Introduction |
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ix | |
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Arbitrage Pricing Theory: The One-Period Model |
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1 | (20) |
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2 | (6) |
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Security-Space Diagram: A Geometric Interpretation of Theorem 1.1 |
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8 | (3) |
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11 | (2) |
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13 | (1) |
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Complete and Incomplete Markets |
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14 | (2) |
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The One-Period Trinomial Model |
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16 | (2) |
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18 | (3) |
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References and Further Reading |
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19 | (2) |
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The Binomial Option Pricing Model |
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21 | (20) |
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Recursion Relation for Pricing Contingent Claims |
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22 | (2) |
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Delta-Hedging and the Replicating Portfolio |
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24 | (2) |
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Pricing European Puts and Calls |
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26 | (2) |
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27 | (1) |
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27 | (1) |
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28 | (1) |
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Relation Between the Parameters of the Tree and the Stock Price Fluctuations |
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28 | (6) |
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Calibration of the Volatility Parameter |
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31 | (1) |
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32 | (1) |
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Implementation of Binomial Trees |
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33 | (1) |
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The Limit of dt → 0: Log-Normal Approximation |
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34 | (1) |
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The Black-Scholes Formula |
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35 | (6) |
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References and Further Reading |
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39 | (2) |
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Analysis of the Black-Scholes Formula |
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41 | (16) |
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42 | (3) |
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44 | (1) |
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45 | (3) |
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Gamma: The Convexity Factor |
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48 | (3) |
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Theta: The Time-Decay Factor |
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51 | (3) |
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The Binomial Model as a Finite-Difference Scheme for the Black-Scholes Equation |
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54 | (3) |
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References and Further Reading |
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55 | (2) |
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Refinements of the Binomial Model |
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57 | (20) |
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Term-Structure of Interest Rates |
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57 | (6) |
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Constructing a Risk-Neutral Measure with Time-Dependent Volatility |
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63 | (3) |
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Deriving a Volatility Term-Structure from Option Market Data |
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66 | (4) |
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Underlying Assets That Pay Dividends |
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70 | (3) |
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Futures Contracts as the Underlying Security |
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73 | (2) |
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Valuation of a Stream of Uncertain Cash Flows |
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75 | (2) |
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References and Further Reading |
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76 | (1) |
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American-Style Options, Early Exercise, and Time-Optionality |
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77 | (16) |
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77 | (1) |
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78 | (2) |
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Pricing American Options Using the Binomial Model: The Dynamic Programming Equation |
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80 | (2) |
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82 | (1) |
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Characterization of the Solution for dt << 1: Free-Boundary Problem for the Black-Scholes Equation |
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82 | (11) |
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References and Further Reading |
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88 | (1) |
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A PDE Approach to the Free-Boundary Condition |
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89 | (1) |
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A Proof of the Free Boundary Condition |
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90 | (3) |
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Trinomial Model and Finite-Difference Schemes |
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93 | (14) |
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93 | (2) |
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95 | (1) |
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96 | (4) |
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``Tree-Trimming'' and Far-Field Boundary Conditions |
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100 | (3) |
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103 | (4) |
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References and Further Reading |
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106 | (1) |
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Brownian Motion and Ito Calculus |
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107 | (20) |
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107 | (2) |
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Elementary Properties of Brownian Paths |
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109 | (2) |
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111 | (6) |
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117 | (3) |
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Ito Processes and Ito Calculus |
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120 | (7) |
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References and Further Reading |
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122 | (1) |
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Properties of the Ito Integral |
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123 | (4) |
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Introduction to Exotic Options: Digital and Barrier Options |
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127 | (34) |
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128 | (11) |
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128 | (7) |
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135 | (4) |
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139 | (7) |
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Pricing Barrier Options Using Trees or Lattices |
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141 | (1) |
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142 | (3) |
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145 | (1) |
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146 | (15) |
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147 | (1) |
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148 | (2) |
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150 | (1) |
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References and Further Reading |
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150 | (1) |
