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

Authors: Kuanhou Tian; Yanfang Li; Guixin Hu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

This 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 formulaϕt,ω is obtained when the T-claim Fω 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 θt=θ0t,θ1t are given explicitly. Finally, some concluding remarks are provided to illustrate the theoretical results.


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## Body

## 1. Introduction

The 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. The classical Black–Scholes model is often described by the following equations:(1)X0t=eρt,dXit=αiXitdt+∑j=1mβijXitdBjt,i=1,2,…,n,where βij is a n×m matrix and Bjt is a Brownian motion. The asset numbers 1,2,…,n are risky because of the presence of their diffusion terms and can be used to represent the stock investments. The 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 ofαi and βij 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 αi and βij 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 [1–4]. This 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 [5–7] 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 X0t and a risky asset X1t. Under the switching noise (Markov chain) and the white noise (Brownian motions), we give an explicit integrand formula ϕt,ω, the price p of the European call, and the self-financing portfolio θt=θ0t,θ1t.


## 2. Black–Scholes Model

Suppose that a market is described byX0t,X1t, where X0t is defined as(2)X0t=eρt.and X1t is an Itô process with the form(3)dX1t=αX1tdt+σdBt,X10=x1.In general, we regularly seek a portfolioθt to hedge the claim Fω=expX1T if ρ, α,andσ are constants (see [8] for more details). A T-claim F is usually given by(4)X0−1TFω=z+∫0Tϕt,ωdB˜t,where X0−1⋅ and ϕt,ω∈Rm satisfy(5)X0−1t,ωθ0t,…,θntσt,ω=ϕt,ω.θ0t can also be chosen by the corresponding formula. The portfolio θt=θ0t,…,θnt is needed to hedge a given claim. It is interesting to find an explicit formula of integrand ϕt,ω for a given T-claim Fω 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 ϕt,ω. To do so, we refer the reader to [9]. There 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 ϕt,ω.The 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 securitiesX0tandX1t, where X0t and X1t are two Itô processes of the form(6)dX0t=ρX0tdt,dX1t=αX1tdt+βX1tdBt.In the literature [8], the authors give the explicit formula for the self-financing portfolio θt=θ0t,θ1t that replicates the T-claim Fω=fX1T,ω explicitly. They also mentioned a model as(7)dX0t=ρt,ωX0tdt,dX1t=αt,ωX1tdt+βt,ωX1tdBt,where Bt is a 1-dimensional Brownian motion and ρt,ω,αt,ω,andβt,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),(8)X0t=X00exp∫0tρs,ωds,X1t=X10exp∫0tβs,ωdBs+∫0tαs,ω−12β2s,ωds,explicitly, if and only if(9)Eexp12∫0Tαs,ω−ρs,ω2β2s,ωds<∞.Suppose that there exists an equivalent martingale measureQ given by(10)dQω=exp−∫0Tut,ωdBt−12∫0Tu2t,ωdtdPω.Under this martingale measureQ, by the Girsanov theorem II, the process(11)B˜t≔∫0tus,ωds+Bt.is a Q-Brownian motion. Thus, equation (7) can be rewritten as(12)dX0t=ρt,ωX0tdt,dX1t=ρt,ωX1tdt+βt,ωX1tdB˜t,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 αt,ω and βt,ω in equation (7) are dependent on the random variable ω∈Ω in an unknown way. The portfolio θt for the T-claim Fω and the price p=pF at t=0 of the European options with T-claim Fω cannot be defined explicitly. But, when ρt,ω=ρt and βt,ω=βt are deterministic, the authors in [8] give the price at t=0 of a European option with payoff given by a contingent T-claim(13)Fω=fX1T,ω,for some lower bounded function f:R⟶R such that(14)EQfX1T<∞.The pricep=pF at time t=0 of a European option with payoff given by a contingent T-claim in equation (13) has the explicit form as(15)p=ξTEQfx1exp∫0TβsdB˜s+∫0Tρs−12β2sds,where ξt=X0−1t and B˜t is a Q-Brownian motion.Moreover, ifρ,α,andβ 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 θt=θ0t,θ1t have been given explicitly in [8]. Considering some practical meanings, we will discuss the case of ρ,α,andβ 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, ρt,ω,αt,ω,andβt,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 X0tandX1t that is formulated by a Black–Scholes model under regime switching.


