1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
|
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2011, 2012, 2013 Andre Miemiec
Copyright (C) 2012 Samuel Tebege
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<http://quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
#include <ql/experimental/swaptions/haganirregularswaptionengine.hpp>
#include <ql/pricingengines/swap/discountingswapengine.hpp>
#include <ql/pricingengines/swaption/blackswaptionengine.hpp>
#include <ql/exercise.hpp>
#include <ql/instruments/swaption.hpp>
#include <ql/math/matrixutilities/svd.hpp>
#include <ql/math/solvers1d/bisection.hpp>
#include <ql/time/daycounters/all.hpp>
#include <ql/cashflows/cashflows.hpp>
#include <ql/cashflows/couponpricer.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/math/interpolations/linearinterpolation.hpp>
#include <ql/math/solvers1d/brent.hpp>
namespace QuantLib {
//////////////////////////////////////////////////////////////////////////
// Implementation of helper class HaganIrregularSwaptionEngine::Basket //
//////////////////////////////////////////////////////////////////////////
HaganIrregularSwaptionEngine::Basket::Basket(boost::shared_ptr<IrregularSwap> swap,
const Handle<YieldTermStructure>& termStructure,
const Handle<SwaptionVolatilityStructure>& volatilityStructure)
:swap_(swap),termStructure_(termStructure),volatilityStructure_(volatilityStructure),targetNPV_(0.0),lambda_(0.0){
engine_ = boost::shared_ptr<PricingEngine>(new DiscountingSwapEngine(termStructure_));
//store swap npv
swap_->setPricingEngine(engine_);
targetNPV_ = swap_->NPV();
//build standard swaps
const Leg& fixedLeg = swap_->fixedLeg();
const Leg& floatLeg = swap_->floatingLeg();
Leg fixedCFS,floatCFS;
for(Size i = 0; i < fixedLeg.size(); ++i)
{
//retrieve fixed rate coupon from fixed leg
boost::shared_ptr<FixedRateCoupon> coupon = boost::dynamic_pointer_cast<FixedRateCoupon>(fixedLeg[i]);
QL_REQUIRE(coupon,"dynamic cast of fixed leg coupon failed.");
expiries_.push_back(coupon->date());
boost::shared_ptr<FixedRateCoupon> newCpn = boost::shared_ptr<FixedRateCoupon> (
new FixedRateCoupon(coupon->date(),
1.0,
coupon->rate(),
coupon->dayCounter(),
coupon->accrualStartDate(),
coupon->accrualEndDate(),
coupon->referencePeriodStart(),
coupon->referencePeriodEnd()));
fixedCFS.push_back(newCpn);
annuities_.push_back(10000*CashFlows::bps(fixedCFS,**termStructure_,true));
floatCFS.clear();
for(Size j = 0; j < floatLeg.size(); ++j){
//retrieve ibor coupon from floating leg
boost::shared_ptr<IborCoupon> coupon = boost::dynamic_pointer_cast<IborCoupon>(floatLeg[j]);
QL_REQUIRE(coupon,"dynamic cast of float leg coupon failed.");
if( coupon->date() <= expiries_[i] ){
boost::shared_ptr<IborCoupon> newCpn = boost::shared_ptr<IborCoupon> (
new IborCoupon(coupon->date(),
1.0,
coupon->accrualStartDate(),
coupon->accrualEndDate(),
coupon->fixingDays(),
coupon->iborIndex(),
1.0,
coupon->spread(),
coupon->referencePeriodStart(),
coupon->referencePeriodEnd(),
coupon->dayCounter(),
coupon->isInArrears()));
if (!newCpn->isInArrears())
newCpn->setPricer(
boost::shared_ptr<FloatingRateCouponPricer>(
new BlackIborCouponPricer()));
floatCFS.push_back(newCpn);
}
}
Real floatLegNPV = CashFlows::npv(floatCFS,**termStructure_,true);
fairRates_.push_back(floatLegNPV/annuities_[i]);
}
}
//computes a replication of the swap in terms of a basket of vanilla swaps
//by solving a linear system of equation
Disposable<Array> HaganIrregularSwaptionEngine::Basket::compute(Rate lambda) const {
//update members
lambda_ = lambda;
Size n = swap_->fixedLeg().size();
//build linear system of equations
Matrix arr(n,n,0.0);
Array rhs(n);
