RobustSOCPOptimizationMulti-PeriodPortfolios2 (2).pdf

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A Robust Optimization Framework for Multi-Period Portfolios with Options
via Second-Order Cone Programming
Article · October 2004
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2. 1
A Robust Optimization Framework for
Multi-Period Portfolios with Options via
Second-Order Cone Programming
Author: Eva-Marie Muller-Stuler
Date: 20 September 2004
Abstract
The classical Markowitz mean-variance portfolio optimization paradigm, while
foundational to modern financial theory, exhibits a critical deficiency: its solutions are
demonstrably unstable and acutely sensitive to statistical errors inherent in the
estimation of market parameters. This often results in portfolios that are "error-
maximized" and sub-optimal in practice. To overcome this fragility, this paper develops a
deterministic, robust optimization framework designed to immunize portfolio selection
against bounded parameter uncertainty.
The primary contribution of this research is the extension of this robust paradigm to
complex, multi-period investment horizons that incorporate financial derivatives,
specifically options with their characteristic non-linear, piecewise-linear payoff
structures. The central mathematical innovation is the application of conic duality to
reformulate the seemingly intractable, semi-infinite optimization problem into a
computationally efficient Second-Order Cone Program (SOCP). This transformation
renders the problem solvable with standard software, bridging the gap between
theoretical robustness and practical implementation.
Numerical experiments, conducted with both simulated and real market data, validate
the proposed model. The resulting robust portfolios consistently exhibit superior stability
and improved risk-adjusted performance, effectively hedging against the worst-case
parameter realizations that cause classical models to fail. This paper provides a tractable,
powerful, and mathematically rigorous blueprint for modern financial risk management
and data-driven decision-making in complex, uncertain environments.

3. 2
1. Introduction
1.1. The Deficiency of Classical Models
The central objective of portfolio management is the maximization of return while
minimizing risk. The seminal work of Markowitz established a mathematical framework
for this task, evaluating a portfolio based on its expected return and its variance. The
optimal portfolio is found by solving a convex quadratic optimization problem. This
mean-variance model and its extensions, such as the Capital Asset Pricing Model
(CAPM), have profoundly influenced economic theory.
However, the practical application of the Markowitz model is limited by its extreme
sensitivity to perturbations in its input data. Since market parameters such as expected
return  μ and covariance  D are derived from historical data, they are fundamentally
uncertain. The classical approach, by assuming these estimators are true values, ignores
this inherent uncertainty and often amplifies estimation errors. While techniques like
resampling or imposing weight constraints can mitigate this issue, they do not offer a
deterministic guarantee of performance under all plausible parameter realizations.
1.2. The Robust Optimization Paradigm
This research employs the paradigm of robust optimization, which provides inherent
immunity to data uncertainty. We model uncertain parameters as unknown but bounded
variables residing within a well-defined uncertainty set. The optimization is then
performed with respect to the worst-case realization of these parameters.
We model the random capital return vector  r such that its natural logarithm follows:
 
lnr=lnμ+ϵ,E(ϵϵ)=
T
D

where
 
μ∈R
n
is the true, unknown mean return vector and  ϵ is a vector of residuals.
Since we only have a statistical estimate
 μ
~
, the core of the robust approach is to define
an ellipsoidal uncertainty set  U around this estimate:

4. 3
 U={μ∣∣∣C(μ−)∣∣≤μ
~
θ}

where the matrix  C is derived from the covariance of the estimator
 μ
~
and the scalar  θ
dictates the size of the set, thereby controlling the level of robustness. The objective is to
optimize for the worst-case vector  μ within this entire set
 U.
 
Figure 1 — Ellipsoidal Uncertainty Set and Worst-case Point
- Illustration of an ellipsoidal uncertainty set  U for mean returns. The red arrow
indicates the direction of the worst-case realization within the set, which the
robust optimization framework accounts for.
The robust analogue of the Markowitz problem is thus a max-min problem: to maximize
the worst-case expected return subject to a constraint on the maximum possible
variance.
 
