Robust SOCPO

1. See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/394407195 A Robust Optimization Framework for Multi-Period Portfolios with Options via Second-Order Cone Programming Article · October 2004 CITATIONS 0 READS 3 1 author: Eva-Marie Muller-Stuler Technical University of Munich 22PUBLICATIONS   14CITATIONS    SEE PROFILE All content following this page was uploaded by Eva-Marie Muller-Stuler on 08 August 2025. The user has requested enhancement of the downloaded file.

2. 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. 1

3. 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: E(ϵϵ ) = T  lnr = lnμ + ϵ, D where  μ ∈ Rn 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: 2

4. μ~  U = {μ ∣ ∣∣C(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 ] max min ϕ μ∈Sm Var[r ] ≤ D∈Sd 1 ϕ = ϕ max s.t. λ, ϕ T 1   3

5. 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 ⊆ Rn 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: ∗ {s ∈ R ∣ n T  K = x s ≥ 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): minc x - Dual (D): maxb y subject to  Ax = b,x ⪰K0 T subject to  A y + T s = c,s ⪰K∗ 0 T 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) Rk (also known as the Lorentz or "ice cream" cone) is A second-order cone C ⊂ defined as: k {[t u] } k−1 ∣ u ∈ R ,t ∈ R,∣∣u∣∣ ≤ t C = k 2   ∗ This cone is self-dual, meaning  C = Ck. k 4

6.   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: T minf x c x + i T   s.t. ∣∣A x + i b ∣∣ ≤ i 2 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 + a ˉi ∣∣P x∣∣ ≤ i bi, is naturally an SOCP constraint. This provides the foundational link between robust optimization and computationally tractable SOCPs. T   5

7. Table 1: Key SOCP Transformations Original Constraint SOCP Reformulation μ~T  min T μ ϕ ϕ − D~1/2 1 − λ) Θ∥Pϕ∥ μ∈U   TD~ (2 ϕ ≤ ϕ  ϕ λ 1 + λ ≤    a X + Dual cone membership  C∗) b ≤ p π − Θ π Vπ T T T Robust cash flow (Sec 5.2) Option duality (Sec 4.2) 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: T max min μ ϕ μ∈Sm ϕ Dϕ ≤ D∈Sd 1 ϕ = 1, ϕ max T s.t. λ, T 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.,  ϕ Dϕ ≤ inner minimization for the worst-case mean return for a fixed portfolio  ϕ within the ellipsoidal set  S = m {μ ∣ μ = + μ ˉ Pu,∣∣u∣∣ ≤ 1} has a closed-form solution: TDˉ. The T ϕ ϕ T μ ˉT min μ ϕ = ϕ − ∣∣Pϕ∣∣   μ∈Sm 6

8. 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: μ ˉT max ϕ − ∣∣Pϕ∣∣ ϕ s.t. 1 ϕ = TDˉ 1, ϕ ≤ ϕ ≥ 0 λ, ϕ T   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  ϕ transformed into a second-order cone constraint. A hyperbolic constraint of the form  z z ≤ xy for  x,y ≥ 0 is equivalent to the SOC ≤ x + yApplying this transformation to the variance constraint TDˉ ϕ ≤ λ can be T (2z x − y) constraint:   and representing the robust return constraint directly yields the final SOCP: Dˉ1/2 1 − λ) (2 ϕ 1 + λ ≤ μ ˉT t s.t. ∣∣Pϕ∣∣ ≤ ϕ −  max t   1 ϕ = 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: μ~ - Estimated returns:  = - Covariance matrix: [0.08,0.12]T 7

9. D~ [0.04 0.01 0.01 0.09] =   D~1/2 - Uncertainty scaling:  (P = - Risk tolerance:  λ = 0.02 / T(with  T = 50historical observations) (μ~T The robust objective  max ϕ − θ∥Pϕ∥ ) ) transforms to SOCP: max t D~1/2 1 − 0.02) Pϕ ϕ − t (2 ( ϕ + ϕ = 1, 1 ϕ s.t. ≤ 1 + 0.02 ) μ~T ≤ ϕ − t + θ μ~T ϕ ≥ 0 2   Solving this SOCP yields weights  ϕ = penalization. ∗ [0.62,0.38]T, balancing return and uncertainty 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,  S1. The return of a call option,  rc  r = s S /S ′, is a piecewise-linear function of the underlying asset's return, 1 0: ′  r = c max{0,a r + b }, c where a > 0,b ≤ c 0 c s c 8

