Constant Proportion Portfolio Insurance under Regime

Switching Exponential Levy Process

Chengguo Weng1

Department of Statistics and Actuarial ScienceUniversity of Waterloo, Waterloo, N2L 3G1, Canada.

Abstract

The constant proportion portfolio insurance is analyzed by assuming thatthe risky asset price follows a regime switching exponential Levy process.Analytical forms of the shortfall probability, expected shortfall and expectedgain are derived. The characteristic function of the gap risk is also obtainedfor further exploration on its distribution. The specific implementation isdiscussed under some popular Levy models including the Merton’s jump-diffusion, Kou’s jump-diffusion, variance gamma and normal inverse Gaus-sian models. Finally, a numerical example is presented to demonstrate theimplication of the established results.

Keywords: constant proportion portfolio insurance, regime switching,exponential Levy process, shortfall, gap risk, matrix exponential.2000 MSC: 91B28, 62P05, 60J75JEL: G11

1. Introduction

Portfolio insurance refers to those managing techniques designed to pro-tect the value of a portfolio. They usually target to provide a guarantee onthe terminal portfolio value by maintaining the portfolio value process notfalling below a preset lower bound, which is called the floor. These techniquesallow the investors to participate in equity market for its potential gains froman upside market move while limit the downside risk. The most prominent

1Corresponding author. Tel: +001(519)888-4567 ext. 31132.Email address: c2weng@uwaterloo.ca.

Preprint submitted to Insurance: Mathematics and Economics

examples among the portfolio insurance strategies are the constant propor-tion portfolio insurance (CPPI) strategy and option-based portfolio insurance(OBPI) strategy. The OBPI combines a position in the risky asset with aput option on this asset; see for example El Karoui et al. (2005) and Lelandand Rurbinstein (1988) among many others.

The CPPI strategy involves no option. It adopts a simplified self-financingstrategy to allocate capital between a risky asset (typically a traded fund orindex) and a reserve asset (typically a bond) dynamically over time. In thismethod, the investor starts by setting a floor equal to the lowest acceptablevalue of the portfolio. Then, the investor computes the cushion as the excessof the portfolio value over the floor and allocates in the risky asset an amountof a constant multiple of the cushion. The constant is called multiplier.The amount allocated to the risky asset is known as the exposure, and theremainders are all invested in the reserve asset.

The CPPI strategy was initially introduced by Perold (1986) (see alsoPerold and Sharpe (1988)) for fixed-income instruments and Black and Jones(1987) for equity instruments. Extensive research has been conducted onCPPI in recent years, often either embedded in a more general framework orcompared with other portfolio insurance strategies. A comparison of OBPIand CPPI (in continuous time) is given in Bertrand and Prigent (2005) andBalder and Mahayni (2010); also see Do (2002) for an empirical investigationof both methods via simulation using Australian data. The performance ofcredit CPPI and constant proportion debt obligation structures are studiedby Garcia et al. (2008) under a dynamic multivariate jump-driven modelfor credit spreads, and an investigation in much more depth under a similarsetting can be found in Joossens and Schoutens (2010). The effect fromprice jumps on the performance of CPPI strategy is studied by Cont andTankov (2009) under a general exponential Levy process. The literature alsodeals with stochastic volatility models and extreme value approaches on theCPPI method; see Bertrand and Prigent (2002, 2003). A general frameworkof CPPI for investment and protection strategies is formulated by Dersch(2010) along with a review on other portfolio insurance techniques. Theinfluence of estimation risk on the performance of CPPI strategies as well asthe mitigation effect of the estimation risk by the robustification of mean-variance efficient portfolios are studied by Schottle and Werner (2010). Theeffectiveness of a CPPI portfolio with proportional trading cost is investigatedby Balder et al. (2009) under the Black-Scholes model and extended by Wengand Xie (2013) to a general exponential Levy model. Moreover, a log-normal

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approximation approach for the gain of CPPI structure is also developed byWeng and Xie (2013).

When the trajectories of the price processes of both risky asset and reserveasset are continuous, the CPPI strategy with continuous trading will lead toa terminal portfolio value no less than the guaranteed value certainly, andhence fully achieve the purpose of portfolio insurance. Nevertheless, it hasbeen widely noticed that there is always possibility for a CPPI portfolio tofall below the floor, leading to the notorious gap risk, which happens whenthe price of the risky asset drop substantially before the portfolio managercan rebalance the portfolio. Obviously, there are two main factors that maycontribute to the gap risk: the illiquidity of the investment assets and thejump in the asset price. In this paper, we focus on the effect from thejump features of the asset price while presume that the investment assets areperfectly liquid.

The present paper is motivated by Cont and Tankov (2009), where, whilethe main results are derived under the exponential Levy model assumption,the authors started with a semimartingale model setup and developed a gen-eral framework for evaluating the CPPI portfolio. The present paper aimsto generalize the main results of Cont and Tankov (2009) to a regime switch-ing exponential Levy model. While the exponential Levy process can wellcapture the jump feature in the price of financial assets, one of its obviouscriticisms is its time homogeneity. In reality, the economic state usuallyshows an obvious feature of transition between two or among several states,and the financial return has quite different characteristics under a differenteconomic state. As such, the regime switching model has been utilized widelynowadays; for its application in finance and actuarial science, see, for exam-ple, Buffington and Elliott (2002), Elliott et al. (1995), Elliott et al. (2005),Hardy (2001), Li et al. (2008), and Siu (2005) among many others.

More specifically, in the present paper the dynamics of the asset pricesare assumed to be governed by distinct exponential Levy processes underdifferent market states (e.g., bull and bear), and the transition from onemarket state to another is supposed to follow a hidden Markov process. Thepresent paper obtain analytical forms for those important risk measures thatare associated with the CPPI portfolio such as the short probability andthe expected shortfall. The characteristic function of the shortfall is alsoobtained in an explicit form so as to make it possible to further exploreon its distribution. In reality, the guarantee for the investors is usuallyprovided by a bank (guarantor), which owns the CPPI portfolio, subject

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to a premium, and thus gap risk is indeed assumed by the bank in exchangefor the premium charged on the investors. Our established results will behelpful not only for the guarantor to conduct an effective evaluation on thegap risk and compute a reasonable level of premium, and but also for theinvestors to develop a good understanding on their risk-and-reward profile ininvesting a CPPI fund; for details, see the beginning part of subsection 3.1.The specific (and its challenges for some models) is investigated for somepopular exponential Levy processes including the Merton’s jump-diffusion,Kou’s jump-diffusion, variance gamma and normal inverse Gaussian models;see section 4. Finally, our numerical example presented in 5 shows that theinitial market state (bull or bear) of the investment will take a critical rolein the resulting risk associated with a CPPI portfolio.

The rest of the paper is organized as follows. Section 2 is the model setup.The main results are collected in section 3, following some preliminaries inthe beginning of the section and followed by a discussion on how to derivethe explicit formulas for the established main results. Section 4 investigateshow to specifically implement main results for some popular exponentialLevy processes including the Merton’s jump-diffusion, Kou’s jump-diffusion,variance gamma and normal inverse Gaussian models. Section 5 presents anumerical example, where the effect of the regime switching feature of themarket is demonstrated. Section 6 concludes the paper. Finally, proof ofsome lemmas and equations are relegated to the appendix.

2. Model setup

Throughout the paper, we suppose that all the random elements involvedare defined on a common probability space (Ω,F ,P), and that the expecta-tion of a random variable Z under P is denoted by E(Z). The transpose ofa matrix H (a vector a) will be denoted by H′ (a′).

Assume that the state of the financial market is described by a finite stateMarkov process X := (Xt)t≥0. In particular, there could be just two statesfor X, respectively, representing bull and bear; if we assume a third marketstate, it is typically interpreted as a normal state. As introduced by Elliottet al. (1995) and Buffington and Elliott (2002), we suppose that the Markovprocess X is generated by an intensity matrix Q with a finite state space ofunit vectors e1, · · · , en, where ek = (0, · · · , 0, 1, 0, · · · , 0)′ ∈ Rn with 1 inits kth coordinate and 0 in all the others. According to Elliott et al. (1995),

4

X has the following semimartingale representation

Xs = X0 +

∫ s

0

XuQ du+Ms, (2.1)

where M is a martingale with respect to the filtration FXt generated by X.Hereafter we assume that the CPPI portfolio is allocated between a stock

index and a zero-coupon bond, and their price processes S and B are respec-tively subject to the following dynamics

dStSt−

= dZt anddBt

Bt−= dRt, (2.2)

for two processes Z and R admitting the following regime switching structure

Zt =n∑j=1

〈ej, Xt〉Z(j)t , and Rt =

n∑j=1

〈ej, Xt〉R(j)t , (2.3)

where 〈a,b〉 denotes the inner product∑n

j=1 ajbj for two vector a ≡ (a1, . . . , an)′

and b ≡ (b1, . . . , bn)′, and all Z(j)t and R

(j)t are some semimartingales. The

above specification implies that S and B are respectively driven by Z(j)t and

R(j)t when the market is in the jth regime state ej, i.e., Xt = ej.