E[r]
ϕ
max
μ∈Sm
minϕ
s.t.Var[r]≤
D∈Sd
max ϕλ,
1ϕ=
T
1

5. 4
2. Mathematical Framework: Conic Programming
The tractability of the robust optimization problem hinges on its reformulation as a
convex program, specifically a Second-Order Cone Program (SOCP).
2.1. Cones and Duality
A set
 
K⊆R
n
is a cone if for any
 x∈K and scalar
 λ>0, it holds that
 λx∈K. A cone
is convex if it is closed under addition.
The dual cone  K
∗
of a cone  K is defined as:
 
K=
∗
{s∈R∣
n
xs≥
T
0 for all x∈K}

A fundamental result is the bipolar relation for a closed convex cone:
 (K)=
∗∗K.
A standard conic linear program (K-LP) has the form:
- Primal (P):
 mincx
T
subject to
 Ax=b,x⪰K0
- Dual (D):
 maxby
T
subject to
 Ay+
T
s=c,s⪰K
∗0
where
 x⪰K0 denotes that  x is in the cone  K. The weak and strong duality theorems of
linear programming extend to conic programs, providing a powerful analytical
framework.
2.2. Second-Order Cone Programming (SOCP)
A second-order cone
 
C⊂kR
k
(also known as the Lorentz or "ice cream" cone) is
defined as:
 
C=k ∣u∈R,t∈R,∣∣u∣∣≤t{[
t
u
]
k−1
2}

This cone is self-dual, meaning
 
C=
k
∗
Ck.

6. 5
 
Figure 2 — Second-Order Cone (SOCP) Feasible Region
- The second-order cone, also known as the Lorentz cone, representing the feasible
region for SOCP constraints. Its self-dual nature enables efficient robust
optimization.
An SOCP is a convex optimization problem of the form:
 
minfx
T
s.t.∣∣Ax+ib∣∣≤i2cx+
i
T
d,i=i 1,...,N

Each constraint requires that an affine transformation of the decision variable  x lies
within a second-order cone. LPs and convex quadratic programs are special cases of
SOCPs. Despite their non-linearity, SOCPs can be solved with high efficiency using
interior-point methods.
The robust linear constraint arising from our ellipsoidal uncertainty set,
 
x+aˉ
i
T
∣∣Px∣∣≤ibi, is naturally an SOCP constraint. This provides the foundational link
between robust optimization and computationally tractable SOCPs.

7. 6
Table 1: Key SOCP Transformations
Original Constraint SOCP Reformulation
 minμϕμ∈U
T

 ϕ−μ
~T
Θ∥Pϕ∥
 ϕϕ≤
TD
~
λ
 
≤(
2ϕD
~
1/2
1−λ
)1+λ
Robust cash flow (Sec 5.2)
 aX+
T
b≤pπ−
T
ΘπVπ
T
Option duality (Sec 4.2) Dual cone membership  C
∗
)
3. The Single-Period Robust Model
We first construct and solve the robust portfolio problem for a single period without
options.
3.1. Problem Transformation
The robust portfolio selection problem is given by:
 
μϕ
ϕ
max
μ∈Sm
min
T
s.t.ϕDϕ≤
D∈Sd
max
T
λ,
1ϕ=
T
1,ϕ≥0

where
 Sm is the ellipsoidal uncertainty set for the mean returns and
 Sd is the uncertainty
set for the diagonal covariance matrix.
The inner maximization over the covariance matrix D simplifies to using the upper
bound of the uncertainty interval for each diagonal element, i.e.,
 ϕDϕ≤
T
ϕϕ
T
Dˉ
. The
inner minimization for the worst-case mean return for a fixed portfolio  ϕ within the
ellipsoidal set
 S=m{μ∣μ=+μˉPu,∣∣u∣∣≤1} has a closed-form solution:
 
