Table of Contents
Foreword xi
Preface xiii
Addendum: A Path to Economic Renaissance xv
PART I BUILDING BLOCKS
1 Interest Rate 3
1-1 Measuring Time 3
1-2 Interest Rate 4
1-2.1 Gross Interest Rate 4
1-2.2 Compounding. Compound Interest Rate 5
1-2.3 Conversion Formula 6
1-2.4 Annualization 6
1-3 Discounting 7
1-3.1 Present Value 7
1-3.2 Discount Rate and Required Return 8
1-4 Problems 8
2 Classical Investment Rules 11
2-1 Rate of Return. Time of Return 11
2-1.1 Gross Rate of Return (ROR) 11
2-1.2 Time of Return (TOR) 11
2-2 Net Present Value (NPV) 12
2-3 Internal Rate of Return (IRR) 13
2-4 Other Investment Rules 14
2-5 Further Reading 15
2-6 Problems 15
3 Fixed Income 19
3-1 Financial Markets 19
3-1.1 Securities and Portfolios 19
3-1.2 Value and Price 19
3-1.3 Financial Markets and Short-selling 20
3-1.4 Arbitrage 21
3-1.5 Price of a Portfolio 22
3-2 Bonds 23
3-2.1 Treasury Bonds 23
3-2.2 Zero-Coupon Bonds 23
3-2.3 Bond Markets 23
3-3 Yield 24
3-3.1 Yield to Maturity 24
3-3.2 Yield Curve 24
3-3.3 Approximate Valuation 26
3-4 Zero-Coupon Yield Curve. Arbitrage Price 27
3-4.1 Zero-Coupon Rate Curve 27
3-4.2 Arbitrage Price of a Bond 28
3-4.3 Zero-Coupon Rate Calculation by Inference: the ‘Bootstrapping’ Method 29
3-5 Further Reading 30
3-6 Problems 30
4 Portfolio Theory 35
4-1 Risk and Return of an Asset 35
4-1.1 Average Return and Volatility 35
4-1.2 Risk-free Asset. Sharpe Ratio 37
4-2 Risk and Return of a Portfolio 38
4-2.1 Portfolio Valuation 38
4-2.2 Return of a Portfolio 38
4-2.3 Volatility of a Portfolio 39
4-3 Gains of Diversification. Portfolio Optimization 41
4-4 Capital Asset Pricing Model 43
4-5 Further Reading 44
4-6 Problems 44
PART II FIRST STEPS IN EQUITY DERIVATIVES
5 Equity Derivatives 49
5-1 Introduction 49
5-2 Forward Contracts 50
5-2.1 Payoff 51
5-2.2 Arbitrage Price 51
5-2.3 Forward Price 53
5-2.4 Impact of Dividends 53
5-2.4.1 Single Cash Dividend 54
5-2.4.2 Single Proportional Dividend 55
5-3 ‘Plain Vanilla’ Options 55
5-3.1 Payoff 56
5-3.2 Option Value 57
5-3.3 Put-Call Parity 57
5-3.4 Option Strategies 58
5-3.4.1 Leverage 58
5-3.4.2 Covered Call 59
5-3.4.3 Straddle 60
5-3.4.4 Butterfly 61
5-4 Further Reading 61
5-5 Problems 61
6 The Binomial Model 65
6-1 One-Step Binomial Model 65
6-1.1 An Example 65
6-1.2 General Formulas 67
6-2 Multi-Step Binomial Trees 67
6-3 Binomial Valuation Algorithm 69
6-4 Further Reading 70
6-5 Problems 70
7 The Lognormal Model 75
7-1 Fair Value 75
7-1.1 Probability Distribution of ST 75
7-1.2 Discount Rate 76
7-2 Closed-Form Formulas for European Options 76
7-3 Monte-Carlo Method 78
7-4 Further Reading 78
7-5 Problems 78
8 Dynamic Hedging 83
8-1 Hedging Option Risks 83
8-1.1 Delta-hedging 84
8-1.2 Other Risk Parameters: the ‘Greeks’ 85
8-1.3 Hedging the Greeks 86
8-2 The P&L of Delta-hedged Options 86
8-2.1 Gamma 86
8-2.2 Theta 87
8-2.3 Option Trading P&L Proxy 88
8-3 Further Reading 89
8-4 Problems 89
PART III ADVANCED MODELS AND TECHNIQUES
9 Models for Asset Prices in Continuous Time 95
