hedgeopt
Allocate optimal hedge for target costs or sensitivities
Syntax
Description
[
allocates an optimal hedge by one of two criteria:PortSens
,PortCost
,PortHolds
] = hedgeopt(Sensitivities
,Price
,CurrentHolds
)
Minimize portfolio sensitivities (exposure) for a given set of target costs.
Minimize the cost of hedging a portfolio given a set of target sensitivities.
Hedging involves the fundamental tradeoff between portfolio insurance and the cost of insurance coverage. This function lets investors modify portfolio allocations among instruments to achieve either of the criteria. The chosen criterion is inferred from the input argument list. The problem is cast as a constrained linear least-squares problem.
[
adds additional optional arguments.PortSens
,PortCost
,PortHolds
] = hedgeopt(___,FixedInd
,NumCosts
,TargetCost
,TargetSens
ConSet
)
Examples
Allocate Optimal Hedge for Target Costs
To illustrate the hedging facility, consider the portfolio HJMInstSet
obtained from the example file deriv.mat
. The portfolio consists of eight instruments: two bonds, one bond option, one fixed-rate note, one floating-rate note, one cap, one floor, and one swap.
In this examples, portfolio target sensitivities are treated as equality constraints during the optimization process. You can use hedgeopt
to specify what sensitivities you want, and hedgeopt
computes what it will cost to get those sensitivities.
load deriv.mat;
Compute the price and sensitivities
warning('off')
[Delta, Gamma, Vega, Price] = hjmsens(HJMTree, HJMInstSet)
Delta = 8×1
-272.6462
-347.4315
-8.0781
-272.6462
-1.0445
294.9700
-47.1629
-282.0465
Gamma = 8×1
103 ×
1.0299
1.6227
0.6434
1.0299
0.0033
6.8526
8.4600
1.0597
Vega = 8×1
0.0000
-0.0397
34.0746
0.0000
0
93.6946
93.6946
0.0000
Price = 8×1
98.7159
97.5280
0.0486
98.7159
100.5529
6.2831
0.0486
3.6923
Extract the current portfolio holdings.
warning('on') Holdings = instget(HJMInstSet, 'FieldName', 'Quantity')
Holdings = 8×1
100
50
-50
80
8
30
40
10
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.
Summarize the portfolio information.
disp([Price Holdings Sensitivities])
1.0e+03 * 0.0987 0.1000 -0.2726 1.0299 0.0000 0.0975 0.0500 -0.3474 1.6227 -0.0000 0.0000 -0.0500 -0.0081 0.6434 0.0341 0.0987 0.0800 -0.2726 1.0299 0.0000 0.1006 0.0080 -0.0010 0.0033 0 0.0063 0.0300 0.2950 6.8526 0.0937 0.0000 0.0400 -0.0472 8.4600 0.0937 0.0037 0.0100 -0.2820 1.0597 0.0000
The first column above is the dollar unit price of each instrument, the second is the holdings of each instrument (the quantity held or the number of contracts), and the third, fourth, and fifth columns are the dollar delta
, gamma
, and vega
sensitivities, respectively.
The current portfolio sensitivities are a weighted average of the instruments in the portfolio.
TargetSens = Holdings' * Sensitivities
TargetSens = 1×3
105 ×
-0.6191 7.8895 0.0485
Maintaining Existing Allocations
To illustrate using hedgeopt
, suppose that you want to maintain your existing portfolio. hedgeopt
minimizes the cost of hedging a portfolio given a set of target sensitivities. If you want to maintain your existing portfolio composition and exposure, you should be able to do so without spending any money. To verify this, set the target sensitivities to the current sensitivities.
FixedInd = [1 2 3 4 5 6 7 8]; [Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,Holdings, FixedInd, [], [], TargetSens)
Sens = 1×3
105 ×
-0.6191 7.8895 0.0485
Cost = 0
Quantity = 1×8
100 50 -50 80 8 30 40 10
Portfolio composition and sensitivities are unchanged, and the cost associated with doing nothing is zero. The cost is defined as the change in portfolio value. This number cannot be less than zero because the rebalancing cost is defined as a nonnegative number.
If Value0
and Value1
represent the portfolio value before and after rebalancing, respectively, the zero cost can also be verified by comparing the portfolio values.
Value0 = Holdings' * Price
Value0 = 2.3675e+04
Value1 = Quantity * Price
Value1 = 2.3675e+04
Partially Hedged Portfolio
Building on this example, suppose you want to know the cost to achieve an overall portfolio dollar sensitivity of [-23000 -3300 3000]
, while allowing trading only in instruments 2
, 3
, and 6
(holding the positions of instruments 1
, 4
, 5
, 7
, and 8
fixed). To find the cost, first set the target portfolio dollar sensitivity.
TargetSens = [-23000 -3300 3000];
Specify the instruments to be fixed.
FixedInd = [1 4 5 7 8];
Use hedgeopt:
[Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,Holdings, FixedInd, [], [], TargetSens)
Sens = 1×3
104 ×
-2.3000 -0.3300 0.3000
Cost = 1.9174e+04
Quantity = 1×8
100.0000 -141.0267 137.2638 80.0000 8.0000 -57.9606 40.0000 10.0000
Recompute Value1
, the portfolio value after rebalancing.