μϕ=
μ∈Sm
min
T
ϕ−μˉ
T
∣∣Pϕ∣∣

8. 7
This is derived by finding the minimum of a linear function over a unit ball, which
occurs on the boundary in the direction opposite to the gradient.
3.2. The Equivalent SOCP Formulation
Substituting these results, the robust problem becomes:
 
ϕ−
ϕ
maxμˉ
T
∣∣Pϕ∣∣
s.t.ϕϕ≤
T
Dˉλ,
1ϕ=
T
1,ϕ≥0

This is a convex optimization problem with a linear objective, a norm term, and a
quadratic constraint. By introducing an auxiliary variable  t for the objective, we can
rewrite this in a standard SOCP format. The quadratic constraint
 ϕϕ≤
T
Dˉλ can be
transformed into a second-order cone constraint. A hyperbolic constraint of the form
 zz≤
T
xy for
 x,y≥0 is equivalent to the SOC
constraint:
 
≤(
2z
x−y
)x+yApplying this transformation to the variance constraint
and representing the robust return constraint directly yields the final SOCP:
 maxt
 s.t.∣∣Pϕ∣∣≤ϕ−μˉ
T
t
 
≤(
2ϕDˉ1/2
1−λ
)1+λ

 1ϕ=
T
1,ϕ≥0

This problem can now be solved efficiently using standard SOCP solvers.
Illustrative Example (Two-Asset Portfolio):
Consider a portfolio with two assets:
- Estimated returns:
 =μ
~
[0.08,0.12]
T

- Covariance matrix:

9. 8
 
=D
~
[
0.04
0.01
0.01
0.09
]

- Uncertainty scaling:
 (P=/D
~
1/2
T(with
 T=50historical observations)
- Risk tolerance:
 λ=0.02

The robust objective
 maxϕ−θ∥Pϕ∥)(μ
~T
)transforms to SOCP:

 
max
s.t.
t
≤1+0.02(
2ϕD
~
1/2
1−0.02
)
≤ϕ−t+θ(
Pϕ
ϕ−tμ~T)μ
~T
ϕ+ϕ=1,ϕ≥012

Solving this SOCP yields weights
 ϕ=
∗[0.62,0.38]
T
, balancing return and uncertainty
penalization.
4. Incorporating Options: A Robust Model with
Non-Linear Payoffs
The inclusion of options significantly complicates the model due to their non-linear,
piecewise-linear payoff structures. This section details how the robust framework is
extended to handle such instruments.
4.1. Modeling Option Payoffs
An option's value at expiry depends on the price of its underlying asset,  S
1
. The return of
a call option,
 
r
c
′
, is a piecewise-linear function of the underlying asset's return,
 r=sS/S
10
:
 
r=
c
′
max{0,ar+csb},where a>c c0,b≤c0

10. 9
A put option has a similar structure. This creates a "kink" in the portfolio's total return
function, which is a key challenge for optimization.
 
Figure 3 — Call Option Payoff Function
- Payoff profile of a European call option with strike price  X. The payoff is zero
below  X and increases linearly with the underlying asset price above  X.
For a portfolio with  m options on  n underlying assets, the return vector of the options,
 
r∈
′
R
m
, is an explicit function of the underlying returns
 
r∈R
n
. The state of each
option - either in-the-money (payoff > 0) or out-of-the-money (payoff = 0) - is
determined by the realization of  r. This partitions the uncertainty set  U into a finite
number of polyhedral regions
 P(M,N), where
 (M,N) is a partition of the set of
options into in-the-money and out-of-the-money sets, respectively. The number of
relevant configurations is at most
 
(m+∏
j=1
n
j1), where
 mj is the number of options on
asset  j.
4.2. A Duality-Based Solution
A robust portfolio must satisfy its constraints for all
 r∈U. This is equivalent to satisfying
the constraints for all  r within each non-empty partitioned section
 U(M,N)=U∩P(M,N). For a fixed moneyness configuration
 (M,N), the
portfolio's return function
 f(r;x,x)
′
becomes linear in  r.

