Markovian regime-switching market completion using ...

IMA Journal of Management Mathematics Advance Access published October 21, 2011 IMA Journal of Management Mathematics Page 1 of 23 doi:10.1093/imaman/dpr018

Markovian regime-switching market completion using additional Markov jump assets

ROBERT J. E LLIOTT School of Mathematical Sciences, University of Adelaide, Adelaide, Australia and Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada [email protected] TAK K UEN S IU Department of Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia [email protected] AND

J UNYI G UO School of Mathematical Sciences and The Key Laboratory of Pure Mathematics and Combinatorics , Nankai University, Tianjin 300071, People’s Republic of China [Received on 15 July 2011; accepted on 4 september 2011] In this paper, we discuss the use of some representation results for double martingales to value and hedge contingent claims in a Markovian regime-switching market. A set of N Markov jump assets is introduced to complete the Markovian regime-switching market. Using a representation for double martingales, we justify the completeness of the enlarged market. An equivalent martingale measure, or price kernel, in the enlarged market is then identified by a measure change. The option pricing formula and the hedging portfolio in the enlarged market is also discussed. Keywords: Markovian regime-switching markets; double martingales; martingale representation; market completion; marked point processes.

1. Introduction Recently, Markovian regime-switching models have become important in economics and finance. The history of Markovian regime-switching model may be traced back to the works of Quandt (1958) and Goldfeld & Quandt (1973), where regime-switching regression models were considered and applied to model non-linear economic data. The idea of probability switching also appeared in the early development of non-linear time series analysis (see Tong, 1977, 1978; Tong & Lim, 1980). Hamilton (1989) c The authors 2011. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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X IN Z HANG∗ School of Mathematical Sciences and The Key Laboratory of Pure Mathematics and Combinatorics , Nankai University, Tianjin 300071, People’s Republic of China and Department of Applied Finance & Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia ∗ Corresponding author: [email protected]

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 dB(t) = B(t)hr, X(t)idt, dS(t) = S(t)dZ (t),

B(0) = 1,

S(0) = s > 0,

(1.1)

where dZ (t) := hμ, X(t)idt + hσ, X(t)idW (t). Here, W := {W (t)|t ∈ T } is a standard Brownian motion; X := {X(t)|t ∈ T } is a continuous-time Markov chain with state space E := {e1 , e2 , . . . , e N }, where ei ∈ < N and the jth component of ei is the Kronecker delta δi j , for each i, j = 1, 2, . . . , N (see Elliott et al., 1994); r := (r1 , . . . , r N )0 , μ := (μ1 , . . . , μ N )0 and σ := (σ1 , . . . , σ N )0 , where 0 denotes the transpose of a vector or matrix and h∙ ∙ ∙ , ∙ ∙ ∙ i is the scalar product in < N . Furthermore, to specify statistical, or probabilistic, properties of the chain, we define the generator A := [ai j ]i, j=1,2,...,N of the chain X under P. This is also called a rate matrix or a Q-matrix. The process (Z (t), X(t)) is called Markovian regime-switching diffusion or a Markov-modulated diffusion. The key feature of this model is that one set of model parameters is in force at any particular time point according to the state of the economy. It switches to another set when there is a transition in the state of the economy. This incorporates some important stylized empirical features of financial returns, such as time-varying volatility and heavy-tailness and asymmetry of unconditional distributions. These features are not captured and explained by the constant-coefficient Black–Scholes–Merton model. This flexibility and generalization do not come without cost. Indeed, due to the presence of an additional source of uncertainty induced by the chain X, the market is incomplete. Consequently, not all contingent claims can be perfectly hedged and there is more than one equivalent martingale measure. In this case, the standard Black–Scholes–Merton option valuation argument cannot be applied. This motivates the quest for some possible methods to solve the valuation problem.

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introduced and popularized Markovian regime-switching model in the economic and econometric literature. These Markovian regime-switching models can incorporate the impact of structural changes in (macro)-economic conditions on the random behaviour of price dynamics. From an empirical perspective, Markovian regime-switching models provide a good fit to many economic and financial time series data. From a theoretical perspective, they describe the stochastic evolution of investment opportunity sets over time. This has some interesting and important economic implications. Nowadays, we witness diverse applications of Markovian regime-switching model in economics and finance. Some works include Elliott & van der Hoek (1997) for asset allocation, Pliska (1997) and Elliott et al. (2001) for short rate models, Elliott & Hinz (2002) for portfolio analysis and chart analysis, Naik (1993), Guo (2001), Buffington & Elliott (2002a,b) and Elliott et al. (2005a, 2007b) for option valuation, Elliott et al. (2003) for volatility estimation, Elliott et al. (2007a) for pricing and hedging volatility swaps and Elliott et al. (2008) for risk measures. In the setting of a continuous-time model for option valuation, a Markovian regime-switching model can be described by considering the following simple situation. Suppose that there is a stochastic basis (Ω, F, F, P) with a finite-time horizon T := [0, T ] and a filtration F = (Ft )t∈[0,T ] satisfying the usual conditions of right continuity and completeness. We assume that (Ω, F, P) is rich enough to describe uncertainties arising from fluctuations of financial prices and structural changes in economic conditions. Suppose there are two primitive securities, namely, a bond and a share. These securities are traded continuously over a finite-time horizon T . The price processes of the bond and the share, denoted by B := {B(t)|t ∈ T } and S := {S(t)|t ∈ T }, respectively, are modelled by the stochastic differential equations as follows:

