup.2]dt.

[[theta].sub.1] and [[theta].sub.2] are the long-run average of [V.sup.(1).sub.t] and [V.sup.(2).sub.t], respertively, [k.sub.1] and [k.sub.2] are the rates of mean reversion, [[sigma].sub.v1] and [[sigma].sub.v2] are the variance of [V.sup.(1).sub.t] and [V.sup.(2).sub.t], respectively, [Z.sub.1] and [Z.sub.2] two exponential stochastic processes with parameters [[mu].sub.v1] and [[mu].sub.v2], [[rho].sub.i] [member of] (-1, 1) for i = 1, 2 are given constants, and process N(dy, dt) is a Poisson random measure with P-compensator v(dy)dt = [lambda]f(y)dydt. Let [??](dy, dt) = N(dy, dt) - v(dy)dt be the compensated Poisson random measure. Moreover, we assume that [[integral].sup.[infinity].sub.-1] - [y.sup.2]v(dy) < [infinity]. In this setting, the locally risk-free floating interest rate [r.sub.t] and the appreciation rate [[mu].sub.t] of the stock price evolve over time depending on the state of the market [X.sub.t], therefore [r.sub.t] = r([X.sub.t]) and [p.sub.t] = [mu]([X.sub.t]) be two functions of [X.sub.t]; that is, [r.sub.t] = r(i) = [r.sub.i] and [[mu].sub.t] = [mu](i) = [[mu].sub.i] when the state of [X.sub.t] is i, i [member of] x Following the description of [2], for i, j [member of] x, i = j, let [[DELTA].sub.ij] be consecutive left closed right open intervals of the real line, each having length [q.sub.ij]. By embedding x in [R.sup.N] by identifying i with [e.sub.i] [member of] [R.sup.N] define a function h : [chi] x R [right arrow] [R.sup.N] by

[mathematical expression not reproducible].


[X.sub.t] = [X.sub.0] + [[integral].sup.t.sub.0][[integral].sub.R] h([X.sub.u-], z)P(dz, du)

where the integration is over the interval (0, T] and P(dz, dt) is a Poisson random measure with intensity m(dz)dt; where m(dz) is the Lebesgue measure on R. P(dz, dt), N(dy, dt), and [X.sub.t] are mutually independent, and independent of [W.sup.1.sub.2], [W.sup.2.sub.t], [W.sup.3.sub.t], and [W.sup.4.sub.t].

The semimartingale [mathematical expression not reproducible] has the following decomposition

[[??].sub.t] = [[??].sub.0] + [M.sub.t] + [A.sub.t]

with [M.sub.t] a square-integrable martingale for which [M.sub.0] = 0, and with [A.sub.t] is a predictable process of finite variation, where

(2.1) [mathematical expression not reproducible],


(2.2) [A.sub.t] = [[integral].sup.t.sub.0] [[??].sub.u-]([[mu].sub.u] - [r.sub.u])du.

3. The Minimal Martingale Measure

Noting that our proposed market is incomplete. More than, one martingale measure exists. In this section, we investigate the minimal martingale measure for presented market.

Definition 3.1. A martingale measure [??] [approximately equal to] P will be called minimal if

[??] = P on [F.sub.0].

and if any square-integrable P-martingale which is orthogonal to M under P remains a martingale under [??].

From [7], for some predictable process [alpha] = [([[alpha].sub.t]).sub.0 [less than or equal to] t [less than or equal to] T] we have

[A.sub.t] = [[integral].sup.t.sub.0] [[alpha].sub.u]d[<M>.sub.u].

Theorem 3.2. [??] exists if and only if

[mathematical expression not reproducible]

is a square-integrable martingale under P; in that case, [??] is giuen by d[??]/dP = [G.sub.T].

Let [mathematical expression not reproducible] denote the P-augmentation of the natural filtrations generated by S, [V.sup.(1)], [V.sup.(2)] and X, respectively. For each t [member of] [0, T], set [mathematical expression not reproducible], to avoid the possibility that the minimal martingale measure becomes a signed measure, we need the following condition.

(3.1) ([[mu].sub.t] - [r.sub.t]/[V.sup.(1).sub.t] + [V.sup.(2).sub.t] + [[integral].sup.[infinity].sub.- 1][y.sup.2]v(dy)) < 1, a.s. for t [member of] [0, T] and y > -1.

From theorem (3.2) we have

[mathematical expression not reproducible]


(3-2) [mathematical expression not reproducible],

for all t [member of] [0, T]. Now we will show that [Z.sub.t] is a square- integrable martingale under P and the measure [??] defined by [mathematical expression not reproducible] satisfies the definition of minimal martingale measure(see Definition 3.1).

