The Pricing and Hedging of an Attainable Claim in a Hybrid Black–Scholes Model under Regime Switching

/is article formulates and dissects a Black–Scholes model with regime switching that can be used to describe the performance of a complete market. An explicit integrand formula φ(t,ω) is obtained when the T-claim F(ω) is given for an attainable claim in this complete market. In addition, some perfect results are presented on how to hedge an attainable claim for this Black–Scholes model, and the price p of the European call and the self-financing portfolio θ(t) � (θ0(t), θ1(t)) are given explicitly. Finally, some concluding remarks are provided to illustrate the theoretical results.


Introduction
e Black-Scholes model (1973), one of the most important models in modern financial theory, is often used to determine the fair prices of various options. Based on the research involving the classical Black-Scholes model, certain empirical phenomena have received considerable attention recently.
e classical Black-Scholes model is often described by the following equations: where β ij is a n × m matrix and B j (t) is a Brownian motion. e asset numbers 1, 2, . . . , n are risky because of the presence of their diffusion terms and can be used to represent the stock investments. e asset number 0 is risk free due to the absence of the diffusion term, and it can be used to represent a bank investment.
A very natural question is: if the values of α i and β ij are random, what will happen to the results? By taking advantage of the ergodic theory of irreducible Markov chain, this paper will provide a perfect result for the case of random α i and β ij according to the switching of Markov chain. As an application of our theoretical results, we will answer this question in Example 1.
It is well known that the adjustments of the interest rates by the central banks can produce large disturbances among various options and asset investments. For this reason, it is necessary to consider a switching noise in the Black-Scholes model. In this paper, we adopt the Markov chain to describe this switching noise as in [1][2][3][4]. is type of noise can be regarded as a significant fluctuation in the models and can be illustrated as a switching between n regimes.
We are motivated by the work of [5][6][7] for the option pricing, and we aim to hedge an attainable claim in a normalized market that is described by a stochastic Black--Scholes model with regime switching between two underlying assets that consist of a bond X 0 (t) and a risky asset X 1 (t). Under the switching noise (Markov chain) and the white noise (Brownian motions), we give an explicit integrand formula ϕ(t, ω), the price p of the European call, and the self-financing portfolio θ(t) � (θ 0 (t), θ 1 (t)).
(2) and X 1 (t) is an Itô process with the form In general, we regularly seek a portfolio θ(t) to hedge the claim F(ω) � exp(X 1 (T)) if ρ, α, and σ are constants (see [8] for more details). A T-claim F is usually given by where X − 1 0 (·) and ϕ(t, ω) ∈ R m satisfy θ 0 (t) can also be chosen by the corresponding formula. e portfolio θ(t) � (θ 0 (t), . . . , θ n (t)) is needed to hedge a given claim. It is interesting to find an explicit formula of integrand ϕ(t, ω) for a given T-claim F(ω) to make the portfolio self-financing. Using a generalized version of the Clark-Ocone theorem of the Malliavin calculus, one can find the explicit expression of ϕ(t, ω). To do so, we refer the reader to [9]. ere is a simpler method; however, for the Markovian case, see [8,10] for instance. However, there are no results for the regime-switching model, so the aim of this paper is to dissect a more practical model for the integrand formula ϕ(t, ω).
e Black-Scholes model assumes that a market consists of at least one risky asset and one riskless asset. Without loss of generality, here we let a market that has only two securities X 0 (t) and X 1 (t), where X 0 (t) and X 1 (t) are two Itô processes of the form dX 0 (t) � ρX 0 (t)dt, In the literature [8], the authors give the explicit formula for the self-financing portfolio θ(t) � (θ 0 (t), θ 1 (t)) that replicates the T-claim F(ω) � f(X 1 (T, ω)) explicitly. ey also mentioned a model as where B(t) is a 1-dimensional Brownian motion and ρ(t, ω), α(t, ω), and β(t, ω) are stochastic processes. (7) belongs to a small class of effectively solvable stochastic differential equations. It is easy to find the solution to equation (7), explicitly, if and only if Suppose that there exists an equivalent martingale measure Q given by Under this martingale measure Q, by the Girsanov theorem II, the process is a Q-Brownian motion. us, equation (7) can be rewritten as dX 0 (t) � ρ(t, ω)X 0 (t)dt, in terms of this Q-Brownian motion B(t). Suppose that the market defined by equation (12) has no arbitrage and it is complete. Moreover, only this information for the European option defined by equation (7) is known. Note that the coefficients α(t, ω) and β(t, ω) in equation (7) are dependent on the random variable ω ∈ Ω in an unknown way. e portfolio θ(t) for the T-claim F(ω) and the price p � p(F) at t � 0 of the European options with T-claim F(ω) cannot be defined explicitly. But, when ρ(t, ω) � ρ(t) and β(t, ω) � β(t) are deterministic, the authors in [8] give the price at t � 0 of a European option with payoff given by a contingent T-claim for some lower bounded function f: R ⟶ R such that e price p � p(F) at time t � 0 of a European option with payoff given by a contingent T-claim in equation (13) has the explicit form as 2 Discrete Dynamics in Nature and Society where ξ(t) � X − 1 0 (t) and B(t) is a Q-Brownian motion. Moreover, if ρ, α, and β are constants and β ≠ 0, it is a very important special case of equation (7); the price p of a European call option and the self-financing portfolio θ(t) � (θ 0 (t), θ 1 (t)) have been given explicitly in [8]. Considering some practical meanings, we will discuss the case of ρ, α, and β in equation (7) dependent on ω in the form of Markov chain. We will give some perfect results by the method of the ergodic theory of an irreducible Markov chain.
In the following part of our paper, we will discuss the Black-Scholes model under regime switching, which is a particular case of equation (7), that is to say, ρ(t, ω), α(t, ω), and β(t, ω) are dependent on ω in the form of Markov chain. At the same time, the model in this paper is an extension of the classical Black-Scholes model and we will give some more perfect results than in [8]. Without loss of generality, we firstly discuss a market X 0 (t) and X 1 (t) that is formulated by a Black-Scholes model under regime switching.