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1 Gerber & Shiu (1994) pioneered the use of the Esscher transform for option valuation in an incomplete market. Their work highlighted the interplay between actuarial and financial pricing. The use of the Esscher transform for option valuation under Markovian regime-switching model was further studied by Siu (2005, 2008) and Elliott et al. (2007a,b).

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Elliott et al. (2005a) introduced the use of a regime-switching version of the Esscher transform to select an equivalent martingale measure for option valuation in a Markovian regime-switching model.1 In this paper, we consider a different approach to value and hedge options in a Markovian regimeswitching market. We discuss the use of some representation results for double martingales developed in Elliott (1976) to give a theoretical analysis for the completion of the Markovian regime-switching market and then introduce a set of N Markov jump assets to complete the Markovian regime-switching market and to hedge a contingent claim. The idea of completing the Markovian regime-switching market is inspired by Corcuera et al. (2005, 2006), Niu (2008) and Guo (2001). In particular, the idea of using the set of Markov jump assets to complete the market is related to that in Corcuera et al. (2005), where a set of power-jump securities was used to complete a L´evy market. Like the power-jump securities in Corcuera et al. (2005), the Markov jump assets are not in an exponential form, and hence, can take negative values with positive probability. However, as explained in Corcuera et al. (2005), the powerjump securities have some advantages. For example, they may be used to hedge the uncertainty arising from the changes in the moments of different orders in a L´evy market. Markov jump securities are more suitable than the power-jump ones in the case of Markovian regime-switching model. The two types of securities have a similar structure, but they are not exactly the same. In addition to using power-jump securities to complete a L´evy market for option valuation, these securities may also be used for optimal portfolio selection (see, e.g. Corcuera et al., 2006; Niu, 2008). The Markov jump assets considered here may entail different interpretations from both theoretical and practical perspectives. From a practical perspective, these assets may be interpreted as an alternative class of assets, which have different risk and return profiles than some traditional assets, such as fixedincome securities and ordinary shares. Of course, in our current context, the Markov jump assets have risk and return profiles that are correlated with the states of an economy or business cycles. These assets can be thought of as proxies of some assets traded in practice, such as energy products, commodities and properties, whose price dynamics are more closely related to transitions of economic states or business cycles (see, e.g. Culot et al., 2006; Blochlinger, 2008). In the Markovian regime-switching market, the presence of the regime-switching effect incorporates the stochastic variation of investment opportunity sets (see, e.g. Jang et al., 2007). The uncertainty attributed to the stochastic variation of investment opportunity sets may induce hedging demand. Traditional assets may not have risk and return characteristics that are suitable for hedging against this additional source of uncertainty. Therefore, we introduce a class of alternative assets whose price dynamics are approximated by those of the Markov jump assets with a view to applying them to hedge against the uncertainty due to the stochastic variation of investment opportunity sets. At a conceptual level, the Markov jump assets possibly provide some insights into developing innovative financial products which might be suitable for hedging the risk attributed to economic transitions. This might suggest some interesting research opportunities for security design and financial innovation. The idea of using a Markov chain model for asset prices has been considered by some authors before (see, e.g. Duffie, 2001; Elliott & Kopp, 2005b). Guo (2001) proposed a valuation method using a set of ‘fictitious’ new securities, namely, the change-of-state contracts, to complete a Markovian regime-switching market. The origin of this idea was discussed in the classic monograph of Duffie (2001). A change-of-state contract is a contingent claim that pays its holder one unit of account at the next time that the Markov chain transits from one state to another. After the payment, the contract has no value and no future dividends. Then another

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2. Representation of double martingales In this section, we give some results on representation of double martingales discussed in Elliott (1976), Boel & Kohlmann (1980) and Last & Brandt (1995). These results will be used for our analysis on completing the Markovian regime-switching market. The key result of double martingales is that any martingales on a σ -field generated by a Brownian motion and an independent jump process can be represented as the sum of a stochastic integral of the Brownian motion and a stochastic integral with respect to the martingale associated with the jump process. This result was first established by Elliott (1976) and further investigated in Boel & Kohlmann (1980). The Markovian regime-switching model presented in the last section has two sources of uncertainty, one generated by the standard Brownian motion and another generated by a Markov chain. Indeed, a Markov chain has an equivalent representation as a marked point process (see, e.g. Elliott et al., 1994; Wu & Zeng, 2003, 2007). Consequently, it is natural to consider a representation for a double martingale generated by a Brownian motion and a jump process, where the jump process is associated to a marked point process. In the sequel, we give some main results of double martingales and the key steps of their proofs. For details of these results and their proofs, we refer readers to Elliott (1976), Boel & Kohlmann (1980) and Last & Brandt (1995). Consider a marked point process corresponding to a random jump process. Write Y := {Y (t)|t ∈ T } for a random jump process on (Ω, F, P) with values in a Lusin space (Y, Y ).2 Suppose {Tn |n = 1, 2, . . .} is an increasing sequence of the jump times of the process Y . That is, T0 ≡ 0, Tn+1 := inf{t > Tn : Y (t) 6= Y (Tn )},

