# Case Study: Hedging Portfolio of Options

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Case study background and problem formulations

Instructions for optimization with PSG Run-File and PSG R.

PROBLEM: problem_cs_hedging_portfolio_of_options
Minimize P(x) (minimizing initial portfolio value)
subject to
Cvar_risk ≤ Const (constraint on hedging error)
Box constraints (non-negativity constraints on positions)
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P(x) = initial price of the hedging portfolio
Cvar_risk = CVaR Risk for Loss
Box constraints = constraints on individual decision variables
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Data and solution in Run-File Environment

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec)
Dataset1 Problem statement Data Solution 121 45,000 73.31675 0.47
Dataset2 Problem statement Data Solution 121 5,000 196.87183 0.04
Data and solution in MATLAB Environment

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 Matlab code Data Solution 121 45,000 73.3169 0.46
Dataset2 Matlab code Data Solution 121 5,000 196.872 0.04
Data and solution in R Environment

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 R code Data 121 45,000 73.3169 0.46
Dataset2 R code Data 121 5,000 196.872 0.04
CASE STUDY SUMMARY

This case study hedges a Portfolio of Options by a Portfolio of Stocks and Options. A similar case study was considered by Rockafellar and Uryasev, 2000. A target portfolio consists of stock options; the hedging portfolio includes indices, stocks, and options on indices. We allow both long and short positions in the hedging portfolio. Long positions are opened at ask prices, short positions at bid prices. Long positions are closed at bid prices, and short at ask prices. The composition of the target portfolio is known. Weights of the hedging portfolio are determined by optimization. We want to build a hedging portfolio with the lowest possible cost at the initial time t = 0 to hedge the target portfolio at the expiration time t = T. The quality of hedging at expiration is controlled through a CVaR-constraint, which bounds the average of (1-alpha)percent of largest underperformances of the hedging portfolio versus the target portfolio. The target portfolio of options is hedged by the hedging portfolio on September 22, 2005. All options in the hedging portfolio and in the target portfolio expire on December 16, 2005. We solve the optimization problem on September 22, 2005 to assure hedging on December 16, 2005.
Optimization problem is solved using two datasets each: Dataset1 for “long case study” including 45,000 scenarios and and Dataset2 for “short case study” including only 5,000 scenarios.
Calculations in the case study are conducted for Const = 530 for “long case study” and Const = 525 for “short case study”.