Bermudans, callable swaps. 1. Introduction. This is part of three related papers: Evaluating and hedging exotic swap instruments via LGM explains the theory. Analytic LGM swaption engine for european exercise. More #include Hagan, Evaluating and hedging exotic swap instruments via LGM. Lichters, Stamm. The evaluation of sensitivities in the Hull White model with respect to changes Evaluating and Hedging Exotic Swap Instruments via LGM.
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Published on Oct View Download Here we break up the methodology for evaluating callable swapsKey words. The theory an methodology for using the LGM model to evaluate Bermudans iscovered in: Evaluating and hedging exotic swap instruments via LGM. In this paper, Callable Range Notes: Procedure breaks instrumfnts the method into a specific procedure and set of algorithms.
In the main body of thepaper, we treat Bermudans on bullet swaps and callable gedging swaps. In the appendix, we indicate2. The coupon leg is defined by: To preform the adjustment, we also need todays value C0i of each coupon making up the range note,V cpn1Vcpn2The inputs to the program are the eective funding leg coupons, 2.
To explain this input, recall that the funding leg is assumed to be a standard floating leg plus a margin rate. As discussed earlier, we approximate the floating legs value by pretending the floating leg schedule is thesame as the fixed leg schedule. Then at any coupon date tjthe value of the floating leg is equivalent to1 paid at tj 2.
We assume that the dates j before today have already been excluded. We also need the array 2. The exercise fees are 2. In the Callable Range Note document we assumed the proxy deal only has 1fixing per period.
Here we allow 2. Payo adjusters caplet vol: Calibration flags and variables.
The value of the Bermudan and the component european options on the range note arecalculated: For payers, one exchanges the receiveds and paids.
Here i,c are vi appropriate digitals.
Calculate the schedule data for each digitalStep 2a. Get the appropriate business day rule, holiday centers, and end of month rule knstruments the k month ratebeing used as an index.
Procedure for Pricing Bermudans and Callable Swaps
ebaluating Also get the appropriate day count basis, basisind, for the k month index rate. Subdivide each interval ti1 to ti into nfix subintervals. Note that the Add function just moves t to a good business day using theappropriate conventions for a k month caplet based on the rate indexStep 2c. Find the end date, fixing date, and day count fraction for each sub-interval.
Get the forward rates and the implied vol for each digital caplet: Calculate discount factors for each coupon date 2. Call characterization routineStep 4. Call the calibration routineStep 5. Call the evaluation routineStep 6. Add t, j,months, bdr, hol1, hol2, hol3, eom ImpliedDigitalVol tex, c, Rfix,vc Use a global Newton chord? User evaluatnig iteration schemeStandard cumulative normal4. Also, input are llgm. Finally, we need to input: In this procedure we develop a set of swaption which have common pay dates However, it greatly simplifies some of the exogic calibration methods, so wemay as well develop it this adn to simplify future work.
In this procedure, we also need quatities which refer to the standard floating leg index such as 3mUS-DLibor and market default parameters for fixed legs opposite these evaluatin legs in single currency swaps.
Well abreviate these arguments as flt and fix. Ensure va the triples are ordered in terms of increasing cjand exclude any date closer than,say, 1 month apart from the previous date. Replace each cj by the nearest whole number of months. Fill in the swaption schedules. Here, the holiday centers, and end-of-month rule are the ones appropriate for fixed legs in the standard swap,and T0 is the first date with 4. If T0 tnow, then replace it with tnow. Calculate the quantities needed for the swaptionsa calculate the coverages day count fractions for each interval using the both the fixed and floating legbases: So for each swaption j, we have tofind the same quantities: ,gm the kink by adding 6.
EavgNhj K mx hj1 k mx pj j1! Routine for generating the integration weights and partial sums detailed below Routine for calculating the payo vector at each j detailed above Routine for calculating the European option values detailed immediately below Standard cumulative normal distributionGaussian density7.
Routine for evaluating the European options. Classically, hedgnig routine is part of the evaluatingthe Bermudan routine7. The todays value V eurj of the jth European option7. Take the maximum of the payo with 0: If there is more than one crossing which there wont be for this deal type choose the crossing with Knearest the midpoint mx. Remember the maxs and mins from the crossing: Standard cumulative normal distributionGaussian density8.
Routine to create integration weights. There are only two inputs hrdging These define the grid of points 8. The weight vector w0, w1, Then use symmetry to get the other valuts 8. Then set re-define the weights by, 8. Pricing and hedging swaps Documents. Swaps pricing and strategies Business.
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