Let Vt denote the value of the CPPI portfolio at time t for t ≥ 0, andwithout loss of any generality, we assume that the floor is set as (Bt)t≥0 sothat the cushion at time t is computed as Ct = Vt − Bt. At any time t withVt > Bt, the portfolio is maintained at an exposure of mCt ≡ m(Vt − Bt) instock, where m > 1 is a constant multiplier; if Vt ≤ Bt, the entire portfoliois invested into the zero-coupon bond.

To analyze the CPPI portfolio, we borrow the same argument Cont andTankov (2009) apply via the tools of stochastic exponential and change ofnumeraire. Let τ := inft ≥ 0 : Vt ≤ Bt be the first time for which the CPPIportfolio value process V := (Vt)t≥0 touches or breaks through the preset floor(Bt)t≥0. To avoid the trivial case with τ = 0, we assume V0 > B0. Since theCPPI strategy is self-financing, up to time τ the portfolio value satisfies

dVt = m(Vt− −Bt−)dStSt−

+ Vt− −m(Vt− −Bt−)dBt

Bt

,

which, by using the equality Ct = Vt −Bt, can be rewritten as

dCtCt−

= mdZt + (1−m)dRt.

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We employ the change of numeraire technique by introducing the followingdiscounted cushion process

C∗t =CtBt

.

It follows from the change-of-variable formula that

dC∗tC∗t−

= m (dZt − d[Z,R]t − dRt + d[R]t) , (2.4)

where [·, ·] and [·] respectively denote the quadratic covariation and thequadratic variation of the corresponding processe(s). Let

Lt := Zt − [Z,R]t −Rt + [R]t. (2.5)

Then, we can rewrite equation (2.4) as

C∗t = C∗0E(mL)t,

where E denotes the stochastic (Doleans-Dade) exponential defined by

dE(mL)tE(mL)t−

= mdLt.

Therefore, the discounted cushion value at a general time t can be expressedas

C∗t = C∗0E(mL)t∧τ , t ≥ 0,

or alternatively

VtBt

= 1 +

(V0

B0

− 1

)E(mL)t∧τ , t ≥ 0.

Hereafter we assume that Z(j)t and R

(j)t in representation (2.3) are all

Levy processes, which are distinct at a different superscript j. We refer toSato (1999), Applebaum (2004) and reference therein for general theory onLevy processes, and Schoutens (2003) and Cont and Tankov (2004) for theirapplications in finance modelling. From (2.5), Lt has the following regimeswitching structure

Lt =n∑j=1

〈ej, X(t)〉L(j)t (2.6)

6

with

L(j)t =

(Z

(j)t −

[Z(j), R(j)

]t−R(j)

t +[R(j)

]t

), (2.7)

which is obviously a Levy process, j = 1, . . . , n. Assume L(j)t has the following

Levy-Ito decomposition

dL(j)t = bjdt+ σjdWt +

∫|x|>1

xNj(dt, dx) +

∫|x|≤1

x[Nj(dt, dx)− vj(dx)dt](2.8)

with a Levy measure vj and a corresponding Poisson random measure Nj

so that Nj(t, A) is a Poisson process with intensity vj(A) for any Borel setA ⊂ R \ 0. The process L can be expressed in a more succinct way asfollows

Lt = 〈LtXt〉 , with Lt =(L

(1)t , · · · , L(n)

t

)′.

3. Gap risk and payoff of CPPI strategies

3.1. Preliminaries

Consider a CPPI portfolio invested over a finite time investment horizon[0, T ] with T > 0, where the investors pay the initial value V0 at time 0 andare guaranteed to receive at least the value of the bond BT at time T . Ifthe portfolio value VT at time T is smaller than BT , a third party pays tothe investor the shortfall amount BT − VT . In practice, this guarantee isindeed usually provided by the bank which owns the CPPI portfolio, subjectto a fee. Consequently, at the expiration date T , the investors will receive apayoff of

maxVT , BT ≡ BT + maxCT , 0 ≡ BT + CT ICT>0,

and the CPPI guarantor will be subject to a gap risk of CT ICT≤0, where I·is the indicator function. The evaluation on both quantities are interesting.An insightful investigation on the payoff can help the investors to establisha comprehensive understanding on their risk-and-reward profile in investingthe CPPI portfolio, and a thorough analysis on the gap risk is necessary forthe CPPI guarantor not only in computing a reasonable premium chargedon the investors but also in their internal risk management. Obviously, boththe gap risk of the guarantor and the payoff of the investors depend on

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the performance of bond, in addition to the stock. While it is interestingto conduct the investigation by taking into account the performance of bothassets, in this paper we focus on the effect from the inherent jump features inthe stock price and the regime switching mechanism of the financial market.Therefore, we measure both quantities at time 0 by discounting them usingthe bond price process B, so that the time 0 value of the payoff for theinvestors is

1 + max

CTBT

, 0

≡ 1 + maxC∗T , 0 (3.9)

and the time 0 value of the gap risk isCTBT

ICT≤0. We further assume that

the trajectories of the bond price process B is continuous so that all of thesethree events τ ≤ T, CT ≤ 0 and C∗T ≤ 0 are eventually the same,all indicating that a gap risk comes up during the investment time horizon[0, T ]. As a result, the time 0 value of the gap risk is the same as

CTBT

ICT /BT≤0 = C∗T IC∗T≤0. (3.10)

From equations (3.9) and (3.10), we note that the time 0 values of boththe payoff and the gap risk only depend on the discounted value of cushionC∗t . We shall focus on this quantity in our subsequent analysis.

To quantify the gap risk and payoff, we shall investigate the followingquantities:

i) the shortfall probability defined by

SF(T ) = P(C∗T < 0),

ii) the unconditional expected shortfall defined by

UES(T ) = E(C∗T1C∗

T≤0),

iii) the expected shortfall defined by

ES(T ) = E (C∗T |C∗T ≤ 0) ,

iv) the unconditional expected gain defined by

UEG(T ) = E(C∗T1C∗

T≤0).

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v) the conditional expected gain defined by

EG(T ) = E (C∗T |C∗T > 0) .

Besides, we will also work towards to the distribution function of C∗T byinvestigating its characteristic function.

Remark 3.1. It is obvious from equations (2.4) and (2.5) that, when thedriving process L is continuous, the loss probability SF(t) will be zero, andthus the gap risk must be resulted from the jump in the asset price if it comesup. This implies that the inherent Levy measures vj in process L will takea critical role in evaluating these measures defined above, as we can also seeshortly.

Remark 3.2. If the investment horizon is not too long (say T=1 (or 2)meaning one (or two) year), we may assume that the return rate associatedwith the bond is a constant r. In this case, the expected shortfall and theexpected gain of the time T quantity CT can be computed by simply multiplyingthe corresponding measure with a factor of erTB0.

Hereafter, for t > 0 and j = 1, . . . , n, we denote

Γj(t) :=

∫ t

0

〈ej, Xu〉 du, (3.11)

the total amount of time which hidden Markov process X spends in state ejover the time horizon [0, t], i.e., the Lebesgue measure of the set u ∈ [0, s] :Xu = ej. Clearly, we have

∑nj=1 Tj(t) = t almost surely, t > 0. Borrowing

the idea used in derivation of the characteristic function of (Γ1(t), . . . ,Γn(t))developed by Buffington and Elliott (2002), we have the following lemma.

Lemma 3.1. Let a := (a1, . . . , an) be a scalar vector, and Xt be the regime

switching process defined in equation (2.1). Denote Yt := exp

(∫ t

0

〈a, Xu〉 du)Xt.

Then, for t ≥ 0,

(a) E[Yt] = E

(exp

(n∑j=1

ajΓj(t)

)Xt

)= (E(X0))′ exp (Q + diag(a)) t,

(b) E(

exp

(∫ t

0

〈a, Xu〉 du))

=⟨1, (E(X0))′ exp (Q + diag(a)) t

⟩,

9

(c) E

(exp

(n∑j=1

ajΓj(t)

))=⟨1, (E(X0))′ exp (Q + diag(a)) t

⟩.

Proof. The proof is completely parallel with that of Lemma A.1 in Buffin-gton and Elliott (2002) and hence omitted.

3.2. Shortfall probability

A CPPI portfolio incurs a shortfall (breaks through the floor), if Vt ≤ Bt

occurs for some t ∈ [0, T ]. Since the trajectories of the bond price processare assumed to be continuous, the event Vt ≤ Bt is equivalent to C∗t ≤ 0.Moreover, since E(Y )t = E(Y )t−(1 + ∆Yt), C

∗t ≤ 0 for some t ∈ [0, T ] if and

only if m∆Lt ≤ −1 for some t ∈ [0, T ]. This leads to the following result.