9-1 Continuously Compounded Interest Rate 95
9-1.1 Fractional Interest Rate 95
9-1.2 Continuous Interest Rate 96
9-2 Introduction to Models for the Behavior of Asset Prices in Continuous Time 96
9-3 Introduction to Stochastic Processes 97
9-3.1 Standard Brownian Motion 98
9-3.2 Generalized Brownian Motion 99
9-3.3 Geometric Brownian Motion 100
9-4 Introduction to Stochastic Calculus 101
9-4.1 Ito Process 101
9-4.2 The Ito-Doeblin Theorem 101
9-4.3 Heuristic Proof of the Ito-Doeblin Theorem 102
9-5 Further Reading 103
9-6 Problems 103
10 The Black-Scholes Model 109
10-1 The Black-Scholes Partial Differential Equation 109
10-1.1 Ito-Doeblin Theorem for the Derivative’s Value 110
10-1.2 Riskless Hedged Portfolio 111
10-1.3 Arbitrage Argument 111
10-1.4 Partial Differential Equation 111
10-1.5 Continuous Delta-hedging 112
10-2 The Black-Scholes Formulas for European Vanilla Options 112
10-3 Volatility 113
10-3.1 Historical Volatility 113
10-3.2 Implied Volatility 114
10-4 Further Reading 114
10-5 Problems 114
11 Volatility Trading 117
11-1 Implied and Realized Volatilities 117
11-1.1 Realized Volatility 117
11-1.2 Implied Volatility 117
11-2 Volatility Trading Using Options 118
11-3 Volatility Trading Using Variance Swaps 119
11-3.1 Variance Swap Payoff 120
11-3.2 Variance Swap Market 120
11-3.3 Variance Swap Hedging and Pricing 120
11-4 Further Reading 123
11-5 Problems 123
12 Exotic Derivatives 127
12-1 Single-Asset Exotics 127
12-1.1 Digital Options 127
12-1.2 Asian Options 127
12-1.3 Barrier Options 128
12-1.4 Lookback Options 129
12-1.5 Forward Start Options 129
12-1.6 Cliquet Options 130
12-1.7 Structured Products 130
12-2 Multi-Asset Exotics 131
12-2.1 Spread Options 131
12-2.2 Basket Options 132
12-2.3 Worst-of and Best-of Options 132
12-2.4 Quanto Options 133
12-2.5 Structured Products 133
12-2.6 Dispersion and Correlation Trading 134
12-3 Beyond Black-Scholes 134
12-3.1 Black-Scholes on Multiple Assets 134
12-3.2 Fitting the Smile 135
12-3.2.1 Stochastic Volatility 135
12-3.2.2 Jumps 135
12-3.2.3 Local Volatility 136
12-3.3 Discrete Hedging and Transaction Costs 137
12-3.3.1 Discrete Hedging 137
12-3.3.2 Transaction Costs 137
12-3.4 Correlation Modeling 138
12-4 Further Reading 139
12-5 Problems 139
SOLUTIONS
Problem Solutions 145
Chapter 1 145
Chapter 2 148
Chapter 3 151
Chapter 4 156
Chapter 5 161
Chapter 6 167
Chapter 7 172
Chapter 8 182
Chapter 9 186
Chapter 10 194
Chapter 11 197
Chapter 12 199
APPENDICES
A Probability Review 205
A-1 States of Nature. Random Variables. Events 205
A-2 Probability. Expectation. Variance 206
A-3 Distribution. Normal Distribution 207
A-4 Independence. Correlation 209
A-5 Probability Formulas 210
A-6 Further Reading 211
B Calculus Review 213
B-1 Functions of Two Variables x and y 213
B-2 Taylor Expansions 214
C Finance Formulas 217
C-1 Rates and Yields 217
C-2 Present Value. Arbitrage Price 217
C-3 Forward Contracts 217
C-4 Options 218
C-5 Volatility 219
C-6 Stochastic Processes. Stochastic Calculus 220
C-7 Greeks etc. 220
Index 223
