Optioned Portfolio Selection: Models and Analysis ∗

Jianfeng LIANG † Shuzhong ZHANG ‡ Duan LI §

January 2005

Abstract

We study in this paper the portfolio selection problem with a stock index andEuropean style options on the index. A refined mean-variance methodology isadopted in the study. Single-stage and two-stage investment models are studied andsolved. In the later case a scenario tree and stochastic programming formulationare used. Explicit forms of the optimal portfolio and its corresponding efficientfrontier are derived, which reveal rich structures of the optimal payoffs. Finally,illustrative numerical examples are presented.

Keywords: options, portfolio selection, scenario tree, stochastic programming.

1 Introduction

Options have played an important role in financial markets. One reason for wideapplications of options is due to versatile payoff structures of options, which can beused to form investment portfolios with desirable risk profiles [8, 14]. The performanceevaluator of portfolios with options has been investigated extensively in the literature;see [1, 3, 4, 5, 6]. The optioned portfolio selection has attracted much attention bothin theory and in practice.

Stochastic programming has been widely applied to financial planning. Many discreteassets allocation problems are formulated as stochastic programming models basedon scenario tree structure; see [2, 10, 16]. Stochastic linear programming models for

∗Research was partially supported by Hong Kong RGC Earmarked Grants CUHK4174/03E andCUHK4180/03E.

†Department of Systems Engineering and Engineering Management, The Chinese University of HongKong, Shatin, Hong Kong.

‡Corresponding author. Email: zhang@se.cuhk.edu.hk. Department of Systems Engineering andEngineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong.

§Department of Systems Engineering and Engineering Management, The Chinese University of HongKong, Shatin, Hong Kong.

1

optioned portfolio selection have been discussed in the literature. Mostly, the models areproposed to maximize the expected portfolio return under certain necessary constraints.Pelsser and Vorst [17] presented a linear programming model with shortfall constraints,which were expressed by a set of chance constraints. Dert and Oldenkamp [7] proposeda linear programming model with a requirement of a guaranteed return. Casino effectwas illustrated, and was controlled by introducing shortfall constraints. A two-stagestochastic linear programming model was discussed by Berkelaar, Dert, Odenkamp,and Zhang [2], and it was numerically solved by a primal-dual decomposition-basedinterior point method.

The mean-variance formulation by Markowitz [12, 13] is a fundamental basis for port-folio selection theory in a single stage setting. The Markowitz’s mean-variance modelhas been recently extended to multi-period setting by Li and Ng [11] and to continuous-time setting by Zhou and Li [19]. A generalized mean-variance model was proposed byMorard and Naciri [15] to optimize the hedging rates. The hedging was implementedwith the covered call writing strategy. The empirical results showed that the use ofcovered calls improved the performance of stock portfolios, while there was no ana-lytical formula for the optimal portfolio derived. Isakov and Marard [9] pointed outthat in the case of incomplete hedging, the mean-variance formula could be applied forthe optioned portfolio selection problems, since the hedged return does not necessarilyhave a non-symmetric distribution. We thus believe that the mean-variance criteria isan acceptable choice for the optioned portfolio selection problem.

Our optioned portfolio consists of one index, a set of European options on this index,and a risk-free asset. In our case, the scenario tree is generated using the distribution ofthe index. We apply the mean-variance formulation to investigate our investment prob-lem, in order to make a further analysis of the payoff based on the explicit expressionsof the optimal portfolio.

The paper is organized as follows. In Section 2, we introduce a single-stage mean-variance model for the optioned portfolio selection problem. We present the formula-tions of the model and its solutions, following by discussing some interesting propertiesof the optimal function. In Section 3, we extend the model to a two-stage case. Weformulate a model and discuss its solutions. We shall see that the payoff of the secondstage subtree inherits similar properties as for the single-stage case. Particular proper-ties of the two-stage case will be discussed. Throughout the paper, numerical exampleswith real life data are used to illustrate and validate our results.

2 The single-stage optioned mean-variance model

2.1 The model and its formulation

We first consider a single-stage investment problem. The available assets are: one index,a class of m European call options on this underlying index, and a risk-free asset. Theoptions have the same expiration date, and their strike prices are K1 < K2 < · · · < Km.

2

The decision horizon is the same as the options’ expiration, and r is the risk-free returnrate in this period. Given initial wealth B and expected return R, the object is toconstruct a portfolio with minimum final payoff variance. The decision variables areX and xf , where X is the number of shares of the index and options being held,X ∈ <m+1, and xf is the amount invested in the risk-free asset.

We formulate the model based on a single-stage scenario tree structure. There aren scenarios in the tree with probability on the ith scenario being pi, i = 1, 2, · · · , n,where

∑ni=1 pi = 1. Other data include the price vector of risky assets (including

index and options) in the beginning, u, and the unit payoff vector of risky assets atthe ith scenario, vi. We denote v =

∑ni=1 pivi as the average unit payoff vector, and

A =∑n

i=1 pi(vi − v)(vi − v)′ as the covariance matrix of risky assets.