Proposition 3.1. The shortfall probability SF(T ) of the CPPI portfolio overan investment time horizon [0, T ] is given by

SF(T ) = 1− A(T ),

where

A(T ) =⟨1, (E(X0))′ exp (Q− diag(µ))T

⟩, t ≥ 0, (3.12)

µ = (µ1, · · · , µn)′ ∈ Rn and µj = vj((−∞,−1/m]), j = 1, 2, · · · , n.Proof. Let M be the number of transitions which the regime switchingprocess X experiences over [0, T ]. Denote η0 = 0 and ηM+1 = t, and let ηj bethe jth transition time of the process for j = 1, . . . ,M . Let τ1, τ2, · · · , τM+1

denote the corresponding inter-transition times so that τj = ηj − ηj−1 forj = 1, 2, · · · ,M + 1. Let ekj denote the state in which the process X staysover period [ηj−1, ηj), j = 1, 2, · · · ,M + 1.

It is well known that the number of jumps of a Levy process with Levymeasure v over a time horizon (a, b] whose sizes fall in (−∞,−1/m) is aPoisson random variable with intensity (b − a)v((−∞,−1/m]). Therefore,the conditional shortfall probability

P (C∗T ≤ 0|Xu, 0 ≤ u ≤ T ) = P (∃t ∈ [0, T ] : Vt ≤ Bt|Xu, 0 ≤ u ≤ T )

=(1− e−τ1µk1

)+ e−τ1µk1

(1− e−τ2µk2

)+ · · ·

+e−τ1µk1−τ2µk2

−···−τMµkM

(1− e−τM+1µk(M+1)

)= 1− exp

(−

M+1∑j=1

τjµkj

).

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Consequently, the unconditional shortfall probability

SF(T ) = P[∃t ∈ [0, T ] : Vt ≤ Bt] = 1− E

exp

(−

M+1∑j=1

τjµkj

). (3.13)

Recall from (3.11) that Γj(T ) denotes the amount of time for which theprocess X spends in state ej over [0, T ]. Thus,

E

exp

(−

M+1∑j=1

τjµkj

)= E

exp

(−

n∑j=1

Γj(T )µj

), (3.14)

which, along with (3.13) and part (c) of Lemma 3.1, yields the desired result.

Corollary 3.1. The density of τ ≡ inft ≥ 0 : Vt ≤ Bt, denoting themoment for the CPPI portfolio value hits or falls below the floor, is given by

fτ (t) = −⟨1, (E(X0))′ exp (Q− diag(µ)) t (Q− diag(µ))

⟩= (E(X0))′ exp (Q− diag(µ)) tµ, for t ≥ 0.

Proof. The proof follows directly from Proposition 3.1 and the fact thatSF(t) = P(τ ≤ t).

Remark 3.3. It is interesting to note that the results in Proposition 3.1and Corollary 3.3 are heuristically obvious, if we recall the definition of theso-called phase-type distribution.

Let (Yt)t≥0 be a Markov jump process with a finite state space E =e1, . . . , en+1,where states e1, . . ., en are transient and state en+1 is absorbing. Then,(Yt)t≥0 has an intensity matrix of the form(

P, p0, 0

),

where P is a matrix of n by n, p is a n dimensional column vector and 0 isthe n dimensional row vector of zeros. Note that we must have p = −P1′

since each row of an intensity matrix must sums to 0. Let ζ be the timeuntil absorption, i.e., ζ = inft ≥ 0, Yt = en+1. The distribution of τ issaid to have a phase-type distribution, written τ ∼ PH(π,P), where π =

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(π1, . . . , πn) with πj = P(Y0 = ej) for j = 1, . . . , n. It is well known that thephase-type distribution has a presentation of

P(τ ≤ t) = 〈1, π exp (Pt)〉 , (3.15)

with a density function

fτ (t) = 〈1, πP exp (Pt)〉 . (3.16)

The phase-type distributions have been widely used as one of the importantmodelling tools in finance and risk theory; see, for example, Bladt (2005) andAsmussen (2000).

In the context of the CPPI portfolio which we analyzed in the above Propo-sition 3.1, we may let Yt be the same as the regime switching process Xt allthe time before the portfolio value breaks the floor, and set it equal to en+1

thereafter. Then, the event Yt = en+1 means that a shortfall happens be-fore or at time t, and thus τ = infYt = en+1. In other words, the portfolioincurs a shortfall if and only if Yt is absorbed by the state en+1. Recall fromthe paragraph right before Proposition 3.1 that the CPPI portfolio incurs ashotfall if and only if a downward jump in the process (Lt)t≥0 happens witha size larger than 1/m. Conditional on the regime ej, the intensity to havea loss of such a size is µj ≡ vj((−∞,−1/m]) for j = 1, . . . , n. This impliesthat Yt has an intensity matrix(

Q− diag(µ), µ0, 0

),

and consequently by applying the presentations in (3.15) and (3.16) we im-mediately recover the results in Proposition 3.1 and Corollary 3.3.

3.3. Expected shortfall and expected gain

For each Levy process L(j), we can always write L(j) = L(j,1)+L(j,2), whereL(j,2) is a process with piecewise trajectories and jumps satisfying ∆L

(j,2)t ≤

−1/m and L(j,1) is a process with jumps satisfying ∆L(j,1)t > −1/m. In other

words, L(j,2) has Levy measure vj(dx)1x≤−1/m, no diffusion component, andno drift. The jump intensity of L(j,2) is

µj = vj((−∞,−1/m])), j = 1, . . . , n. (3.17)

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Let φj(·; t) be the characteristic function of the Levy process ln E(mL(j,1))tand ψj(u) = 1

tlnφj(u; t) denoting the corresponding Levy exponent. For

presentation convenience, we will use the following notation

θj := ψj(1) =1

tlnE

[E(mL(j,1))t

], j = 1, . . . , n. (3.18)

Further denote L(1)t ; = (L

(1,1)t , . . . , L

(n,1)t )′ and L

(2)t = (L

(1,2)t , . . . , L

(n,2)t )′. Let

L(1)t and L

(2)t respectively denote

⟨L

(1)t , Xt

⟩and

⟨L

(2)t , Xt

⟩so that Lt =

L(1)t + L

(2)t , and use ∆L(2) to denote L

(2)t − L

(2)

t− , the jump of L(2)t at time t.

Moreover, we use κj to denote E(

1 +m∆L(j,2)t

)so that

κj = 1 +m

µj

∫x≤−1/m

x vj(dx), j = 1, . . . , n. (3.19)

Finally, let κ := (κ1, · · · , κn).To proceed, we recall some useful properties of Levy process as below.

Lemma 3.2. (a). Let (U)t≥0 be a real valued Levy process with Levy triplet(σ2, ν, ω) and Y = E(U) its stochastic exponential. If Y > 0 a.s., then

there exists another Levy process (U)t≥0 satisfying Yt = eUt and

Ut = Ut −σ2t

2+∑

0≤s≤t

ln(1 + ∆Us)−∆Us .

The Levy triplet (σ2, ν, ω) of U is given by

σ = σ,

ν(A) = ν (x : ln(x+ 1) ∈ A) =

∫IA(ln(1 + x))ν(dx),

ω = ω − σ2

2+

∫ (ln(x+ 1)) I[−1,1] (ln(x+ 1))− xI[−1,1](x)

ν(dx).

(b). Let (U)t≥0 be a real valued Levy process with Levy triplet (σ2, ν, ω) and

Yt = eUt its exponential. Then there exists another Levy process (U)t≥0

with Yt = E(U)t and

Ut = Ut +σ2t

2+∑

0≤s≤t

1 + ∆Us − e∆Us

.

13

The Levy triplet (σ2, ν, ω) of U is given by

σ = σ,

ν(A) = ν (x : ex − 1 ∈ A) =

∫IA(ex − 1)ν(dx),

ω = ω +σ2

2+

∫ (ex − 1) I[−1,1] (ex − 1)− xI[−1,1](x)

ν(dx).

(c). E[E(mL(j,1))t

]= φj(−i; t) = etθj , where

θj = mbj +m

∫x>−1/m

x vj(dx), j = 1, . . . , n, (3.20)

where the constant bj and Levy measure vj are from equation (2.8).Proof. For the proof of (a) and (b), see Cont and Tankov (2004, chapter 9)or Goll and Kallsen (2000); also see part 1 of Proposition A.1 from Cont andTankov (2009). (c) follows from part (a) and Theorem 25.17 in Sato (1999,p. 165).

