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
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
|
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2013 Gary Kennedy
Copyright (C) 2015, 2024 Peter Caspers
Copyright (C) 2017 Klaus Spanderen
Copyright (C) 2020 Marcin Rybacki
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
<https://www.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 "toplevelfixture.hpp"
#include "utilities.hpp"
#include <ql/pricingengines/blackformula.hpp>
#include <cmath>
using namespace QuantLib;
using namespace boost::unit_test_framework;
BOOST_FIXTURE_TEST_SUITE(QuantLibTests, TopLevelFixture)
BOOST_AUTO_TEST_SUITE(BlackFormulaTests)
BOOST_AUTO_TEST_CASE(testBachelierImpliedVol) {
BOOST_TEST_MESSAGE("Testing Bachelier implied vol...");
Real forward = 1.0;
Real bpvol = 0.01;
Real tte = 10.0;
Real stdDev = bpvol * std::sqrt(tte);
Option::Type optionType = Option::Call;
Real discount = 0.95;
Real d[] = {-3.0, -2.0, -1.0, -0.5, 0.0, 0.5, 1.0, 2.0, 3.0};
for (Real i : d) {
Real strike = forward - i * bpvol * std::sqrt(tte);
Real callPrem = bachelierBlackFormula(optionType, strike, forward, stdDev, discount);
Real impliedBpVol = bachelierBlackFormulaImpliedVolChoi(optionType, strike, forward, tte,
callPrem, discount);
if (std::fabs(bpvol - impliedBpVol) > 1.0e-12) {
BOOST_ERROR("Failed, expected " << bpvol << " realised " << impliedBpVol);
}
Real impliedBpVolExact =
bachelierBlackFormulaImpliedVol(optionType, strike, forward, tte, callPrem, discount);
if (std::fabs(bpvol - impliedBpVolExact) > 1.0e-15) {
BOOST_ERROR("Failed, expected " << bpvol << " realised " << impliedBpVolExact);
}
}
}
BOOST_AUTO_TEST_CASE(testChambersImpliedVol) {
BOOST_TEST_MESSAGE("Testing Chambers-Nawalkha implied vol approximation...");
Option::Type types[] = {Option::Call, Option::Put};
Real displacements[] = {0.0000, 0.0010, 0.0050, 0.0100, 0.0200};
Real forwards[] = {-0.0010, 0.0000, 0.0050, 0.0100, 0.0200, 0.0500};
Real strikes[] = {-0.0100, -0.0050, -0.0010, 0.0000, 0.0010, 0.0050,
0.0100, 0.0200, 0.0500, 0.1000};
Real stdDevs[] = {0.10, 0.15, 0.20, 0.30, 0.50, 0.60, 0.70,
0.80, 1.00, 1.50, 2.00};
Real discounts[] = {1.00, 0.95, 0.80, 1.10};
Real tol = 5.0E-4;
for (auto& type : types) {
for (Real& displacement : displacements) {
for (Real& forward : forwards) {
for (Real& strike : strikes) {
for (Real& stdDev : stdDevs) {
for (Real& discount : discounts) {
if (forward + displacement > 0.0 && strike + displacement > 0.0) {
Real premium = blackFormula(type, strike, forward, stdDev, discount,
displacement);
Real atmPremium = blackFormula(type, forward, forward, stdDev,
discount, displacement);
Real iStdDev = blackFormulaImpliedStdDevChambers(
type, strike, forward, premium, atmPremium, discount,
displacement);
Real moneyness = (strike + displacement) / (forward + displacement);
if(moneyness > 1.0) moneyness = 1.0 / moneyness;
Real error = (iStdDev - stdDev) / stdDev * moneyness;
if(error > tol)
BOOST_ERROR("Failed to verify Chambers-Nawalkha "
"approximation for "
<< type << " displacement=" << displacement
<< " forward=" << forward << " strike=" << strike
<< " discount=" << discount << " stddev=" << stdDev