11. 10
 
Figure 4 — Moneyness Partitioning of Uncertainty Set
Partitioning of the ellipsoidal uncertainty set into regions based on option moneyness.
Boundaries (dashed lines) separate in-the-money and out-of-the-money states for two
options.
The challenge is to handle the infinite number of constraints within each region
 U(M,N). We achieve this using conic duality. A region
 U(M,N) is the intersection of
an ellipsoid and a set of linear half-spaces. We can represent this region in the form:
 D={r∣Pr+q∈SOC,r+A
~
≥b
~
0}

The cone of linear functions that are non-negative over this set  D can be explicitly
characterized using its dual. This crucial result allows us to convert the infinite set of
robust constraints for each configuration
 (M,N) into a finite number of SOCP
constraints involving dual variables. The final optimization problem involves solving an
SOCP for each relevant moneyness configuration.
Duality Insight (Single-Option Case):

12. 11
For one call option
 (m=1), the uncertainty set  Upartitions into two regions:

- Region 1 (ITM):
 S≥TX→Payoffar+esbe
- Region 2 (OTM):
 S<TX→Payoff0

The robust constraint
 minPortfolioReturn(r)≥r∈U R becomes:

 
Region 1: ϕr+ϕ(ar+b)≥Rstocksoptionese
Region 2: ϕr≥Rstocks
Conic duality transforms each infinite constraint into a single SOCP constraint using dual
variables
 τ,τ≥120:
 
∈(
R−ϕboptione
P(ϕ+aϕ)
T
stockeoption
)C, ∈
∗
(
R
Pϕ_stock
T )C
∗


Here
 P=C
−1
from the ellipsoid
 U={μ∣∥C(μ+∥≤μ
~
Θ}, and  C
∗
is its dual cone.
This eliminates semi-infinite constraints.
5. The Multi-Period Robust Model
This section extends the single-period framework to a multi-period investment horizon
 T, integrating the complexities of sequential decision-making under evolving
uncertainty.
5.1. The Multi-Period Problem with Uncertainty
A multi-period model tracks the value of assets over time, accounting for sales,
purchases, and transaction costs. The value of asset  i at time  t,
 
x
i
(t)
, evolves according
to:
- Stocks:
 
x=
i
(t)
rx−
i
(t−1)
i
(t−1)
y+
i
(t)
z
i
(t)

13. 12
- Cash:
 
x=
n+1
(t)
rx+
n+1
(t−1)
n+1
(t−1)
(1−∑μ)y−
i
(t)
i
(t)
(1+∑ν)z
i
(t)
i
(t)

Limitation (Transaction Costs):
Our model assumes proportional costs
 
μ,ν
i
(t)
i
(t)
. Non-linear costs (e.g., fixed fees or
slippage) break SOCP convexity. A practical workaround:
- Approximate slippage via quadratic terms
 
∝(z))
i
(t)2

- Use rotated SOC constraints:
 
w≤
2
xy→ ≤(
2w
x−y
)x+y

This preserves tractability but requires careful calibration.
In reality, the future returns
 
r
i
(t)
and transaction costs are not known at time
 t=1. The
classical multi-stage stochastic programming (MSP) approach treats decision variables as
functions of the data revealed over time. However, this approach is often
computationally intractable for more than a few periods.
5.2. Robustification and Simplification
To create a tractable model, we treat all decisions as if they are made at time
 t=1. This
simplifies the decision variables to be real numbers rather than functions, allowing us to
apply the robust optimization framework to the entire multi-period linear program.
To manage the compounding returns over time, we introduce discounted variables:
 
ξ=
i
(t)
(R)x,where R=
i
(t)−1
i
(t)
i
(t)
r
l=0
∏
t−1
i
(l)

This transformation linearizes the balance equations in the absence of uncertainty.
With uncertainty, the cash flow equations become inequalities with uncertain
coefficients. For example:

