N.H. Bingham and Rüdiger Kiesel
Risk-Neutral
Valuation
Pricing and Hedging of Financial Derivatives
W) Springer
Contents
1.
Derivative Background
1.1 Financial Markets and Instruments
1.1.1 Derivative Instruments
1.1.2 Underlying securities
1.1.3 Markets
1.1.4 Types of Traders
1.1.5 Modelling Assumptions
1.2 Arbitrage
1.3 Arbitrage Relationships
1.3.1 Fundamental Determinants of Option Values
1.3.2 The Put-Call Parity
1.3.3 Arbitrage Bounds
1.4 Single-Period Market Models
Exercises
1
2
2
4
5
6
7
8
11
11
13
16
20
30
2.
Probability Background
2.1 Measure
2.2 Integral
2.3 Probability
2.4 Equivalent Measures and Radon-Nikodym Derivatives
2.5 Conditional Expectations
2.6 Properties of Conditional Expectation
2.7 Modes of Convergence
2.8 Convolution and Characteristic Functions
2.9 The Central Limit Theorem
Exercises
33
34
38
41
47
48
51
53
56
60
63
3.
Stochastic Processes in Discrete Time
3.1 Information and Filtrations
3.2 Discrete-Parameter Stochastic Processes
3.3 Discrete-Parameter Martingales
3.3.1 Definition and Simple Properties
3.3.2 Martingale Convergence
3.3.3 Doob Decomposition
67
67
68
70
70
72
73
xii
Contents
3.4 Martingale Transforms
3.5 Stopping Times and Optional Stopping
3.6 The Snell Envelope
Exercises
73
75
78
80
4.
Mathematical Finance in Discrete Time
4.1 The Model
4.2 Existence of Equivalent Martingale Measures
4.2.1 The No-Arbitrage Condition
4.2.2 Risk-Neutral Pricing
4.3 Complete Markets
4.4 Risk-Neutral Valuation
4.5 The Cox-Ross-Rubinstein Model
4.5.1 Model Structure
4.5.2 Risk-Neutral Pricing
4.5.3 Hedging
4.5.4 Comparison With the General Arbitrage Bounds
4.6 Binomial Approximations
4.6.1 Model Structure
4.6.2 The Black-Scholes Option Pricing Formula
4.6.3 Further Limiting Models
4.7 Multifactor Models
4.7.1 Extended Binomial Model
4.7.2 Multinomial Models
4.8 Further Contingent Claim Valuation in Discrete Time
4.8.1 American Options
4.8.2 Barrier Options
4.8.3 Lookback Options
4.8.4 A Three-Period Example
Exercises
83
83
87
87
93
96
100
103
103
105
108
110
Ill
112
113
118
121
121
122
123
123
126
127
128
130
5.
S t o c h a s t i c P r o c e s s e s in Continuous Time
5.1 Filtrations; Finite-Dimensional Distributions
5.2 Classes of Processes
5.3 Brownian Motion
5.4 Quadratic Variation of Brownian Motion
5.5 Stochastic Integrals; Ito Calculus
5.6 Itö's Lemma
5.6.1 Geometric Brownian Motion
5.7 Stochastic Differential Equations
5.8 Stochastic Calculus for Black-Scholes Models
5.9 Weak Convergence of Stochastic Processes
5.9.1 The Spaces Cd and Dd
5.9.2 Definition and Motivation
5.9.3 Basic Theorems of Weak Convergence
133
133
134
138
141
144
149
152
154
157
161
161
162
164
Contents
5.9.4
Exercises
xiii
Weak Convergence Results for Stochastic Integrals. . . . 165
167
6.
Mathematical Finance in Continuous Time
6.1 Continuous-time Financial Market Models
6.1.1 The Financial Market Model
6.1.2 Equivalent Martingale Measures
6.1.3 Risk-neutral Pricing
6.1.4 Changes of Numeraire
6.2 The Generalised Black-Scholes Model
6.2.1 The Model
6.2.2 Pricing and Hedging Contingent Claims
6.2.3 The Greeks
6.2.4 Volatility
6.3 Further contingent claim valuation
6.3.1 American Options
6.3.2 Asian Options
6.3.3 Barrier Options
6.3.4 Lookback Options
6.3.5 Binary Options
6.4 Discrete- vs. Continuous-Time Models
6.4.1 Convergence Reconsidered
6.4.2 Finite Market Approximations
6.4.3 Examples of Finite Market Approximations
6.5 Further Applications
6.5.1 Futures Markets
6.5.2 Currency Markets
Exercises
171
171
171
174
177
180
184
184
192
195
197
199
199
202
204
207
210
211
211
212
214
221
221
224
226
7.
Incomplete Markets
7.1 Pricing in Incomplete Markets
7.1.1 A General Option Pricing Formula
7.1.2 The Esscher Measure
7.2 Hedging in Incomplete Markets
7.2.1 Variance Minimising Hedging
7.2.2 Risk-Minimising Hedging
7.3 Stochastic Volatility Models
229
229
229
232
235
235
239
241
8.
Interest R a t e T h e o r y
8.1 The Bond Market
8.1.1 The Term Structure of Interest Rates
8.1.2 Mathematical Modelling
8.1.3 Bond Pricing,
8.2 Short Rate Models
8.2.1 The Term Structure Equation
245
246
246
247
252
254
255
xiv
Contents
8.2.2 Martingale Modelling
8.2.3 Parameter Estimation
8.3 Heath-Jarrow-Morton Methodology
8.3.1 The Heath-Jarrow-Morton Model Class
8.3.2 Forward Risk-Neutral Martingale Measures
8.3.3 Completeness
8.4 Pricing and Hedging Contingent Claims
8.4.1 Short Rate Models
8.4.2 Gaussian HJM Framework
8.4.3 Swaps
8.4.4 Caps
Exercises
256
260
262
262
265
267
268
268
269
271
272
274
A. Hubert Space
277
B.
Projections and Conditional Expectations
279
C.
The S e p a r a t i n g Hyperplane T h e o r e m
281
Bibliography
283
Index
293