(2.1)

where inf{∅} := ∞ by convention. 2 In general, one may consider (Y, Y ) as a Blackwell space (see, e.g. Elliott, 1982). However, for financial applications, a Lusin space is more than enough.

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new change-of-state contract is issued and pays at the next time the state of the chain changes and so on. Then a complete market results and a unique price for a square-integrable contingent claim is obtained. Using a representation for a double martingale, we show that the enlarged market augmented by N Markov jump assets is complete. We employ a measure change for both the diffusion and the jumps in the Markovian regime-switching market to determine the unique martingale measure in the enlarged market. This measure change is specified by a product of two density processes, one for the diffusion and another one for the jumps. A related measure change was considered in Elliott & Siu (2010). The present work here is different to that in Guo (2001) in two major respects. Firstly, the Markov jump securities used here are different from the change-of-state contracts in Guo (2001). Secondly, we adopt the double martingale representation in Elliott (1976) to provide a theoretical justification for the proposed method. The rest of the paper is structured as follows. Section 2 gives some results on the representation of double martingales that are used to complete the market. In Section 3, we discuss the completion of the Markovian regime-switching market and the hedging of a contingent claim. Section 4 gives the justification that the enlarged market is arbitrage-free and complete. In Section 5, we discuss the option pricing formula and obtain the hedging portfolio in an explicit form.

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Suppose that Y is non-explosive, i.e. T∞ := limn→∞ Tn = ∞, P-a.s. Define a sequence {Yn |n = 0, 1, . . .} by Yn := Y (Tn ),

(2.2)

Φ(B) := card{n: (Tn , Yn ) ∈ B}.

(2.3)

Let Φ(t) := Φ([0, t] × Y) be the number of jump times Tn less than or equal to t, for each t > 0. Then, the following lemma gives the relationship between the information set generated by the jump process Y and that generated by the marked point process Φ. L EMMA 2.1 Let FY (t) = σ {Y (s)|s 6 t}. Write F Y := {FY (t)|t ∈ T }. Then, for all t > 0, FY (t) = σ {Y (0), Φ(t)}.

(2.4)

The above lemma was established in Last & Brandt (1995, Lemma 2.5.5). Now, for any A ∈ Y , write Φ(t, A) for the number of jumps whose sizes are in the region A in the time interval [0, t], i.e. X I{Tn 6t,Yn ∈ A} , Φ(t, A) := Φ([0, t] × A) = n >1

where I E is the indicator function of an event E. Indeed, Φ(t, A) is a special case of a random measure (see, e.g. Elliott, 1982, Chapter 15). Suppose φ(t, A) is a unique predictable projection of Φ(t, A), for each t ∈ T and A ∈ Y . Then, a compensated ˜ A) of Φ(t, A) is defined by version Φ(t, ˜ A) := Φ(t, A) − φ(t, A). Φ(t,

(2.5)

˜ A)|t ∈ T } is an (F Y , P)-martingale. The following martingale Note that for each A ∈ Y , {Φ(t, representation was originally due to Elliott (1976) and then further developed by Boel & Kohlmann (1980). T HEOREM 2.1 Let G := {G(t)|t ∈ T } be the right-continuous complete filtration generated by the standard Brownian motion {W (t), t ∈ T } and the marked point process {Y (t), t ∈ T }. Then, for any (G, P)-local martingale M := {M(t)|t ∈ T }, there exist unique G-predictable processes h 0 := {h 0 (t)|t ∈ T } and h 1 (y) := {h 1 (t, y)|t ∈ T }, y ∈ Y, such that Z tZ Z t 0 ˜ h (s)dW (s) + h 1 (s, y)Φ(ds, M(t) = M(0) + dy). (2.6) 0

0

Y

R EMARK 2.1 Note that the first integral is continuous and of unbounded variation P-a.s., while the second is purely discontinuous (i.e. a compensated sum of jumps). For a comprehensive treatment of

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where Yn represents the value of the jump process Y at the nth jump epoch Tn . Consequently, the sequence Φ := {(Tn , Yn )|n = 1, 2, . . .} is a marked point process with marked space (Y, Y ) (see Last & Brandt, 1995, Example 1.8.5). Indeed, Φ is the marked point process generated by the jump process Y . For any B ∈ B( 0, P − a.s., we must have 1 + ψ 1 (s, Y (s)) > 0,

P-a.s.