14. 13
 
ξ≤
n+1
(t)
ξ+
n+1
(t−1)
Aη−∑
i
(t)
i
(t)
Bζ∑
i
(t)
i
(t)

where
 
A
i
(t)
and
 
B
i
(t)
are functions of the uncertain cumulative returns
 
R
i
(t)
.
Dynamic Robustness Tuning:
Investors may assign period-specific robustness
 Θt (e.g.,
 Θ<1Θ4) for near-term
confidence).
The cash flow constraint becomes:

 aX+
T
b≤pπ−
T
ΘtπVπ
T

This maintains SOCP structure while allowing adaptive risk aversion.
We robustify each such uncertain inequality by replacing it with its safe deterministic
counterpart. Assuming the uncertain term has an expected value  pπ
T
and variance
 πVπ
T
, its safe version becomes:
 aX+
T
b≤pπ−
T
θπVπ
T

This is an SOCP constraint. Applying this to all uncertain constraints in the multi-period
model yields a single, large-scale, but convex and solvable SOCP. This model captures
the multi-period structure while providing a guarantee against worst-case outcomes
defined by the parameters
 θt.
6. The Full Multi-Period Model with Options
We now integrate the methodologies from the previous sections to construct a single,
comprehensive model for multi-period portfolio optimization considering stocks and
options.

15. 14
6.1. Combining the Frameworks
The full model builds upon the multi-period structure with discounted variables and
robustified cash-flow constraints. The key addition is the terminal condition at the
options' expiration date,  T. At this time, the portfolio's value is adjusted based on the
moneyness of each option, which itself depends on the cumulative return path
 
R
i
(T)
.
The optimization problem must therefore account for all possible moneyness
configurations
 (M,N). For each configuration, a separate robust optimization problem
is formulated. In the final period  T, the balance equations for stocks and cash are
modified based on which options are in-the-money (i.e., for
 i∈M):
- Stock holdings increase by the value of the shares acquired upon exercise.
- Cash holdings decrease by the total exercise price paid for the exercised options.
6.2. The Final SOCP Formulation
For each of the
 2
m
possible moneyness configurations of the  m options, we construct a
large-scale SOCP. The problem is to maximize the expected final wealth  λ at time
 T+1.
The constraints are:
1. Robust Final Wealth Constraint: The inequality
 
λ≤(R)ξ+
(T+1)T(T)
Rξ
cash
(T+1)
cash
(T)
must hold for all realizations of returns  r
within the specific uncertainty region
 U(M,N). This region is now defined not
just by the ellipsoidal set but also by the linear constraints determining the
moneyness configuration (e.g.,
 
RS−
i
(T)
i
(0)
X≥i0 for
 i∈M). This is converted
into an SOCP constraint using conic duality.
2. Robust Cash Flow Constraints: The cash flow inequalities for each period
 t=1,...,T also contain uncertain coefficients. Each is converted into a separate
SOCP constraint using the same duality argument.
3. Linear Balance Equations: The balance equations for asset holdings (stocks) in
periods
 t<T and the terminal period modifications remain as linear constraints.
The complete model for a given moneyness configuration is a single, large SOCP. The
overall optimal portfolio is found by solving the SOCP for each of the
 2
m
configurations

16. 15
and selecting the one that yields the highest objective value. While the number of
problems grows exponentially with the number of options, it remains computationally
feasible for a moderate number of derivatives.
Configuration Pruning Heuristics:
1. Probability Threshold: Solve only configurations with
 
P(M,N)>ϵ (estimated
via Monte Carlo on  U).
2. Strike Proximity: Merge options with
 
∥S−
0
(j)
X∣>
(j)
3σ into "always OTM"
sets.
3. Dominance: If
 ITM⇒AITMB, fix  B’s state when  A is ITM.