## 3. Pricing and Hedging of an Attainable Claim

Throughout the paper, unless otherwise specified, letΩ,ℱ,ℱt,P be a complete probability space with a filtration ℱtt≥0 satisfying the usual conditions (it is right-continuous and increasing while ℱ0 contains all P-null sets). Let Bt 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(16)dX0t=ρrtX0tdt,X00=1,dX1t=αrtX1tdt+βrtX1tdBt,X10=x1>0,where rt is a right-continuous Markov chain taking values in a finite state space S=1,2. The generator Γ=γijN×N of rt is given by(17)ℙrt+δ=j|rt=i=γijδ+oδ,ifi≠j,1+γijδ+oδ,ifi=j,where δ>0 and γij≥0 are the transition rate from i to j satisfying γij>0 if i≠j while γii=−∑j≠iγij.We also assume that the Markov chain is irreducible which means that Markov chainrt has a unique stationary (probability) distribution π=π1,π2,…,πN∈R1×N that can be determined by solving the following linear equation:(18)πΓ=0,subject to(19)∑k=1Nπk=1,πk>0,for∀k∈S.Therefore, equation (16) can be regarded as the results of the following equations:(20)dX0t=ρ1X0tdt,dX1t=α1X1tdt+β1X1tdBt,dX0t=ρ2X0tdt,dX1t=α2X1tdt+β2X1tdBt,switching from one to the other according to the movement of Markov chain rt. Suppose that the stationary (probability) distribution of Markov chain is π=π1,π2 and the initial distribution of rt is also π=π1,π2. Then, for any t≥0, the Markov chain rt has a stationary distribution π=π1,π2 because it is irreducible. Note that the solution of equation (16) is(21)X0t=exp∫0tρrsds,X1t=x1exp∫0tβrsdBs+∫0tαrs−12β2rsds.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 letrt=1,ρrt=ρ1,αrt=α1 and βrt=β1. We can take the value rt=2,ρrt=ρ2,αrt=α2 and βrt=β2 to represent the state in contraction. More generally, we can use the state space S=1,2,…,N for the value of rt to model more complex business cycle structures. In this section, without loss of generality, we consider only two states for a market X0t,X1t by using equation (16).Theorem 1 (see [8]).
Suppose that a marketXt in terms of B˜t has the following form:(22)dX0t=ρXtX0tdt,dXit=ρXtXitdt+σiXtX1tdB˜t,1≤i≤n.And, assume that h0:Rn+1⟶R is a given function such that(23)∂∂xiEQxξT−th0XT−ti=1n,exists and(24)EQx∫0Tϕ2t,ωdt<∞,where(25)ϕt,ω≔∑i=1n∂∂xiEQxξT−th0YT−tx=XtσiXt.
Then, we have the Ito ̂representation formula(26)ξTh0XT=EQξTh0Xt+∫0Tϕt,ωdB˜t.Remark 1.
Note that the solution of equation (16) is Markovian and this makes it possible to apply the result of Theorem 1 to find ϕt,ω in equation (4).Lemma 1.
Letrt be a stationary Markov chain taking value in a finite state space S=1,2,…,N. Then, Yt=∫0tβrsdBs is a Gaussian process for any t>0.Proof.
We assume that the initial distribution ofrs is π=π1,π2,…,πN; then,(27)Yt=∫0tβrsdBs=∑i=1NπiβiBt.
Recall thatBt is a Gaussian process, so it is easy to see that for any t0≥0, the random variable Yt0 is normally distributed with mean 0 and variance ∑i=1Nπiβi2t0; hence, Yt is a Gaussian process.Remark 2.
By Lemma1, 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 X0t be a risk free asset and X1t 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.Theorem 2.
Suppose that a market is described byXt=X0t,X1t which is given by equation (16) with ρrt,αrt,βrt satisfying(28)Eexp12∫0Tαrs−ρrs2β2rsds<∞,for rs taking values in 1,2. Then, we have the following:(i)
The marketXt is no arbitrage and complete, and the price at time t=0 of the European T-claim Fω=fX1T,ω is(29)p=ξTΔ2π∫Rfx1expy+∫0Tρrs−12β2rsdsexp−y22Δ2dy,whereEQfX1T,ω<∞ and with(30)ξT=e−∫0Tρrsds=e−π1ρ1+π2ρ2T,Δ2=∫0Tβ2rsds=π1β21+π2β22T.(ii)