10. 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 ∈ 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   j asset  j. ′ Rm, is an explicit function of the underlying returns  r ∈ Rn. The state of each n (m + 1), where  mj is the number of options on ∏j=1 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. 9

11.   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: A~ b~  D = {r ∣ Pr + q ∈ SOC, r + ≥ 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): 10

12. For one call option  (m = 1), the uncertainty set  Upartitions into two regions: - Region 1 (ITM): S ≥ - Region 2 (OTM): S < X → Payoffa r + X → Payoff0 be T e s T The robust constraint  min PortfolioReturn(r) ≥ R becomes: r∈U Region 1: ϕ r + ϕ r ≥ R (a r + b ) ≥ R e s stock s option e   Region 2: ϕ stock s Conic duality transforms each infinite constraint into a single SOCP constraint using dual variables  τ ,τ ≥ 1 2 0: R − ϕ ( option) ( ) b R option e + a ϕ ∗ C∗ ∈ C , ∈ P (ϕ ) P ϕ_stock T T stock e   Here  P = C−1 from the ellipsoid  U = {μ∣∥C(μ + ∥ ≤ This eliminates semi-infinite constraints. μ~ Θ}, and  C∗ is its dual cone. 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,  xi to: (t), evolves according (t) (t−1) (t−1) (t) (t) = − + - Stocks: x r x y zi i i i i 11

13. (t) (t−1) (t−1) (t) (t) (t) (t) (1 − (1 + ∑ ∑ = + μ )y i − )z - Cash: x r x ν n+1 n+1 n+1 i i i Limitation (Transaction Costs): (t) (t). Non-linear costs (e.g., fixed fees or Our model assumes proportional costs  μ ,ν slippage) break SOCP convexity. A practical workaround: i i (t) 2 - Approximate slippage via quadratic terms  ∝ (z ) ) - Use rotated SOC constraints: i (2w x − y) 2 w ≤ xy → ≤ x + y   This preserves tractability but requires careful calibration. (t) and transaction costs are not known at time  t = 1. The In reality, the future returns  ri 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: t−1 ∏ (l) (t) (t) −1 (t) (t) = (R ) i x , i where R = r ξ i i i l=0   This transformation linearizes the balance equations in the absence of uncertainty. With uncertainty, the cash flow equations become inequalities with uncertain coefficients. For example: 12

14. ∑ ∑ (t) (t−1) (t) (t) (t) (t) ≤ + −  ξ ξ A η i B ζ i n+1 n+1 i i (t) and  Bi (t) are functions of the uncertain cumulative returns  Ri (t). where  Ai Dynamic Robustness Tuning: Investors may assign period-specific robustness  Θt (e.g.,  Θ < confidence). Θ4) for near-term 1 The cash flow constraint becomes: T T  a X + b ≤ p π − Θt π Vπ 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  p π  π Vπ , its safe version becomes: T and variance T T T  a X + b ≤ p π − T θ π Vπ 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. 13

15. 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  Ri (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  2m 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 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.,  R S i i into an SOCP constraint using conic duality. (T+1) (T) must hold for all realizations of returns  r (T+1) T (T) ) ξ + R ξ cash cash (T) (0) X ≥ i 0 for  i ∈ M). This is converted − 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  2m configurations 14

16. 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 sets. 3. Dominance: If  ITM ⇒ A ITMB, fix  B’s state when  A is ITM. (j) (j) ∣ > 3σ into "always OTM" − X 0 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. 15

17. 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. 16

18. 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: Robustness Level ( Θ) Scenario Description Achieved Return 15 Stocks, 16 Weekly Measurements 1.6 (Lower Robustness) 3.6997 15 Stocks, 16 Weekly Measurements 2.5 (Higher Robustness) 2.4556 3 Stocks, 4 Monthly Measurements 1.6 (Lower Robustness) 1.4596 3 Stocks, 4 Monthly Measurements 2.5 (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 17

19. 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, 18

20. 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 19

21. 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. 20 View publication stats