Option pricing in incomplete markets : modeling based on geometric Lévy processes and minimal entropy martingale measures / Yoshio Miyahara.

Format:

Book

Author/Creator:

Miyahara, Yoshio, 1944-

Series:

Series in quantitative finance ; v. 3.

Series in quantitative finance ; v. 3

Language:

English

Subjects (All):

Options (Finance)--Prices.

Options (Finance).

Stock options.

Physical Description:

xiv, 185 pages : illustrations ; 24 cm.

Place of Publication:

Singapore : World Scientific ; Hackensack, NJ : Distributed by World Scientific Pub., 2012.

Summary:

This volume offers the reader practical methods to compute the option prices in the incomplete asset markets. The [GLP & MEMM] pricing models are clearly introduced, and the properties of these models are discussed in great detail. It is shown that the geometric Lévy process (GLP) is a typical example of the incomplete market, and that the MEMM (minimal entropy Martingale measure) is an extremely powerful pricing measure.

This volume also presents the calibration procedure of the [GLP & MEMM] model that has been widely used in the application of practical problems. Book jacket.

Contents:

1 Basic Concepts in Mathematical Finance 1

1.1 Price Processes 1

1.2 No-arbitrage and Martingale Measures 1

1.3 Complete and Incomplete Markets 2

1.4 Fundamental Theorems 2

1.5 The Black-Scholes Model 3

1.6 Properties of the Black-Scholes Model 4

1.6.1 Distribution of log returns 4

1.6.2 Historical volatility and implied volatility 4

1.7 Generalization of the Black-Scholes Model 5

1.7.1 Geometric Lévy process models 5

1.7.2 Stochastic volatility models 5

Notes 6

2 Lévy Processes and Geometric Lévy Process Models 7

2.1 Lévy Processes 7

2.1.1 Definitions and properties 7

2.1.2 Infinitely divisible distributions 8

2.1.3 Canonical representation of Lévy processes 11

2.2 Geometric Lévy Process Models 12

2.2.1 The geometric Brownian motion model 13

2.2.2 Geometric compound Poisson models 13

2.2.3 Jump-diffusion models 15

2.2.4 Geometric variance gamma models 15

2.2.5 Geometric stable process models 16

2.2.6 Geometric CGMY models 17

2.3 Doléans-Dade Exponential 18

Notes 20

3 Equivalent Martingale Measures 21

3.1 Equivalent Martingale Measure Methods 21

3.2 Equivalent Martingale Measures for Geometric Lévy Processes 22

3.2.1 Candidates for suitable equivalent martingale measure 22

3.3 Methods for Construction of Martingale Measures 23

3.3.1 Variance optimal martingale measure (VOMM) 23

3.3.2 Minimal entropy martingale measure (MEMM) 25

Notes 26

4 Esscher-Transformed Martingale Measures 29

4.1 Esscher Transformation 29

4.2 Esscher-Transformed Martingale Measure for Geometric Lévy Processes 30

4.2.1 Simple-return process and compound-return process 30

4.2.2 Two kinds of Esscher-transformed martingale measure 32

4.3 Existence Theorems of P^(ESSMM) and P̊^(ESSMM) for Geometric Lévy Processes 33

4.3.1 Existence theorem of P(ESSMM) 33

4.3.2 Existence theorem of P̊(ESSMM) 35

4.4 Comparison of P(ESSMM) and P̊(ESSMM) 39

4.5 Other Examples of Esscher-Transformed Martingale Measures 40

Notes 40

5 Minimax Martingale Measures and Minimal Distance Martingale Measures 41

5.1 Utility Function, Duality, and Minimax Martingale Measures 41

5.2 Distance Function Corresponding to Utility Function 42

5.3 Minimal Distance Martingale Measures 45

Notes 45

6 Minimal Distance Martingale Measures for Geometric Lévy Processes 47

6.1 Minimal Distance Problem 47

6.2 The Minimal Variance Equivalent Martingale Measure (MVEMM) 53

6.2.1 Deterministic problem 54

6.2.2 Existence theorem of the MVEMM 58

6.2.3 Generating triplet of Z_t under MVEMM 62