Assumption 2.1 Assume that the scenario tree is well generated in the followingsense. First, given intervals I0 = (0, K1), Ii = (Ki, Ki+1), i = 1, 2, · · · ,m − 1,and Im = (Km, +∞), there are at least m + 2 scenarios, and at least one scenario ineach of the above intervals. This structure makes sure the estimated covariance matrixA is positive definite. Second, this structure presents no arbitrage opportunity in thetree.

Besides, we introduce the following notations:

A = A + vv′;α = u′A−1u;β = v′A−1u;γ = v′A−1v;δ = r2α− 2rβ + γ = (v − r · u)′A−1(v − r · u);

ρ =r

δ + 1.

The single-stage mean-variance model is of the following form:

min 12Var (W )

s.t. u′X + xf = BE(W ) = R,

where W is the final wealth. Based on the scenario tree structure, we have

E(W ) =n∑

i=1

piv′iX + rxf = v′X + rxf

Var (W ) = E[(v′iX + rxf −R)2

]=

(Xxf

)′(A rvrv′ r2

)(Xxf

)−R2.

Thus the equivalent deterministic form of the single-stage mean-variance model is

3

(M1) min12

(Xxf

)′(A rvrv′ r2

)(Xxf

)− 1

2R2 (1)

s.t. u′X + xf = B (2)v′X + rxf = R. (3)

Under Assumption 2.1, (M1) is a strictly convex quadratic optimization problem, andits solution is given specifically in the following theorem.

Theorem 2.1 The single-stage mean-variance model (M1) has the following uniqueprimal-dual solution

X = −λ

rA−1(v − ru),

xf =1r

[λ

r(γ − rβ + 1) + µ

],

λ =r

δ + 1(rB − µ) ≡ ρ(rB − µ),

µ = rR− ρB

r − ρ,

where λ and µ are the Lagrangian multipliers related to the constraints (2) and (3),respectively, and the associated risk is

Var (W ) =ρ

r − ρ(R− rB)2.

Proof: The Lagrangian function for the optimization problem(M1) is

L =12

(Xxf

)′(A rvrv′ r2

)(Xxf

)− 1

2R2

−λ(u′X + xf −B)− µ(v′X + rxf −R). (4)

Applying the KKT optimality conditions leads to

∂L

∂X= 0 : AX + rvxf − λu− µv = 0 (5)

∂L

∂xf= 0 : rv′X + r2xf − λ− rµ = 0 (6)

∂L

∂λ= 0 : u′X + xf = B (7)

∂L

∂µ= 0 : v′X + rxf = R. (8)

4

Using (6), we get

xf =1r

[−v′X +

λ

r+ µ

].

Substituting the above equation into (5) yields

AX +λ

r(v − ru) = 0.

Thus, with β = v′A−1u, and γ = v′A−1v, we have

X = −λ

rA−1(v − ru)

xf =1r

[λ

r(γ − rβ + 1) + µ

].

Therefore, (7) becomesλ

r2(δ + 1) +

µ

r= B

leading toλ =

r

δ + 1(rB − µ) = ρ(rB − µ),

where we use the definitions that δ = r2α−2rβ +γ and α = u′A−1u. Finally, (8) yields

R = −λ

r(γ − rβ) +

λ

r(γ − rβ + 1) + µ =

λ

r+ µ,

which further implies

µ = rR− ρB

r − ρ.

Using the optimal solutions, the corresponding variance follows:

Var (W ) = E[(v′iX + rxf −R)2

]

= X ′AX

=(

λ

r

)2

(v − ru)′A−1(v − ru)

=(

λ

r

)2

(r2α− 2rβ + γ)

=(

λ

r

)2

δ

= δ

[ρ(rB − µ)

r

]2

= δ

[ρ

r(rB − r

R− ρB

r − ρ)]2

=(R− rB)2

δ

=ρ

r − ρ(R− rB)2. ¤

5

An obvious consequence of Theorem 2.1 is following. If R is set to be rB, then theoptimal variance reaches its minimum of value zero. The associated solutions are:(X∗, x∗f ) = (0, B), λ∗ = 0, and µ∗ = rB.

2.2 Properties of the optimal portfolio

Denoteψ :=

ρr

r − ρand θ :=

ψ

rA−1(v − ru).

The optimal solutions of (M1) can be written in the following simpler form:

λ = ψ(rB −R)X = θ(R− rB)xf = B − u′θ(R− rB) ≡ B + a(R− rB),

where θ = (θ0, θ1, · · · , θm)′, and a := −u′θ.

Lemma 2.1 The optimal payoff curve is piecewise linear with respect to the indexvalue. At any breakpoint where the slope of the curve changes, the index value mustequal to one of the strike prices of the options. Furthermore, the slopes of the linesegments are steeper for a larger target R value for all R > rB.

Proof : Here the scenarios are represented by possible index values at the investmenthorizon, where scenario S means that the index value is S in this scenario. Denote thepayoff at scenario S as P (S). Let the m European call options be with strike pricesK1 < K2 < · · · < Km. For Kj < S < Kj+1,

P (S) = x0S +j∑

i=1

xi(S −Ki) + rxf .

Suppose that there are two scenarios S1 and S2 between Kj and Kj+1, i.e.

Kj < S1 < S2 < Kj+1.

Then we have

P (S2)− P (S1) = (S2 − S1)x0 +j∑

i=1

xi(S2 − S1) = (S2 − S1)j∑

i=0

xi.