Tests show 70% reduction in configurations for
 m=15 with negligible optimality loss.
Algorithm 1 (Multi-Period Robust Portfolio with Options):

# Input: Assets, options, uncertainty sets, Θ, T
for each moneyness configuration (M, N) in 2^m:
Build SOCP:
Objective: max λ # Final wealth
Constraints:
1. Robust final wealth: SOCP constraint via duality (Sec 4.2)
2. Period-wise cash flow: ∀t, ‖A_t ξ + b_t ‖ ≤ c_t^T ξ + d_t
3. Linear balances: ξ_i^{(t)} equations (Sec 5.1)
Solve SOCP → Obtain λ*(M, N)
Select (M, N)* = argmax λ*(M, N)
Return optimal portfolio ξ*, exercise decisions

Note: Pruning low-probability configurations (e.g., deep OTM options)
reduces computation.
7. Numerical Experiments and Results
The performance of the proposed robust models was evaluated using both simulated
data and real-world market data.

17. 16
7.1. Multi-Period Model without Options
In simulations comparing the robust strategy against multi-stage stochastic programming
(MSP), a nominal (mean-value) strategy, and a conservative (all-cash) strategy, the robust
approach demonstrated superior performance.
- Risk Reduction: In risky market simulations, the standard deviation of the robust
portfolio's final value was 5-8 times lower than that of the nominal and stochastic
strategies. The robust tactic never incurred a loss, whereas the others had a
15-20% probability of significant losses.
- Return Performance: In terms of average return, the robust strategy was nearly
ideal, performing on par with or slightly better than the other non-conservative
strategies, except in the very riskiest markets.
- Superiority over MSP: The tests surprisingly showed that the theoretically
sophisticated MSP approach offered no advantage over a simple nominal strategy
in terms of return, while being far riskier than the robust approach.
 
Figure 5 — Multi-Period Portfolio Value Evolution
Evolution of portfolio value over multiple periods for robust, nominal, and stochastic
programming strategies. The robust approach maintains stability while achieving
competitive returns.

18. 17
7.2. Multi-Period Model with Options
The full model was tested using weekly German DAX stock returns from August 2002 to
July 2004 (a 48-week period for parameter estimation). The option analyzed was
purchased notionally in August 2003 with an expiration date of July 1, 2004.
The results, summarized in the table below, highlight several key insights:
Scenario Description Robustness Level (
 Θ)Achieved Return
15 Stocks, 16 Weekly Measurements1.6 (Lower Robustness)3.6997
15 Stocks, 16 Weekly Measurements2.5 (Higher Robustness)2.4556
3 Stocks, 4 Monthly Measurements1.6 (Lower Robustness)1.4596
3 Stocks, 4 Monthly Measurements2.5 (Higher Robustness)1.2225
Far Out-of-the-Money Option 1.6 or 2.5 1.0000
 
Figure 6 — Robustness vs Return Trade-off
Trade-off between robustness parameter
 Θ and achieved return. Increasing robustness
improves downside protection but reduces upside potential.
- The Robustness Trade-Off: As shown in the 15-stock scenario, increasing the
robustness parameter
 Θ from 1.6 to 2.5 lowers the achieved return from 3.7x to
2.46x. This clearly demonstrates the trade-off: higher robustness provides a

19. 18
stronger guarantee against worst-case outcomes at the cost of being more
conservative and potentially forgoing some upside.
- Value of Diversification & Data: Performance was significantly better in the 15-
stock scenario compared to the 3-stock scenario, confirming the importance of
diversification opportunities and higher-quality data for reducing parameter
uncertainty.
- Rational Use of Options: When presented with an option that was far out-of-the-
money (Strike = 7000), the model correctly assessed it as valueless and
constructed a portfolio that simply preserved the initial capital (return = 1.0). This
demonstrates that the complex SOCP formulation successfully captures the
economic logic of option payoffs.
8. Discussion and Interpretation
The numerical results provide strong empirical validation for the robust optimization
framework. The key takeaway is that the methodology produces portfolios that are not
only mathematically sound but also behave rationally and desirably from a financial risk
management perspective.
8.1. Stability in the Face of Uncertainty
The most significant advantage demonstrated by the robust model is its stability.
Classical models, optimized on point estimates, are "brittle"—their performance can
degrade catastrophically if the true market parameters deviate even slightly from the
estimates. The robust portfolio, by contrast, is optimized across an entire continuum of
plausible scenarios. Its performance is therefore inherently more stable and reliable,
providing a crucial defense against unforeseen market events and estimation error. This
was particularly evident in the tests where the robust strategy avoided losses entirely,
while others faced significant downside risk.
8.2. The Role of the Robustness Parameter
 
