Introduction

Hedging is an investment to reduce the risk of adverse price movements in an asset. Financial Instruments Toolbox™ 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:

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. So, 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, ittsens, or sttsens 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 Equity Instrument Sensitivities.

Hedging with hedgeopt

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

FixedInd = [1 2 3 4 5 6 7 8];
[Sens, Cost, Quantity] = hedgeopt(Sensitivities, Price,Holdings, FixedInd, [], [], TargetSens)

Holdings =

        100.00
         50.00
        -50.00
         80.00
          8.00
         30.00
         40.00
         10.00


Sens =

     -61910.22     788946.21       4852.91


Cost =

             0


Quantity =

  Columns 1 through 6

        100.00         50.00        -50.00         80.00          8.00         30.00

  Columns 7 through 8

         40.00         10.00

Value0 = Holdings' * Price

Value0 =

     23674.62

Value1 = Quantity * Price

Value1 =

     23674.62

TargetSens = [-23000 -3300 3000];

FixedInd = [1 4 5 7 8];

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

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

Value1 = Quantity * Price

Value1 =

      4500.60

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

Sens =

         -0.00          -0.00          -0.00

Cost =

      23055.90

Quantity' =

        100.00
       -182.36
        -19.55
         80.00
          8.00
        -32.97
         40.00
         10.00

Value1 = Quantity * Price

Value1 =

      618.72

MaxCost = [0:1000:50000];

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

plot(MaxCost/1000, Cost/1000, 'red'), grid
xlabel('Funds Available for Rebalancing ($1000''s)')
ylabel('Actual Rebalancing Cost ($1000''s)')
title ('Rebalancing Cost Profile')

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)

Self-Financing Hedges with hedgeslf

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)

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.

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.