PENETRACIONES 1. Penetraciones Estáticas

where x = (x1, . . . , xn) is the variable vector whose components xi denote

the proportion of the total funds invested in security i, µ and Σ are the expected return vector and the covariance matrix, and A, b, C, and d are the coefficients of the linear equality and inequality constraints that define feasible portfolios. The objective is to maximize the expected excess return while limiting the portfolio tracking error to a predefined value of TE.

Unlike the formulations (8.1) and (8.4) that have only linear constraints, this formulation is not in standard quadratic programming form and there- fore can not be solved directly using efficient and widely available QP algo- rithms. The reason for this is the existence of a nonlinear constraint, namely the constraint limiting the portfolio tracking error. So, if all MVO formula- tions are essentially equivalent as we argued before, why would anyone use the “harder” formulations with the risk constraint?

As Jorion observes [40], ex post returns are “enormously noisy measures of expected returns” and therefore investors may not be able or willing to determine minimum acceptable expected return levels, or risk-aversion constants–inputs required for problems (8.1) and (8.4)–with confidence. Jo- rion [40] notes that “it is much easier to constrain the risk profile, either be- fore or after the fact–which is no doubt why investors give managers tracking error constraints.”

Fortunately, the tracking error constraint is a convex quadratic con- straint which means that we can rewrite this constraint in conic form as we saw in the previous chapter. If the remaining constraints are linear as in (10.2), the resulting problem is a second-order cone optimization problem that can be solved with specialized methods.

Furthermore, in situations where the control of multiple measures of risk is desired the conic reformulations can become very useful. In [40], Jo- rion observes that MVO with only a tracking error constraint may lead to

portfolios with high overall variance. He considers a model where a variance constraint as well as a tracking error constraint is imposed for optimizing the portfolio. When no additional constraints are present, Jorion is able to solve the resulting problem since analytic solutions are available. His approach, however, does not generalize to portfolio selection problems with additional constraints such as no-shorting limitations, or exposure limitations to such factors as size, beta, sectors or industries. The strength of conic optimiza- tion models, and in this particular case, of second-order cone programming approaches is that the algorithms developed for them will work for any com- bination of linear equality, linear inequality, and convex quadratic inequality constraints. Consider, for example, the following generalization of the mod- els in [40]: maxx _√ µTx xT_{Σx ≤ σ} q (x− xBM)TΣ(x− xBM) ≤ T E Ax = b Cx ≥ d. (10.3)

This problem can be rewritten as a second-order cone programming problem using the conversions outlined in Section 9.2.2. Since Σ is positive semidef- inite, there exists a matrix R such that Σ = RRT. Defining

y = RTx

z = RTx− RTxBM

we see that the first two constraints of (10.3) are equivalent to (y0, y)∈ Cq,

(z0, z) ∈ Cq with y0 = σ and z0 = T E. Thus, (10.3) is equivalent to the

following second-order cone program: maxx µTx Ax = b Cx ≥ d RTx_{− y = 0} RT_{x− z = R}T_x BM y0 = σ z0 = T E (y0, y)∈ Cq, (z0, z)∈ Cq (10.4)

Exercise 10.1 Second-order cone formulations can also be used for model- ing a tracking error constraint under different risk models. For example, if we had k alternative estimates of the covariance matrix denoted by Σ1, . . . , Σk

and wanted to limit the tracking error with respect to each estimate we would have a sequence of constraints of the form

q

(x_{− x}BM)TΣi(x− xBM)≤ T Ei, i = 1, . . . , k.

Show how these constraints can be converted to second-order cone con- straints.

Exercise 10.2 Using historical returns of the stocks in the DJIA, estimate their mean µi and covariance matrix. Let R be the median of the µis. Find

an expected return maximizing long-only portfolio of Dow Jones constituents that has (i) a tracking error of 10% or less, and (ii) a volatility of 20% or less.