Iff∈C1R, then the self-financing portfolio θt=θ0t,θ1t needed to replicate the T-claim Fω=fX1T,ω is given by(31)θ1t=e∫0Tρrsds2πT−t∫Rf′X1t,ωexpπ1β1+π2β2x+∫0T−tρrs−12β2rsds⋅expπ1β1+π2β2x+∫0T−tρrs−12β2rsds⋅exp−x22T−tdx,andθ0t,ω is determined by(32)θ0t=Vθ0+ξtAt+∫0tρrsξsds,whereAt=∫0tθ1sdX1s+θ1tX1t and Vθ0=p.Proof.
(i)
It is easy to observe that there exists a processut,ω which satisfies(33)βrtX1tut,ω=αrtX1t−ρrtX1t.Then, equation (28) implies that(34)Eexp12∫0Tu2s,ωds<∞.Define the measureQ on ℱT by(35)dQω=exp−∫0Tut,ωdBt−12∫0Tu2t,ωdtdPω.Then,Q∼P and by the Girsanov theorem II (see [8, 11, 12]), the process(36)B˜t≔∫0tus,ωds+Bt.is aQ-Brownian motion. By Theorem 12.1.8 and Theorem 12.2.5 of [8], the market is complete with no arbitrage opportunity. Therefore, the price at t=0 of the European option with payoff given by a contingent T-claim Fω=fX1T,ω is(37)pF=EQξTF.That is,(38)p=ξTEQfx1exp∫0TβrsdB˜s+∫0Tρrs−12β2rsds.By Lemma1, under the measure Q, the random variable Y=∫0TβrsdB˜s is normally distributed with mean 0 and variance:(39)Δ2=∫0Tβ2rsds=π1β21+π2β22T.By the definition of the expectation of the function of random variables,p can be expressed explicitly as equation (29).(ii)
In terms ofB˜t, we rewrite the second equation of equation (16) as(40)dX1t=ρrtX1tdt+βrtX1tdB˜t.So, we seek the portfolio as(41)θ1t,ω=X0tβrtX1t,ω−1ϕt,ω,where(42)ϕt,ω≔∂∂xEQxξT−th0X1T−tx=X1tβrtX1t,with h0y=fy and(43)X1t=x1exp∫0tβrsdB˜s+∫0tρrs−12β2rsds=x1expπ1β1+π2β2B˜t+π1ρ1−12β21+π2ρ2−12β22t.Hence,(44)θ1t,ω=exp∫0tρrsds∂∂xEQx1exp∫0T−tρrsdsfX1T−tx=X1t=exp∫0tρrsds∂∂xEQx1fx1exp∫0T−tβrsdB˜s+∫0T−tρrs−12β2rsdsx1=X1t=exp∫0tρrsdsEQx1f′x1exp∫0T−tβrsdB˜s+∫0T−tρrs−12β2rsdsexp∫0T−tβrsdB˜s+∫0T−tρrs−12β2rsdsx1=X1t=exp∫0tρrsds2πT−t∫Rf′X1t,ωexpπ1β1+π2β2x+∫0T−tρrs−12β2rsdsexpπ1β1+π2β2x+∫0T−tρrs−12β2rsdsexp−x22T−tdx,which is assertion equation (31) and this completes the proof.Considering the irreducible of the Markov chainrt, we can get(45)∫0T−tρrs−12β2rsds=π1ρ1−12β21+π2ρ2−12β22T−t,∫0Tρrsds=π1ρ1+π2ρ2T.So, for the self-financing portfolioθt of a market described by a stochastic Black–Scholes model with Markovian switching, we find a perfect result than assertion equation (31) as(46)θ1t=eπ1ρ1+π2ρ2T2πT−t∫Rf′X1t,ωexpπ1β1+π2β2x+π1ρ1−12β21+π2ρ2−12β22T−t⋅expπ1β1+π2β2x+π1ρ1−12β21+π2ρ2−12β22T−t⋅exp−x22T−tdx.Remark 3.
Whenρ1=ρ2=ρ, α1=α1=α, and β1=β2=β, Theorem 2 reduces to the classical stochastic Black–Scholes formula. The T-claim Fω is given in Theorem 2 according to the movement of Markov chain. The results indicated that T-claim Fω 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 theT-claims of the European call option are(47)Fω=X1T,ω−K+,where the exercise price K>0 is a constant. Then, the price p at time 0 is(48)p=x1Φη+12π1β1+π2β2T−Ke−π1ρ1+π2ρ2TΦη−12π1β1+π2β2T,where(49)η=π1β1+π2β2−1T−1/2lnx1K+π1ρ1+π2ρ2T,Φy=12π∫−∞ye−1/2x2dx,y∈R,is the standard normal distribution function. Moreover, the replicating portfolio θt=θ0t,θ1t for the claim Fω in (47) is given by(50)θ1t,ω=Φπ1β1+π2β2−1T−t−1/2lnX1tK+π1ρ1+π2ρ2T−t+12π1β1+π2β23T−t.
θ1t,ω>0,t∈0,T means that we can replicate the European call without short selling. For the European put options with T-claims Fω=K−X1T,ω+, it should be θ1t,ω<0,t∈0,T which means that we have to short sell to replicate the European put option.
In fact, the pricep at time 0 given in equation (48) follows by applying Theorem 2 to the function(51)fx=x−K+.