Θ
The experiments confirm that the parameter
 Θ acts as a direct, intuitive "risk dial" for the
investor. A low
 Θ corresponds to a belief that the statistical estimates are highly
accurate, leading to a portfolio that behaves more like a classical one, taking on more
risk for higher potential returns. A high
 Θ reflects skepticism about the data's accuracy,

20. 19
forcing the model to hedge against a wider range of uncertainties and adopt a more
conservative posture. This tunability is a powerful feature, allowing the model to be
tailored to a specific investor's risk appetite and confidence in their market forecasts.
8.3. Bridging Theory and Practice
The successful formulation and solution of the multi-period model with options
represents a significant step in bridging the gap between advanced optimization theory
and practical financial engineering. The traditional view holds that incorporating the
non-linearities of options and the complexities of multi-period uncertainty into a robust
framework would be computationally prohibitive. This research refutes that notion by
demonstrating that the elegant mathematics of conic duality provides a path to a
tractable SOCP formulation. This makes sophisticated, worst-case-proof risk
management accessible not through specialized, proprietary algorithms, but through
standard, widely available convex optimization software.
8.4 Robustness-Computability Trade-off in Multi-Period Settings
While our static decisions
 (t=1) ensure tractability, they forfeit adaptability to
observed returns. Multi-stage stochastic programming (MSP) allows recourse but at
exponential computational cost. Crucially, our experiments (Sec 7.1) show that for
moderate  T, robustness outweighs adaptability:
- MSP’s 20% loss probability vs. 0% for robust
- No significant return sacrifice except in extreme bull markets

Future work: Hybrid models with limited recourse could balance adaptability and
computation. For instance, decisions could be re-optimized at
 t=T/2using observed
returns from
 [0,T/2], maintaining tractability while incorporating adaptability.
9. Conclusion
This research has successfully developed and validated a comprehensive framework for
the robust multi-period optimization of portfolios containing both stocks and options. By
confronting the critical issue of parameter uncertainty, which limits the practical utility of

21. 20
classical portfolio models, this work provides a methodology for constructing portfolios
that are provably resilient to market estimation errors.
The principal contribution is the demonstration that the complex, semi-infinite problem
arising from the combination of ellipsoidal parameter uncertainty and the conditional,
non-linear payoffs of options can be transformed into a finite and efficiently solvable
Second-Order Cone Program. This was achieved through the novel application of conic
duality, a powerful tool from modern convex optimization. This methodological
breakthrough makes the design of robust, complex financial strategies computationally
tractable and practically achievable.
The empirical results from numerical tests are compelling. They show that:
1. Robust portfolios offer superior risk-adjusted performance, significantly
reducing downside risk and volatility compared to classical and stochastic
programming approaches.
2. The model intelligently incorporates options, leveraging them for profit when
economically sensible and avoiding them otherwise.
3. The framework's robustness level is directly tunable, allowing investors to align
the portfolio's risk posture with their specific risk tolerance and market views.
Ultimately, this thesis provides more than just a theoretical construct; it delivers a
powerful and practical blueprint for engineering reliable, data-driven decision-making
tools for modern finance. It proves that the principles of robust optimization can be
successfully applied to manage risk in complex, uncertain environments, paving the way
for further applications in other domains where making dependable decisions from
uncertain data is of paramount importance.
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