Assume that there exists a minimal martingale measure, and let us denote it by [P.sup.*]. Define [Z.sub.t] by

[Z.sub.t] = E [d[P.sup.*]/dP]| [A.sub.t]].

Under [P.sup.*], the Doob-Meyer decomposition of M is given by

[M.sub.t] = [[??].sub.t] - [[??].sub.0] + (-[A.sub.t]).

But the theory of the Girsanov transformation shows that the predictable process of bounded variation can also be computed in terms of [P.sup.*]

-[A.sub.t] = [[integral].sup.t.sub.0] 1/[Z.sub.s-] d[<(M,Z)>.sub.s].

By Kunita-Watanabe decomposition, we have

[Z.sub.t] = 1 + [[integral].sup.t.sub.0] [[beta].sub.s]d[M.sub.s] + [L.sub.t],

where L is a square-integrable martingale under P orthogonal to M, and [beta] = [([[beta].sub.t]).sub.0[less than or equal to]t[less than or equal to]T] is a predictable process with

E[[[integral].sup.T.sub.0] [[beta].sup.2.sub.0]d <M>] < [infinity].

Since [P.sup.*] is a minimal martingale measure, we can easily obtain that L is [P.sup.*] martingale and that LZ is a P martingale. Then we have

<L, L> = <L, Z> = 0,

hence L [equivalent to] 0, [Z.sub.t] = 1 + [[integral].sup.t.sub.0] [[beta].sub.s][M.sub.s], and d[A.sub.t] = - [[beta].sub.t]/[Z.sub.t-] d <M, M>, so

[Z.sub.t] = 1 - [[integral].sup.t.sub.0] [Z.sub.s-] [d[A.sub.s]/d[<M>.sub.s]] d[M.sub.s].

Let d[Y.sub.s] = - d[A.sub.s]/d[<M>.sub.s]. From (2.1) and (2.2), we get

(3.3) [mathematical expression not reproducible].

Noting that there is a unique solution of (3.3), the minimal martingale measure is unique if it exists. We can get

[mathematical expression not reproducible],

from the formula of the Doleans-Dade exponential. Under conditions (3.1) and (3.2), Z is a square-integrable P martingale.

First, we can see that [??] is an equivalent martingale measure to P. Next, let L' be a P martingale and let it be orthogonal to M; that is, <L', M> = 0.

[mathematical expression not reproducible].

By the Girsanov-Meyer theorem, L' is a [??]-martingale. Hence, [??] is the unique minimal martingale measure of S.

From the Girsanov theorem we have

[mathematical expression not reproducible]

are standard [??]-Brownian motions.

Remark 3.3. Given [G.sub.T], under [??], the compensator of N(dy, dt) is

[mathematical expression not reproducible].

4. Option pricing

In this section, we derive the options pricing by Local risk minimization method. The price at time t of the European call option with strike price K and time to expiration T is given by

[mathematical expression not reproducible],

We set [V.sup.(1).sub.t] = [[alpha].sub.t] and [V.sup.(2).sub.t] = [[alpha]'.sub.t], and let

[mathematical expression not reproducible].

In the sequel, we apply Ito's formula for C(t, [S.sub.t], [[alpha].sub.t], [[alpha]'.sub.t], [X.sub.t]) and find its dynamics.

[mathematical expression not reproducible]

where [??](dz, dt) = P(dy, dt) - m(dz)dt is the compensated Poisson random measure and [??](dy, dt) = N(dy, dt) - [??](dy)dt. Since C(t, [S.sub.t], [[alpha].sub.t], [[alpha]'.sub.t], [X.sub.t]) is a [??] martingale, the drift term must be identical to zero. Hence, we have

[mathematical expression not reproducible],

with the terminal condition V(T, [S.sub.T], [[sigma].sub.T], [[sigma]'.sub.T], [X.sub.T]) = [([S.sub.T] - K).sup.+].

5. Locally risk-minimizing strategies

In this section we obtain an optimal hedging strategy in terms of local risk minimization.

Let H be the contingent claim with H [member of] [L.sup.2] ([OMEGA], A, P) at time T and [phi] = ([theta], [alpha]) be a portfolio, where [theta] = [([[theta].sub.t]).sub.0[less than or equal to]t[less than or equal to]T] is the amount of risky asset and [alpha] = [([[alpha].sub.t]).sub.0[less than or equal to]t[less than or equal to]T] the amount of risk less asset. The discounted portfolio valuation at time t is

[V.sub.t] = [[theta].sub.t][[??].sub.t] + [[alpha].sub.t].