It does not matter that the functionfx is not C1 because an approximation argument shows that equation (31) or (46) still holds if we represent f′ by(52)f′x=χK,∞x.To illustrate our theoretical results, we provide the following example for the pricing, hedging, and an appropriate portfolio of a given attainable claim.Example 1.
Suppose that a market isX0t,X1t, where X0t obeys that(53)dX0t=ρrtX0tdt,X00=1,and X1t is an Ornstein–Uhlenbeck process with Markovian switching as(54)dX1t=αrtX1tdt+σrtdBt,X10=x1>0,where rt is an irreducible Markov chain taking values in S=1,2. Let σ1=σ1,σ2=σ1, ρ1=ρ1,andρ2=ρ2. We can seek that the portfolio θt=θ0t,θ1t is(55)θ1t,ω=X0tσrt−1ϕt,ω,where ϕt,ω and V0=z are uniquely given by(56)ξTFω=z+∫0Tϕt,ωdB˜t,and the attainable claim is(57)Fω=expX1T=zeπ1ρ1+π2ρ2T+∫0Tϕ0t,ωdB˜t,with ϕ0t,ω=eπ1ρ1+π2ρ2Tϕt,ω. Then, by Theorem 1, we can find ϕt,ω and the portfolio θt explicitly.
In fact, we can rewrite equation (54) in terms of B˜t as(58)dX1t=ρrtX1tdt+σrtdBt,X10=x1.
Then, the solution to equation (58) can be explicitly expressed as(59)X1t=x1e∫0tρrsds+∫0tσrse∫0tρξτdτdB˜s=x1eπ1ρ1+π2ρ2t+π1σ1+π2σ2∫0teπ1ρ1+π2ρ2t−sdB˜s.
From equation (59), we know that the solution to equation (58) obeys the normal distribution for any t≥0. The mean of Xt is(60)EXt=Exeπ1ρ1+π2ρ2t=eπ1ρ1+π2ρ2tEx=0,and the variance is(61)VXt=EX2t−EXt2=EX2t=π1σ1+π2σ222π1ρ1+π2ρ2.
So,Xt obeys the normal distribution N0,π1σ1+π2σ22/2π1ρ1+π2α2 if the initial value X0=x obeys the normal distribution N0,π1σ1+π2σ22/2π1ρ1+π2ρ2.
Now, if we chooseh0x1=ex1, we can get that(62)EQx1h0X1T−t=EQx1expX1T−t=EQexpx1eπ1ρ1+π2ρ2T−t+π1σ1+π2σ2∫0T−teπ1ρ1+π2ρ2T−t−sdB˜s=expx1eπ1ρ1+π2ρ2T−t+π1σ1+π2σ224π1ρ1+π2ρ2e2π1ρ1+π2ρ2T−t−1,for π1ρ1+π2ρ2≠0. So, Theorem 1 implies that(63)ϕ0t,ω=ddx1EQx1h0X1T−tx1=X1tπ1σ1+π2σ2=π1σ1+π2σ2eπ1ρ1+π2ρ2T−texpX1teπ1ρ1+π2ρ2T−t+π1σ1+π2σ224π1ρ1+π2ρ2e2π1ρ1+π2ρ2T−t−1,and from ϕ0t,ω=eπ1ρ1+π2ρ2Tϕt,ω, we get the expression of ϕt,ω. Therefore,(64)θ1t=eπ1ρ1+π2ρ2T−texpX1teπ1ρ1+π2ρ2T−t+π1σ1+π2σ224π1ρ1+π2ρ2e2π1ρ1+π2ρ2T−t−1,ifπ1ρ1+π2ρ2≠0,expX1t+π1σ1+π2σ222T−t,ifπ1ρ1+π2ρ2=0.Remark 5.
We provide the results for the pricing and hedging of a EuropeanT-claim defined by the Ornstein–Uhlenbeck process with Markovian switching in example 2. If we let the generator of Markov chain rt be Γ=−112−2, then by solving linear equation (18), we have the result π1=2/3 and π2=1/3 that is the unique stationary (probability) distribution of rt. Taking the values of ρ1, ρ2, σ1, and σ2, we can explicitly get the expression of ϕt,ω and the portfolio θt of T-claim Fω.


## 4. Concluding Remarks

This paper mainly studied the Black–Scholes model with Markovian switching. The 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 formulaϕt,ω and p and θt when T-claim Fω 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 chainrt 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. This will lead to the back-and-forth conversion of the option pricing between several models. The 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. Therefore, we carry out the pricing and hedging of an attainable claim for the European call options in a Black–Scholes model with Markovian switching.The present paper is the first attempt, to our knowledge, to investigate the stochastic option pricing model with regime switching modulated by an irreducible Markov chainrt. 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.



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