Source : https://github.com/LechGrzelak/Computational-Finance-Course
책 자체는 600페이지 이내로 최병선 교수님(약 700페이지)보다는 얇고 김창기 교수님(약 500 페이지) 교재보다는 두껍다. 그런데 목차 구성이 매우 디테일하고 풍부해서 공부에 많은 도움이 될 듯.
공부 순서는 김창기 교수님 책을 가장 먼저 읽는게 맞고, 그 뒤는 아무거나 봐도 상관없을 듯.
Lecture Slides
Lecture 1- Introduction and Overview of Asset Classes
Lecture 2- Stock, Options and Stochastics
Lecture 3- Option Pricing and Simulation in Python
Lecture 4- Implied Volatility
Lecture 5- Jump Processes
Lecture 6- Affine Jump Diffusion Processes
Lecture 7- Stochastic Volatility Models
Lecture 8- Fourier Transformation for Option Pricing
Lecture 9- Monte Carlo Simulation
Lecture 10- Monte Carlo Simulation of the Heston Model
Lecture 11- Hedging and Monte Carlo Greeks
Lecture 12- Forward Start Options and Model of Bates
Lecture 13- Exotic Derivatives
Lecture 14- Summary
Book
1 Basics about Stochastic Processes 1
1.1 Stochastic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Density function, expectation, variance . . . . . . . . . . . . 1
1.1.2 Characteristic function . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Cumulants and moments . . . . . . . . . . . . . . . . . . . 4
1.2 Stochastic processes, martingale property . . . . . . . . . . . . . . 9
1.2.1 Wiener process . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.3 Iterated expectations (Tower property) . . . . . . . . . . . . 13
1.3 Stochastic integration, Itô integral . . . . . . . . . . . . . . . . . . 14
1.3.1 Elementary processes . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Itô isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.3 Martingale representation theorem . . . . . . . . . . . . . . 20
1.4 Exercise set .
2 Introduction to Financial Asset Dynamics 27
2.1 Geometric Brownian motion asset price process . . . . . . . . . . . 27
2.1.1 Itô process . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.2 Itô’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.3 Distributions of S(t) and log S(t) . . . . . . . . . . . . . . . 34
2.2 First generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.1 Proportional dividend model . . . . . . . . . . . . . . . . . 38
2.2.2 Volatility variation . . . . . . . . . . . . . . . . . . . . . . . 39
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2.2.3 Time-dependent volatility . . . . . . . . . . . . . . . . . . . 39
2.3 Martingales and asset prices . . . . . . . . . . . . . . . . . . . . . . 40
2.3.1 P-measure prices . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.2 Q-measure prices . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.3 Parameter estimation under real-world measure P . . . . . 44
2.4 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 The Black-Scholes Option Pricing Equation 51
3.1 Option contract definitions . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Option basics . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.2 Derivation of the partial differential equation . . . . . . . . 56
3.1.3 Martingale approach and option pricing . . . . . . . . . . . 60
3.2 The Feynman-Kac theorem and the Black-Scholes model . . . . . . 61
3.2.1 Closed-form option prices . . . . . . . . . . . . . . . . . . . 63
3.2.2 Green’s functions and characteristic functions . . . . . . . . 66
3.2.3 Volatility variations . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Delta hedging under the Black-Scholes model . . . . . . . . . . . . 73
3.4 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 Local Volatility Models 81
4.1 Black-Scholes implied volatility . . . . . . . . . . . . . . . . . . . . 81
4.1.1 The concept of implied volatility . . . . . . . . . . . . . . . 82
4.1.2 Implied volatility; implications . . . . . . . . . . . . . . . . 86
4.1.3 Discussion on alternative asset price models . . . . . . . . . 86
4.2 Option prices and densities . . . . . . . . . . . . . . . . . . . . . . 89
4.2.1 Market implied volatility smile and the payoff . . . . . . . . 89
4.2.2 Variance swaps . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 Non-parametric local volatility models . . . . . . . . . . . . . . . . 102
4.3.1 Implied volatility representation of local volatility . . . . . 105
4.3.2 Arbitrage-free conditions for option prices . . . . . . . . . . 107
4.3.3 Advanced implied volatility interpolation . . . . . . . . . . 111
