## A Benchmark Approach to Quantitative FinanceIn recent years products based on ?nancial derivatives have become an ind- pensabletoolforriskmanagersandinvestors. Insuranceproductshavebecome part of almost every personal and business portfolio. The management of - tual and pension funds has gained in importance for most individuals. Banks, insurance companies and other corporations are increasingly using ?nancial and insurance instruments for the active management of risk. An increasing range of securities allows risks to be hedged in a way that can be closely t- lored to the speci?c needs of particular investors and companies. The ability to handle e?ciently and exploit successfully the opportunities arising from modern quantitative methods is now a key factor that di?erentiates market participants in both the ?nance and insurance ?elds. For these reasons it is important that ?nancial institutions, insurance companies and corporations develop expertise in the area of quantitative ?nance, where many of the as- ciated quantitative methods and technologies emerge. This book aims to provide an introduction to quantitative ?nance. More precisely, it presents an introduction to the mathematical framework typically usedin?nancialmodeling,derivativepricing,portfolioselectionandriskm- agement. It o?ers a uni?ed approach to risk and performance management by using the benchmark approach, which is di?erent to the prevailing paradigm and will be described in a systematic and rigorous manner. This approach uses the growth optimal portfolio as numeraire and the real world probability measure as pricing measure. |

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worst book I've ever seen, misleading, easily getting distracted with too many useless notations, this is not a student book for sure.

### Contents

1 | |

11 | |

13 Moments of Random Variables | 22 |

14 Joint Distributions and Random Vectors | 39 |

15 Copulas | 50 |

16 Exercises for Chapter 1 | 53 |

Statistical Methods | 55 |

22 Confidence Intervals | 63 |

85 European Put Option | 295 |

86 Hedge Simulation | 298 |

87 Squared Bessel Processes | 304 |

88 Exercises for Chapter 8 | 317 |

Various Approaches to Asset Pricing | 319 |

92 Actuarial Pricing | 329 |

93 Capital Asset Pricing Model | 332 |

94 Risk Neutral Pricing | 337 |

23 Estimation Methods | 70 |

24 Maximum Likelihood Estimation | 78 |

25 Normal Variance Mixture Models | 81 |

26 Distribution of Index LogReturns Estimation of LogReturns | 84 |

27 Convergence of Random Sequences | 92 |

28 Exercises for Chapter 2 | 98 |

Modeling via Stochastic Processes | 99 |

32 Certain Classes of Stochastic Processes | 106 |

33 Discrete Time Markov Chains | 110 |

34 Continuous Time Markov Chains | 113 |

35 Poisson Processes | 120 |

36 Levy Processes | 126 |

37 Insurance Risk Modeling | 128 |

38 Exercises for Chapter 3 | 131 |

Diffusion Processes | 133 |

42 Examples for Continuous Markov Processes | 136 |

43 Diffusion Processes | 141 |

44 Kolmogorov Equations | 145 |

45 Diffusions with Stationary Densities | 154 |

46 MultiDimensional Diffusion Processes | 159 |

47 Exercises for Chapter 4 | 161 |

Martingales and Stochastic Integrals | 163 |

52 Quadratic Variation and Covariation | 174 |

53 Gains from Trade as Stochastic Integral | 187 |

54 Ito Integral for Wiener Processes | 193 |

55 Stochastic Integrals for Semimartingales | 197 |

56 Exercises for Chapter 5 | 203 |

The Ito Formula | 204 |

62 Multivariate Ito Formula | 209 |

63 Some Applications of the ItoFormula | 213 |

64 Extensions of the Ito Formula | 222 |

65 Levys Theorem | 227 |

66 A Proof of the Ito Formula | 230 |

67 Exercises for Chapter 6 | 234 |

Stochastic Differential Equations | 237 |

72 Linear SDE with Additive Noise | 241 |

73 Linear SDE with Multiplicative Noise | 243 |

74 Vector Stochastic Differential Equations | 246 |

75 Constructing Explicit Solutions of SDEs | 248 |

76 Jump Diffusions | 254 |

77 Existence and Uniqueness | 261 |

78 Markovian Solutions of SDEs | 272 |

79 Exercises for Chapter 7 | 275 |

Introduction to Option Pricing | 276 |

82 Options under the BlackScholes Model | 281 |

83 The BlackScholes Formula | 288 |

84 Sensitivities for European Call Option | 290 |

95 Girsanov Transformation and Bayes Rule | 345 |

96 Change of Numeraire | 350 |

97 FeynmanKac Formula | 356 |

98 Exercises for Chapter 9 | 364 |

Continuous Financial Markets | 367 |

102 Growth Optimal Portfolio | 372 |

103 Supermartingale Property | 375 |

104 Real World Pricing | 378 |

105 GOP as Best Performing Portfolio | 386 |

106 Diversified Portfolios in CFMs | 389 |

107 Exercises for Chapter 10 | 402 |

Portfolio Optimization | 403 |

111 Locally Optimal Portfolios | 404 |

112 Market Portfolio and GOP | 415 |

113 Expected Utility Maximization | 419 |

114 Pricing Nonreplicable Payoffs | 427 |

115 Hedging | 430 |

116 Exercises for Chapter 11 | 437 |

Modeling Stochastic Volatility | 438 |

122 Modified CEV Model | 444 |

123 Local Volatility Models | 461 |

124 Stochastic Volatility Models | 472 |

125 Exercises for Chapter 12 | 481 |

Minimal Market Model | 483 |

132 Stylized Minimal Market Model | 488 |

133 Derivatives under the MMM | 496 |

134 MMM with Random Scaling | 502 |

135 Exercises for Chapter 13 | 511 |

Markets with Event Risk | 512 |

142 Diversified Portfolios | 523 |

143 MeanVariance Portfolio Optimization | 532 |

144 Real World Pricing for Two Market Models | 536 |

145 Exercises for Chapter 14 | 549 |

Numerical Methods | 551 |

152 Scenario Simulation | 558 |

153 Classical Monte Carlo Method | 570 |

154 Monte Carlo Simulation for SDEs | 578 |

155 Variance Reduction of Functionals of SDEs | 587 |

156 Tree Methods | 591 |

157 Finite Difference Methods | 600 |

158 Exercises for Chapter 15 | 611 |

Solutions for Exercises | 614 |

667 | |

Author Index | 682 |

689 | |

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### Common terms and phrases

approximate asset price benchmark approach conditional expectation converges corresponding covariation deﬁned Deﬁnition denote diﬀerent diﬀerential diﬀusion coeﬃcient diﬀusion process discounted GOP distributed random variable drift eﬀect equation equivalent risk neutral estimate European call option Feynman-Kac formula ﬁnance ﬁnite ﬁrst geometric Brownian motion given hedge ICAPM implied volatility initial value Itˆo integral Ito formula jump locally optimal portfolio log-returns Markov martingale matrix neutral probability measure nonnegative Note numeraire obtain option price parameter payoﬀ Platen Poisson process primary security account probability measure quadratic variation real world pricing risk neutral risk neutral probability satisﬁes savings account Sect sequence short rate solution squared Bessel process standard Wiener process stationary density stochastic process stochastic volatility strictly positive portfolio supermartingale Theorem transition density underlying security variance vector Wiener process zero coupon bond