Then, by Elliott (1982) and Jacod & Shiryaev (2003), under Q, t

ψ 0 (s)ds,

0

t ∈ T,

is a standard Brownian motion, and Φ˜ Q(t, A) := Φ(t, A) −

Z tZ 0

A

(1 + ψ 1 (s, y))φ(ds, dy),

t ∈ T,

(2.12)

is a (G, Q)-martingale, i.e. the unique predictable projection of Φ(t, A) under Q is given by Q

φ (t, A) =

Z tZ 0

A

(1 + ψ 1 (s, y))φ(ds, dy).

(2.13)

3. Enlarging the Markovian regime-switching market In this section, we introduce Markov jump assets to enlarge the Markovian regime-switching market in Section 1 and verify the completeness of the enlarged financial market.

3.1

Markov jump assets

Suppose (E, E) is a discrete marked space, where E = {e1 , e2 , . . . , e N } and E is the power set of E. Let Φ(t, j) be the number of jumps into the state e j up to and including time t, i.e. Φ(t, j) := Φ([0, t] × e j ) =

X

n >1

1(Tn 6t,Xn =e j ) ,

(3.1)

where {Tn |n = 1, 2, . . . , N } are the jump epochs of the chain X and Xn := X(Tn ). To obtain the unique predictable projection of Φ(t, j), we follow the results of Elliott (1994) and Elliott et al. (1994). Elliott et al. (1994) obtained the following semi-martingale dynamics for the chain X: X(t) = X(0) +

Z

t 0

A0 X(u)du + M(t),

where {M(t)|t ∈ [0, T ]} is an R N -valued, (F X , P)-martingale. Here, F X := {FX (t)|t ∈ T } is the right-continuous, P-completed, filtration generated by the chain X.

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Z

B Q(t) := B(t) −

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For each i, j = 1, 2, . . . , N , with i 6= j, and t ∈ [0, T ], let J i j (t) be the number of jumps from state ei to state e j up to time t. Then X J i j (t) := hX(s−), ei ihX(s), e j i 0 0, i, j = 1, . . . , N , i 6= j, are the constraints for constructing the Markov jump securities. It is worth mentioning that what we need for the construction of the Markov jump assets (4.1) is just a set of positive numbers aˉ i j which does not depend on any equivalent martingale measure. In what follows, we shall show that the enlarged market (1.1) augmented by the N Markov

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Z

t

j=1 0

= B(t)M(t) − M(0) − −

Z

MARKOVIAN REGIME-SWITCHING MARKET COMPLETION

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E0 (eα Z (t) ; X(t)) = etK(α) ,

(4.2)

where Ez (Y ; X(τ )) is the matrix with the (i, j)-element E z,i (Y ; X(τ ) = e j ); E z,i is expectation under Pz,i := P(∙|Z (0) = z, X(0) = ei ). From Asmussen (2000), we obtain the expression of K (α) as K(α) := A + diag(k1 (α), k2 (α), . . . , k N (α)),

(4.3)

where ki (α) := μi α +

σi2 α 2 . 2

For more information about the Markovian regime-switching diffusion, we refer to Asmussen (2000, Chapter II.5). P ˉ by the matrix (aˉ i j ). We now In what follows, we let aˉ ii := − j6=i aˉ i j , i = 1, . . . , N and denote A give a theorem for a measure change of the Markovian regime-switching diffusion. The proof of the theorem involves the use of the matrix cumulant generating function K (α). T HEOREM 4.1 Suppose ({Z (t), X(t)}, P) is a Markovian regime-switching diffusion with parameters (μi , σi2 , A). Then the following two statements are equivalent: 1. There is a probability measure Q equivalent to P such that ({Z (t), X(t)}, Q) is Markovian ˉ regime-switching diffusion with parameters (μˉ i , σˉ i2 , A). 2. All the following conditions hold (a) σˉ i = σi ; (b) there exists a vector ψ 0 := (ψ10 , ψ20 , . . . , ψ N0 ) such that μˉi = μi + ψi0 σi ;

(c) there exist vectors ψ N −1 := (ψ1 , . . . , ψ j−1 , ψ j+1 , . . . , ψ N )0 ∈ < N −1 for each j = 1, 2, . . . , N such that for i 6= j j

j

j

j

j

j

aˉ i j = (1 + ψi )ai j ,

and aˉ ii = −

X

j

(4.4)

(1 + ψi )ai j .

(4.5)

Proof. Suppose that P and Q are equivalent probability measures. Let   dQ L(t) := E F(t) . dP

(4.6)

j6=i

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jump assets (4.1) admits a unique equivalent martingale measure, so the arbitrage-free and completeness of the enlarged market are justified. By the fundamental theorem of asset pricing, the absence of arbitrage is ‘essentially’ equivalent to the existence of an equivalent martingale measure Q under which the discounted share price process and the discounted price processes of the set of N Markov jump assets are martingales. Therefore, we only need to find an equivalent martingale measure to ensure that the enlarged market is arbitrage-free. To achieve this, we need the following theorem on the measure change of the Markovian regime-switching diffusion. The matrix cumulant generating function, denoted by K (α), is an important characterization for the probability law of the Markovian regime-switching diffusion (Z (t), X(t)). K (α) is determined by

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Applying Theorem 2.2 to a Markov chain, we obtain that there exist predictable processes ψ 0 (s), ψ 1 (s), . . . , ψ N (s) such that L(t) = 1 +

Z

t 0

L(s−)ψ 0 (s)dW (s) +

N Z X

j=1 0

t

˜ L(s−)ψ j (s)Φ(ds, j).