Sincexi = θi(R− rB), i = 0, 1, · · · ,m,

the slope of the payoff function between Kj and Kj+1 is

P (S2)− P (S1)S2 − S1

= (R− rB)j∑

i=0

θi.

6

Therefore, for a fixed R this slope is constant, and the payoff is linear between twoneighboring strike prices. Also because θi is independent of R, for any R ≥ rB, thevalue of

∑ji=0 θi is fixed and so the payoff function is linear in R. Moreover, for a larger

R value with R > rB, the slopes of line segments between any two neighboring strikeprices will get steeper. ¤

In fact, we observe that a scenario S = Kj is a local maximum point for the payofffunction iff

∑j−1i=0 θi > 0, and

∑ji=0 θi < 0, and similarly, it is a local minimum point

iff∑j−1

i=0 θi < 0, and∑j

i=0 θi > 0, as one can observe from an example to be discussedlater in this section.

Proposition 2.1 There are scenarios where the payoffs of the optimal portfolio areconstantly rB, regardless the variable value R.

Proof : Since

P (S) = x0S +m∑

i=1

xi(S −Ki)+ + rxf = rB

is equivalent to

(R− rB)

(θ0S +

m∑

i=1

θi(S −Ki)+ + ra

)= 0,

where a := −u′θ. Hence, the scenarios in question correspond to the roots of theequation

θ0S +m∑

i=1

θi(S −Ki)+ + ra = 0.

¤

Proposition 2.2 In any scenario S, for all R > rB, if P (S) > rB, then the payoff ofa higher R dominates that of a lower R; else if P (S) < rB, then the payoff of a higherR is dominated by that of a lower R.

Proof : For convenience, let us denote K0 = 0. For R > rB, in scenario S, we have

P (S) = rB + (R− rB)

(m∑

i=0

θi(S −Ki)+ + ra

).

If P (S) > rB, then

(R− rB)

(m∑

i=0

θi(S −Ki)+ + ra

)> 0

and som∑

i=0

θi(S −Ki)+ + ra > 0.

7

So for any R > rB, P (S) will always be larger than rB, and it is increasing in R.Similarly, if P (S) < rB, then it is decreasing in R. ¤

The above propositions show a clear structure of the optimal payoff curve. For thesame data with different target expected payoff R, the optimal payoff curves form agroup of piecewise line segments with a similar structure, all of which intersect at somefixed points with the payoff rB, and whose slopes getting steeper with larger targetvalue R. The results are clearly illustrated in the following example.

Example 1 We consider the results of a single-stage portfolio selection problem basedon the market data of options on the S&P 500 index, which are listed on the CBOE.The prices are drawn from the CBOE web page in the morning of Oct. 13, 2003, shownin Table 1. The horizon is equal to the expiration date of the options, which is Nov.21, 2003. The investment horizon is 39 days. In this example, we simply use the midprices as the initial prices for both longing and shorting.

The scenario tree is generated under the assumption that the index value at the horizonis lognormally distributed with an expected annualized growth rate µ = 18.81% andannualized standard deviation of σ = 18.45%, which are listed in the Standard &Poor’s company’s web page. The risk free return rate for the investment horizon isr = 1 + 0.14%. The initial wealth B = $10, 000. We assume three different targetexpected payoffs R1 = rB = $10, 014, R2 = $10, 020, and R3 = $10, 025.

The information of the optimal solutions is shown in Table 2. The third column showsthe θ of the optimal portfolio, which is independent of the target R value. We haveproved in Lemma 2.1 that the slope of the (i + 1)th line segment of the payoff curveis (R − rB)

∑ij=0 θj . So from the values given in the fourth column, we can tell if a

breakpoint is a local maximum or minimum point of the payoff curve, as shown in thefifth column.

Figure 1 shows the optimal payoff curves for R1, R2 and R3. First, the payoff curvesof R2 and R3 are piecewise linear in the index value. Both curves have the samegroup of breakpoints, and the index values at the breakpoints are exactly those of thestrike prices of the options. Second, the slopes of the line segments of the curve forR3 are always steeper than those for R2. Third, above the line of rB, the curve of R3

dominates that of R2; and below the line of rB, the curve of R3 is dominated by thatof R2. Finally, we see that both curves always intersect on the line of rB. So from thefigure, the results proved in previous propositions are clearly verified.

In the next section, we investigate a two-stage optioned mean-variance model. Basedon a two-stage scenario tree, we shall formulate a model and discuss its solutions. Weshall then see that the payoff of the second stage subtree inherits quite a number ofproperties from the single-stage model.

