Hedging with Constrained Portfolios
Both hedging functions cast the optimization as a constrained linear least-squares problem.
(See the function
lsqlin for details.) In particular,
lsqlin attempts to minimize the constrained linear least squares problem
where C, A, and Aeq are matrices, and d, b, beq, lb, and ub are vectors. For Financial Instruments Toolbox™ software, x is a vector of asset holdings (contracts).
Depending on the constraint and the number of assets in the portfolio, a solution to a particular problem may or may not exist. Furthermore, if a solution is found, it may not be unique. For a unique solution to exist, the least squares problem must be sufficiently and appropriately constrained.
Example: Fully Hedged Portfolio
you to allocate an optimal hedge by one of two goals:
Minimize the cost of hedging a portfolio given a set of target sensitivities.
Minimize portfolio sensitivities for a given set of maximum target costs.
As an example, reproduce the results for the fully hedged portfolio example.
TargetSens = [0 0 0]; FixedInd = [1 4 5 7 8]; [Sens,Cost,Quantity] = hedgeopt(Sensitivities, Price,... Holdings, FixedInd, , , TargetSens);
Sens = -0.00 -0.00 -0.00 Cost = 23055.90 Quantity' = 98.72 -182.36 -19.55 80.00 8.00 -32.97 40.00 10.00
This example finds a unique solution at a cost of just over
$23,000. The matrix
C (formed internally by
hedgeopt and passed to
lsqlin) is the asset
expressed as a row vector.
C = Price' == [98.72 97.53 0.05 98.72 100.55 6.28 0.05 3.69]
d is the current portfolio value
= 23674.62. The example maintains, as closely as possible,
a constant portfolio value subject to the specified constraints.
In the absence of any additional constraints, the least squares objective involves a single equation with eight unknowns. This is an under-determined system of equations. Because such systems generally have an infinite number of solutions, you need to specify additional constraints to achieve a solution with practical significance.
The additional constraints can come from two sources:
User-specified equality constraints
Target sensitivity equality constraints imposed by
The example in Fully Hedged Portfolio specifies five equality constraints
associated with holding assets 1, 4, 5, 7, and 8 fixed. This reduces
the number of unknowns from eight to three, which is still an under-determined
system. However, when combined with the first goal of
hedgeopt, the equality constraints associated
with the target sensitivities in
an additional system of three equations with three unknowns. This
additional system guarantees that the weighted average of the delta,
gamma, and vega of assets 2, 3, and 6, together with the remaining
assets held fixed, satisfy the overall portfolio target sensitivity
Combining the least-squares objective equation with the three portfolio sensitivity equations provides an overall system of four equations with three unknown asset holdings. This is no longer an under-determined system, and the solution is as shown.
If the assets held fixed are reduced, for example,
= [1 4 5 7],
a no cost, fully hedged portfolio (
Sens = [0 0 0] and
If you further reduce
FixedInd (for example,
[1 4], or even
hedgeopt always returns a no cost, fully
hedged portfolio. In these cases, insufficient constraints result
in an under-determined system. Although
no cost, fully hedged portfolios, there is nothing unique about them.
These portfolios have little practical significance.
Constraints must be sufficient and appropriately defined. Additional constraints having no effect on the optimization are called dependent constraints. As a simple example, assume that parameter Z is constrained such that . Furthermore, assume that you somehow add another constraint that effectively restricts . The constraint now has no effect on the optimization.
Example: Minimize Portfolio Sensitivities
To illustrate using
minimize portfolio sensitivities for a given maximum target cost,
specify a target cost of $20,000 and determine the new portfolio sensitivities,
holdings, and cost of the rebalanced portfolio.
MaxCost = 20000; [Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,... Holdings, [1 4 5 7 8], , MaxCost);
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 example corresponds to the $20,000 point along the cost axis in the figures Rebalancing Cost Profile, Funds Available for Rebalancing, and Rebalancing Cost.
When minimizing sensitivities, the maximum target cost is treated
as an inequality constraint; in this case,
the most you are willing to spend to hedge a portfolio. The least-squares
C is the matrix transpose of the
input asset sensitivities
C = Sensitivities'
8 matrix in this
d is a
vector of zeros,
[0 0 0]'.
Without any additional constraints, the least-squares objective
results in an under-determined system of three equations with eight
unknowns. By holding assets 1, 4, 5, 7, and 8 fixed, you reduce the
number of unknowns from eight to three. Now, with a system of three
equations with three unknowns,
the solution shown.
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
Note also that no solution exists whenever constraints are inconsistent.