(4.7)

2. the unique predictable projection of Φ(t, j) under Q is given by X (1 + ψ j (t))I{X(t−) = ei } ai j dt, φ(dt, j) =

(4.8)

i6= j

and therefore, Φ˜ Q(t, j) = Φ(t, j) −

XZ i6= j

t 0

(1 + ψ j (s))I{X(s−) = ei } ai j ds,

j = 1, 2, . . . , N ,

(4.9)

are basic martingales of the chain X under Q and 3. Under Q, Z (t) satisfies the equation Z t Z t Z t Z (t) − μ(s)ds − ψ 0 (s)σ (s)ds = σ (s)dWˉ (s). 0

0

(4.10)

0

From the expression for Z (t) in (4.10) and the fact that ({Z (t), X(t)}, Q) is a Markovian regimeswitching diffusion, ψ 0 (s) must be a function of X(s), i.e. there exists a vector ψ 0 = (ψ10 , . . . , ψ N0 )0 such that ψ 0 (s) = hψ 0 , X(s)i.

(4.11)

Consequently, the drift and the volatility of Z (t) with X(t) = ei under Q are μˉ i := μi + ψi0 σi and σˉ i := σi , respectively. ˉ the predictable projection of the marked Due to the fact that X is a Markov chain withP rate matrix A, point process Φ(t, j) under the measure Q is i6= j I{X(t−) = ei } aˉ i j dt. By the uniqueness of the predictable projection of Φ(t, j) and (4.8), we must have X X (1 + ψ j (t))I{X(t−) = ei } ai j dt = I{X(t−) = ei } aˉ i j dt, (4.12) i6= j

i6= j

i.e. (1 + ψ j (t))I{X(t−) = ei } ai j = I{X(t−) = ei } aˉ i j ,

i 6= j, i = 1, 2, . . . , N .

(4.13)

Consequently, ψ j (t) must be a function of X(t−), i.e. there exists a vector ψ N −1 = (ψ1 , . . . , ψ j−1 , ψ j+1 , . . . , ψ N ) ∈ < N −1 j

j

j

j

j

(4.14)

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By the generalized Girsanov’s theorem in VanSchuppen & Wong (1974) and the fact that the representation property is preserved if P is replaced by an equivalent probability measure Q, which was shown in Jacod & M´emin (1976), we have the following results: Rt 1. Wˉ := {Wˉ (t)|t ∈ T } defined by Wˉ (t) := W (t) − 0 ψ 0 (s)ds is an (G, Q)-Brownian motion;

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such that for each i 6= j, Note that

PN

ˉi j j=1 a

j

aˉ i j = (1 + ψi )ai j .

= 0, and, so aˉ ii = −

X j6=i

(4.15)

j

(1 + ψi )ai j .

(4.16)

L(t) = 1 +

Z

t 0

L(s−)hX(s), ψ 0 iW (s) +

N Z X

t

j=1 0

˜ L(s−)hX(s−), ψ j iΦ(ds, j),

(4.17)

where ψ 0 is given in Statement 2 of Theorem 4.1 and j

j

j

j

j

ψ j := (ψ1 , . . . , ψ j−1 , ψ j , ψ j+1 , . . . , ψ N ) ∈ < N j

(4.18)

j

is generated from ψ N −1 by adding any constant ψ j as the jth element of ψ j . Note that the impact of j ˜ ψ on the right side of (4.17) is just when X(s−) = e j , however, in this case Φ(ds, j) = 0. Hence, j j ψj

j

has no impact on the value of the right side of (4.17). The reason for replacing ψ N −1 by ψ j is that we wish to simplify the notation in our analysis. Thus, from (4.17), we obtain that {L(t)|t ∈ T } is a (F, P)-martingale, and, hence E[L(t)] = 1. For each t ∈ T , we define Qt (B) := E[L(t)I B ], for any B ∈ F(t). Now, we wish to show that the matrix cumulant generating function of Z (t) under the measure Q t is the same as that under the measure Q. In fact, applying Itˆo’s differentiation rule to M(t) := eα Z (t) δ k (t) with δ k (t) := hX(t), ek i, we get ! Z t 2σ 2 α k ds M(t) = hX(0), ek i + eα Z (s) hX(s), ek i αμk + 2 0 +

Z

+

N Z X

t 0

eα Z (s) hX(s−), Aek ids +

j=1 0

t

Z

t 0

eα Z (s) hX(s), ek iασk dW (s)