6.3 The Minimal L^q Equivalent Martingale Measure 63

6.3.1 The case of q > 1 64

6.3.2 The case of 0 < q <1 65

6.3.3 The case of q < 0 66

6.4 Minimal Entropy Martingale Measures 67

6.5 Convergence of ML^qEMM to MEMM (as q ↓ 1) 69

Notes 73

7 The [GLP & MEMM] Pricing Model 75

7.1 The Model 75

7.1.1 Sufficient condition for the existence of MEMM 76

7.1.2 Properties of geometric Lévy processes under MEMM 78

7.2 Examples of [GLP & MEMM] Pricing Model 79

7.2.1 Geometric (Brownian motion + compound Poisson) model (or jump-diffusion model) ([GJD & MEMM]) 79

7.2.2 Geometric variance gamma model QGVG & MEMM]) 80

7.2.3 Geometric stable process model ([GSP & MEMM]) 82

7.2.4 Geometric CGMY model ([GCGMY & MEMM]) 84

7.3 Why the Geometric Lévy Process? 85

7.4 Why the MEMM? 85

7.5 Comparison of Equivalent Martingale Measures for Geometric Lévy Processes 88

7.5.1 Corresponding risk process relating to Esscher-transformed MM 88

7.5.2 Integrability condition of Lévy measures for the existence of martingale measures 88

7.5.3 Corresponding utility function 90

7.6 The Explicit Form of Lévy Measure of Z_t under an Equivalent Martingale Measure 91

7.6.1 General form of Lévy measure of Z_t under equivalent martingale measures 91

7.6.2 Geometric variance gamma model 92

7.6.3 Geometric stable process model 94

7.6.4 Geometric CGMY model 94

7.6.5 Merton model (jump-diffusion model) 96

Notes 97

8 Calibration and Fitness Analysis of the [GLP & MEMM] Model 99

8.1 The Physical World and the MEMM World 99

8.1.1 From the physical world to the MEMM world 99

8.1.2 From the MEMM world to the physical world 100

8.1.3 Calibration problem and fitness analysis 101

8.2 Reproducibility of Volatility Smile/Smirk Property of the [GLP & MEMM] Model 101

8.2.1 Implied volatility of the model 102

8.2.2 [Geometric Variance Gamma Process & MEMM] model 103

8.2.3 [Geometric CGMY Process & MEMM] model 103

8.2.4 [Geometric Stable Process & MEMM] model 104

8.3 Calibration of [GLP & MEMM] Pricing Model 105

8.3.1 Pricing error 106

8.3.2 Minimization problem 107

8.3.3 Procedure of calibration 107

8.4 Fitness Analysis 109

8.4.1 Procedure of fitness analysis 109

Notes 110

9 The [GSP & MEMM] Pricing Model 111

9.1 The Physical World and the MEMM World 111

9.1.1 From the physical world to the MEMM world (GSP case) 112

9.1.2 From the MEMM world to the physical world (GSP case) 113

9.2 Calibration by the [GSP & MEMM] Pricing Model 113

9.2.1 Calibration in the physical world 114

9.2.2 Calibration in the MEMM world 114

9.3 Application of the Calibrated Process to Dollar-Yen Currency Options 115

9.3.1 Fact: the implied volatility curve of currency options 115

9.3.2 In-sample analysis 116

9.3.3 Out-of-sample analysis 117

9.3.4 Volatility-based calibration 117

Notes 119

10 The Multi-Dimensional [GLP & MEMM] Pricing Model 121

10.1 Multi-Dimensional Lévy Processes 121

10.2 Multi-Dimensional Geometric Lévy Processes 123

10.3 Esscher MM and MEMM 124

10.3.1 Equivalent martingale measures 125

10.3.2 Esscher martingale measures 125

10.3.3 Minimal entropy martingale measures 126

10.3.4 [GLP & MEMM] pricing model 127

10.4 Application to Portfolio Evaluation 127

10.4.1 Geometric Lévy process portfolio model 127

10.4.2 Risk-sensitive value measure 128

10.4.3 Risk-sensitive evaluation of portfolio 129

10.4.4 Re-balancing portfolios 131

10.5 Risk-Sensitive Evaluation of Growth Rate 132

10.5.1 Risk-sensitive evaluation of RIRR 133

10.5.2 Risk-sensitive evaluation of re-balancing portfolios 135

10.5.3 Risk-sensitive evaluation of a single asset 139.

Notes:

Includes bibliographical references and index.

ISBN:

9781848163478

1848163479

OCLC:

751836206