//fill the matrix describing the linear system of equations by looping over rows
for(Size r = 0; r < n; ++r)
{
boost::shared_ptr<FixedRateCoupon> cpn_r = boost::dynamic_pointer_cast<FixedRateCoupon>(swap_->fixedLeg()[r]);
QL_REQUIRE(cpn_r,"Cast to fixed rate coupon failed.");
//looping over columns
for(Size c = r; c < n; ++c){
//set homogenous part of lse
arr[r][c] = ( fairRates_[c] + lambda_ ) * cpn_r->accrualPeriod();
}
// add nominal repayment for i-th swap
arr[r][r] += 1;
}
for(Size r = 0; r < n; ++r)
{
boost::shared_ptr<FixedRateCoupon> cpn_r = boost::dynamic_pointer_cast<FixedRateCoupon>(swap_->fixedLeg()[r]);
// set inhomogenity of lse
Real N_r = cpn_r->nominal();
if(r < n - 1){
boost::shared_ptr<FixedRateCoupon> cpn_rp1 = boost::dynamic_pointer_cast<FixedRateCoupon>(swap_->fixedLeg()[r+1]);
Real N_rp1 = cpn_rp1->nominal();
rhs[r] = N_r * (cpn_r->rate()) * cpn_r->accrualPeriod() + (N_r - N_rp1);
} else {
rhs[r] = N_r * (cpn_r->rate()) * cpn_r->accrualPeriod() + N_r;
}
}
SVD svd(arr);
Disposable<Array> weights = svd.solveFor(rhs);
return weights;
}
Real HaganIrregularSwaptionEngine::Basket::operator()(Rate lambda) const {
Disposable<Array> weights = compute(lambda);
Real defect = -targetNPV_;
for(Size i=0; i< weights.size();++i) defect -= swap_->type()*lambda*weights[i]*annuities_[i];
return defect;
}
//creates a standard swap by deducing its conventions from market data objects
boost::shared_ptr<VanillaSwap> HaganIrregularSwaptionEngine::Basket::component(Size i) const {
boost::shared_ptr<IborCoupon> iborCpn = boost::dynamic_pointer_cast<IborCoupon>(swap_->floatingLeg()[0]);
QL_REQUIRE(iborCpn,"dynamic cast of float leg coupon failed. Can't find index.");
boost::shared_ptr<IborIndex> iborIndex = iborCpn->iborIndex();
Period dummySwapLength = Period(1,Years);
boost::shared_ptr<VanillaSwap> memberSwap_ = MakeVanillaSwap(dummySwapLength,iborIndex)
.withType(VanillaSwap::Type(swap_->type()))
.withEffectiveDate(swap_->startDate())
.withTerminationDate(expiries_[i])
.withRule(DateGeneration::Backward)
.withDiscountingTermStructure(termStructure_);
Real stdAnnuity = 10000*CashFlows::bps(memberSwap_->fixedLeg(),**termStructure_,true);
//compute annuity transformed rate
Rate transformedRate = (fairRates_[i]+lambda_)*annuities_[i]/stdAnnuity;
memberSwap_ = MakeVanillaSwap(dummySwapLength,iborIndex,transformedRate)
.withType(VanillaSwap::Type(swap_->type()))
.withEffectiveDate(swap_->startDate())
.withTerminationDate(expiries_[i])
.withRule(DateGeneration::Backward)
.withDiscountingTermStructure(termStructure_);
return memberSwap_;
}
///////////////////////////////////////////////////////////
// Implementation of class HaganIrregularSwaptionEngine //
///////////////////////////////////////////////////////////
HaganIrregularSwaptionEngine::HaganIrregularSwaptionEngine(const Handle<SwaptionVolatilityStructure>& volatilityStructure,
const Handle<YieldTermStructure>& termStructure)
: GenericEngine<IrregularSwaption::arguments,IrregularSwaption::results>(),
termStructure_(termStructure),
volatilityStructure_(volatilityStructure)
{
registerWith(termStructure_);
registerWith(volatilityStructure_);
}
void HaganIrregularSwaptionEngine::calculate() const {
//check exercise type
boost::shared_ptr<Exercise> exercise_ = this->arguments_.exercise;
QL_REQUIRE(exercise_->type() == QuantLib::Exercise::European,"swaption must be european");
//extract the underlying irregular swap
boost::shared_ptr<IrregularSwap> swap_ = this->arguments_.swap;
//Reshuffle spread from float to fixed (, i.e. remove spread from float side by finding the adjusted fixed coupon
//such that the NPV of the swap stays constant).
