## Specifying Constraints with `ConSet`

### Introduction

Both `hedgeopt` and `hedgeslf` accept an optional input argument, `ConSet`, that allows you to specify a set of linear inequality constraints for instruments in your portfolio. The examples in this section are brief. For additional information regarding portfolio constraint specifications, refer to Analyzing Portfolios.

### Setting Constraints

For the first example of setting constraints, return to the fully hedged portfolio example that used `hedgeopt` to determine the minimum cost of obtaining simultaneous delta, gamma, and vega neutrality (target sensitivities all `0`). Recall that when `hedgeopt` computes the cost of rebalancing a portfolio, the input target sensitivities you specify are treated as equality constraints during the optimization process. The situation is reproduced next for convenience.

```TargetSens = [0 0 0]; [Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,... Holdings, FixedInd, [], [], TargetSens); ```

The outputs provide a fully hedged portfolio

```Sens = -0.00 -0.00 -0.00 ```

at an expense of over \$23,000.

```Cost = 23055.90 ```

The positions required to achieve this fully hedged portfolio are

```Quantity' = 100.00 -182.36 -19.55 80.00 8.00 -32.97 40.00 10.00 ```

Suppose now that you want to place some upper and lower bounds on the individual instruments in your portfolio. You can specify these constraints, along with a variety of general linear inequality constraints, with function `portcons`.

As an example, assume that, in addition to holding instruments 1, 4, 5, 7, and 8 fixed as before, you want to bound the position of all instruments to within +/- 180 contracts (for each instrument, you cannot short or long more than 180 contracts). Applying these constraints disallows the current position in the second instrument (short 182.36). All other instruments are currently within the upper/lower bounds.

You can generate these constraints by first specifying the lower and upper bounds vectors and then calling `portcons`.

```LowerBounds = [-180 -180 -180 -180 -180 -180 -180 -180]; UpperBounds = [ 180 180 180 180 180 180 180 180]; ConSet = portcons('AssetLims', LowerBounds, UpperBounds); ```

To impose these constraints, call `hedgeopt` with `ConSet` as the last input.

```[Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,... Holdings, FixedInd, [], [], TargetSens, ConSet); ```

Examine the outputs and see that they are all set to `NaN`, indicating that the problem, given the constraints, is not solvable. Intuitively, the results mean that you cannot obtain simultaneous delta, gamma, and vega neutrality with these constraints at any price.

To see how close you can get to portfolio neutrality with these constraints, call `hedgeslf`.

```[Sens, Value1, Quantity] = hedgeslf(Sensitivities, Price,... Holdings, FixedInd, ConSet);```
```Sens = -352.43 21.99 -498.77 Value1 = 855.10 Quantity = 100.00 -180.00 -37.22 80.00 8.00 -31.86 40.00 10.00 ```

`hedgeslf` enforces the lower bound for the second instrument, but the sensitivity is far from neutral. The cost to obtain this portfolio is

`Value0 - Value1`
```ans = 22819.52 ```

### Portfolio Rebalancing

As a final example of user-specified constraints, rebalance the portfolio using the second hedging goal of `hedgeopt`. Assume that you are willing to spend as much as \$20,000 to rebalance your portfolio, and you want to know what minimum portfolio sensitivities you can get for your money. In this form, recall that the target cost (\$20,000) is treated as an inequality constraint during the optimization process.

For reference, start up `hedgeopt` without any user-specified linear inequality constraints.

```[Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,... Holdings, FixedInd, [], 20000);```
```Sens = -4345.36 295.81 -6586.64 Cost = 20000.00 Quantity' = 100.00 -151.86 -253.47 80.00 8.00 -18.18 40.00 10.00 ```

This result corresponds to the \$20,000 point along the Portfolio Sensitivities Profile shown in the figure Rebalancing Cost.

Assume that, in addition to holding instruments 1, 4, 5, 7, and 8 fixed as before, you want to bound the position of all instruments to within +/- 150 contracts (for each instrument, you cannot short more than 150 contracts and you cannot long more than 150 contracts). These bounds disallow the current position in the second and third instruments (-151.86 and -253.47). All other instruments are currently within the upper/lower bounds.

As before, you can generate these constraints by first specifying the lower and upper bounds vectors and then calling `portcons`.

```LowerBounds = [-150 -150 -150 -150 -150 -150 -150 -150]; UpperBounds = [ 150 150 150 150 150 150 150 150]; ConSet = portcons('AssetLims', LowerBounds, UpperBounds); ```

To impose these constraints, again call `hedgeopt` with `ConSet` as the last input.

```[Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,... Holdings,FixedInd, [], 20000, [], ConSet);```
```Sens = -8818.47 434.43 -4010.79 Cost = 19876.89 Quantity' = 100.00 -150.00 -150.00 80.00 8.00 -28.32 40.00 10.00 ```

With these constraints, `hedgeopt` enforces the lower bound for the second and third instruments. The cost incurred is \$19,876.89.