<< " result=" << iStdDev
<< " exceeds maximum error tolerance");
}
}
}
}
}
}
}
}
BOOST_AUTO_TEST_CASE(testRadoicicStefanicaImpliedVol) {
BOOST_TEST_MESSAGE(
"Testing Radoicic-Stefanica implied vol approximation...");
const Time T = 1.7;
const Rate r = 0.1;
const DiscountFactor df = std::exp(-r*T);
const Real forward = 100;
const Volatility vol = 0.3;
const Real stdDev = vol * std::sqrt(T);
const Option::Type types[] = { Option::Call, Option::Put };
const Real strikes[] = {
50, 60, 70, 80, 90, 100, 110, 125, 150, 200, 300 };
const Real tol = 0.02;
for (Real strike : strikes) {
for (auto type : types) {
const ext::shared_ptr<PlainVanillaPayoff> payoff(
ext::make_shared<PlainVanillaPayoff>(type, strike));
const Real marketValue = blackFormula(payoff, forward, stdDev, df);
const Real estVol = blackFormulaImpliedStdDevApproximationRS(
payoff, forward, marketValue, df) / std::sqrt(T);
const Real error = std::fabs(estVol - vol);
if (error > tol) {
BOOST_ERROR("Failed to verify Radoicic-Stefanica"
"approximation for "
<< type
<< "\n forward :" << forward
<< "\n strike :" << strike
<< "\n discount :" << df
<< "\n implied vol :" << vol
<< "\n result :" << estVol
<< "\n error :" << error
<< "\n tolerance :" << tol);
}
}
}
}
BOOST_AUTO_TEST_CASE(testRadoicicStefanicaLowerBound) {
BOOST_TEST_MESSAGE("Testing Radoicic-Stefanica lower bound...");
// testing lower bound plot figure 3.1 from
// "Tighter Bounds for Implied Volatility",
// J. Gatheral, I. Matic, R. Radoicic, D. Stefanica
// https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2922742
const Real forward = 1.0;
const Real k = 1.2;
for (Real s=0.17; s < 2.9; s+=0.01) {
const Real strike = std::exp(k)*forward;
const Real c = blackFormula(Option::Call, strike, forward, s);
const Real estimate = blackFormulaImpliedStdDevApproximationRS(
Option::Call, strike, forward, c);
const Real error = s - estimate;
if (std::isnan(estimate) || std::fabs(error) > 0.05) {
BOOST_ERROR("Failed to lower bound Radoicic-Stefanica"
"approximation for "
<< "\n forward :" << forward
<< "\n strike :" << k
<< "\n stdDev :" << s
<< "\n result :" << estimate
<< "\n error :" << error);
}
if (c > 1e-6 && error < 0.0) {
BOOST_ERROR("Failed to verify Radoicic-Stefanica is lower bound"
<< "\n forward :" << forward
<< "\n strike :" << k
<< "\n stdDev :" << s
<< "\n result :" << estimate
<< "\n error :" << error);
}
}
}
BOOST_AUTO_TEST_CASE(testImpliedVolAdaptiveSuccessiveOverRelaxation) {
BOOST_TEST_MESSAGE("Testing implied volatility calculation via "
"adaptive successive over-relaxation...");
const DayCounter dc = Actual365Fixed();
const Date today = Date(12, July, 2017);
Settings::instance().evaluationDate() = today;
const Date exerciseDate = today + Period(15, Months);
const Time exerciseTime = dc.yearFraction(today, exerciseDate);
const ext::shared_ptr<YieldTermStructure> rTS = flatRate(0.10, dc);
const ext::shared_ptr<YieldTermStructure> qTS = flatRate(0.06, dc);
const DiscountFactor df = rTS->discount(exerciseDate);
const Volatility vol = 0.20;
const Real stdDev = vol * std::sqrt(exerciseTime);
const Real s0 = 100;
const Real forward= s0 * qTS->discount(exerciseDate)/df;