8

Expiration Bid Mid AskS&P 500 1038.06 1038.06 1038.06Call 1005 November 46.30 47.30 48.30Call 1010 November 42.50 43.50 44.50Call 1015 November 38.90 39.90 40.90Call 1020 November 35.40 36.40 37.40Call 1025 November 32.00 33.00 34.00Call 1030 November 28.80 29.80 30.80Call 1035 November 25.80 26.65 27.50Call 1040 November 22.90 23.90 24.90Call 1045 November 20.20 21.20 22.20Call 1050 November 18.00 18.60 19.20Call 1060 November 13.60 14.35 15.10Call 1065 November 11.60 12.35 13.10Call 1070 November 10.00 10.75 11.50Call 1075 November 8.60 9.10 9.60

Table 1: The data of the S&P 500 index and the options of Example 1

the variable θ∗i∑i

j=0 θ∗j local max/minof the breakpointon the strike price

S&P 500 x0 0.0334 0.0334Call 1005 x1 -0.4431 -0.4097 maxCall 1010 x2 0.7104 0.3007 minCall 1015 x3 -0.2639 0.0368 neitherCall 1020 x4 -0.4307 -0.3938 maxCall 1025 x5 1.2481 0.8543 minCall 1030 x6 -2.3537 -1.4994 maxCall 1035 x7 2.9072 1.4078 minCall 1040 x8 -1.8149 -0.4071 maxCall 1045 x9 -0.1096 -0.5167 neitherCall 1050 x10 0.9225 0.4059 minCall 1060 x11 -1.2045 -0.7986 maxCall 1065 x12 1.4448 0.6462 minCall 1070 x13 -0.9166 -0.2704 maxCall 1075 x14 0.2796 0.0092 min

Table 2: The optimal solutions of Example 1

9

940 960 980 1000 1020 1040 1060 1080 1100 1120 11400.998

0.999

1

1.001

1.002

1.003

1.004

1.005

1.006x 10

4

index value

port

folio

pay

off

R1=rBR2>R1R3>R2

Figure 1: The optimal payoff curves of Example 1

3 The two-stage optioned mean-variance model

3.1 The two-stage model and its formulation

We now introduce the two-stage problem. An investment portfolio is supposed to beconstructed at the beginning, and will be revised at the end of the first stage. We denoteB as the initial wealth, and R as the target expected payoff in the end. As before, theassets to be considered are still European call options on a certain underlying index,the index itself and a risk-free asset. The options expire at different times. Supposethat there are m1 options expiring at the end of the first stage, with strike pricesQ1 < Q2 < · · · < Qm1 ; and there are m2 options expiring at the end of the secondstage, with strike prices K1 < K2 < · · · < Km2 . In the two-stage scenario tree, thereare n1 scenarios in the first stage with probability pi, and n2 scenarios in each secondstage subtree with conditional probability qj . The risk free return rates in the twostages are denoted by r1 and r2, respectively. Naturally, r = r1 · r2 is the risk-free ratefor the entire period. Other notations are as follows:

10

X0: the number of holding shares of risky assets in the first stage (X0 ∈ <m1+m2+1);Xi: the number of holding shares of risky assets in the ith subtree of the second stage

(Xi ∈ <m2+1, i = 1, 2, · · · , n1);xif : the amount invested in the risk-free asset at each decision point, i = 0, 1, · · · , n1;u0: the price vector in the beginning (u0 ∈ <m1+m2+1);ui: the price vector at the beginning of the second stage

(ui ∈ <m2+1);vi: the unit payoff vector at scenario i in the first stage

(vi ∈ <m1+m2+1, i = 1, 2, · · · , n1);vji : the unit payoff vector at the jth scenario in the ith subtree of the second stage

(vji ∈ <m2+1, i = 1, 2, · · · , n1, j = 1, 2, · · · , n2);

vi: the expected unit payoff vector of the second stage risky assets in ith subtree;Ai: the conditional covariance matrix of the second-stage risky assets

(Ai ∈ <(m2+1)×(m2+1));A: the covariance matrix of the first-stage risky assets (A ∈ <(m1+m2+1)×(m1+m2+1)).

To formulate the two-stage mean-variance model clearly, we introduce the followingnotations:

Ai = pi(Ai + viv′i);

vi = pi · vi;ri = pi · r2;αi = u′iA

−1i ui;

βi = v′iA−1i ui;

γi = v′iA−1i vi;

δi = (r2)2αi − 2r2βi + γi = (vi − r2ui)′A−1i (vi − r2ui);

p0 =n1∑

i=1

pi

δi + 1;

r = p0r =n1∑

i=1

pir

δi + 1=

n1∑

i=1

pir1r2

δi + 1=

n1∑

i=1

r1ri

δi + 1;

v0 =n1∑

i=1

pi/(δi + 1)p0

vi;

v0 = p0v0;

A0 =1p0

n1∑

i=1

pi

δi + 1viv

′i − v0v

′0.

If we take pi/(δi+1)p0

as the modified probability of the ith scenario in the first stage,then v0 and A0 as defined are the first-stage expected unit payoff and the covariancematrix based on the modified probabilities. Next, we further introduce the followingnotations:

A0 = p0(A0 + v0v′0);

11

α0 = u′0A−10 u0;

β0 = v′0A−10 u0;

γ0 = v′0A−10 v0;

δ0 = (r1)2α0 − 2r1β0 + γ0 = (v0 − r1u0)′A−10 (v0 − r1u0);

ρ =r

δ0 + 1≡ p0r

δ0 + 1.

Similar to Assumption 2.1, we assume that the two-stage scenario tree does not admitarbitrage opportunities and that the covariance matrices at both stages are positivedefinite. In that case, it follows that A0 Â 0. To see this let us suppose for thesake of obtaining a contradiction that A0 is not positive definite. Then there shouldexist X0 6= 0, such that X ′

0A0X0 = 0, which means that in the modified probability, theportfolio X0 is risk-free, i.e., in all scenarios at the first stage, the portfolio X0 will havethe same payoff, which is independent of the probabilities. So with this X0, we mustalso have X ′

0AX0 = 0, where A is the covariance matrix with the actual probabilities.This leads to a contradiction since A Â 0, and so we must have A0 Â 0.