4.3.4 Simulation of local volatility model . . . . . . . . . . . . . . 114
4.4 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5 Jump Processes 121
5.1 Jump diffusion processes . . . . . . . . . . . . . . . . . . . . . . . . 121
5.1.1 Itô’s lemma and jumps . . . . . . . . . . . . . . . . . . . . . 124
5.1.2 PIDE derivation for jump diffusion process . . . . . . . . . 127
5.1.3 Special cases for the jump distribution . . . . . . . . . . . . 128
5.2 Feynman-Kac theorem for jump diffusion process . . . . . . . . . . 130
5.2.1 Analytic option prices . . . . . . . . . . . . . . . . . . . . . 131
5.2.2 Characteristic function for Merton’s model . . . . . . . . . 133
5.2.3 Dynamic hedging of jumps with the Black-Scholes
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3 Exponential Lévy processes . . . . . . . . . . . . . . . . . . . . . . 139
5.3.1 Finite activity exponential Lévy processes . . . . . . . . . . 142
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5.3.2 PIDE and the Lévy triplet . . . . . . . . . . . . . . . . . . 143
5.3.3 Equivalent martingale measure . . . . . . . . . . . . . . . . 145
5.4 Infinite activity exponential Lévy processes . . . . . . . . . . . . . 146
5.4.1 Variance Gamma process . . . . . . . . . . . . . . . . . . . 146
5.4.2 CGMY process . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.4.3 Normal inverse Gaussian process . . . . . . . . . . . . . . . 155
5.5 Discussion on jumps in asset dynamics . . . . . . . . . . . . . . . . 156
5.6 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6 The COS Method for European Option Valuation 163
6.1 Introduction into numerical option valuation . . . . . . . . . . . . 164
6.1.1 Integrals and Fourier cosine series . . . . . . . . . . . . . . 164
6.1.2 Density approximation via Fourier cosine expansion . . . . 165
6.2 Pricing European options by the COS method . . . . . . . . . . . 169
6.2.1 Payoff coefficients . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2.2 The option Greeks . . . . . . . . . . . . . . . . . . . . . . . 173
6.2.3 Error analysis COS method . . . . . . . . . . . . . . . . . . 174
6.2.4 Choice of integration range . . . . . . . . . . . . . . . . . . 177
6.3 Numerical COS method results . . . . . . . . . . . . . . . . . . . . 182
6.3.1 Geometric Brownian Motion . . . . . . . . . . . . . . . . . 183
6.3.2 CGMY and VG processes . . . . . . . . . . . . . . . . . . . 184
6.3.3 Discussion about option pricing . . . . . . . . . . . . . . . . 187
6.4 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7 Multidimensionality, Change of Measure, Affine Processes 193
7.1 Preliminaries for multi-D SDE systems . . . . . . . . . . . . . . . . 193
7.1.1 The Cholesky decomposition . . . . . . . . . . . . . . . . . 194
7.1.2 Multi-D asset price processes . . . . . . . . . . . . . . . . . 197
7.1.3 Itô’s lemma for vector processes . . . . . . . . . . . . . . . . 198
7.1.4 Multi-dimensional Feynman-Kac theorem . . . . . . . . . . 200
7.2 Changing measures and the Girsanov theorem . . . . . . . . . . . . 201
7.2.1 The Radon-Nikodym derivative . . . . . . . . . . . . . . . . 202
7.2.2 Change of numéraire examples . . . . . . . . . . . . . . . . 204
7.2.3 From P to Q in the Black-Scholes model . . . . . . . . . . . 206
7.3 Affine processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.3.1 Affine diffusion processes . . . . . . . . . . . . . . . . . . . 211
7.3.2 Affine jump diffusion processes . . . . . . . . . . . . . . . . 216
7.3.3 Affine jump diffusion process and PIDE . . . . . . . . . . . 217
7.4 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8 Stochastic Volatility Models 223
8.1 Introduction into stochastic volatility models . . . . . . . . . . . . 224
8.1.1 The Schöbel-Zhu stochastic volatility model . . . . . . . . . 224
8.1.2 The CIR process for the variance . . . . . . . . . . . . . . . 225
8.2 The Heston stochastic volatility model . . . . . . . . . . . . . . . . 231
8.2.1 The Heston option pricing partial differential equation . . . 233
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8.2.2 Parameter study for implied volatility skew and smile . . . 236
8.2.3 Heston model calibration . . . . . . . . . . . . . . . . . . . 238
8.3 The Heston SV discounted characteristic function . . . . . . . . . . 242