const Option::Type types[] = { Option::Call, Option::Put };
const Real strikes[] = { 50, 60, 70, 80, 90, 100, 110, 125, 150, 200 };
const Real displacements[] = { 0, 25, 50, 100};
const Real tol = 1e-8;
for (Real strike : strikes) {
for (auto type : types) {
const ext::shared_ptr<PlainVanillaPayoff> payoff(
ext::make_shared<PlainVanillaPayoff>(type, strike));
for (Real displacement : displacements) {
const Real marketValue = blackFormula(payoff, forward, stdDev, df, displacement);
const Real impliedStdDev = blackFormulaImpliedStdDevLiRS(
payoff, forward, marketValue, df, displacement,
Null<Real>(), 1.0, tol, 100);
const Real error = std::fabs(impliedStdDev - stdDev);
if (error > 10*tol) {
BOOST_ERROR("Failed to calculated implied volatility"
" with adaptive successive over-relaxation"
<< "\n forward :" << forward
<< "\n strike :" << strike
<< "\n stdDev :" << stdDev
<< "\n displacement:" << displacement
<< "\n result :" << impliedStdDev
<< "\n error :" << error
<< "\n tolerance :" << tol);
}
}
}
}
}
void assertBlackFormulaForwardDerivative(
Option::Type optionType,
const std::vector<Real> &strikes,
Real bpvol)
{
Real forward = 1.0;
Real tte = 10.0;
Real stdDev = bpvol * std::sqrt(tte);
Real discount = 0.95;
Real displacement = 0.01;
Real bump = 0.0001;
Real epsilon = 1.e-10;
std::string type = optionType == Option::Call ? "Call" : "Put";
for (Real strike : strikes) {
Real delta = blackFormulaForwardDerivative(optionType, strike, forward, stdDev, discount,
displacement);
Real bumpedDelta = blackFormulaForwardDerivative(
optionType, strike, forward + bump, stdDev, discount, displacement);
Real basePremium = blackFormula(
optionType, strike, forward, stdDev, discount, displacement);
Real bumpedPremium = blackFormula(
optionType, strike, forward + bump, stdDev, discount, displacement);
Real deltaApprox = (bumpedPremium - basePremium) / bump;
/*! Based on the Mean Value Theorem, the below inequality
should hold for any function that is monotonic in the
area of the bump.
*/
bool success =
(std::max(delta, bumpedDelta) + epsilon > deltaApprox)
&& (deltaApprox > std::min(delta, bumpedDelta) - epsilon);
if (!success)
{
BOOST_ERROR("Failed to calculate the derivative of the"
" Black formula w.r.t. forward"
<< "\n option type :" << type
<< "\n forward :" << forward
<< "\n strike :" << strike
<< "\n stdDev :" << stdDev
<< "\n displacement :" << displacement
<< "\n analytical delta :" << delta
<< "\n approximated delta:" << deltaApprox);
}
}
}
BOOST_AUTO_TEST_CASE(testBlackFormulaForwardDerivative) {
BOOST_TEST_MESSAGE("Testing forward derivative of the Black formula...");
std::vector<Real> strikes;
strikes.push_back(0.1);
strikes.push_back(0.5);
strikes.push_back(1.0);
strikes.push_back(2.0);
strikes.push_back(3.0);
const Real vol = 0.1;
assertBlackFormulaForwardDerivative(Option::Call, strikes, vol);
assertBlackFormulaForwardDerivative(Option::Put, strikes, vol);
}
BOOST_AUTO_TEST_CASE(testBlackFormulaForwardDerivativeWithZeroStrike) {
BOOST_TEST_MESSAGE("Testing forward derivative of the Black formula "
"with zero strike...");
std::vector<Real> strikes;
strikes.push_back(0.0);
const Real vol = 0.1;
assertBlackFormulaForwardDerivative(Option::Call, strikes, vol);
