Mathematics for Finance
An Introduction to Financial Engineering

by Marek Capi´ nski and Tomasz Zastawniak

1. Introduction: A Simple Market Model ...................... 1
1.1 Basic Notions and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 No-ArbitragePrinciple .................................... 5
1.3 One-StepBinomialModel ................................. 7
1.4 RiskandReturn ......................................... 9
1.5 ForwardContracts........................................ 11
1.6 Call and Put Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Managing Risk with Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2. Risk-Free Assets ............................................ 21
2.1 TimeValueofMoney ..................................... 21
2.1.1 Simple Interest..................................... 22
2.1.2 Periodic Compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.3 Streams of Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.4 Continuous Compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.5 How to Compare Compounding Methods . . . . . . . . . . . . . . 35
2.2 MoneyMarket ........................................... 39
2.2.1 Zero-CouponBonds ................................ 39
2.2.2 CouponBonds ..................................... 41
2.2.3 MoneyMarketAccount ............................. 43
3. Risky Assets ................................................ 47
3.1 DynamicsofStockPrices.................................. 47
3.1.1 Return............................................ 49
3.1.2 ExpectedReturn................................... 53
3.2 BinomialTreeModel...................................... 55
3.2.1 Risk-Neutral Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.2 Martingale Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 OtherModels ............................................ 63
3.3.1 TrinomialTreeModel............................... 64
3.3.2 Continuous-TimeLimit ............................. 66
4. Discrete Time Market Models .............................. 73
4.1 StockandMoneyMarketModels........................... 73
4.1.1 Investment Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.2 The Principle of No Arbitrage . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.3 Application to the Binomial Tree Model . . . . . . . . . . . . . . . 81
4.1.4 Fundamental Theorem of Asset Pricing . . . . . . . . . . . . . . . 83
4.2 ExtendedModels......................................... 85
5. Portfolio Management ...................................... 91
5.1 Risk .................................................... 91
5.2 TwoSecurities ........................................... 94
5.2.1 Risk and Expected Return on a Portfolio . . . . . . . . . . . . . . 97
5.3 SeveralSecurities.........................................107
5.3.1 Risk and Expected Return on a Portfolio . . . . . . . . . . . . . . 107
5.3.2 EfficientFrontier ...................................114
5.4 CapitalAssetPricingModel ...............................118
5.4.1 CapitalMarketLine ................................118
5.4.2 BetaFactor........................................120
5.4.3 SecurityMarketLine ...............................122
6. Forward and Futures Contracts ............................. 125
6.1 ForwardContracts........................................125
6.1.1 ForwardPrice......................................126
6.1.2 Value of a Forward Contract . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.2 Futures .................................................134
6.2.1 Pricing............................................136
6.2.2 Hedging with Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7. Options: General Properties ................................ 147
7.1 Definitions...............................................147
7.2 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3 Bounds on Option Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.3.1 European Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3.2 European and American Calls on Non-Dividend Paying
Stock .............................................157
7.3.3 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.4 VariablesDeterminingOptionPrices........................159
7.4.1 European Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.4.2 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.5 Time Value of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8. Option Pricing.............................................. 173
8.1 EuropeanOptions intheBinomialTreeModel ...............174
8.1.1 OneStep..........................................174
8.1.2 TwoSteps.........................................176
8.1.3 General N-StepModel ..............................178
8.1.4 Cox–Ross–RubinsteinFormula .......................180
8.2 AmericanOptions intheBinomialTreeModel ...............181
8.3 Black–ScholesFormula ....................................185
9. Financial Engineering ....................................... 191
9.1 HedgingOptionPositions..................................192
9.1.1 DeltaHedging .....................................192
9.1.2 GreekParameters ..................................197
9.1.3 Applications .......................................199
9.2 HedgingBusinessRisk ....................................201
9.2.1 ValueatRisk ......................................202
9.2.2 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.3 SpeculatingwithDerivatives...............................208
9.3.1 Tools .............................................208
9.3.2 CaseStudy........................................209
10. Variable Interest Rates ..................................... 215
10.1 Maturity-IndependentYields...............................216
10.1.1 Investment inSingleBonds ..........................217
10.1.2 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
10.1.3 PortfoliosofBonds .................................224
10.1.4 Dynamic Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
10.2 GeneralTermStructure ...................................229
10.2.1 ForwardRates .....................................231
10.2.2 MoneyMarketAccount .............................235
11. Stochastic Interest Rates ................................... 237
11.1 BinomialTreeModel......................................238
11.2 Arbitrage Pricing of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
11.2.1 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
11.3 InterestRateDerivativeSecurities..........................253
11.3.1 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
11.3.2 Swaps ............................................255
11.3.3 CapsandFloors....................................258
11.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Solutions ....................................................... 263
Bibliography .................................................... 303
Glossary of Symbols ............................................ 305
Index ........................................................... 307