˜ j). eα Z (s) he j − X(s−), ek iΦ(ds,

(4.19)

Thus, applying the Itˆo product rule to N (t) := L(t)M(t) gives Z t Z t N (t) = M(0)L(0) + L(s−)dM(s) + M(s−)dL(s) + [M, L](t) 0

= hX(0), ek i +

Z

0

t

L(s−)e 0

α Z (s)

hX(s), ek i

α(μk + σk ψk0 ) +

α 2 σk2 2

!

ds

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Now suppose that Statement 2 in Theorem 4.1 is satisfied. We then prove that P and Q are equivalent. Suppose that L(t) is given by the stochastic integration equation

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Z

+

N Z X

+

N Z X

t 0

L(s−)eα Z (s) hX(s−), Aek ids + t

j=1 0

j=1 0

t

Z

t 0

L(s−)eα Z (s) hX(s), ek i(ασk + ψk0 )dW (s)

j
˜ j) L(s−)eα Z (s) he j − X(s−), ek i + hX(s−), ek iψk Φ(es,

L(s−)eα Z (s) he j − X(s−), ek ihX(s−), ψ j iφ(ds, j).

(4.20)

Note that N X j=1

he j − X(s−), ek ihX(s−), ψ j iφ(ds, j) = =

N X j=1

he j − X(s−), ek ihX(s−), ψ j i

N X X j=1 i6= j

X i6= j

I{X(s−) = ei } ai j ds

he j − X(s−), ek iI{X(s−) = ei } (aˉ i j − ai j )ds

ˉ k ids − hX(s−), Aek ids. = hX(s−), Ae

(4.21)

Thus, substituting the above equation into (4.20) and taking expectation yield E 0,i [L(t)eα Z (t) hX(t), ek i] = I{k=i} + +

Z

t 0

t 0

α 2 σk2 E 0,i [L(s−)eα Z (s) hX(s), ek i] α(μk + σk ψk0 ) + 2

!

ds

ˉ k i]ds E 0,i [L(s−)eα Z (s) hX(s), Ae

= I{k=i} +

+

Z

Z

Z tX N 0 j=1

t

α Z (s)

E 0,i [L(s−)e 0

αˉ 2 σk2 hX(s), ek i] α μˉ k + 2

E 0,i [L(s−)eα Z (s) hX(s), e j iaˉ jk ]ds.

!

ds

(4.22)

2 2

ˉ + diag(kˉ1 (α), . . . , kˉ N (α)) with kˉi (α) = μˉ i α + σˉ i α . Then, we can write (4.22) in ˉ Let K(α) := A 2 the matrix form as Z t ˉ E0 [L(s)eα Z (s) IX(s) ]K(α)ds, E0 [L(t)eα Z (t) IX(t) ] = I + (4.23) 0

where I is the (n × n)-identity matrix. Solving the above equation shows that the matrix cumulant generating function of Z (t) under the ˉ measure Qt is K(α), and, therefore {(Z (s), X(s))}s 6t is a Markovian regime-switching diffusion under

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the measure Qt with the same parameters as those under the measure Q. So Q and Qt are identical on F(t). Thus, by the definition of the Qt , we can easily get dQ = L(t), (4.24) dP F(t)

so Q and P are equivalent probability measures.

2. The intensity matrix A := (aˉ i j )16i, j 6 N of transitions of the state of the chain X under Q is j

aˉ i j = (1 + ψi )ai j , and aˉ ii = −

X j6=i

i 6= j, j

(1 + ψi )ai j ;

3. Under such Q given in the Theorem 4.1, Z (t) is governed by Z t Z t 0 Z (t) = (μ(s) + ψ (s)σ (s))ds + σ (s)dWˉ (s), 0

(4.25)

(4.26)

(4.27)

0

where ψ 0 (s) := hX(s), ψ 0 i. Now, we wish to find an equivalent martingale measure Q under which the discounted price process S˜ is a martingale. By the above Theorem 4.1, under such Q, Z (t) is given by (4.27) and therefore,   Z t ˜ : = exp − r (s)ds S(t) S(t) 0

= S(0) exp

Z t" 0

! # Z t σ (s)2 ˉ σ (s)dW (s) . μ(s) + σ (s)ψ (s) − − r (s) ds + 2 0 0

(4.28)

Then, the necessary and sufficient condition for S˜ to be a (G, Q)-martingale is the existence of ψ 0 := (ψ10 , . . . , ψ N0 ) such that μi + σi ψi0 − ri = 0,

i = 1, 2, . . . , N ,

(4.29)

i.e. ψi0 =

ri − μi , σi

i = 1, 2, . . . , N .

