Mathematical formulation
The set S contains the available assets, and the index s∈S represents an asset from this set. The market price of the asset s is noted PRICEs, and its variability range is given by VARs. For each asset s∈S, the variable xs represents the fraction of the budget spent to buy asset s. The variable w is used to represent the worst case value of the portfolio. The variable devs describes the possible variation of the value of the asset s. The value N describes the worst case the trader will consider and corresponds to the maximum decrease of the asset value expressed as a multiple of its standard deviation.
Highest protection
A conservative trader will want to maximize the worst case value of her portfolio. If she wants to get the best protection against worst case realization, then she will try to maximize the worst case value of her portfolio. In our example, this worst case realizes when the value of every asset decreases by N times its standard deviation.
s.t. w ≤ ∑s∈S (PRICEs + N·devs)·xs (with devs = - VARs)
∑s∈S xs = 1
0 ≤ xs ≤ 1
Budgeted protection
As we will see in the discussion of the results (Section Results), achieving a high protection against worst case realization incurs an important loss of value in the average case. And hence, this strategy may not be such a clever choice because average cases are more likely to happen. The trader therefore might want to improve her model by adding a control parameter to limit the protection level. The overall variation budget G is defined by the following formula:
| ∑ |
| s∈S |
The ellipsoidal uncertainty set
Let k be the percentage value representing the protection level. The possible asset value decrease is controlled by the uncertainty set U(k) that is defined as follows:
| ∑ |
| s∈S |
Robust constraints
The resulting robust optimization problem can then be stated as:
s.t. w ≤ ∑s∈S (PRICEs + es) · xs (∀ e∈U(k))
∑s∈S xs = 1
0 ≤ xs ≤ 1
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