Proposition 3.2. Denote θ := (θ1, . . . , θn)′, where θj is given in (3.20).Then, we have the following results regarding the expected shortfall and ex-pected gain of the CPPI portfolio.

(a). The unconditional expected shortfall is

E[C∗T IC∗

T≤0]

=

∫ T

0

B(t)C(t)dt,

and hence, the expected shortfall is

ES(T ) = E [C∗T |C∗T ≤ 0] =

∫ T0B(t)C(t)dt

1− A(T ), (3.21)

where

B(t) =⟨κ, (E(X0))′ exp (Q + diag(θ)) t

⟩, (3.22)

and

C(t) = (E(X0))′ exp (Q− diag(µ)) tµ. (3.23)

14

(b). The unconditional expected gain

E[C∗T IC∗

T>0]

=⟨1, (E(X0))′ exp (Q + diag (θ)) t

⟩A(T ),

and the (conditional) expected gain

EG(T ) = E[C∗T |C∗T > 0] =⟨1, (E(X0))′ exp (Q + diag (θ))T

⟩.(3.24)

Proof. (a). For any time t ∈ [0, T ], the discounted cushion satisfies

C∗s = E(mL(1)

)τ∧s

(1 +m∆L(2)

τ Iτ≤s)

= E(mL(1)

)tIτ>t + E

(mL(1)

)τ

(1 +m∆L(2)

τ

)Iτ≤t. (3.25)

As in proof of Proposition 3.1, we let M be the number of transitions whichthe regime switching process X experiences over [0, t], let η0 = 0 and ηM+1 =t, and let ηj denote the jth transition time of the regime switching process forj = 1, . . . ,M . Let Ej denote the state in which the processX stay over period[ηj−1, ηj), j = 1, 2, · · · ,M + 1. Let τ1, τ2, · · · , τM+1 denote the correspondinginter-transition times so that τj = ηj − ηj−1 for j = 1, 2, · · · ,M + 1.

Then, we have the following expression for the stochastic exponential:

E(mL(1))t =M+1∏j=1

E(mL(1))ηj

E(mL(1))ηj−1

. (3.26)

where (mL(1))t is identical to the Levy process mL(j,1) for t ∈ [ηj−1, ηj),j = 1, · · · ,M + 1. According to Lemma 3.2, for each j, there exits a Levy

process V ∗(j,1) such that E(mL(j,1)) = eV∗(j,1)

so that all the items in theproduct (3.26) are independent conditional on FXs and

E

(E(mL(1))ηj

E(mL(1))ηj−1

∣∣∣∣∣FXs)

= E

(E(mL(1))ηj

E(mL(1))ηj−1

∣∣∣∣∣ ηj−1, ηj, Ej

)= exp

(V ∗(Kj ,1)ηj

− V ∗(Kj ,1)ηj−1

), (3.27)

where Kj is a random variable such that Ej = eKj. Moreover, by the property

of the Levy process, E(mL(1)

)t

is independent of ∆L(2)t conditional on FXt ,

which is the σ-algebra generated by Xu, 0 ≤ u ≤ t, and indeed the latterquantity only depends on Xt.

15

The above analysis implies that

E(E(mL(1))t

(1 +m∆L

(2)t

)∣∣∣FXt )= E

(E(mL(1))t

∣∣∣FXt )E(1 +m∆L(2)t

∣∣∣FXt )= E

(M+1∏j=1

E(mL(1))ηj

E(mL(1))ηj−1

∣∣∣∣∣FXt)E(

1 +m∆L(2)t

∣∣∣Xt

)=

M+1∏j=1

E

(E(mL(1))ηj

E(mL(1))ηj−1

∣∣∣∣∣FXt)· 〈κ, Xt〉

= exp

(M+1∑j=1

(V ∗(Kj ,1)ηj

− V ∗(Kj ,1)ηj−1

))· 〈κ, Xt〉 . (3.28)

Note that, conditional on FXt , the increment V∗(Kj ,1)ηj −V ∗(Kj ,1)

ηj−1 has the same

distribution as V∗(Kj ,1)τj , and all of these increaments are independent to each

other. Thus, from (3.28) it follows that

E(E(mL(1))t

(1 +m∆L

(2)t

))= E

(E

(exp

M+1∑j=1

(V ∗(Kj ,1)ηj

− V ∗(Kj ,1)ηj−1

)· 〈κ, Xt〉

∣∣∣∣∣FXt))

= E

(M+1∏j=1

E

exp(V ∗(Kj ,1)τj

)∣∣∣FXt · 〈κ, Xt〉

)

= E

(M+1∏j=1

E(E(mL(Kj ,1)

)τj

∣∣∣FXt ) · 〈κ, Xt〉

)

= E

(M+1∏j=1

eτjθKj 〈κ, Xt〉

), (3.29)

where the last step is due to part (b) of Lemma 3.2. We further note thefollowing inequality

M+1∑j=1

τj θKj=

n∑j=1

Γj(t)θj.

16

Thus, by applying part (a) of Lemma 3.1, we derive from (3.29) that

E(E(mL(1)

)t

(1 +m∆L

(2)t

))=

⟨κ, E

(Xt ·

M+1∏j=1

exp(τjθKj

))⟩

=

⟨κ, E

(exp

(n∑j=1

Γj(t)θj

)Xt

)⟩=

⟨κ, (E(X0))′ exp (Q + diag(θ)) t

⟩,

(3.30)

and consequently, it follows from the density of τ given in Corollary andrepresentation of C∗t in (3.25) that

E[C∗T IC∗T≤0] = E[C∗T Iτ≤T]

= EE(mL1)τ

(1 +m∆L(2)

τ

)Iτ≤T

=

∫ T

0

EE(mL1)t

(1 +m∆L

(2)t

)· fτ (t)dt

=

∫ T

0

B(t)C(t) dt.

The expected shortfall given in (3.21) follows immediately from the aboveestablished result and Proposition 3.1.

(b). We first note that, the same analysis as in the proof of part (a) leadsto

E(E(mL(1)

)t

)= E

(exp

(n∑j=1

Γj(t) θj

))=

⟨1, (E(X0))′ exp (Q + diag (θ)) t

⟩.

By (3.25), we obviously have

E(C∗T IC∗T>0) = E(C∗T Iτ>T) = E

E(mL(1)

)T

P(τ > T ),

and

E(C∗T |C∗T > 0) = E(C∗T |τ > T ) = EE(mL(1)

)T

.

Finally, by noticing that P(τ > T ) = A(T ), we obtain the desired results.

17

3.4. Loss distribution

Let φj(u) :=1

µj

∫ −1/m

−∞eiu ln(−1−mx)vj(dx) denoting the characteristic func-

tion of ln(−1−m∆L(j,2)

), where we recall that µj denotes vj((−∞,−1/m]),

the jump intensity of L(j,2), j = 1, . . . , n. Denote φ(u) :=(φ1(u), . . . , φn(u)

)′.

Also recall that ψj(u) denotes the characteristic exponent of ln E(mL(j,1)

)t

so that E(eiu ln E(mL(j,1))

t

)= eψj(u)t for j = 1, . . . , n. Let

ψ(u) := (ψ1(u), . . . , ψn(u))′ , u ∈ R.

The next Proposition provides a possible way to explore the distributionof the gap risk by using the Fourier inversion technique.

Proposition 3.3. (a). The characteristic function of the shortfall (−C∗T )conditional on the event C∗T < 0 is given by

ψ(u) := E(eiu ln(−C∗

T )∣∣C∗T < 0

)=

∫ T0C(t)D(t, u)dt

1− A(T ),

where

D(t, u) =⟨φ(u), (E(X0))′ exp (Q + diag(ψ(u)))t

⟩. (3.31)

(b). Choose a random variable X∗ with a characteristic function φ∗ such

that E|X∗| <∞ and|φ∗(u)|1 + |u|

is integrable. If, additionally,

1

1 + |u|

∣∣∣∣∫ T

0

C(t)D(t, u)dt

∣∣∣∣is integrable, and

∫|x|≥ε (| ln(|1 +mx|)|) vj(dx) < ∞ for sufficiently

small ε, j = 1, . . . , n, then for every x < 0,

P(C∗T ≤ x|C∗T < 0)

= P(−eX∗

< x)

+1

2π

∫Re−iu ln(−x)

(∫ T0C(t)D(t, u)dt

iu(1− A(T ))− φ∗(u)

iu

)du.

18

Proof. (a). By (3.25) and Proposition 3.1, we have

E(eiu ln(−C∗

T )∣∣ τ ≤ T

)= E

(eiu ln(−C∗

T )∣∣C∗T ≤ 0

)=

E(T )

1− A(T ), (3.32)

where

E(T ) =

∫ T

0

E(eiu ln

−E(mL(1))t·

(1+m∆L

(2)t

))fτ (t)dt. (3.33)

Using the conditional independence between E(mL(1))t and L(2)t , we can cal-

culate the expectation in the above integral as follows.