8.3.1 Stochastic volatility as an affine diffusion process . . . . . . 242
8.3.2 Derivation of Heston SV characteristic function . . . . . . . 244
8.4 Numerical solution of Heston PDE . . . . . . . . . . . . . . . . . . 247
8.4.1 The COS method for the Heston model . . . . . . . . . . . 248
8.4.2 The Heston model with piecewise constant parameters . . . 250
8.4.3 The Bates model . . . . . . . . . . . . . . . . . . . . . . . . 251
8.5 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
9 Monte Carlo Simulation 257
9.1 Monte Carlo basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.1.1 Monte Carlo integration . . . . . . . . . . . . . . . . . . . . 260
9.1.2 Path simulation of stochastic differential equations . . . . . 265
9.2 Stochastic Euler and Milstein schemes . . . . . . . . . . . . . . . . 266
9.2.1 Euler scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 266
9.2.2 Milstein scheme: detailed derivation . . . . . . . . . . . . . 269
9.3 Simulation of the CIR process . . . . . . . . . . . . . . . . . . . . . 274
9.3.1 Challenges with standard discretization schemes . . . . . . 274
9.3.2 Taylor-based simulation of the CIR process . . . . . . . . . 276
9.3.3 Exact simulation of the CIR model . . . . . . . . . . . . . . 278
9.3.4 The Quadratic Exponential scheme . . . . . . . . . . . . . . 279
9.4 Monte Carlo scheme for the Heston model . . . . . . . . . . . . . . 283
9.4.1 Example of conditional sampling and integrated
variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
9.4.2 The integrated CIR process and conditional sampling . . . 285
9.4.3 Almost exact simulation of the Heston model . . . . . . . . 288
9.4.4 Improvements of Monte Carlo simulation . . . . . . . . . . 292
9.5 Computation of Monte Carlo Greeks . . . . . . . . . . . . . . . . . 294
9.5.1 Finite differences . . . . . . . . . . . . . . . . . . . . . . . . 295
9.5.2 Pathwise sensitivities . . . . . . . . . . . . . . . . . . . . . . 297
9.5.3 Likelihood ratio method . . . . . . . . . . . . . . . . . . . . 301
9.6 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
10 Forward Start Options; Stochastic Local Volatility Model 309
10.1 Forward start options . . . . . . . . . . . . . . . . . . . . . . . . . 309
10.1.1 Introduction into forward start options . . . . . . . . . . . . 310
10.1.2 Pricing under the Black-Scholes model . . . . . . . . . . . . 311
10.1.3 Pricing under the Heston model . . . . . . . . . . . . . . . 314
10.1.4 Local versus stochastic volatility model . . . . . . . . . . . 316
10.2 Introduction into stochastic-local volatility model . . . . . . . . . . 319
10.2.1 Specifying the local volatility . . . . . . . . . . . . . . . . . 320
10.2.2 Monte Carlo approximation of SLV expectation . . . . . . . 327
10.2.3 Monte Carlo AES scheme for SLV model . . . . . . . . . . 330
10.3 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
11 Short-Rate Models 339
11.1 Introduction to interest rates . . . . . . . . . . . . . . . . . . . . . 339
11.1.1 Bond securities, notional . . . . . . . . . . . . . . . . . . . . 340
11.1.2 Fixed-rate bond . . . . . . . . . . . . . . . . . . . . . . . . 341
11.2 Interest rates in the Heath-Jarrow-Morton framework . . . . . . . 343
11.2.1 The HJM framework . . . . . . . . . . . . . . . . . . . . . . 343
11.2.2 Short-rate dynamics under the HJM framework . . . . . . . 347
11.2.3 The Hull-White dynamics in the HJM framework . . . . . . 349
11.3 The Hull-White model . . . . . . . . . . . . . . . . . . . . . . . . . 352
11.3.1 The solution of the Hull-White SDE . . . . . . . . . . . . . 352
11.3.2 The HW model characteristic function . . . . . . . . . . . . 353
11.3.3 The CIR model under the HJM framework . . . . . . . . . 356
11.4 The HJM model under the T-forward measure . . . . . . . . . . . 359
11.4.1 The Hull-White dynamics under the T-forward measure . . 360
11.4.2 Options on zero-coupon bonds under Hull-White model . . 362
11.5 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
12 Interest Rate Derivatives and Valuation Adjustments 367
12.1 Basic interest rate derivatives and the Libor rate . . . . . . . . . . 368
12.1.1 Libor rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
12.1.2 Forward rate agreement . . . . . . . . . . . . . . . . . . . . 370
