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Financial Instruments Toolbox™ software offers two functions for assessing the fundamental hedging tradeoff, hedgeopt and hedgeslf.
The first function, hedgeopt, addresses the most general hedging problem. It allocates an optimal hedge to satisfy either 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.
hedgeopt allows investors to modify portfolio allocations among instruments according to either of the goals. The problem is cast as a constrained linear least-squares problem. For additional information about hedgeopt, see Hedging with hedgeopt.
The second function, hedgeslf, attempts to allocate a self-financing hedge among a portfolio of instruments. In particular, hedgeslf attempts to maintain a constant portfolio value consistent with reduced portfolio sensitivities (that is, the rebalanced portfolio is hedged against market moves and is closest to being self-financing). If hedgeslf cannot find a self-financing hedge, it rebalances the portfolio to minimize overall portfolio sensitivities. For additional information on hedgeslf, see Self-Financing Hedges with hedgeslf.
The examples in this section consider the delta, gamma, and vega sensitivity measures. In this toolbox, when you work with interest-rate derivatives, delta is the price sensitivity measure of shifts in the forward yield curve, gamma is the delta sensitivity measure of shifts in the forward yield curve, and vega is the price sensitivity measure of shifts in the volatility process. See bdtsens or hjmsens for details on the computation of sensitivities for interest-rate derivatives.
For equity exotic options, the underlying instrument is the stock price instead of the forward yield curve. Consequently, delta now represents the price sensitivity measure of shifts in the stock price, gamma is the delta sensitivity measure of shifts in the stock price, and vega is the price sensitivity measure of shifts in the volatility of the stock. See crrsens, eqpsens, or ittsens for details on the computation of sensitivities for equity derivatives.
For examples showing the computation of sensitivities for interest-rate based derivatives, see Computing Instrument Sensitivities. Likewise, for examples showing the computation of sensitivities for equity exotic options, see Computing Instrument Sensitivities.
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.
Both hedging functions require some common inputs, including the current portfolio holdings (allocations), and a matrix of instrument sensitivities. To create these inputs, load the example portfolio into memory
load deriv.mat;
compute price and sensitivities
[Delta, Gamma, Vega, Price] = hjmsens(HJMTree, HJMInstSet);
Warning: Not all cash flows are aligned with the tree. Result will be approximated.
and extract the current portfolio holdings.
Holdings = instget(HJMInstSet, '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.
To summarize the portfolio information
disp([Price Holdings Sensitivities])
98.72 100.00 -272.65 1029.90 0.00 97.53 50.00 -347.43 1622.69 -0.04 0.05 -50.00 -8.08 643.40 34.07 98.72 80.00 -272.65 1029.90 0.00 100.55 8.00 -1.04 3.31 0 6.28 30.00 294.97 6852.56 93.69 0.05 40.00 -47.16 8459.99 93.69 3.69 10.00 -282.05 1059.68 0.00
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 = -61910.22 788946.21 4852.91
To illustrate using hedgeopt, suppose that you want to maintain your existing portfolio. The first form of 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.
[Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,... Holdings, [], [], [], TargetSens)
Sens = -61910.22 788946.21 4852.91 Cost = 0 Quantity' = 100.00 50.00 -50.00 80.00 8.00 30.00 40.00 10.00
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 = 23674.62
Value1 = Quantity * Price
Value1 = 23674.62
Building on the example in Maintaining Existing Allocations, 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];
Then, specify the instruments to be fixed.
FixedInd = [1 4 5 7 8];
Finally, call hedgeopt
[Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,... Holdings, FixedInd, [], [], TargetSens);
and again examine the results.
Sens = -23000.00 -3300.00 3000.00 Cost = 19174.02 Quantity' = 100.00 -141.03 137.26 80.00 8.00 -57.96 40.00 10.00
Recompute Value1, the portfolio value after rebalancing.
Value1 = Quantity * Price
Value1 = 4500.60
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 have been changed.
The example in Partially Hedged Portfolio illustrates 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 0s and call hedgeopt again. The following example uses data from Hedging with hedgeopt.
TargetSens = [0 0 0]; [Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price, ... Holdings, FixedInd, [], [], TargetSens);
Examining the outputs reveals that you have obtained a fully hedged portfolio
Sens = -0.00 -0.00 -0.00
but at an expense of over $20,000.
Cost = 23055.90
The positions required to achieve a fully hedged portfolio
Quantity' = 100.00 -182.36 -19.55 80.00 8.00 -32.97 40.00 10.00
result in the new portfolio value
Value1 = Quantity * Price
Value1 = 618.72
The examples in Fully Hedged Portfolio illustrate how to use hedgeopt to determine the minimum cost of hedging a portfolio given a set of target sensitivities. In these examples, portfolio target sensitivities are treated as equality constraints during the optimization process. You tell hedgeopt what sensitivities you want, and it tells you what it will cost to get those sensitivities.