E(eiu ln

−E(mL(1))t·

(1+m∆L

(2)t

))= E

E(eiu ln

[−E(mL(1))t·

(1+m∆L

(2)t

)]∣∣∣∣FXt )= E

E(eiu ln E(mL(1))t

∣∣∣FXt ) · E(eiu ln[−(

1+m∆L(2)t

)]∣∣∣∣FXt ) ,where, using the same technique we applied in the proof of Proposition 3.2,we obtain

E(eiu ln E(mL(1))t

∣∣∣FXt ) =n∏j=1

E(eiu ln E(mL(j,1))Γj(t)

∣∣∣Γ1(t), . . . ,Γn(t))

=n∏j=1

eψj(u)Γj(t)

= exp

(n∑j=1

ψj(u)Γj(t)

)

and

E(eiu ln

[−(

1+m∆L(2)t

)]∣∣∣∣FXt ) = E(eiu ln

(−(

1+m∆L(2)t

))∣∣∣∣Xt

)=⟨φ(u), Xt

⟩.

19

Thus, it follows from part (a) of Lemma 3.1 that

E(eiu ln

[−E(mL(1))t·

(1+m∆L

(2)t

)])= E

[⟨φ(u), exp

(n∑j=1

ψj(u)Γj(t)

)Xt

⟩]

=

⟨φ(u), E

(exp

(n∑j=1

ψj(u)Γj(t)

)Xt

)⟩=

⟨φ(u), (E(X0))′ · exp (Q + diag(ψ(u)))t

⟩. (3.34)

Combining (3.32), (3.33) and (3.34) immediately yields the desired result.(b). The proof follows directly from applying Lemma B.1 of Cont and

Tankov (2009). Also see the proof of Theorem 3.1 in Cont and Tankov (2009).

3.5. Analysis under a diagonalizable intensity matrix

The results established in the previous subsections depend on these func-tions A(t), B(t), C(t) and D(t, u) defined in Propositions 3.1, 3.2 and 3.3,where matrix exponentials are involved; see equations (3.12), (3.22), (3.23)and (3.31). To evaluate the CPPI portfolio by using these results, we have tomake sure that these functions are computationally amenable. While thereare many ways to calculate or approximate the matrix exponentials, seeMoler and Loan (2003) for example, we adopt the Putzer’s spectral formula(see Putzer, 1996). Indeed, because the intensity matrix Q associated witha regime switching process is always diagonalizable with distinct eigenvaluesto be meaningful, Putzer’s spectral formula turns out to be very friendly indeveloping explicit forms of the functions A(t), B(t), C(t), D(t, u) and thoserelated measures we target.

As we can see shortly, Putzer’s spectral formula is an accurate approach incalculating a matrix exponential, though the computational efforts increaseexponentially along with the dimension of the exponent matrix. Therefore,it is an ideal way only in dealing with low-dimensional case. Fortunately, inthe financial context, a very small number of regime states is often sufficientto capture the state transition of a financial market. Indeed, the two-stateregime switching model is the most common one with states respectivelyrepresenting a “bear” market and a “bull” market. As mentioned in the very

20

beginning of section 2, when three states are assumed in the model, the thirdone means a “normal” state of the market. Hereafter we assume that theintensity matrix Q has distinct eigenvalues r1, . . ., rn.

Lemma 3.3. Let λ1, . . . , λn be the eigenvalues of a matrix H of n by n,which are not necessarily distinct, in some arbitrary but specified order. LetI denote the identity matrix of an appropriate dimension. Then,

eHt =n−1∑j=0

rj+1(t)Pj, (3.35)

where P0 = I, Pj =

j∏k=1

(H− λkI), j = 1, . . . , n, and (r1(t), . . . , rn(t)) is the

solution to the following triangular system

d

dtr1(t) = λ1r1(t),

d

dtrj(t) = rj−1(t) + λjrj(t), j = 2, . . . , n,

r1(0) = 1, rj(0) = 0, j = 2, . . . , n.

(3.36)

Proof. See Putzer (1996). Obviously, system (3.36) can be solved explicitly provided that all the

eigenvalues λj are distinct, although Putzer (1996) refrains from doing so.We summarize its explicit solutions in the following lemmas.

Lemma 3.4. When all the eigenvalues λ1, . . . , λn of H are distinct, the so-lutions to the system (3.36) are as follows:

rn(t) =n∑j=1

(eλjt

n∏k=1,k 6=j

1

λj − λk

), n ≥ 1. (3.37)

Proof. See Appendix. By applying Lemma 3.3 and Lemma 3.4, we can obtain the explicit form

for the matrix exponential in the case of distinct eigenvalues, as shown in thefollowing Lemma.

21

Lemma 3.5. Assume that a matrix H of n by n has distinct eigenvaluesλ1, . . . , λn. Then, the matrix exponential eHt has the following representation:

eHt =n∑j=1

[eλjt

n∏k 6=j,k=1

H− λkIλj − λk

](3.38)

Proof. See Appendix.

Now we can readily derive the explicit forms for functions A(t), B(t),C(t) and D(t, u), which are respectively defined in (3.12), (3.22), (3.23) and(3.31).

Proposition 3.4. Suppose that the intensity matrix Q associated with theregime process Xt in (2.1) has n distinct eigenvalues λ1, . . . , λn. For j =1, . . . , n, denote

(βj,1, . . . , βj,n) := (E(X0))′n∏

k 6=j,k=1

Q′ − λkIλj − λk

, j = 1, . . . , n.

Then,

A(t) =n∑i=1

n∑j=1

βi,je(λi−µj)t,

B(t) =n∑i=1

n∑j=1

βi,jκje(λi+θj)t,

C(t) =n∑i=1

n∑j=1

βi,jµje(λi−µj)t,

D(t, u) =n∑j=1

n∑k=1

βi,jφj(u)e(λi−ψk(u))t.

Remark 3.4. (a). From the above Proposition 3.4, we can easily obtain an

expression for the integrals

∫ t

0

B(s)C(s)ds and

∫ t

0

C(s)D(s)ds that are

22

involved in Propositions 3.2 and 3.3, as shown below.∫ t

0

B(s)C(s)ds

=n∑

i1=1

n∑i2=1

n∑i3=1

n∑i4=1

βi1,i2κi2βi3,i4µi4H (t;λi1 + θi2 + λi3 − µi4) ,

and ∫ t

0

C(s)D(s)ds

=n∑

i1=1

n∑i2=1

n∑i3=1

n∑i4=1

βi1,i2µi2βi3,i4φi4(u)H(t, λi1 − µi2 + λi3 − ψi4(u)),

where

H(t;x) :=

1

x

(ext − 1

), if x 6= 0;

t, if x = 0.

(b). By comparing equation (3.12) with equation 3.24, we notice that expres-sions for function A and the expected gain EG are in the same form.Thus, the expression for A(t) established in Proposition 3.4 implies

EG(T ) =n∑i=1

n∑j=1

βi,je(λi+θj)T .

4. Implementation under some popular exponential Levy models

The results we established in Propositions 3.1 and 3.2 are expressed interms of functions A(t), B(t) and C(t), which are respectively defined in(3.12), (3.22) and (3.23). As shown in the previous section, these threefunctions further depend on quantities µj, κj and θj, which are respectivelygiven in (3.17), (3.19) and (3.20). Therefore, to implement our results, weessentially need to calculate these three quantities for a given exponentialLevy process of the asset price. In this section, we will study their calcu-lation for some popular exponential Levy processes including the Merton’sjump-diffusion, Kou’s jump-diffusion, variance gamma, and normal inverse

23

Gaussian models. As we will see shortly, the calculation is rather amenablefor the first two models, but it is quite computational demanding for the lasttwo.

Prevailing financial models with jumps can be basically categorized intotwo groups. In the first group are the so-called jump-diffusion models, wherethe “normal” evolution of prices is given by a diffusion process and punctu-ated by jumps at random intervals. The Merton’s model and Kou’s modelare the most popular two among this group. The second group consists of theso-called infinite activity models as they admit infinite number of jumps in ev-ery interval of time. Many models from the second group can be constructedvia Brownian subordination, and both the variance gamma model and thenormal inverse Gaussian model fall into this group. We refer to Schoutens(2003), Cont and Tankov (2004) and references therein for an introduction toa variety of interesting Levy processes as well as their applications in finance.

For simplicity, hereafter we assume that the bond price process Bt = B0ert

with a constant interest rate r so that the driven process Rt = rt, t ≥ 0;see (2.3). Furthermore, we assume that, conditional on event Xt = ej, t ∈(u, v], the stock index admits the following dynamics

Sv = Su exp(M (j)

v −M (j)u

)(4.39)

for n independent Levy processes M (j)t , t ≥ 0, j = 1, . . . , n. We shall

investigate how to compute µj, κj and θj for a given Levy process M(j)t with

known Levy triplet.