10.2

Approximating Covariance Matrices

The covariance matrix of a vector of random variables is one of the most important and widely used statistical descriptors of the joint behavior of these variables. Covariance matrices are encountered frequently is financial mathematics, for example, in mean-variance optimization, in forecasting, in time-series modeling, etc.

Often, true values of covariance matrices are not observable and one must rely on estimates. Here, we do not address the problem of estimat- ing covariance matrices and refer the reader, e.g., to Chapter 16 in [48]. Rather, we consider the case where a covariance matrix estimate is already provided and one is interested in determining a modification of this estimate that satisfies some desirable properties. Typically, one is interested finding the smallest distortion of the original estimate that achieves these desired properties.

Symmetry and positive semidefiniteness are structural properties shared by all “proper” covariance matrices. A correlation matrix satisfies the addi- tional property that its diagonal consists of all ones. Recall that a symmetric and positive semidefinite matrix M _{∈ IR}n×n satisfies the property that

xTM x≥ 0, ∀x ∈ IRn.

This property is equivalently characterized by the nonnegativity of the eigen- values of the matrix M .

In some cases, for example when the estimation of the covariance matrix is performed entry-by-entry, the resulting estimate may not be a positive semidefinite matrix, that is it may have negative eigenvalues. Using such an estimate would suggest that some linear combinations of the underlying random variables have negative variance and possibly result in disastrous results in mean-variance optimization. Therefore, it is important to correct such estimates before they are used in any financial decisions.

Even when the initial estimate is symmetric and positive semidefinite, it may be desirable to modify this estimate without compromising these prop- erties. For examples, if some pairwise correlations or covariances appear counter-intuitive to a financial analyst’s trained eye, the analyst may want to modify such entries in the matrix. All these variations of the problem of obtaining a desirable modification of an initial covariance matrix estimate can be formulated within the powerful framework of semidefinite optimiza- tion and can be solved with standard software available for such problems.

We start the mathematical treatment of the problem by assuming that we have an estimate ˆΣ _{∈ S}n of a covariance matrix and that ˆΣ is not

necessarily positive semidefinite. Here, _Sn _{denotes the space of symmetric}

n_{× n matrices. An important question in this scenario is the following:} What is the “closest” positive semidefinite matrix to ˆΣ? For concreteness, we use the Frobenius norm of the distortion matrix as a measure of closeness:

dF(Σ, ˆΣ) :=

s X

i,j

(Σij − ˆΣij)2.

Now we can state the closest covariance matrix problem as follows: Given ˆ Σ_{∈ S}n, minΣ dF(Σ, ˆΣ) Σ∈ Cn s (10.5) where C_sn is the cone of n× n symmetric and positive semidefinite matrices as defined in (9.9). Notice that the decision variable in this problem is represented as a matrix rather than a vector as in all previous optimization formulations we considered.

Furthermore, introducing a dummy variable t, we can rewrite the last problem above as:

min t

dF(Σ, ˆΣ) ≤ t

Σ_{∈ C}n s.

It is easy to see that the inequality dF(Σ, ˆΣ)≤ t can be written as a second-

order cone constraint, and therefore, the formulation above can be trans- formed into a conic optimization problem.

Variations of this formulation can be obtained by introducing additional linear constraints. As an example, consider a subset E of all (i, j) covariance pairs and lower/upper limits lij, uij∀(i, j) ∈ E that we wish to impose on

these entries. Then, we would need to solve the following problem: min dF(Σ, ˆΣ)

lij ≤ Σij ≤ uij,∀(i, j) ∈ E

Σ ∈ Cn

s.

(10.6)

When E consists of all the diagonal (i, i) elements and lii= uii = 1,∀i,

we get the correlation matrix version of the original problem. For example, three-dimensional correlation matrices have the following form:

Σ =    1 x y x 1 z y z 1   , Σ∈ C 3 s.

The feasible set for this instance is shown in Figure 10.2.