assertBlackFormulaForwardDerivative(Option::Put, strikes, vol);
}
BOOST_AUTO_TEST_CASE(testBlackFormulaForwardDerivativeWithZeroVolatility) {
BOOST_TEST_MESSAGE("Testing forward derivative of the Black formula "
"with zero volatility...");
std::vector<Real> strikes;
strikes.push_back(0.1);
strikes.push_back(0.5);
strikes.push_back(1.0);
strikes.push_back(2.0);
strikes.push_back(3.0);
const Real vol = 0.0;
assertBlackFormulaForwardDerivative(Option::Call, strikes, vol);
assertBlackFormulaForwardDerivative(Option::Put, strikes, vol);
}
void assertBachelierBlackFormulaForwardDerivative(
Option::Type optionType,
const std::vector<Real> &strikes,
Real bpvol)
{
Real forward = 1.0;
Real tte = 10.0;
Real stdDev = bpvol * std::sqrt(tte);
Real discount = 0.95;
Real bump = 0.0001;
Real epsilon = 1.e-10;
std::string type = optionType == Option::Call ? "Call" : "Put";
for (Real strike : strikes) {
Real delta =
bachelierBlackFormulaForwardDerivative(optionType, strike, forward, stdDev, discount);
Real bumpedDelta = bachelierBlackFormulaForwardDerivative(
optionType, strike, forward + bump, stdDev, discount);
Real basePremium = bachelierBlackFormula(
optionType, strike, forward, stdDev, discount);
Real bumpedPremium = bachelierBlackFormula(
optionType, strike, forward + bump, stdDev, discount);
Real deltaApprox = (bumpedPremium - basePremium) / bump;
/*! Based on the Mean Value Theorem, the below inequality
should hold for any function that is monotonic in the
area of the bump.
*/
bool success =
(std::max(delta, bumpedDelta) + epsilon > deltaApprox)
&& (deltaApprox > std::min(delta, bumpedDelta) - epsilon);
if (!success)
{
BOOST_ERROR("Failed to calculate the derivative of the"
" Bachelier Black formula w.r.t. forward"
<< "\n option type :" << type
<< "\n forward :" << forward
<< "\n strike :" << strike
<< "\n stdDev :" << stdDev
<< "\n analytical delta :" << delta
<< "\n approximated delta:" << deltaApprox);
}
}
}
BOOST_AUTO_TEST_CASE(testBachelierBlackFormulaForwardDerivative) {
BOOST_TEST_MESSAGE("Testing forward derivative of the "
"Bachelier Black formula...");
std::vector<Real> strikes;
strikes.push_back(-3.0);
strikes.push_back(-2.0);
strikes.push_back(-1.0);
strikes.push_back(-0.5);
strikes.push_back(0.0);
strikes.push_back(0.5);
strikes.push_back(1.0);
strikes.push_back(2.0);
strikes.push_back(3.0);
const Real vol = 0.001;
assertBachelierBlackFormulaForwardDerivative(Option::Call, strikes, vol);
assertBachelierBlackFormulaForwardDerivative(Option::Put, strikes, vol);
}
BOOST_AUTO_TEST_CASE(testBachelierBlackFormulaForwardDerivativeWithZeroVolatility) {
BOOST_TEST_MESSAGE("Testing forward derivative of the Bachelier Black formula "
"with zero volatility...");
std::vector<Real> strikes;
strikes.push_back(-3.0);
strikes.push_back(-2.0);
strikes.push_back(-1.0);
strikes.push_back(-0.5);
strikes.push_back(0.0);
strikes.push_back(0.5);
strikes.push_back(1.0);
strikes.push_back(2.0);
strikes.push_back(3.0);
const Real vol = 0.0;
assertBachelierBlackFormulaForwardDerivative(Option::Call, strikes, vol);
assertBachelierBlackFormulaForwardDerivative(Option::Put, strikes, vol);
}
BOOST_AUTO_TEST_SUITE_END()
BOOST_AUTO_TEST_SUITE_END()
|