Pricing and Hedging of an Attainable Claim
roughout the paper, unless otherwise specified, let (Ω, F, F t , P) be a complete probability space with a filtration F t t ≥ 0 satisfying the usual conditions (it is rightcontinuous and increasing while F 0 contains all P-null sets). Let B(t) be a 1-dimensional standard Brownian motion defined on a complete probability space.
If we consider switching noise (Markov chain) in the classical Black-Scholes model, we can find a Black-Scholes model under regime switching that has the form where We also assume that the Markov chain is irreducible which means that Markov chain r(t) has a unique stationary (probability) distribution π � (π 1 , π 2 , . . . , π N ) ∈ R 1×N that can be determined by solving the following linear equation: erefore, equation (16) can be regarded as the results of the following equations: switching from one to the other according to the movement of Markov chain r(t). Suppose that the stationary (probability) distribution of Markov chain is π � (π 1 , π 2 ) and the initial distribution of r(t) is also π � (π 1 , π 2 ). en, for any t ≥ 0, the Markov chain r(t) has a stationary distribution π � (π 1 , π 2 ) because it is irreducible. Note that the solution of equation (16) is We all know that a business cycle is often divided into two or more different states, called "expansion" and "contraction" in financial economics. A growing economy is frequently described as being in expansion. For this, we can let r(t) � Discrete Dynamics in Nature and Society 1, ρ(r(t)) � ρ(1), α(r(t)) � α(1) and β(r(t)) � β (1). We can take the value r(t) � 2, ρ(r (t)) � ρ (2), α(r(t)) � α (2) and β(r(t)) � β (2) to represent the state in contraction. More generally, we can use the state space S � 1, 2, . . . , N { } for the value of r(t) to model more complex business cycle structures. In this section, without loss of generality, we consider only two states for a market (X 0 (t), X 1 (t)) by using equation (16).
Theorem 1 (see [8]). Suppose that a market X(t) in terms of B(t) has the following form: And, assume that h 0 : R n+1 ⟶ R is a given function such that exists and where en, we have the Itorepresentation formula Remark 1. Note that the solution of equation (16) is Markovian and this makes it possible to apply the result of eorem 1 to find ϕ(t, ω) in equation (4).