Leg fixedLeg = swap_->fixedLeg();
Real fxdLgBPS = CashFlows::bps(fixedLeg,**termStructure_,true);
Leg floatLeg = swap_->floatingLeg();
Real fltLgNPV = CashFlows::npv(floatLeg,**termStructure_,true);
Real fltLgBPS = CashFlows::bps(floatLeg,**termStructure_,true);
Leg floatCFS,fixedCFS;
floatCFS.clear();
for(Size j = 0; j < floatLeg.size(); ++j){
//retrieve ibor coupon from floating leg
boost::shared_ptr<IborCoupon> coupon = boost::dynamic_pointer_cast<IborCoupon>(floatLeg[j]);
QL_REQUIRE(coupon,"dynamic cast of float leg coupon failed.");
boost::shared_ptr<IborCoupon> newCpn = boost::shared_ptr<IborCoupon> (
new IborCoupon(coupon->date(),
coupon->nominal(),
coupon->accrualStartDate(),
coupon->accrualEndDate(),
coupon->fixingDays(),
coupon->iborIndex(),
coupon->gearing(),
0.0,
coupon->referencePeriodStart(),
coupon->referencePeriodEnd(),
coupon->dayCounter(),
coupon->isInArrears()));
if (!newCpn->isInArrears())
newCpn->setPricer(
boost::shared_ptr<FloatingRateCouponPricer>(
new BlackIborCouponPricer()));
floatCFS.push_back(newCpn);
}
Real sprdLgNPV = fltLgNPV - CashFlows::npv(floatCFS,**termStructure_,true);
Rate avgSpread = sprdLgNPV/fltLgBPS/10000;
Rate cpn_adjustment = avgSpread*fltLgBPS/fxdLgBPS;
fixedCFS.clear();
for(Size i = 0; i < fixedLeg.size(); ++i)
{
//retrieve fixed rate coupon from fixed leg
boost::shared_ptr<FixedRateCoupon> coupon = boost::dynamic_pointer_cast<FixedRateCoupon>(fixedLeg[i]);
QL_REQUIRE(coupon,"dynamic cast of fixed leg coupon failed.");
boost::shared_ptr<FixedRateCoupon> newCpn = boost::shared_ptr<FixedRateCoupon> (
new FixedRateCoupon(coupon->date(),
coupon->nominal(),
coupon->rate()-cpn_adjustment,
coupon->dayCounter(),
coupon->accrualStartDate(),
coupon->accrualEndDate(),
coupon->referencePeriodStart(),
coupon->referencePeriodEnd()));
fixedCFS.push_back(newCpn);
}
//this is the irregular swap with spread removed
swap_ = boost::shared_ptr<IrregularSwap>(new IrregularSwap(arguments_.swap->type(),fixedCFS,floatCFS));
//Sets up the basket by implementing the methodology described in
//P.S.Hagan "Callable Swaps and Bermudan 'Exercise into Swaptions'"
Basket basket(swap_,termStructure_,volatilityStructure_);
///////////////////////////////////////////////////////////////////////////////////////////////////
//find lambda //
///////////////////////////////////////////////////////////////////////////////////////////////////
Bisection s1d;
Rate minLambda = -0.5;
Rate maxLambda = 0.5;
s1d.setMaxEvaluations(10000);
s1d.setLowerBound(minLambda);
s1d.setUpperBound(maxLambda);
s1d.solve(basket,1.0e-8,0.01, minLambda, maxLambda);
/////////////////////////////////////////////////////////////////////////////////////////////////
// compute the price of the irreg swaption as the sum of the prices of the regular swaptions //
/////////////////////////////////////////////////////////////////////////////////////////////////
results_.value = HKPrice(basket,exercise_);
}
/////////////////////////////////////////////////////////////////////////////////////////
// Computes irregular swaption price according to P.J. Hunt, J.E. Kennedy: //
// "Implied interest rate pricing models", Finance Stochast. 2, 275293 (1998) //
/////////////////////////////////////////////////////////////////////////////////////////
Real HaganIrregularSwaptionEngine::HKPrice(Basket& basket,boost::shared_ptr<Exercise>& exercise) const {
// Black 76 Swaption Engine: assumes that the swaptions exercise date equals the swap start date
boost::shared_ptr<PricingEngine> blackSwaptionEngine =
boost::shared_ptr<PricingEngine>(new BlackSwaptionEngine(termStructure_,volatilityStructure_));
//retrieve weights of underlying swaps
Disposable<Array> weights = basket.weights();
Real npv = 0.0;
for(Size i=0; i<weights.size(); ++i)
{
boost::shared_ptr<VanillaSwap> pvSwap_ = basket.component(i);
Swaption swaption = Swaption(pvSwap_,exercise);
swaption.setPricingEngine(blackSwaptionEngine);
npv += weights[i]*swaption.NPV();
}
return npv;
}
}
|