Inconsistent constraints create an infeasible solution space; the outputs are all
Example: Portfolio Constraints with
The other hedging function,
attempts to minimize portfolio sensitivities such that the rebalanced
portfolio maintains a constant value (the rebalanced portfolio is
hedged against market moves and is closest to being self-financing).
If a self-financing hedge is not found,
to rebalance a portfolio to minimize sensitivities.
From a least-squares systems approach,
attempts to minimize cost in the same way that
hedgeopt does. If it cannot solve this
problem (a no cost, self-financing hedge is not possible),
hedgeslf proceeds to minimize sensitivities
hedgeopt. Thus, the discussion
of constraints for
directly applicable to
To illustrate this hedging facility using equity exotic options,
consider the portfolio
CRRInstSet obtained from
the example MAT-file
deriv.mat. The portfolio consists
of eight option instruments: two stock options, one barrier, one compound,
two lookback, and two Asian.
The hedging functions require inputs that include the current portfolio holdings (allocations) and a matrix of instrument sensitivities. To create these inputs, start by loading the example portfolio into memory
Next, compute the prices and sensitivities of the instruments in this portfolio.
[Delta, Gamma, Vega, Price] = crrsens(CRRTree, CRRInstSet);
Extract the current portfolio holdings (the quantity held or the number of contracts).
Holdings = instget(CRRInstSet, 'FieldName', 'Quantity');
For convenience place the delta, gamma, and vega sensitivity measures into a matrix of sensitivities.
Sensitivities = [Delta Gamma Vega];
Each row of the
Sensitivities matrix is associated
with a different instrument in the portfolio and each column with
a different sensitivity measure.
disp([Price Holdings Sensitivities])
8.29 10.00 0.59 0.04 53.45 2.50 5.00 -0.31 0.03 67.00 12.13 1.00 0.69 0.03 67.00 3.32 3.00 -0.12 -0.01 -98.08 7.60 7.00 -0.40 -45926.32 88.18 11.78 9.00 -0.42 -112143.15 119.19 4.18 4.00 0.60 45926.32 49.21 3.42 6.00 0.82 112143.15 41.71
The first column contains the dollar unit price of each instrument, the second contains the holdings of each instrument, and the third, fourth, and fifth columns contain the delta, gamma, and vega dollar sensitivities, respectively.
Suppose that you want to obtain a delta, gamma, and vega neutral
[Sens, Value1, Quantity]= hedgeslf(Sensitivities, Price, ... Holdings)
Sens = 0.00 -0.00 0.00 Value1 = 313.93 Quantity = 10.00 7.64 -1.56 26.13 9.94 3.73 -0.75 8.11
hedgeslf returns the
portfolio dollar sensitivities (
Sens), the value
of the rebalanced portfolio (
Value1) and the new
allocation for each instrument (
the portfolio value before and after rebalancing, respectively, you
can verify the cost by comparing the portfolio values.
Value0= Holdings' * Price
Value0 = 313.93
In this example, the portfolio is fully hedged (simultaneous
delta, gamma, and vega neutrality) and self-financing (the values
of the portfolio before and after balancing (
are the same.
Suppose now that you want to place some upper and lower bounds on the individual instruments
in your portfolio. By using function
portcons, you can specify these constraints, along with various general linear
As an example, assume that, in addition to holding instrument 1 fixed as before, you want to bound the position of all instruments to within +/- 20 contracts (for each instrument, you cannot short or long more than 20 contracts). Applying these constraints disallows the current position in the fourth instrument (long 26.13). 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
LowerBounds = [-20 -20 -20 -20 -20 -20 -20 -20]; UpperBounds = [20 20 20 20 20 20 20 20]; ConSet = portcons('AssetLims', LowerBounds, UpperBounds);
To impose these constraints, call
the last input.
[Sens, Cost, Quantity1] = hedgeslf(Sensitivities, Price, ... Holdings, 1, ConSet)
Sens = -0.00 0.00 0.00 Cost = 313.93 Quantity1 = 10.00 5.28 10.98 20.00 20.00 -6.99 -20.00 9.39
the upper bound on the fourth instrument, and the portfolio continues
to be fully hedged and self-financing.
- Portfolio Creation Using Functions
- Adding Instruments to an Existing Portfolio Using Functions
- Instrument Constructors
- Creating Instruments or Properties
- Searching or Subsetting a Portfolio
- Pricing a Portfolio Using the Black-Derman-Toy Model
- Pricing and Hedging a Portfolio Using the Black-Karasinski Model
- Specifying Constraints with ConSet
- Portfolio Rebalancing