r(s))dB(s) is a Gaussian process for any t > 0.
Proof. We assume that the initial distribution of r(s) is π � (π 1 , π 2 , . . . , π N ); then, Recall that B(t) is a Gaussian process, so it is easy to see that for any t 0 ≥ 0, the random variable Y(t 0 ) is normally distributed with mean 0 and variance [ N i�1 (π i β(i))] 2 t 0 ; hence, Y(t) is a Gaussian process. □ Remark 2. By Lemma 1, we are able to study the hedging of an attainable claim of a European option defined by a Black-Scholes model with Markovian switching. In the following, we consider a situation where a market has just two securities; we let X 0 (t) be a risk free asset and X 1 (t) a risky asset that is an Itô process with the form of equation (16). We have the following result for the hedging of an attainable claim for this situation.
(33) en, equation (28) implies that Define the measure Q on F T by en, Q ∼ P and by the Girsanov theorem II (see [8,11,12]), the process is a Q-Brownian motion. By eorem 12.1.8 and eorem 12.2.5 of [8], the market is complete with no arbitrage opportunity. erefore, the price at t � 0 of the European option with payoff given by a contingent T-claim F(ω) � f(X 1 (T, ω)) is at is, By Lemma 1, under the measure Q, the random variable Y � T 0 β(r(s))dB(s) is normally distributed with mean 0 and variance: By the definition of the expectation of the function of random variables, p can be expressed explicitly as equation (29).
(ii) In terms of B(t), we rewrite the second equation of equation (16) as

t)dB(t). (40)
So, we seek the portfolio as where with h 0 (y) � f(y) and Hence, Discrete Dynamics in Nature and Society which is assertion equation (31) and this completes the proof.

□
Considering the irreducible of the Markov chain r(t), we can get So, for the self-financing portfolio θ(t) of a market described by a stochastic Black-Scholes model with Markovian switching, we find a perfect result than assertion equation (31) as Remark 3. When ρ(1) � ρ(2) � ρ, α(1) � α(1) � α, and β(1) � β(2) � β, eorem 2 reduces to the classical stochastic Black-Scholes formula. e T-claim F(ω) is given in eorem 2 according to the movement of Markov chain. e results indicated that T-claim F(ω) is dependent on the randomness of Markov chain, which extends the classical Black-Scholes model without Markovian switching.

Remark 4.
Applications to the pricing and hedging of the European call option: we know that the T-claims of the European call option are where the exercise price K > 0 is a constant. en, the price p at time 0 is where is the standard normal distribution function. Moreover, the replicating portfolio θ(t) � (θ 0 (t), θ 1 (t)) for the claim F(ω) in (47) is given by (50) θ 1 (t, ω) > 0, t ∈ [0, T] means that we can replicate the European call without short selling. For the European put options with T-claims F(ω) � (K − X 1 (T, ω)) + , it should be θ 1 (t, ω) < 0, t ∈ [0, T] which means that we have to short sell to replicate the European put option.
In fact, the price p at time 0 given in equation (48) follows by applying eorem 2 to the function It does not matter that the function f(x) is not C 1 because an approximation argument shows that equation (31) or (46) still holds if we represent f ′ by To illustrate our theoretical results, we provide the following example for the pricing, hedging, and an appropriate portfolio of a given attainable claim.

Concluding Remarks
is paper mainly studied the Black-Scholes model with Markovian switching. e hedging of an attainable claim of a European option defined by this model is discussed. Under the assumption that Markov chain is irreducible, we obtained the explicit formula ϕ(t, ω) and p and θ(t) when T-claim F(ω) is given. An example of a market defined by an Ornstein-Uhlenbeck process is used to illustrate our theoretical results.
A business cycle is often divided into two or more different states, called "expansion" and "contraction" in financial economics. In this paper, we used a regime switching modulated by an irreducible Markov chain r(t) to describe a business cycle. As described in the introduction, the adjustment of interest rates for the central bank will affect the operation of the economy and produce large economic fluctuations. For example, an interest rate increase will prompt investors to move their capitals towards the bank deposits. As a result, investments in stocks, options, and bonds will fall off. In contrast, decreased interest rates will cause capitals to flow into equities, options, and bonds. is will lead to the back-and-forth conversion of the option pricing between several models. e regime switching can be described by a Markov chain. For these reasons, it is necessary to consider the hedging and replication of an option pricing model under regime switching. erefore, we carry out the pricing and hedging of an attainable claim for the European call options in a Black-Scholes model with Markovian switching. e present paper is the first attempt, to our knowledge, to investigate the stochastic option pricing model with regime switching modulated by an irreducible Markov chain r(t). We believe that parts of methods and results appearing in this paper are also available for other option pricing models, such as American option pricing model, Parisian option pricing models, and currency option pricing models. We leave this additional work for our future research.

Data Availability
Data are available upon reasonable request to the corresponding author.

Conflicts of Interest
e authors declare no conflicts of interest.