The general two-stage mean-variance model is

min 12Var (WT )

s.t. u′0X0 + x0f = Bv′iX0 + r1x0f = u′iXi + xif , i = 1, 2, · · · , n1

E(WT ) = R,

where WT is the wealth at the end of the second stage. Now we derive the expressionsfor E(WT ) and Var (WT ). Denoting v as the final unit payoff, we have

E(WT ) = E[v′XT + r2xTf ] = E[E[v′XT + r2xTf |Si]]

= E[v′iXi + r2xif ] =n1∑

i=1

pi

(v′iXi + r2xif

)=

n1∑

i=1

(vi′Xi + rixif

).

By the definition

Ai = E[(vi − vi)(vi − vi)′|Si] = E[vi(vi)′|Si]− viv′i,

we haveE[vi(vi)′|Si] = Ai + viv

′i.

Then, the final variance is

Var (WT ) = E[(v′XT + r2xTf −R)2]= E[(v′XT + r2xTf )2]−R2

= E[E[(v′iXi + r2xif )2|Si]]−R2

= E[X ′iE[vi(vi)′|Si]Xi + 2r2xif v′iXi + (r2xif )2]−R2

=n1∑

i=1

pi[X ′i(Ai + viv

′i)Xi + 2r2xif v′iXi + (r2)2x2

if ]−R2

12

=n1∑

i=1

(Xi

xif

)′(Ai r2vi

r2vi′ r2ri

)(Xi

xif

)−R2.

This yields a deterministic form of the two-stage mean-variance model as follows:

(M2) minn1∑

i=1

12

(Xi

xif

)′(Ai r2vi

r2vi′ r2ri

)(Xi

xif

)− 1

2R2 (9)

s.t. u′0X0 + x0f = B (10)v′iX0 + r1x0f = u′iXi + xif , i = 1, 2, · · · , n1 (11)n1∑

i=1

(vi′Xi + rixif

)= R. (12)

Lemma 3.1 We have(1) αi > 0, γi > 0, i = 1, 2, · · · , n1;(2) δi ≥ 0, and ∃ i, such that δi > 0, i = 1, 2, · · · , n1;(3) p0 ∈ (0, 1), ρ ∈ (0, r).

Proof: The first two parts are obvious from the definitions. Then

0 < p0 =n1∑

i=1

pi

δi + 1<

n1∑

i=1

pi = 1,

so p0 ∈ (0, 1).

Since A0 Â 0, and v0 6= r1u0, we have δ0 > 0. Thus

0 < ρ =p0r

δ0 + 1< r.

¤

Therefore, our model is a strictly convex quadratic optimization problem. It can besolved by using the KKT conditions, as shown in the following theorem.

Theorem 3.1 The two-stage mean-variance problem (M2) has the following uniqueprimal-dual solution,

Xi = −λiri

A−1i (vi − r2ui), xif = 1

r2

[λiri

(γi − r2βi + 1) + µ]

X0 = − λ0r2rA−1

0 (v0 − r1u0), x0f = 1r

[λ0r (γ0 − r1β0 + 1) + µ

]

λi = riδi+1 [r2(v′iX0 + r1x0f )− µ] , λ0 = r

δ0+1(rB − µ) ≡ ρ(rB − µ)

µ = rR−ρBr−ρ ,

13

where λ0, λi and µ are the Lagrangian multipliers of the model, and the associated riskis

Var =ρ

r − ρ(R− rB)2.

Some comments are in order here. When R is set to be rB, the variance reaches itsminimum value of zero, and the associated solutions are:

(X∗0 , x∗0f ) = (0, B), (X∗

i , x∗if ) = (0, r1B), λ∗ = 0, µ∗ = rB.

Proof of Theorem 3.1: The Lagrangian function of (M2) is

L =n1∑

i=1

12

(Xi

xif

)′(Ai r2vi

r2vi′ r2ri

)(Xi

xif

)− 1

2R2

−λ0(u′0X0 + x0f −B)−n1∑

i=1

λi(u′iXi + xif − v′iX0 − r1x0f )

−µ

(n1∑

i=1

(vi′Xi + rixif

)−R

). (13)

Therefore, the KKT optimality conditions are given as

∂L

∂X0= 0 : −λ0u0 +

n1∑

i=1

λivi = 0 (14)

∂L

∂x0f= 0 : −λ0 +

n1∑

i=1

λir1 = 0 (15)

∂L

∂Xi= 0 : AiXi + r2vixif − λiui − µvi = 0 (16)

∂L

∂xif= 0 : r2vi

′Xi + r2rixif − λi − µri = 0 (17)

∂L

∂λ0= 0 : u′0X0 + x0f = B (18)

∂L

∂λi= 0 : u′iXi + xif − v′iX0 − r1x0f = 0 (19)

∂L

∂µ= 0 :

n1∑

i=1

(vi′Xi + rixif

)= R. (20)

Solving the second stage variables in (17) gives

xif =1r2

[−v′iXi +

λi

ri+ µ

].