4.1. Merton’s model

Merton’s model is the first proposed to address option pricing problemwhen the underlying asset admits price jumps; see Merton (1976). Merton’smodel (with the regime switching feature under consideration) assumes the

following form for the logarithmic price M(j)t :

M(j)t =

(aj −

σ2j

2

)t+ σjW (t) +

Nj(t)∑i=1

Y(j)i , (4.40)

where aj ∈ R, σj > 0, W (t) is a standard Brownian motion, Nj(t) is a

Poisson process with intensity rate λ∗j , andY

(j)1 , Y

(j)2 , . . .

is a sequence of

24

independent and identically distributed normal random variables subject toa density function

fj(x) =1√2πδ2

j

exp

−(x− cj)2

2δ2j

,

with mean cj and variance δ2j .

Applying part (b) of Lemma 3.2 yields the following form for the driving

process Z(j)t of the stock index process:

Z(j)t = ajt+ σjW (t) +

Nj(t)∑i=1

(eY

(j)i − 1

); (4.41)

see (2.2) and (2.3) for definition of Z(j)t . Consequently, the driving process

L(j)t defined in (2.8) has a drift bj = aj−r, a volatility σj and a Levy measure

vj(dx) =λ∗

(x+ 1)√

2πδ2j

exp

(−(ln(x+ 1)− cj)2

2δ2j

)dx, x > −1.

From the Levy triplet(σ2j , vj, bj

)obtained in the above for L

(j)t , we can use

(3.17), (3.19) and (3.20), and integrate directly to obtain expressions of µj,κj and θj as given below:

µj =

∫x≤−1/m

vj(dx) = λ∗h(cj, δj),

κj = 1 +m

µj

∫ −1/m

−1

x vj(dx) = mecj+δ2j /2h(cj + δ2

j , δj)

h(cj, δj)− (m− 1)

θj = mbj +m

∫x>−1/m

x vj(dx)

= mbj +mλ∗ecj+δ2

j /2(1− h

(cj + δ2

j , δj))− (1− h (cj, δj))

,

where

h(c, δ) := Φ

(ln(1− 1/m)− c

δ

), c ∈ R, δ > 0,

and Φ(·) denotes the standard normal distribution function.

25

4.2. Kou’s model

The Kou’s model, also known as double exponential jump-diffusion model,admits the same structure as Merton’s model introduced in the last subsec-tion. The only difference lies in the assumption on jump distribution. Whilethe jumps are assumed to have a normal distribution in Merton’s model,they have a double exponential distribution in Kou’s model. Specifically, theKou’s model assumes that the logarithmic price process M

(j)t of the stock

index is as defined in (4.40) with all the other components being the same

as in the Merton’s model while the jumpsY

(j)1 , Y

(j)2 , . . .

being a sequence

of independent and identically distributed random variables subject to thefollowing double exponential density function

fj(x) =pjη+j

exp(−x/η+

j

)Ix≥0 +

qjη−j

exp(−|x|/η−j

)Ix<0, (4.42)

where parameters η±j > 0 and pj + qj = 1 for j = 1, . . . , n. The memorylessproperty of the exponential distribution makes the Kou’s model much moretractable relative to the other jump-diffusion models in many situations. Assuch, Kou’s model has been widely applied in literature for option pricingand hedging; see, for example, Kou (2002) and Kou and Wang (2004) amongmany others.

Following the same reasoning stream as we did for the Merton’s model inthe last subsection, we can easily show that the driving process L

(j)t defined

in (2.8) has a drift bj = aj − r, a volatility σj and a Levy measure

vj(dx) = δ+j c

+j (1 + x)−1−δ+

j Ix>0 + δ−j c−j (1 + x)−1+δ−j I−1<x≤0dx,

where δ±j = 1/η±j , c+j = pjλ

∗j , and c−j = qjλ

∗j . From the Levy triplet(

σ2j , vj, bj

)obtained in the above for L

(j)t , we can use (3.17), (3.19) and

(3.20), and integrate directly to obtain expressions of µj, κj and θj as given

26

below:

µj =

∫x≤−1/m

vj(dx) = c−j

(1− 1

m

)δ−j, (4.43)

κj = 1 +m

µj

∫ −1/m

−1

x vj(dx) = −m− 1

δ−j + 1, (4.44)

θj = mbj +m

∫x>−1/m

x vj(dx)

= mbj +mc+j

δ+j − 1+

mc−j

δ−j + 1

(1 + δ−j )

(1− 1

m

)δ−j− 1− δ−j

(1− 1

m

)δ−j +1.

(4.45)

4.3. Variance gamma process

The variance gamma (VG) model is one of the most popular infiniteactivity models with finite variation but relatively low activity of small jumps.It was introduced by Madan and Seneta (1990) as a model for stock returns;for its use in option pricing, we refer to Madan and Milne (1991) and Madan,et al. (1998) among many others. It is an important special case of the CGMYdistribution proposed by Carr et al. (2002); also see Schoutens (2003) andCont and Tankov (2004). The VG model assumes the following dynamic forthe logarithmic price processes

M(j)t = mj t+ ωjΓ(t; 1, δj) + σjW (Γ(t; 1, δj)), t ≥ 0, (4.46)

where mj ∈ R, ωj ∈ R, σj > 0, Γ(t;u, v) denotes a gamma process withmean rate u and variance rate v, and W is a standard Brownian motion. Agamma process is a Levy process with independent gamma increments overnon-overlapping intervals of time.

M(j)t defined in (4.46) has a Levy triplet of (0, v∗j ,mj) with Levy measure

v∗j (dx) =1

δj|x|exp (Ajx−Bj|x|) dx, x ∈ R \ 0, (4.47)

and characteristic function

E(eiuM

(j)t

)= eiumjt

(1− iuωjδj +

u2

2σ2j δj

)−t/δj(4.48)

27

where i =√−1 is the unit imaginary number,

Aj =ωjσ2j

and Bj =

√ω2j + 2σ2

j/δj

σ2j

;

for its derivation, see for example Revuz and Yor (1999), Schoutens (2003)and Cont and Tankov (2004). We note that Bj > Aj, and moreover, (4.48)

implies that E[eM

(j)t

]<∞ if and only if 1 > ωjδj + σ2

j δj/2, which is equiva-

lent to the condition Bj > Aj + 1. We will have to impose this condition forthe value of θj; otherwise, θj in (3.18) is ill-defined.

Using part (b) of Lemma 3.2, we can easily see that diffusion coefficient

for L(j)t is zero and its Levy measure (which is the same as that of Z

(j)t ) is as

follows

vj(dx) =1

v(1 + x)Aj−1 e

−Bj | ln(1+x)|

| ln(1 + x)|dx, x > −1. (4.49)

The drift coefficient for L(j)t should be computed by the following formula

bj = mj − r +

∫ (ex − 1) I[−1,1] (ex − 1)− xI[−1,1](x)

v∗j (dx). (4.50)

By resorting to the zero truncation function for calculating the integral in(4.50), we can obtain an expression for bj analytically as follows

bj = mj − r −1

δj

(2Aj

B2j −A2

j

+e−(Bj+Aj)

Bj +Aj− e−(Bj−Aj)

Bj −Aj− Ei((Aj −Bj) ln 2)

)+ Cj ,(4.51)

for a constant

Cj =

1

δjln

(B2j −A2

j

|Bj −Aj − 1|(Bj +Aj + 1)

)+

1

δjEi((1 +Aj −Bj) ln 2), if Bj 6= Aj + 1,

1

δjln

(B2j −A2

j

Bj +Aj + 1

)+ γ + ln ln 2, if Bj = Aj + 1,

where

Ei(x) = −∫ ∞−x

e−t

tdt (4.52)

28

is known as the exponential integral function, and γ is the Euler-Mascheroniconstant (also known as Euler’s constant) defined by

γ = limn→∞

(n∑k=1

1

k− ln(n)

),

approximately equal to 0.57721. The proof of equation (4.51) is relegated toAppendix, and it involves applying the zero truncation function for evaluat-ing the integral in (4.50).

From the Levy triple of L(j)t we obtained in the above, we apply formulas

(3.17), (3.19), and (3.20) to obtain expressions for µj, κj and θj as follows:

µj =

∫x≤−1/m

vj(dx) = −1

vEi

((Aj +Bj) ln

(1− 1

m

)),

κj = 1 +m

µj

∫x≤−1/m

x vj(dx)

= 1 +m

vµj

Ei

((Aj +Bj + 1) ln

(1− 1

m

))− Ei

((Aj +Bj) ln

(1− 1

m

)),

θj = mbj +m

∫x>−1/m

x vj(dx)

= mbj +m

δj

ln

(B2j − (Aj + 1)2

B2j − A2

j

)+ H(Aj +Bj)− H(Aj +Bj + 1)

,

where

H(x) = Ei(x · ln(1− 1/m)),

and Bj > Aj + 1 is assumed for calculating θj. The derivation of the aboveexpressions for µj, κj and θj is similar to that of (4.51), and therefore omitted.

