LECTURE NOTES IN MATHEMATICAL
FINANCE
X. Sheldon Lin
Department of Statistics & Actuarial Science
University of Iowa
Iowa City, IA 52242
Phone: (319)335-0730
Fax: (319)335-3017
Email:
Comments are welcome!
c X. Sheldon Lin, 1996.
1
,Contents
I Discrete-Time Finance Models 4
1 Basic Concepts and One Time-Period Models 5
1.1 The Basic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Characterisation of No-Arbitrage Strategies . . . . . . . . . . . . . . . . . 8
1.4 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Risk Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Discrete-Time Stochastic Processes and Lattice Models 16
2.1 Discrete-Time Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 General Lattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 No-Arbitrage Valuation 28
3.1 No-Arbitrage Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Risk-Neutral Probability Measures . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Binomial Models of Option Pricing . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Binomial Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6 Multinomial/Multifactor Interest Rate Models . . . . . . . . . . . . . . . . 46
2
,II Continuous-Time Finance Models 51
4 Stochastic Calculus 52
4.1 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Wiener Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Re ection Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Stochastic(Ito) Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Stochastic Di erential Equations and Ito's Lemma . . . . . . . . . . . . . . 67
4.6 Feynman-Kac Formula and Other Applications . . . . . . . . . . . . . . . . 72
4.7 Option Pricing: Dynamic Hedging Approach . . . . . . . . . . . . . . . . . 75
4.8 Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.9 Multi-Dimensional Ito Processes . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Continuous-Time Finance Models 89
5.1 Security Markets and Valuation . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Digital and Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3 Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4 Swaps and Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A Probability Theory 106
B Functional Analysis 109
3
, Part I
Discrete-Time Finance Models
4
FINANCE
X. Sheldon Lin
Department of Statistics & Actuarial Science
University of Iowa
Iowa City, IA 52242
Phone: (319)335-0730
Fax: (319)335-3017
Email:
Comments are welcome!
c X. Sheldon Lin, 1996.
1
,Contents
I Discrete-Time Finance Models 4
1 Basic Concepts and One Time-Period Models 5
1.1 The Basic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Characterisation of No-Arbitrage Strategies . . . . . . . . . . . . . . . . . 8
1.4 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Risk Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Discrete-Time Stochastic Processes and Lattice Models 16
2.1 Discrete-Time Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 General Lattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 No-Arbitrage Valuation 28
3.1 No-Arbitrage Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Risk-Neutral Probability Measures . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Binomial Models of Option Pricing . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Binomial Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6 Multinomial/Multifactor Interest Rate Models . . . . . . . . . . . . . . . . 46
2
,II Continuous-Time Finance Models 51
4 Stochastic Calculus 52
4.1 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Wiener Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Re ection Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Stochastic(Ito) Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Stochastic Di erential Equations and Ito's Lemma . . . . . . . . . . . . . . 67
4.6 Feynman-Kac Formula and Other Applications . . . . . . . . . . . . . . . . 72
4.7 Option Pricing: Dynamic Hedging Approach . . . . . . . . . . . . . . . . . 75
4.8 Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.9 Multi-Dimensional Ito Processes . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Continuous-Time Finance Models 89
5.1 Security Markets and Valuation . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Digital and Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3 Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4 Swaps and Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A Probability Theory 106
B Functional Analysis 109
3
, Part I
Discrete-Time Finance Models
4