Value1 = Quantity * Price
Value1 = 4.5006e+03
As expected, the cost, $19174.02, is the difference between Value0
and Value1
, $23674.62 — $4500.60. Only the positions in instruments 2
, 3
, and 6
are changed.
Fully Hedged Portfolio
The example has illustrated a partial hedge, but perhaps the most interesting case involves the cost associated with a fully hedged portfolio (simultaneous delta
, gamma
, and vega
neutrality). In this case, set the target sensitivity to a row vector of 0
s and call hedgeopt
again.
TargetSens = [0 0 0]; [Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price, Holdings, FixedInd, [], [], TargetSens)
Sens = 1×3
10-10 ×
0.1091 0.5821 0.0045
Cost = 2.3056e+04
Quantity = 1×8
100.0000 -182.3615 -19.5501 80.0000 8.0000 -32.9674 40.0000 10.0000
Examining the outputs reveals that you have obtained a fully hedged portfolio but at an expense of over $20,000 and Quantity
defines the positions required to achieve a fully hedged portfolio.
The resulting new portfolio value is
Value1 = Quantity * Price
Value1 = 618.7168
Input Arguments
Sensitivities
— Sensitivities of each instrument
matrix
Sensitivities of each instrument, specified as a number of instruments
(NINST
) by number of sensitivities (NSENS
)
matrix of dollar sensitivities. Each row represents a different instrument. Each column
represents a different sensitivity.
Data Types: double
Price
— Instrument prices
vector
Instrument prices, specified as an NINST
-by-1
vector.
Data Types: double
CurrentHolds
— Contracts allocated to each instrument
vector
Contracts allocated to each instrument, specified as an
NINST
-by-1
vector.
Data Types: double
FixedInd
— Number of fixed instruments
[ ]
(default) | vector
(Optional) Number of fixed instruments, specified as an
NFIXED
-by-1
vector of indices of instruments to
hold fixed. For example, to hold the first and third instruments of a 10 instrument
portfolio unchanged, set FixedInd = [1 3]
. Default =
[]
, no instruments held fixed.
Data Types: double
NumCosts
— Number of points generated along the cost frontier
10
(default) | integer
(Optional) Number of points generated along the cost frontier when a vector of
target costs (TargetCost
) is not defined, specified as an integer.
The default is 10 equally spaced points between the point of minimum cost and the point
of minimum exposure. When specifying TargetCost
, enter
NumCosts
as an empty matrix []
.
Data Types: double
TargetCost
— Target cost values along the cost frontier
[ ]
(default) | vector
(Optional) Target cost values along the cost frontier, specified as a vector. If
TargetCost
is empty, or not entered, hedgeopt
evaluates NumCosts
equally spaced target costs between the minimum
cost and minimum exposure. When specified, the elements of TargetCost
should be positive numbers that represent the maximum amount of money the owner is
willing to spend to rebalance the portfolio.
Data Types: double
TargetSens
— Target sensitivity values of the portfolio
[ ]
(default) | vector
(Optional) Target sensitivity values of the portfolio, specified as a
1
-by-NSENS
vector containing the target
sensitivity values of the portfolio. When specifying TargetSens
,
enter NumCosts
and TargetCost
as empty
matrices []
.
Data Types: double
ConSet
— Additional conditions on the portfolio reallocations
[ ]
(default) | matrix
(Optional) Additional conditions on the portfolio reallocations, specified as a
number of constraints (NCONS
) by number of instruments
(NINST
) matrix of additional conditions on the portfolio
reallocations. An eligible NINST
-by-1
vector of
contract holdings, PortWts
, satisfies all the inequalities
A*PortWts <= b
, where A = ConSet(:,1:end-1)
and b = ConSet(:,end)
.
Note
The user-specified constraints included in ConSet
may be
created with the functions pcalims
or portcons
. However, the portcons
default PortHolds
positivity constraints
are typically inappropriate for hedging problems since short-selling is usually
required.
NPOINTS
, the number of rows in PortSens
and PortHolds
and the length of PortCost
,
is inferred from the inputs. When the target sensitivities,
TargetSens
, is entered, NPOINTS = 1
;
otherwise NPOINTS = NumCosts
, or is equal to the length of the
TargetCost
vector.
Not all problems are solvable (for example, the solution space may be infeasible
or unbounded, or the solution may fail to converge). When a valid solution is not
found, the corresponding rows of PortSens
,
PortHolds
, and the elements of PortCost
are padded with NaN
s as placeholders.
Data Types: double
Output Arguments
PortSens
— Portfolio dollar sensitivities
matrix
Portfolio dollar sensitivities, returned as a number of points
(NPOINTS
-by-NSENS
) matrix. When a perfect hedge
exists, PortSens
is zeros. Otherwise, the best hedge possible is
chosen.
Note
Not all problems are solvable (for example, the solution space may be
infeasible, unbounded, or insufficiently constrained), or the solution may fail to
converge. When a valid solution is not found, the corresponding rows of
PortSens
and PortHolds
and elements of
PortCost
are padded with NaN
's as
placeholders. In addition, the solution may not be unique.
PortCost
— Total portfolio costs
vector
Total portfolio costs, returned as a
1
-by-NPOINTS
vector.
PortHolds
— Contracts allocated to each instrument
matrix
Contracts allocated to each instrument, returned as an
NPOINTS
-by-NINST
matrix. These are the
reallocated portfolios.
Version History
Introduced before R2006a
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