12.1.3 Floating rate note . . . . . . . . . . . . . . . . . . . . . . . 371
12.1.4 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
12.1.5 How to construct a yield curve . . . . . . . . . . . . . . . . 375
12.2 More interest rate derivatives . . . . . . . . . . . . . . . . . . . . . 378
12.2.1 Caps and floors . . . . . . . . . . . . . . . . . . . . . . . . . 378
12.2.2 European swaptions . . . . . . . . . . . . . . . . . . . . . . 383
12.3 Credit Valuation Adjustment and Risk Management . . . . . . . . 386
12.3.1 Unilateral Credit Value Adjustment . . . . . . . . . . . . . 392
12.3.2 Approximations in the calculation of CVA . . . . . . . . . . 395
12.3.3 Bilateral Credit Value Adjustment (BCVA) . . . . . . . . . 396
12.3.4 Exposure reduction by netting . . . . . . . . . . . . . . . . 397
12.4 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
13 Hybrid Asset Models, Credit Valuation Adjustment 405
13.1 Introduction to affine hybrid asset models . . . . . . . . . . . . . . 406
13.1.1 Black-Scholes Hull-White (BSHW) model . . . . . . . . . . 406
13.1.2 BSHW model and change of measure . . . . . . . . . . . . . 408
13.1.3 Schöbel-Zhu Hull-White (SZHW) model . . . . . . . . . . . 413
13.1.4 Hybrid derivative product . . . . . . . . . . . . . . . . . . . 416
13.2 Hybrid Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . 417
13.2.1 Details of Heston Hull-White hybrid model . . . . . . . . . 418
13.2.2 Approximation for Heston hybrid models . . . . . . . . . . 420
13.2.3 Monte Carlo simulation of hybrid Heston SDEs . . . . . . . 428
13.2.4 Numerical experiment, HHW versus SZHW model . . . . . 431
13.3 CVA exposure profiles and hybrid models . . . . . . . . . . . . . . 433
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13.3.1 CVA and exposure . . . . . . . . . . . . . . . . . . . . . . . 434
13.3.2 European and Bermudan options example . . . . . . . . . . 434
13.4 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
14 Advanced Interest Rate Models and Generalizations 445
14.1 Libor market model . . . . . . . . . . . . . . . . . . . . . . . . . . 446
14.1.1 General Libor market model specifications . . . . . . . . . . 446
14.1.2 Libor market model under the HJM framework . . . . . . . 449
14.2 Lognormal Libor market model . . . . . . . . . . . . . . . . . . . . 451
14.2.1 Change of measure in the LMM . . . . . . . . . . . . . . . . 452
14.2.2 The LMM under the terminal measure . . . . . . . . . . . . 453
14.2.3 The LMM under the spot measure . . . . . . . . . . . . . . 454
14.2.4 Convexity correction . . . . . . . . . . . . . . . . . . . . . . 457
14.3 Parametric local volatility models . . . . . . . . . . . . . . . . . . . 460
14.3.1 Background, motivation . . . . . . . . . . . . . . . . . . . . 460
14.3.2 Constant Elasticity of Variance model (CEV) . . . . . . . . 461
14.3.3 Displaced diffusion model . . . . . . . . . . . . . . . . . . . 467
14.3.4 Stochastic volatility LMM . . . . . . . . . . . . . . . . . . . 470
14.4 Risk management: The impact of a financial crisis . . . . . . . . . 475
14.4.1 Valuation in a negative interest rates environment . . . . . 476
14.4.2 Multiple curves and the Libor rate . . . . . . . . . . . . . . 479
14.4.3 Valuation in a multiple curves setting . . . . . . . . . . . . 484
14.5 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
15 Cross-Currency Models 489
15.1 Introduction into the FX world and trading . . . . . . . . . . . . . 490
15.1.1 FX markets . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
15.1.2 Forward FX contract . . . . . . . . . . . . . . . . . . . . . . 491
15.1.3 Pricing of FX options, the Black-Scholes case . . . . . . . . 493
15.2 Multi-currency FX model with short-rate interest rates . . . . . . . 495
15.2.1 The model with correlated, Gaussian interest rates . . . . . 496
15.2.2 Pricing of FX options . . . . . . . . . . . . . . . . . . . . . 498
15.2.3 Numerical experiment for the FX-HHW model . . . . . . . 505
15.2.4 CVA for FX swaps . . . . . . . . . . . . . . . . . . . . . . . 508
15.3 Multi-currency FX model with interest rate smile . . . . . . . . . . 510
15.3.1 Linearization and forward characteristic function . . . . . . 514
15.3.2 Numerical experiments with the FX-HLMM model . . . . . 517
15.4 Exercise set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
