Financial Derivatives Toolbox

Example: Under-Determined System

Reducing the number of assets held fixed creates an under-determined system with meaningless solutions. For example, see what happens with only four assets constrained.

• FixedInd = [1 4 5 7];
  [Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,... 
  Holdings, FixedInd, [], MaxCost);
  
  Sens =
  
           -0.00          -0.00          -0.00
  
  Cost =
  
        20000.00
  
  Quantity' =
  
          100.00
          -149.31
          -14.91
           80.00
            8.00
          -34.64
           40.00
          -32.60
  

You have spent $20,000 (all the funds available for rebalancing) to achieve a fully-hedged portfolio.

With an increase in available funds to $50,000, you still spend all available funds to get another fully-hedged portfolio.

• MaxCost  = 50000;
  [Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,... 
  Holdings, FixedInd, [],MaxCost);
  
  Sens =
  
           -0.00          0.00          0.00
  Cost =
  
        50000.00
  
  Quantity' =
  
          100.00
         -473.78
          -60.51
           80.00
            8.00
          -18.20
           40.00
          385.60
  

All solutions to an under-determined system are meaningless. You buy and sell various assets to obtain zero sensitivities, spending all available funds every time. If you reduce the number of fixed assets any further, this problem is insufficiently constrained, and you find no solution (the outputs are all NaN).

Note also that no solution exists whenever constraints are inconsistent. Inconsistent constraints create an infeasible solution space; the outputs are all NaN.

Example: Minimize Portfolio Sensitivities

Portfolio Constraints with hedgeslf