Example 10.1 We consider the following estimate of the correlation matrix of 4 securities: ˆ Σ =      1.0 0.8 0.5 0.2 0.8 1.0 0.9 0.1 0.5 0.9 1.0 0.7 0.2 0.1 0.7 1.0      . (10.7)

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x y z

Figure 10.1: The feasible set of the nearest correlation matrix problem in 3-dimensions

This, in fact, is not a valid correlation matrix; its smallest eigenvalue is negative: λmin =−0.1337. Note, for example, the high correlations between

assets 1 and 2 as well as assets 2 and 3. This suggests that 1 and 3 should be highly correlated as well, but they are not. Which entry should one adjust to find a valid correlation matrix?

We can approach this problem using formulation (10.6) with E consisting of all the diagonal (i, i) elements and lii = uii = 1,∀i. Solving the result-

ing problem, for example, using SDPT3 [74], we obtain (approximately) the following nearest correction to ˆΣ:

Σ =      1.00 0.76 0.53 0.18 0.76 1.00 0.82 0.15 0.53 0.82 1.00 0.65 0.18 0.15 0.65 1.00      .

Exercise 10.3 Use a semidefinite optimization software package to verify that Σ given above is the solution to (10.5) when ˆΣ is given by (10.7). Exercise 10.4 Resolve the problem above, this time imposing the con- straint that Σ23= Σ32≥ 0.85.

One can consider several variations on the “plain vanilla” version of the nearest correlation matrix problem. For example, if we would rather keep some of the entries of the matrix ˆΣ constant, we can expand the set E to contain those elements with matching lower and upper bounds. Another possibility is to weight the changes in different entries, for example if esti- mates of some entries are more trust-worthy than others.

Another important variation of the original problem is obtained by plac- ing lower limits on the smallest eigenvalue of the correlation matrix. Even when we have a valid (positive semidefinite) correlation matrix estimate, having small eigenvalues in the matrix can be undesirable as they lead to unstable portfolios. Indeed, the valid correlation matrix we obtained above has a positive but very small eigenvalue, which would in fact be exactly zero in exact arithmetic. Hauser and Zuev consider models where minimum eigenvalue of the covariance matrix is maximized and use the matrices in a robust optimization setting [37].

Exercise 10.5 We want to find the nearest symmetric matrix to ˆΣ in (10.7) whose smallest eigenvalue is at least 0.25. Express this problem as a semidef- inite optimization problem. Solve it using an SDP software package.

All these variations are easily handled using semidefinite programming formulations and solved using semidefinite optimization software. As such, semidefinite optimization presents a new tool for asset managers that was not previously available at this level of sophistication and flexibility. While these tools are not yet available as commercial software packages, many academic products are freely available; see the link given in Section 9.4.

10.3

Recovering Risk-Neural Probabilities from Op-

tions Prices

In this section, we revisit our study of the risk-neutral density estimation problem in Section 8.4. Recall that the objective of this problem is to estimate an implied risk-neutral density function for the future price of an underlying security using the prices of options written on that security. Rep- resenting the density function using cubic splines to ensure its smoothness, and using a least-squares type objective function for the fit of the estimate with the observed option prices, we formulated an optimization problem in Section 8.4.

One issue that we left open in Section 8.4 is the rigorous enforcement of the nonnegativity of the risk-neutral density estimate. While we heuristi- cally handled this issue by enforcing the nonnegativity of the cubic splines at the knots, it is clear that a cubic function that is nonnegative at the end- points of an interval can very well become negative in between and therefore, the heuristic technique of Section 8.4 may be inadequate. Here we discuss an alternative formulation that is based on necessary and sufficient conditions for ensuring the nonnegativity of a single variable polynomial in intervals. This characterization is due to Bertsimas and Popescu [10] and is stated in the next proposition.