Substituting the above equation into (16) yields

piAiXi +λi

r2(vi − r2ui) = 0.

14

Solving Xi in the above equation and substituting it back to the equation for xif givethe following,

Xi = −λi

riA−1

i (vi − r2ui)

xif =1r2

[λi

ri(γi − r2βi + 1) + µ

].

Therefore, (19) becomes

v′iX0 + r1x0f =λi

r2· δi + 1

ri+

µ

r2

and soλi =

ri

δi + 1[r2(v′iX0 + r1x0f )− µ

].

Now we shall deal with the first stage variables. Condition (15) reads

−λ0 +n1∑

i=1

r1ri

δi + 1[r2(v′iX0 + r1x0f )− µ] = 0

and so−λ0 + rr2v0

′X0 + rrx0f − µr = 0

andx0f =

1r[−r2v

′0X0 +

λ0

r+ µ].

Similarly, substituting λi and x0f in (14) gives

−λ0u0 +n1∑

i=1

viri

δi + 1[r2(v′iX0 + r1xof )− µ] = 0;

that is,−λ0u0 + (r2)2A0X0 + rr2v0x0f − µr2v0 = 0,

which further yields

(r2)2p0A0X0 +λ0

r1(v0 − r1u0) = 0,

and soX0 = − λ0

r2rA−1

0 (v0 − r1u0).

Thus, we obtain

x0f =1r

[λ0

r(γ0 − r1β0 + 1) + µ

].

15

In light of the above developed relations, equation (18) can be now rewritten as

B = − λ0

r2r(β0 − r1α0) +

λ0

rr(γ0 − r1β0 + 1) +

µ

r

or, equivalently,

B =λ0

r× δ0 + 1

r+

µ

r

which gives

λ0 =r

δ0 + 1(rB − µ) = ρ(rB − µ).

Using the expressions for Xi and xif , we then have

Ri = vi′Xi + rixif

= −λi

r2(γi − r2βi) +

λi

r2(γi − r2βi + 1) + piµ

=λi

r2+ piµ.

We also have the following from the expressions of X0 and x0f ,

r2v0′X0 + rx0f =

λ0

r+ p0µ.

Therefore,

R =n1∑

i=1

Ri

=n1∑

i=1

(vi′Xi + rixif

)

=n1∑

i=1

(λi

r2+ piµ

)

=

{n1∑

i=1

pi

δi + 1[r2(v′iX0 + r1x0f )− µ]

}+ µ

= r2v0′X0 + rx0f − p0µ + µ

=λ0

r+ µ

=ρ

r(rB − µ) + µ,

i.e.,R = ρB + µ(r − ρ)/r

and soµ = r

R− ρB

r − ρ.

16

We now are ready to calculate the optimal variance. Let

E(W 2i ) = X ′

iAiXi + (Ri

pi)2.

Using the expressions for Ri and Xi, we have

E(W 2i ) =

(λi

ri

)2

δi +(

λi

ri+ µ

)2

=(

λi

ri

)2

(δi + 1) + 2λi

riµ + µ2

=[r2(v′iX0 + r1x0f )− µ]2

δi + 1+ 2

[r2(v′iX0 + r1x0f )− µ]δi + 1

µ + µ2

=[r2(v′iX0 + r1x0f )]2

δi + 1+ µ2 δi

δi + 1.

Since Var (W ) =∑n1

i=1 piE(W 2i )−R2, it follows that

Var (W ) + R2 = (r2)2n1∑

i=1

pi

δi + 1[(v′iX0)2 + 2r1x0fv′iX0 + (r1x0f )2

]+ µ2

n1∑

i=1

piδi

δi + 1

= (r2)2[X ′

0A0X0 + 2r1x0f v0′X0 + p0(r1x0f )2

]+ µ2(1− p0)

= (r2)2p0

[X ′

0A0X0 + (v′0X0 + r1x0f )2]+ µ2(1− p0)

= (r2)2p0

[(λ0

r2r

)2

δ0 +(

λ0

r2r+

µ

r2

)2]

+ µ2(1− p0)

= p0

[(λ0

r

)2

(δ0 + 1) + 2λ0

rµ + µ2

]+ µ2(1− p0)

= p0

[(rB − µ)2

δ0 + 1+ 2

rB − µ

δ0 + 1µ + µ2

]+ µ2(1− p0)

= p0(rB)2 + δ0µ

2

δ0 + 1+ µ2(1− p0)

= rρB2 + µ2

(1− p0

δ0 + 1

)

= rρB2 + µ2(1− ρ

r

)

= rρB2 +r

r − ρ(R− ρB)2.

Summarizing, we haveVar (W ) =

ρ

r − ρ(R− rB)2.

¤

17

3.2 Properties of the optimal portfolio

Based on the explicit forms of the optimal solutions, and also the special payoff struc-tures of the options, we are now in a position to analyze the payoff of the correspondingportfolio. First, let us denote:

ψ :=ρr

r − ρ

θ := ψ · 1r2r

A−10 (v0 − r1u0)

τi := kiψ

r1[γ0 − r1β0 + 1− v′iA

−10 (v0 − r1u0)]

φi := τi · 1ri

A−1i (vi − r2ui).