(4.30)

Now, we also require that the discounted price processes of the N Markov jump assets are (G, Q)martingales. From (4.1) together with the fact that the intensity matrix A of transition of {X(t)|t ∈ T }

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Theorem 4.1 gives the characterization of structure preserving the equivalence between the measures Q and P. The equivalent conditions in Theorem 4.1 imply that Rt 1. Wˉ := {Wˉ (t)|t ∈ T } defined by Wˉ (t) := W (t)− 0 hX(s), ψ 0 ids is an (G, Q)-Brownian motion;

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under Q is given by (4.25) and (4.26), we must have the following equations to ensure the discounted price processes of the N Markov jump assets are (G, Q)-martingales: j

aˉ i j = (1 + ψi )ai j , aˉ ii = −

N X j=1

i 6= j, j

(1 + ψi )ai j ,

j

ψi =

aˉ i j − 1. ai j

(4.31)

Thus, from Theorem 4.1 and the above analysis, the equivalent martingale measure Q such that the discounted share price and the discounted price processes of the N Markovian regime-switching assets are martingales is unique and given by dQ = L(t), (4.32) dP G (t)

where L(t) is determined by L(t) = 1 +

Z

t 0

0

L(s−)hX(s), ψ idW (s) +

N Z X

j=1 0

t

˜ L(s−)hX(s−), ψ j iΦ(ds, j),

(4.33)

j

with ψi0 for 1 6 i 6 N and ψi for i 6= j are given by (4.30) and (4.31), respectively. The reason that j we do not need to know the value of ψ j in (4.33) is the same as that in the proof of Theorem 4.1. Note that, by applying Remark 2.2 to the case of Markov chain, we must have 1 + hX(s−), ψ j i > 0,

a.s. j = 1, . . . , N .

From (4.31), in order to ensure the above inequality holds, we need to set j

1 + ψi =

aˉ i j > 0, ai j

i 6= j, i, j = 1, . . . , N ,

which implies that aˉ i j > 0. That is why we need aˉ i j > 0 to ensure that the enlarged market is arbitragefree and has a unique equivalent martingale measure. The above analysis yields the following theorem. T HEOREM 4.2 If we assume that the constants aˉ i j > 0, i 6= j, i, j = 1, . . . , N and construct a set of N Markov jump assets whose price processes are given by (4.1), then the enlarged Markovian regimeswitching market (augmented by N Markov jump assets) has a unique equivalent martingale measure Q given by dQ = L(t), (4.34) dP G (t)

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i.e

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MARKOVIAN REGIME-SWITCHING MARKET COMPLETION

where L(t) is determined by L(t) = 1 +

Z

t

0

L(s−)hX(s), ψ 0 idW (s) +

N Z X

j=1 0

t

˜ j), L(s−)hX(s−), ψ j iΦ(ds,

j

with ψi0 for 1 6 i 6 N and ψi for i 6= j are given by (4.30) and (4.31), respectively. Therefore, the enlarged Markovian regime-switching market is arbitrage-free and complete.

In the previous section, we have justified the arbitrage-free and completeness of the enlarged market (1.1) augmented by the set of N Markov jump assets (4.1). In other words, there is a unique equivalent martingale measure Q which is determined by (4.34). Furthermore, from the last section, the price process of the share under (4.34) is modelled by the stochastic differential equation dS(t) = S(t)(r (s)ds + σ (s)dWˉ (s)),

(5.1)

and the intensity matrix of the Markov chain X(t) is given by Aˉ := {aˉ i j }. Based on these results, we shall derive, in this section, the pricing formula and the hedging portfolio. We first consider a contingent claim whose pay-off is only a function of the value at maturity of the stock price (i.e. X = f (S(T ))). Note that the value of the contingent claim at time t is given by    Z T  (5.2) F(t, S, ei ) = E Q exp − r (s)ds f (S(T ))|S(t) = S, X(t) = ei . t

Write F := (F(t, S, e1 ), . . . , F(t, S, e N

))0 .

Then applying Itˆo’s differentiation rule to F gives  T  1 2 2 ˉ Ft + r (u)S(u)FS + σ (u) S(u) FSS + hF, AX(u)i du F(T, S(T ), X(T )) = F(t, S(t), X(t))+ 2 t Z T FS σ (u)S(u)dWˉ (u) + Z

t

+

We now write

N Z X

j=1 t

T

[F(u, S(u), e j ) − F(u, S(u), X(u−))]Φ˜ Q(du, j). (5.3)

 Z t  V (t, S, ei ) = exp − r (s)ds F(t, S, ei ). 0

Then, {V (t, S(t), X(t))|t ∈ T } is a (G, Q)-martingale. From the relationship between V (t, S(t), X(t)) and F(t, S(t), X(t)) as well as the integral expression for F(t, S(t), X(t)) in (5.3), we must have the following partial differential equation to ensure that {V (t, S(t), X(t))|t ∈ T } is a (G, Q)-martingale. That is, 1 ˉ Ft + r (u)S(u)FS + σ (u)2 S(u)2 FSS + hF, AX(u)i = r (u)F, 2

∀u ∈ T.

(5.4)

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5. Option pricing and hedging portfolios

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The above equation reduces to the following system of N coupled partial differential equations with X(u) = e1 , . . . , e N , 1 i ˉ i i = 0, + hF, Ae − ri F i + Fti + ri S FSi + σi2 S 2 FSS 2

i = 1, 2, . . . , N ,

(5.5)

where F i := F(t, S, ei ). Moreover, the terminal condition at maturity time T is then given by i = 1, 2, . . . , N .