Proposition 10.1 (Proposition 1 (d),[10]) The polynomial g(x) =Pk

r=0yrxr

semidefinite matrix X = [xij]i,j=0,...,k such that X i,j:i+j=2`−1 xij = 0, ` = 1, . . . , k, (10.8) X i,j:i+j=2` xij = ` X m=0 k+m−` X r=m yr <sub>m</sub>r ! k− r `_{− m} ! ar−mbm, (10.9) ` = 0, . . . , k, (10.10) X  0. (10.11)

In the statement of the proposition above, the notation r m

!

stands for

r!

m!(r−m)! and X 0 indicates that the matrix X is symmetric and positive

semidefinite. For the cubic polynomials fs(x) = αsx3+ βsx2+ γsx + δs that

are used in the formulation of Section 8.4, the result can be simplified as follows:

Corollary 10.1 The polynomial fs(x) = αsx3+ βsx2+ γsx + δs satisfies

fs(x) ≥ 0 for all x ∈ [xs, xs+1] if and only if there exists a 4× 4 matrix

Xs= [xs_ij]i,j=0,...,3 such that

xs ij = 0, if i + j is 1 or 5, xs 03+ xs12+ xs21+ xs30 = 0, xs₀₀ = αsx3s+ βsx2s+ γsxs+ δs, xs₀₂+ xs₁₁+ xs₂₀ = 3αsxs2xs+1+ βs(2xsxs+1+ x2s) + γs(xs+1+ 2xs) + 3δs, xs₁₃+ xs₂₂+ xs₃₁ = 3αsxsx2s+1+ βs(2xsxs+1+ x2s+1) + γs(xs+ 2xs+1) + 3δs, xs 33 = αsx3s+1+ βsx2s+1+ γsxs+1+ δs, Xs _{ 0.} (10.12)

Observe that the positive semidefiniteness of the matrix Xsimplies that the first diagonal entry xs

00 is nonnegative, which corresponds to our earlier

requirement fs(xs) ≥ 0. In light of Corollary 10.1, we see that introducing

the additional variables Xs and the constraints (10.12), for s = 1, . . . , ns,

into the earlier quadratic programming problem in Section 8.4, we obtain a new optimization problem which necessarily leads to a risk-neutral prob- ability distribution function that is nonnegative in its entire domain. The new formulation has the following form:

min

y,X1_,...,Xns E(y) s.t. (8.19), (8.20), (8.21), (8.22), (8.25), [(10.12), s = 1, . . . , ns].

(10.13) All constraints in (10.13), with the exception of the positive semidefi- niteness constraints Xs  0, s = 1, . . . , ns, are linear in the optimization

variables (αs, βs, γs, δs) and Xs, s = 1, . . . , ns. The positive semidefiniteness

constraints are convex constraints and thus the resulting problem can be re- formulated as a convex semidefinite programming problem with a quadratic objective function.

For appropriate choices of the vectors c, fi, gks, and matrices Q and Hks,

we can rewrite problem (10.13) in the following equivalent form:

min_y,X1_,...,Xns cTy +1₂yTQy s.t. fT i y = bi, i = 1, . . . , 3ns, H_ks• Xs_{= 0, k = 1, 2, s = 1, . . . , n} s, (gs k)Ty + Hks• Xs= 0, k = 3, 4, 5, 6, s = 1, . . . , ns, Xs 0, s = 1, . . . , ns, (10.14) where• denotes the trace matrix inner product.

We should note that standard semidefinite optimization software such as SDPT3_{[74] can solve only problems with linear objective functions. Since} the objective function of (10.14) is quadratic in y a reformulation is necessary to solve this problem using SDPT3 or other SDP solvers. We can replace the objective function with min t where t is a new artificial variable and impose the constraint t ≥ cT_{y +} 1

2yTQy. This new constraint can be expressed as

a second-order cone constraint after a simple change of variables; see, e.g., [49]. This final formulation is a standard form conic optimization problem — a class of problems that contain semidefinite programming and second- order cone programming as special classes. Since SDPT3 can solve standard form conic optimization problems we used this formulation in our numerical experiments.

Exercise 10.6 Express the constraint t _{≥ c}T_{y +} 1

2yTQy using linear con-

straints and a second-order cone constraint.

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