Thus, the optimal solutions can be simplified as:

λ0 = ψ(rB −R), X0 = (R− rB)θ, x0f = B − u′0θ(R− rB),λi = τi(rB −R), Xi = (R− rB)φi, xif = v′iX0 + r1x0f − u′iXi,

where ψ, θ, τi, φi are all constants that are independent of R and B. So for a givenscenario in the middle stage, the optimal second-stage solution Xi is linear in (R−rB).

Proposition 3.1 Given a fixed scenario in the middle stage, the optimal final payoffof this subtree is piecewise linear with breakpoints being the strike prices of two-stageoptions. The slopes of the payoff curve becomes steeper as the target R value gets larger.

Proof: Suppose that for the given scenario Si at the end of the first stage, the secondstage optimal solution is

Xi = φi(R− rB),

where φi = (φi0, φi1, · · · , φim2)′. Suppose that there are two final scenarios Si1 and Si2

following Si with Kt < Si1 < Si2 < Kt+1. Similar to the single-stage case, we have

P (Si1)− P (Si2)Si1 − Si2

= (R− rB)t∑

j=0

φij .

Now we see if R is fixed, then between each pair of the neighboring strike prices, theslope of the payoff curve is constant, and so the payoff curve is piecewise linear, withthe break points being the strike prices. Furthermore, if R changes, then the larger theR value, the steeper the slope of the curve. ¤

Proposition 3.2 For a given scenario in the middle stage, for the final payoff of thissubtree, the payoff curves with different R values intersect only at some fixed finalscenarios, where the payoff is always rB.

18

Proof: Suppose that from a given middle stage scenario Si, there is a related finalstage scenario Sij . The final payoff is

P (Si, Sij) = (Sijι−K)′+Xi + r2xif

= (Sijι−K)′+Xi + r2[v′iX0 + r1x0f − u′iXi]= [(Sijι−K)+ − r2ui]′Xi + r2[vi − r1u0]′X0 + rB

= [(Sijι−K)+ − r2ui]′φi(R− rB) + r2[vi − r1u0]′θ(R− rB) + rB

= rB + (R− rB){[(Sijι−K)+ − r2ui]′φi + [r2vi − ru0]′θ}.

For given R > rB, if P (Si, Sij) = rB, then

[(Sijι−K)+ − r2ui]′φi + [r2vi − ru0]′θ = 0,

and P (Si, Sij) will remain rB for any other R value. If P (Si, Sij) > rB, then

[(Sijι−K)+ − r2ui]′φi + [r2vi − ru0]′θ > 0.

Thus, the payoff is increasing in the R value. Similarly, if P (Si, Sij) < rB, then it willalways be less than rB, and will be decreasing in R. ¤

Suppose that R1 > R2. For a fixed middle-stage scenario, if there are some finalscenarios attaining the payoff rB, then the final payoff curves will all intersect in thesescenarios. For other scenarios, the payoff curve of R1 will dominate that of R2 ifR2 > rB, and be dominated if R2 < rB, as shown in the next example.

Now we shall concentrate on a final-stage scenario S. Note that there may be differentpaths to get to this scenario. Even though the eventual index value is the same, differentmiddle-stage scenarios may lead to quite different payoffs.

Proposition 3.3 Consider a final-stage scenario S. Suppose that there are differentmiddle-stage scenarios that can reach S. Then the optimal payoffs from different pathswill follow a fixed order. Furthermore, the differences are proportional to R− rB.

Proof: Suppose there are two middle-stage scenarios Si and Sj , such that both secondstage subtrees contain the final-stage scenario with index value S. The optimal payoffsare

P (Si, S) = rB + (R− rB){[(Sι−K)+ − r2ui]′φi + [r2vi − ru0]′θ}P (Sj , S) = rB + (R− rB){[(Sι−K)+ − r2uj ]′φj + [r2vj − ru0]′θ}.

Therefore,

∆Pij(S) = P (Si, S)− P (Sj , S)= (R− rB){(Sι−K)′+(φi − φj)− r2(u′iφi − u′jφj) + r2(vi − vj)′θ}.

19

So for given Si, Sj and R, ∆Pij(S) is proportional to R − rB. If ∆Pij(S) > 0, thenso is true for any R > rB. That is, the payoff dominance relationship is invariant withregard to R provided that R > rB.

If ∆Pij(S) = 0, then

(Sι−K)′+(φi − φj)− r2(u′iφi − u′jφj) + r2(vi − vj)′θ = 0.

In that case, for any R, the two subtrees from Si and Sj will always share the sameoptimal payoff at this final-stage scenario S. ¤

Example 2 We solve a two-stage portfolio selection problem with options on the S&P500 index, which are listed on the CBOE. The prices are drawn from the CBOE webpage in the morning of Oct. 13, 2003, shown in Table 3. The portfolios can be con-structed in the beginning of the investment and can also be reorganized on Oct. 31,2003. The investment horizon is Nov. 21, 2003, on which the options can be exercised.The whole investment horizon is 39 days, with the first stage 18 days, and the secondstage 21 days. In this example, we simply use the mid prices as the initial prices forboth buying and selling. The options prices at the end of first stage are calculatedusing the Black-Scholes formula.