4.4. Normal inverse Gaussian process

The normal inverse Gaussian (NIG) process was introduced by Barndorff-Nielsen (1997, 1998), and studied further by Barndorff-Nielsen and Shephard(2001). It is one of the most popular infinite activity models with infinitevariation in any finite interval of time, and it belongs to the class of general-ized hyperbolic Levy processes; see Schoutens (2003) and Cont and Tankov

29

(2004) for more details. The NIG model assumes the following logarithmicprice process

M(j)t = aj t+ βjδ

2j It + δjWIt ,

where W is a standard Brownian motion and I is an inverse Gaussian process

with parameters 1 and δj√α2j − β2

j for αj > 0, −αj < βj < αj and δj > 0.

The Levy measure for the NIG process is given by

v∗j (dx) =δjαjπ

eβj xK1(αj|x|)|x|

dx, x ∈ R \ 0,

where Kλ(x) denotes the modified Bessel function of the third kind withindex λ; see Appendix A in Schoutens (2003) for the definition. Using part

(b) of Lemma 3.2, we can obtain the following Levy measure for L(j)t (which

is the same as that of Z(j)t ):

vj(dx) =δjαjπ

(1 + x)βj−1K1(αj| ln(1 + x)|)| ln(1 + x)|

dx, x > −1, (4.53)

where K1 denotes the modified Bessel function of the third kind with index1.

In principe, we can follow the same steps as we illustrated for VG processin the last subsection to calculate the values of µj, κj and θj. However,due to the complexity of the Levy measure vj as demonstrated in (4.53), itseems impossible to obtain analytical expressions for these three quantitiesin the NIG case. We may have to resort to some numerical approach for theirvalues, and we leave this for future exploration.

5. Numerical example

In this section, we shall illustrate how to use Propositions 3.1 and 3.2to calculate these quantities including shortfall probability, expected short-fall and expected gain under a specific risk asset price model. We considerthe Kou’s model, which is also called the double exponential jump-diffusionmodel, as an example; see subsection 4.2 for more details. As we can seeshortly, the existence of the regime switching feature has a significant effecton the performance of CPPI portfolio. A different market state at the incep-tion of the investment will lead to a quite different risk-reward profile of aCPPI portfolio.

30

Under Kou’s model, the logarithmic price process M(j)t and the driving

processes Z(j)t respectively admit a form as in (4.40) and (4.41), with double

exponentially distributed jumps Y (j)i as explained in subsection 4.2. The

density function of the jumps is given in (4.42). Analytical expressions forµj, κj and θj are respectively given in (3.17), (3.19) and (3.20).

We let n = 2 meaning that the market are switching between two states——bull and bear, and the intensity matrix of the regime switching Markovprocess is assumed to be

Q =

(−0.25, 0.25

0.25, −0.25

), (5.54)

which implies that it is equally likely for the market to change from a bullto a bear and from a bear to a bull. Under each market regime, parametervalues of the model are set as shown in Table 1, where the two regimes aredistinguished by a positive expected return in a bull market and a negativeone in a bear market. Moreover, we intentionally set the values of parameterspj in such a way that the stock price is more likely to jump upwards thandownwards in a bull market while it behaviors on an oppose way in a bearmarket. Furthermore, the values of η− imply that the downward jump sizeis much larger than the upward jump size in a bear market. All of theseassumptions are made in accordance to our intuitive observations on thestock market.

Table 1: Parameter values

Parameters aj r bj(= aj − r) σj λ∗j pj η+j η−j

Bull (j = 1) 0.14 0.04 0.1 0.16 50 0.55 0.013 0.016Bear(j = 2) -0.01 0.04 -0.05 0.20 100 0.45 0.013 0.02

Department of Mathematics, Suzhou University, Suzhou 215006, PR ChinaTo demonstrate the effect from the regime switching feature of the market,

we calculate the shortfall probability, (unconditional) expected shortfall and(unconditional) expected gain of the CPPI portfolio in two distinct scenarios:X0 = (1, 0) and X0 = (0, 1), which respectively means that the investmentstarts with a bull and bear market state, respected called “bull-start portfo-lio” and “bear-start portfolio”. We set T = 2, meaning an investment of two

31

2 3 4 5 6 7

0.000

0.010

0.020

0.030

start in a bull marketstart in a bear market

Figure 1: Shortfall probability over T = 2 year as a function of the multiplier.

2 3 4 5 6 7

0.0

0.5

1.0

1.5

2.0

2.5

3.0

start in a bull marketstart in a bear market

2 3 4 5 6 7

020

4060

80100

120 start in a bull market

start in a bear market

Figure 2: Expected shortfall over T = 2 year as a function of the multiplier, for C∗0 =

$1, 000; on the left is the unconditional expected shortfall while, on the right, is theexpected shortfall conditional on a shortfall having occurred.

32

2.0 3.0 4.0 5.0 6.0 7.0

0500

1000

1500

2000

2500

3000

start in a bull marketstart in a bear market

2.0 3.0 4.0 5.0 6.0 7.0

010

2030

4050

start in a bull marketstart in a bear market

Figure 3: Expected gain over T = 2 year as a function of the multiplier, for C∗0 = $1, 000;

on the left is the unconditional expected gain while, on the right, is the expected gainconditional on no shortfall having occurred.

Table 2: Comparison between an initial market state of bull and bear.

m 4 5 6 7initialmarketstate

SF (%) 0.0020 0.0523 0.4144 1.7256

Bu

llUES 0.0008 0.0314 0.3663 2.1532ES 38.3959 60.0430 88.4145 124.7812

UEG 0.0328 1.0525 10.3496 53.6167EG 1,625.5057 2,013.1046 2,497.7835 3,107.0973

SF (%) 0.0043 0.1085 0.8377 3.3919

BearUES 0.0014 0.0481 0.4794 2.4747

ES 33.4657 44.3524 57.2314 72.9600UEG 0.0325 1.0125 9.6769 48.7133EG 758.5772 932.8299 1,155.1486 1,436.1573

33

year. The results are reported in Figures 1-3 and Table 2. In addition to theincreasing trend of all these quantities as a function of the multiplier m, wehave the following important observations:

(a). First, the shortfall probability is substantially larger for a bull-startportfolio than a bear-start portfolio; see Figure 1. In particular, whenthe multiplier m is as large as 7, there is a probability of 3.3919%to realize a shortfall within the investment of one year if the CPPIportfolio starts in a bear market, while it is only about 1.7256% if itstarts in a bull market.

(b). Second, the unconditional expected shortfall is larger for a bear-startportfolio than a bull-start portfolio for all the chosen multiplier val-ues (m = 1, 2, . . . , 7); see the left graph in Figure 2. Nevertheless, theconditional expected shortfall does not show such a strict ordering phe-nomenon between the bull-start portfolio and the bear-start portfolio.The right graph in Figure 2 indicates that the conditional expectedshortfall is slightly smaller for the bull-start portfolio than the bear-start portfolio when the multiplier m is as small as 3. For a multiplierof no less than 4, however, the conditional expected shortfall is alwayslarger for the bull-start portfolio than the bear-start portfolio, and thegap increases rapidly as the multiplier m increases. These observationsindicate that the risk on the guarantor substantially differs from a bull-start portfolio to a bear-start portfolio. All in all, there is more riskon the CPPI fund guarantor if the investors choose to enter the fundin a bull market than a bear market, which in turn implies that morepremium should be charged on the investors if they choose to enter thefund in a bull market.

(c). Third, the risk-award profile for the investors can also be quite differentbetween starting the investment in a bull market and a bear market;see Figure 3. Consider the case with m = 6 for an example; see Table2. If an investor entered the CPPI fund in a bull market, there is aprobability of 0.4144% for him/her to obtain a payoff at the guaranteelevel, and a very large probability of 99.5856% to receive an expectedgain of $2, 497.78. On the contrary, if the investor started in a bearmarket, the probability for him/her to receive a payoff at the guar-antee level is 0.8377%, and given no shortfall having occurred his/herexpected gain will be $1, 155.15. Although the unconditional expected

34

gain is closed, the risk-reward profile is quite different between thesetwo types of CPPI portfolios.