A related problem involves minimizing portfolio sensitivities for a given set of maximum target costs. For this goal, the target costs are treated as inequality constraints during the optimization process. You tell hedgeopt the most you are willing spend to insulate your portfolio, and it tells you the smallest portfolio sensitivities you can get for your money.
To illustrate this use of hedgeopt, compute the portfolio dollar sensitivities along the entire cost frontier. From the previous examples, you know that spending nothing replicates the existing portfolio, while spending $23,055.90 completely hedges the portfolio.
Assume, for example, you are willing to spend as much as $50,000, and want to see what portfolio sensitivities will result along the cost frontier. Assume that the same instruments are held fixed, and that the cost frontier is evaluated from $0 to $50,000 at increments of $1000.
MaxCost = [0:1000:50000];
Now, call hedgeopt.
[Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price, ... Holdings, FixedInd, [], MaxCost);
With this data, you can plot the required hedging cost versus the funds available (the amount you are willing to spend)
plot(MaxCost/1000, Cost/1000, 'red'), grid xlabel('Funds Available for Rebalancing ($1000''s)') ylabel('Actual Rebalancing Cost ($1000''s)') title ('Rebalancing Cost Profile')
Rebalancing Cost Profile
and the portfolio dollar sensitivities versus the funds available.
figure plot(MaxCost/1000, Sens(:,1), '-red') hold('on') plot(MaxCost/1000, Sens(:,2), '-.black') plot(MaxCost/1000, Sens(:,3), '--blue') grid xlabel('Funds Available for Rebalancing ($1000''s)') ylabel('Delta, Gamma, and Vega Portfolio Dollar Sensitivities') title ('Portfolio Sensitivities Profile') legend('Delta', 'Gamma', 'Vega', 0)
Funds Available for Rebalancing
The figures Rebalancing Cost Profile and Funds Available for Rebalancing indicate that there is no benefit because the funds available for hedging exceed $23,055.90, the point of maximum expense required to obtain simultaneous delta, gamma, and vega neutrality. You can also find this point of delta, gamma, and vega neutrality using hedgeslf.
[Sens, Value1, Quantity] = hedgeslf(Sensitivities, Price,... Holdings, FixedInd);
Sens = -0.00 -0.00 -0.00 Value1 = 618.72 Quantity = 100.00 -182.36 -19.55 80.00 8.00 -32.97 40.00 10.00
Similar to hedgeopt, hedgeslf returns the portfolio dollar sensitivities and instrument quantities (the rebalanced holdings). However, in contrast, the second output parameter of hedgeslf is the value of the rebalanced portfolio, from which you can calculate the rebalancing cost by subtraction.
Value0 - Value1
ans = 23055.90
In this example, the portfolio is clearly not self-financing, so hedgeslf finds the best possible solution required to obtain zero sensitivities.
There is, in fact, a third calling syntax available for hedgeopt directly related to the results shown above for hedgeslf. Suppose, instead of directly specifying the funds available for rebalancing (the most money you are willing to spend), you want to simply specify the number of points along the cost frontier. This call to hedgeopt samples the cost frontier at 10 equally spaced points between the point of minimum cost (and potentially maximum exposure) and the point of minimum exposure (and maximum cost).
[Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,... Holdings, FixedInd, 10)
Sens = -32784.46 2231.83 -49694.33 -29141.74 1983.85 -44172.74 -25499.02 1735.87 -38651.14 -21856.30 1487.89 -33129.55 -18213.59 1239.91 -27607.96 -14570.87 991.93 -22086.37 -10928.15 743.94 -16564.78 -7285.43 495.96 -11043.18 -3642.72 247.98 -5521.59 0.00 -0.00 0.00 Cost = 0.00 2561.77 5123.53 7685.30 10247.07 12808.83 15370.60 17932.37 20494.14 23055.90
Now plot this data.
figure plot(Cost/1000, Sens(:,1), '-red') hold('on') plot(Cost/1000, Sens(:,2), '-.black') plot(Cost/1000, Sens(:,3), '--blue') grid xlabel('Rebalancing Cost ($1000''s)') ylabel('Delta, Gamma, and Vega Portfolio Dollar Sensitivities') title ('Portfolio Sensitivities Profile') legend('Delta', 'Gamma', 'Vega', 0)
Rebalancing Cost
In this calling form, hedgeopt calls hedgeslf internally to determine the maximum cost needed to minimize the portfolio sensitivities ($23,055.90), and evenly samples the cost frontier between $0 and $23,055.90.
Note that both hedgeopt and hedgeslf cast the optimization problem as a constrained linear least squares problem. Depending on the instruments and constraints, neither function is guaranteed to converge to a solution. In some cases, the problem space may be unbounded, and additional instrument equality constraints, or user-specified constraints, may be necessary for convergence. See Hedging with Constrained Portfolios for additional information.