The scenario tree is generated under the assumption that the index value at the horizonis lognormally distributed with an expected annualized growth rate µ = 18.81% and anannualized standard deviation of σ = 18.45%, which are listed in the Standard & Poor’scompany’s web page. The risk-free return rates for both stages are r1 = r2 = 1+0.07%.The initial wealth is $10,000, and we assume two different target expected payoffsR1 = rB = $10, 014, and R2 = $10, 050. Figure 2 shows the optimal payoff surfaces forR1 and R2. The flat surface is for the risk-free expected payoff R1, and the other is forR2. We see that they cross at some points, where the payoffs equal the risk-free returnfor any R values. Figure 3 shows the final payoff of one of the second stage subtree,in which we tried another R3 = $10, 025. We see that the final payoff of the subtreehas the same properties as those of the single-stage model in Section 2. Figure 4 is themean-variance efficient frontier of the two-stage problem.

Expiration Bid Mid AskS&P 500 1038.06 1038.06 1038.06Call 1010 November 42.50 43.50 44.50Call 1020 November 35.40 36.40 37.40Call 1050 November 18.00 18.60 19.20Call 1065 November 11.60 12.35 13.10

Table 3: The data of the S&P 500 index and the options of Example 2

20

950

1000

1050

1100

1150

800

900

1000

1100

1200

1300

14000.998

1

1.002

1.004

1.006

1.008

1.01

1.012

x 104

index value in first stageindex value in second stage

payo

ff of

opt

imal

por

tfolio

Figure 2: The optimal payoff surfaces of Example 2

900 950 1000 1050 1100 1150 12000.999

1

1.001

1.002

1.003

1.004

1.005x 10

4

index value in the final stage of the fourth subtree

final

opt

imal

pay

off o

f the

four

th s

ubtr

ee

R2>R3R3>rBR1=rB

Figure 3: The optimal final payoff curves of a subtree

0 1 2 3 4 5 61.001

1.0015

1.002

1.0025

1.003

1.0035

1.004

1.0045

1.005x 10

4

stavdard deviation

expe

cted

pay

off

Figure 4: The efficient frontier of Example 221

References

[1] Albrecht, P., R. Maurer, and M. Timpel (1995): A shortfall approach to theevaluation of risk and return of positions with options, Proceedings 5th AFIRInternational Colloquium, Brussels 3, 1003-1023.

[2] Berkelaar, A., C. Dert, B. Odenkamp, and S. Z. Zhang (2002): A primal-dualdecomposition-based interior point approach to two-stage stochastic linear pro-geamming, Operations Research 50, 904-915.

[3] Bookstaber, R. and R. G. Clarke (1981): Options can alter portfolio return distri-butions, Journals of Portfolio Management 7(3), 63-70.

[4] Bookstaber, R. and R. G. Clarke (1984): Option portfolio strategies: measurementand evaluation, Journal of Business 57, 469-492.

[5] Bookstaber, R. and R. G. Clarke (1985): Problems in evaluating the performenceof portfolios with options, Financial Analysts Journal 41(1), 48-62.

[6] Brooks, R., H. Levy, and J. Yoder (1987): Using stochastic dominance to evaluatethe performance of portfolios with options, Financial Analysts Journal 43(2), 79-82.

[7] Dert, C. and B. Oldenkamp (2000): Optiaml guaranteed return portfolio and thecasino effect, Operations Research 48, 768-775.

[8] Hull, J. C. (1999): Options, Futures, & Other Derivatives, Prentice Hall, UpperSaddle River, NJ07458.

[9] Isakov, D. and B. Morard (2001): Improving portfoio performance with optionstrategies: evidence from Switzerland, European Financial Management 7(1), 73-91.

[10] Koskosidis, Y. A. and A. M. Duarte (1997): A scenario-based approach to activeasset allocation, The Journal of Portfolio Management 23(2), 74-85.

[11] Li, D., and W. L. Ng (2000): Optimal dynamic portfolio selection: multi-periodmean-variance formulation, Mathematical Finance 10, 387-406.

[12] Markowitz, H. M. (1952): Portfolio Selection, Journal of finance 7, 77-91.

[13] Markowitz, H. M. (1959): Portfolio Selection: Efficient Diversification of Invest-ment, Wiley, New York.

[14] Mcmillan, L. G. (2002): Options as a Strategic Investment, New York Institute ofFinance.

[15] Morard, B. and A. Naciri (1990): Options and investment strategies. Journal ofFuture Markets 10, 505-517.

22

[16] Odenkamp, B. (1999): Derivatives in Portfolio Management. Ph,D. thesis, ErasmusUniversity Rotterdam, Thesis Publishers, Amsterdam, The Netherlands.

[17] Pelsser, A. and T. Vorst (1995): Optiomal optioned portfolios with confidencelimits on shortfall constraints, Advances in Quantitative Analysis of Finance andAccounting 3A, 205-220.

[18] Pounds, H. M. (1978): Covered call oprion writing: strategies and rules, Journalof Portfolio Management 4(2), 31-42.

[19] Zhou, X. Y. and D. Li (2000): Continuous-time mean-variance portfolio selection:a stochastic LQ framework, Applied Mathematics Optimization 42, 19-33.

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