(5.6)

The above analysis yields the following theorem. T HEOREM 5.1 Let F(t, S, ei ) be the value of the contingent claim X = f (S(T )) at time t in the enlarged market (1.1) augmented by the set of N Markov jump assets (4.1). That is, F(t, S, ei ) is given by (5.2). Then, F(t, S, ei ) satisfies the partial differential equation (5.5) with the terminal condition (5.6). The following theorem then gives an explicit form of the hedging portfolio for the claim X := f (S(T )). T HEOREM 5.2 Let F(t, S, ei ) be the value of the contingent claim X = f (S(T )) determined by (5.5) and (5.6) in the enlarged market (1.1) augmented by the N Markov jump assets (4.1). Then the selffinancing portfolio replicating the contingent claim X := f (S(T )) at time t is given by number of bonds = π r (t) = B(t)−1 [F(t, S(t), X(t)) − FS (t, S(t), X(t))S(t)] −B(t)−1

N X [F(t, S(t), e j ) − F(t, S(t), X(t−))]H j (t) j=1

B(t)

,

number of share = π 0 (t) = FS (t, S(t), X(t)), F(t, S(t), e j ) − F(t, S(t), X(t−)) , B(t)

number of Markov jump assets = π j (t) =

j = 1, 2, . . . , N .

Proof. Note that the value of the claim F(t, S(t), X(t)) satisfies the partial differential equation (5.4) and substituting this into the Itˆo expression of F(t, S(t), X(t)) in (5.3) yields the equation

F(T, S(T ), X(T )) = F(t, S(t), X(t)) + +

N Z X

j=1 t

T

Z

t

T

r (u)F(u, S(u), X(u))du +

Z

T t

FS σ (u)S(u)dWˉ (u)

[F(u, S(u), e j ) − F(u, S(u), X(u−))]Φ˜ Q(du, j).

(5.7)

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F(T, S, ei ) = f (S),

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We now observe that under the equivalent martingale measure Q, the price processes of the share and the Markov jump assets are modelled by (5.1) and (4.1), respectively. Consequently, we can rewrite the j above equation in the terms of S(t) and Ht as F(T, S(T ), X(T )) − F(t, S(t), X(t)) =

T F(u, S(u), X(u)) −

FS (u, S(u), X(u))S(u) − B(u)

t

+

Z

T t

FS (u, S(u), X(u))dS(u) +

N Z X

j=1 t

T

PN

j=1

[F(u,S(u),e j )−F(u,S(u),X(u−))]H j (u) B(u)

dB(u)

F(u, S(u), e j ) − F(u, S(u), X(u−)) dH j (u). B(u)

Hence, the result follows. Acknowledgements The authors would like to thank the referees and associate editor for their helpful comments. Funding National Natural Science Foundation of China (11001139), the Specialized Research Fund for the Doctoral Program of Higher Education (20100031120002), the Fundamental Research Funds for the Central Universities (65010771) and the Discovery grant from the Australian Research Council (DP1096243). R EFERENCES A SMUSSEN , S. (2000) Ruin Probabilities. Singapore: World Scientific. B LOCHLINGER , L. (2008) Power prices—a regime-switching spot/forward price model with Kim filter estimation. Ph.D. Thesis, University of St. Gallen, St. Gallen, Switzerland. B OEL , R. & KOHLMANN , M. (1980) Semimartingale models of stochastic optimal control, with applications to double martingales. SIAM J. Control Optim., 18, 511–533. B UFFINGTON , J. & E LLIOTT, R. (2002a) American options with regime switching. Int. J. Theor. Appl. Financ., 5, 497–514. B UFFINGTON , J. & E LLIOTT, R. (2002b) Regime switching and European options. Stochastic Theory and Control, Proceedings of a Workshop, Lawrence, KS, New York: Springer, pp. 73–81. C ORCUERA , J. M., N UALART, D. & S CHOUTENS , W. (2005) Completion of a L´evy market by power-jump assets. Financ. Stoch., 9, 109–127. C ORCUERA , J. M., N UALART, D. & S CHOUTENS , W. (2006) Optimal investment in a L´evy market. Appl. Math. Optim., 53, 279–309. C ULOT, M., G OFFIN , V., L AWFORD , S., DE M ENTEN , S. & S MEERS , Y. (2006) An affine jump diffusion model for electricity. Working Paper, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium. D UFFIE , D. (2001) Dynamic Asset Pricing Theory. Princeton, NJ: Princeton University Press. E LLIOTT, R.(1976) Double martingales. Probab. Theory Relat. Fields, 34, 17–28. E LLIOTT, R. (1982) Stochastic Calculus and Applications. New York: Springer. E LLIOTT, R. (1994) Exact adaptive filters for Markov chains observed in Gaussian noise. Automatica, 30, 1399– 1408.

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