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Proofs of Lemmas 8.1 and 8.2 |
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151 | (1) |
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A Consequence of the Invariance of Brownian Motion Under Reflections |
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151 | (2) |
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153 | (2) |
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Closed-Form Solutions for Double-Barrier Options |
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155 | (1) |
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Exit Probabilities of a Brownian Trajectory from a Strip -B < Z < A |
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155 | (3) |
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Applications to Pricing Barrier Options |
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158 | (3) |
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Ito Processes, Continuous-Time Martingales, and Girsanov's Theorem |
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161 | (10) |
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Martingales and Doob-Meyer Decomposition |
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161 | (2) |
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163 | (2) |
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165 | (6) |
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References and Further Reading |
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168 | (1) |
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169 | (2) |
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Continuous-Time Finance: An Introduction |
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171 | (12) |
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171 | (2) |
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173 | (3) |
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176 | (7) |
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References and Further Reading |
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181 | (2) |
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Valuation of Derivative Securities |
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183 | (16) |
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183 | (2) |
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185 | (4) |
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Dynamic Hedging and Dynamic Completeness |
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189 | (4) |
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Fokker-Planck Theory: Computing Expectations Using PDEs |
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193 | (6) |
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References and Further Reading |
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196 | (1) |
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Proof of Proposition 11.5 |
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197 | (2) |
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Fixed-Income Securities and the Term-Structure of Interest Rates |
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199 | (30) |
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199 | (7) |
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206 | (3) |
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Term Rates, Forward Rates, and Futures-Implied Rates |
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209 | (3) |
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212 | (5) |
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217 | (1) |
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Swaptions and Bond Options |
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218 | (3) |
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Instantaneous Forward Rates: Definition |
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221 | (3) |
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Building an Instantaneous Forward-Rate Curve |
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224 | (5) |
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References and Further Reading |
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227 | (2) |
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The Health-Jarrow-Morton Theorem and Multidimensional Term-Structure Models |
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229 | (24) |
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The Health-Jarrow-Morton Theorem |
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230 | (4) |
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234 | (3) |
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Mean Reversion: The Modified Vasicek or Hull-White Model |
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237 | (2) |
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Factor Analysis of the Term-Structure |
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239 | (6) |
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Example: Construction of a Two-Factor Model with Parametric Components |
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245 | (3) |
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More General Volatility Specifications in the HJM Equation |
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248 | (5) |
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References and Further Reading |
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251 | (2) |
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Exponential-Affine Models |
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253 | (26) |
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A Characterization of EA Models |
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255 | (3) |
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Gaussian State-Variables: General Formulas |
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258 | (3) |
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Gaussian Models: Explicit Formulas |
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261 | (3) |
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Square-Root Processes and the Non-Central Chi-Squared Distribution |
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264 | (4) |
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One-Factor Square-Root Model: Discount Factors and Forward Rates |
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268 | (11) |
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References and Further Reading |
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272 | (1) |
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Behavior of Square-Root Processes for Large Times |
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273 | (2) |
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Characterization of the Probability Density Function of Square-Root Processes |
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275 | (2) |
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The Square-Root Diffusion with v = 1 |
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277 | (2) |
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279 | (34) |
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279 | (3) |
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279 | (3) |
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Commodity Options with Stochastic Interest Rate |
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282 | (1) |
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Options on Zero-Coupon Bonds |
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283 | (2) |
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Money-Market Deposits with Yield Protection |
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285 | (4) |
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Forward Rates and Forward Measures |
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286 | (3) |
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289 | (10) |
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289 | (3) |
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Cap Pricing with Gaussian Models |
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292 | (1) |
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Cap Pricing with Square-Root Models |
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293 | (4) |
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Cap Pricing and Implied Volatilities |
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297 | (2) |
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Bond Options and Swaptions |
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299 | (9) |
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General Pricing Relations |
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299 | (2) |
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301 | (2) |
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303 | (5) |
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Epilogue: The Brace-Gatarek-Musiela model |
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308 | (5) |
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References and Further Reading |
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312 | (1) |
Index |
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313 | |