6. Conclusion

The regime switching framework for modeling econometric series offers atransparent and intuitive way to capture market behavior through differenteconomic conditions. It has been widely used in econometrics since the pio-neering work of Hamilton (1989). The present paper developed a frameworkfor studying the performance of CPPI portfolio in the presence of jumpsin the asset price and the regime switching feature of the financial mar-ket. Analytically tractable expressions for the shortfall probability, expectedshortfall, expected gain and the characteristic function of the “gap risk” arederived under a general Markov regime switching exponential Levy model.These results can not only help the investors to develop a comprehensiveunderstanding on their risk-and-reward profiles, but also offer an effectiveframework for CPPI fund guarantors to conduct stress tests by adjustinginput parameters. Our results show that the behavior of the CPPI portfoliocan differ significantly with a different market state at the inception of theinvestment. Indeed, from a perspective of the guarantor, it is much riskier ifthe portfolio starts in a bull market than a bear market.

Acknowledgements

The author acknowledges the funding support from the University of Wa-terloo Start-up Grant, and the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC).

Appendix

Proof of Lemma 3.4. Clearly, the first equation in (3.36) with the initialcondition r1(t) = 1 implies r1(t) = eλ1t. Substituting r1(t) = eλ1t into its

35

second equationd

dtr2(t) = r1(t) + λr2(t), and solving it for r2(t) with the

initial condition r2(0) = 1, we obtain

r2(t) = eλ1t1

λ1 − λ2

+ eλ2t1

λ2 − λ1

.

This implies that the expression (3.37) holds for n = 2. We will completethe proof by the induction principle.

Suppose (3.37) is true for some positive integer n. Then, from (3.36),rn+1(t) satisfies the following differential equation:

d

dtrn+1(t) =

n∑j=1

(eλjt

n∏k=1,k 6=j

1

λj − λk

)+ λn+1rn+1(t). (A.1)

Thus, it remains to show that

rn+1(t) =n+1∑j=1

(eλjt

n+1∏k=1,k 6=j

1

λj − λk

)(A.2)

satisfies equation (A.1). In fact, on the one hand, (A.2) implies

d

dtrn+1(t) =

n+1∑j=1

(λje

λjt

n+1∏k=1,k 6=j

1

λj − λk

). (A.3)

36

On the other hand, the right hand side of (A.1)

n∑j=1

(eλjt

n∏k=1,k 6=j

1

λj − λk

)+ λn+1rn+1(t)

=n∑j=1

(eλjt

n∏k=1,k 6=j

1

λj − λk

)+ λn+1

n+1∑j=1

(eλjt

n+1∏k=1,k 6=j

1

λj − λk

)

=n∑j=1

[eλjt

(n∏

k=1,k 6=j

1

λj − λk+ λn+1

n+1∏k=1,k 6=j

1

λj − λk

)]

+λn+1

(eλn+1t

n+1∏k=1,k 6=j

1

λj − λk

)

=n∑j=1

[eλjt

((λj − λn+1)

n+1∏k=1,k 6=j

1

λj − λk+ λn+1

n+1∏k=1,k 6=j

1

λj − λk

)]

+λn+1eλn+1t

n+1∏k=1,k 6=j

1

λj − λk

=n∑j=1

(λje

λjt

n+1∏k=1,k 6=j

1

λj − λk

)+ λn+1e

λn+1t

n+1∏k=1,k 6=j

1

λj − λk)

=n+1∑j=1

(λje

λjt

n+1∏k=1,k 6=j

1

λj − λk

). (A.4)

Combining (A.3) with (A.4) leads to (A.2), and thus the proof is complete.

Proof of Lemma 3.5. Let Υn to denote the right hand side of (3.38).Obviously, Lemmas 3.3 and 3.4 imply that eHt = λ1e

λ1t = Υ1(t) for n = 1and

eHt = r1(t)I + r2(t)(H− λ1I)

= eλ1tI +

(eλ1t

1

λ1 − λ2

+ eλ2t1

λ2 − λ1

)(H− λ1I)

= eλ1tH− λ1I

λ1 − λ2

+ eλ2tH− λ2I

λ2 − λ1

= Υ2(t)

37

for n = 2. This implies that the desired result is true for n = 1 and n = 2.Next, we assume that the representation (3.38) is true for some positiveinteger n, and move to investigate the case for n+ 1. In fact, from Lemmas3.3 and 3.4,

Υn+1(t) = Υn(t) + rn+1(t)Pn

=n∑j=1

[eλjt

n∏k 6=j,k=1

H− λkIλj − λk

]

+

[n+1∑j=1

(eλjt

n+1∏k=1,k 6=j

1

λj − λk

)]·

[n∏k=1

(H− λkI)

]

=n∑j=1

eλjt

[n∏

k 6=j,k=1

H− λkIλj − λk

+

(n+1∏

k=1,k 6=j

1

λj − λk

)·

(n∏k=1

(H− λkI)

)]

+eλn+1t

n∏k=1

H− λkIλj − λk

=n∑j=1

[eλjt

n∏k 6=j,k=1

H− λkIλj − λk

(1 +

H− λjIλj − λn+1

)]

+eλn+1t

n+1∏k 6=n+1,k=1

H− λkIλj − λk

=n∑j=1

[eλjt

n+1∏k 6=j,k=1

H− λkIλj − λk

]+ eλn+1t

n+1∏k 6=n+1,k=1

H− λkIλj − λk

=n+1∑j=1

[eλjt

n+1∏k 6=j,k=1

H− λkIλj − λk

],

which implies that the desired result holds for n+ 1 as well, and thus, by theinduction principle, the proof is complete.

Proof of Proposition 3.4. We only show the expression for function A, asthe others can be derived in a similar way. We first note that e[Q−diag(µ)]t =

38

eQte−diag(µ)t, and for a diagonal matrix,

exp (−diag(µ)t) =

e−µ1t, 0, · · · 0

0, e−µ2t, · · · 0...

.... . .

...0, 0, · · · e−µnt

Thus, combining equation (3.12) and Lemma 3.5 yields

A(t) =⟨1, (E(X0))′ e[Q−diag(µ)]t

⟩= (E(X0))′ eQte−diag(µ)t1

= (E(X0))′ eQt

e−µ1t

...e−µnt

= (E(X0))′

(n∑j=1

eλjtn∏

k 6=j,k=1

Q′ − λkIλj − λk

) e−µ1t

...e−µnt

=

n∑j=1

eλjt

((E(X0))′

n∏k 6=j,k=1

Q′ − λkIλj − λk

) e−µ1t

...e−µnt

=

n∑j=1

eλjt (βj,1, . . . , βj,n)

e−µ1t

...e−µnt

=

n∑j=1

n∑k=1

βj,ke(λj−µk)t,

as desired.

Proof of equation (4.51). To save lines, we only show the result for thecase with B > A+ 1, and the results follow in a completely parallel way forthe other cases. From (4.47) and (4.50), we have

bj = mj − r + limε→0+

[I11(ε) + I12(ε)] + I2, (A.5)

39

where

I11(ε) = −∫ −ε−∞

(ex − 1)1

δjxe(Aj+Bj)xdx,

I12(ε) =

∫ ln 2

ε

(ex − 1)1

δjxe−(Bj−Aj)xdx,

and

I2 =

∫ 1

−1

x

δj|x|eAjx−Bj |x|dx.

Using the fact that Bj > Aj, we can directly calculate the integral for I2 toobtain

I2 =1

δj

2Aj

B2j − A2

j

+

(e−(Bj+Aj)

Bj + Aj− e−(Bj−Aj)

Bj − Aj

)(A.6)

To proceed, we recall the definition of exponential integral function Ei in(4.52) and the following expression for Ei:

Ei(x) = γ + ln |x|+∞∑k=1

xk

(k)(k!), x 6= 0,

where γ is the Euler’s constant. We also note that limx→0

∑∞k=1

xk

(k)(k!)= 0.

Consequently, by applying these facts and the change-of-variable formula forintegral, we obtain

limε→0+

I11(ε) = limε→0+

(∫ ∞(Bj+Aj+1)ε

e−u

udu−

∫ ∞(Bj+Aj)ε

e−u

udu

)= lim

ε→0+−Ei(−(Bj + Aj + 1)ε) + Ei(−(Bj + Aj)ε)

= ln

(Bj + Aj

Bj + Aj + 1

), (A.7)

40

and

limε→0+

I12(ε) = limε→0+

∫ (Bj−Aj−1) ln 2

(Bj−Aj−1)ε

e−u

udu−

∫ (Bj−Aj) ln 2

(Bj−Aj)ε

e−u

udu

= lim

ε→0+−Ei(−(Bj − Aj − 1)ε) + Ei(−(Bj − Aj)ε)

+Ei(−(Bj − Aj − 1) ln 2)− Ei(−(Bj − Aj) ln 2)

= ln

(Bj − Aj

Bj − Aj − 1

)+Ei(−(Bj − Aj − 1) ln 2)− Ei(−(Bj − Aj) ln 2). (A.8)

Combining (A.5), (A.6), (A.7) and (A.8) yields the desired result.

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