Suppose [[alpha].sub.t] adapted process with E([[alpha].sup.2]) < [infinity], [theta] is predictable process and

(5.1) [mathematical expression not reproducible].

Our market is incomplete, so we find an admissible portfolio [phi] which minimizes, at each time t, the residual risk, given by

[R.sub.t]([phi]) = E [[([C.sub.T]([phi]) - [C.sub.t]([phi])).sup.2]|[A.sub.t]], t [less than or equal to] T

over all admissible portfolio. [C.sub.t]([phi]) = [V.sub.t]([theta]) - [[integral].sup.t.sub.0] [[theta].sub.s][[??].sub.s] is the discounted cost accumulated up to time t.

We have the following definitions from [16].

Definition 5.1 (Small Perturbation). A trading strategy [DELTA] = ([delta], [epsilon]) is called a small perturbation if it satisfies the following conditions:

1. [delta] is bounded,

2. [[integral].sup.T.sub.] [absolute value of ([[delta].sub.u]d[A.sub.u])] is bounded,

3. [delta]T = [epsilon]T = 0.

Definition 5.2 (Locally Risk Minimizing). For a trading strategy [phi], a small perturbation [DELTA], and a partition [tau] of [0, T], the risk quotient [r.sup.[tau]][[phi], [DELTA]] is defined as

[mathematical expression not reproducible].

A trading strategy [phi] is called locally risk minimizing if

[mathematical expression not reproducible],

for every small perturbation [DELTA] and every increasing sequence ([[tau].sub.n]) of the partition of [0, T] tending to the identity.

Definition 5.3 (Pseudo Locally Risk-Minimizing Hedging Strategy). A strategy is called pseudo locally risk minimizing, or equivalently pseudo optimal risk minimizing, if the associated cost process C([phi]) is a martingale under P and orthogonal to Mt.

Definition 5.4 (Follmer-Schweizer Decomposition).

[mathematical expression not reproducible],

is the Follmer-Schweizer Decomposition of the discounted contingent claim [mathematical expression not reproducible] satisfies formula (5.1) and if is a square-integrable P-martingale orthogonal to [M.sub.t], with [L.sup.H.sub.0] = 0. The associated optimal strategy given by [mathematical expression not reproducible] is locally risk minimizing.

We also need the following assumptions in [16]:

(1) For P-almost all [omega], the measure on [0,T] induced by <M> ([omega]) has the whole interval [0,T] as its support. This means that <M> should be P-almost surely strictly increasing on the whole interval [0,T].

(2) A is continuous.

(3) A is absolutely continuous with respect to <M> with density [alpha] satisfying E [[absolute value of ([alpha] [ln.sup.+]] [absolute value of ([alpha])])]] < [infinity]. A sufficient condition is that E [<[integral][alpha]dM>] < [infinity].

By [16], the pseudo locally risk-minimizing hedging strategy is the locally risk-minimizing strategy if assumptions (1)-(3) are satisfied. So, for [[??].sub.t], we check conditions (1)-(3).

[mathematical expression not reproducible].

[[??].sup.2.sub.u-] ([V.sup.(1).sub.u] + [V.sup.(2).sub.u] + [[integral].sup.[infinity].sub.-1] [y.sup.2]v(dy)) du > 0, [<M>.sub.t] is strictly increasing for every t [member of] [0, T]. Assumption (1) is verified. Note that

[A.sub.t] = [[integral].sup.t.sub.0][[??].sub.u-]([[mu].sub.u] - [r.sub.u])du,

is continuous, assumption (2) is satisfied. Also we have

[mathematical expression not reproducible]


[mathematical expression not reproducible]


[mathematical expression not reproducible]

then (5.2) is finite. So, assumption (3) is satisfied.

Now we derive the locally risk-minimizing strategy for the associated discounted portfolio. The Follmer-Schweizer decomposition of the associated discounted portfolio is

(5.3) V([phi]) = [V.sub.0]([phi]) + [[integral].sup.t.sub.0] [phi](s, u)d[[??].sub.u] + [L.sub.t],

So, we have

[mathematical expression not reproducible]

Since [L.sub.t] is a P martingale, the integrands with respect to du on the right-hand side should vanish. This gives us the following equation:

[mathematical expression not reproducible].

Received April 4, 2017


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Department of Mathematical Science, Shahrood University of Technology, Shahrood,

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Author:Dastranj, Elham; Namazi, Shiva
Publication:Dynamic Systems and Applications
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Date:Mar 1, 2017

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