لایب سان

Author(s): Gerhard Larcher

Series: Springer Texts in Business and Economics

Publisher: Springer, Year: 2023

ISBN: 3031238729,9783031238727

Contents 1 Basic Products and Interest Calculations . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1 1.1 Basic Properties of Bonds .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3 1.2 Example of a Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4 1.3 Issuance of a Bond and Initial Issue Price . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7 1.4 Chart of the Bond and Factors Influencing Price Developments . . . . 8 1.5 Discrete and Continuous Interest Compounding .. . . . . . . . . . . . . . . . . . . . 10 1.6 Continuous Compounding at a Time-Varying Interest Rate . . . . . . . . . 19 1.7 Euribor, Libor, Swap Rates, and Key Interest Rates . . . . . . . . . . . . . . . . . 22 1.8 Two Comments Regarding Loans: Calculation of Continuously Redeemable Loan Instalments as Well as Foreign Currency Loans and Interest Rate Parity Theory . . . . . . . . . . . 28 1.9 Bond Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 34 1.10 Forward Interest Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 39 1.11 The Fair Value of a Future Payment, Discounting . . . . . . . . . . . . . . . . . . . 43 1.12 The Fair Value of a Bond .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 45 1.13 Some Examples of Junk Bonds . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 48 1.14 Stock (Basics) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 50 1.15 Stock Market Dynamics .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 54 1.16 Stock Indices .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 58 1.17 Trading in Indices Through Index Certificates . . .. . . . . . . . . . . . . . . . . . . . 63 1.18 The ShortDAX and Index Certificates on the ShortDAX .. . . . . . . . . . . 63 1.19 The S&P500 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 66 1.20 The S&P500 on the Black Monday of 19 October 1987 . . . . . . . . . . . . 72 1.21 11 September 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 74 1.22 The S&P500 Index Around 10 October 2008, Overnight Gaps . . . . . 76 1.23 The “Flash Crash” on 6 May 2010.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 77 1.24 Price Forecasting and Run Analyses of Stock Prices and Index Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 79 1.25 Notes on a Simple Trading Strategy by Signals Using Exponential Moving Averages . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 85 2 Derivatives and Trading in Derivatives, Basic Concepts and Strategies.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 91 2.1 What Is a Derivative? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 92 xv xvi Contents 2.2 European Plain-Vanilla Options, Definition and Basic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 92 2.3 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101 2.4 Any Strategy Is Better than No Strategy and “The Secretary Problem” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 105 2.5 How Do You Trade Options? Trading Through a Bank . . . . . . . . . . . . . 113 2.6 How Do You Trade Options? Trading Through an Electronic Trading Platform .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 117 2.7 Who Trades in Options? Long Positions in Call Options, Leverage.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 124 2.8 Who Trades in Options? Long Positions in Put Options, Protective Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 130 2.9 Who Trades in Options? Short Positions in Put Options, Selling Insurance, Put Spreads .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 138 2.10 Who Trades in Options? Short Positions in Call Options, Covered Call Strategies.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 140 2.11 Discount Certificates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 142 2.12 Who Trades in Options? Long Straddle, Short Straddle .. . . . . . . . . . . . 145 2.13 Relationship Between the Payoffs of Puts, Calls, and Underlying Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 152 2.14 More Option Combinations . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 155 2.15 Margin Rules for Short Positions in (CBOE S&P500) Options .. . . . 160 2.16 CBOE-Traded Options on the S&P500 Index, Market-Maker System, Settlement of SPX Options . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 165 2.17 Futures, Basic Characteristics, Trading, Margin .. . . . . . . . . . . . . . . . . . . . 167 2.18 Long and Short Trades of Underlying Assets with Futures.. . . . . . . . . 172 2.19 The “Euro-Bund Future” . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 177 2.20 More Comments on Futures Contracts (Rolling, Futures Options, Forwards) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 181 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 187 3 Basics of Derivative Valuation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 189 3.1 Frictionless Markets and the No-Arbitrage Principle . . . . . . . . . . . . . . . . 190 3.2 Application of the NA Principle: Put-Call Parity Equation .. . . . . . . . . 196 3.3 Simple Conclusions from the Put-Call Parity Equation . . . . . . . . . . . . . 203 3.4 Another Application of the NA Principle: The “Fair” Strike Price of a Futures Contract (on a Dividend-Free/Cost-Free Underlying Asset) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 208 3.5 Valuation of Futures for Underlying Assets with Payouts or Costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 212 3.6 The Put-Call Parity Equation for Underlying Assets with Payouts or Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 216 3.7 Basics of Derivative Valuation and Pricing Models .. . . . . . . . . . . . . . . . . 219 3.8 The One-Step Binomial Model and Derivative Valuation in the One-Step Binomial Model: Part I . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 222 Contents xvii 3.9 The One-Step Binomial Model and Derivative Valuation in the One-Step Binomial Model: Part II .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 227 3.10 Derivative Valuation in the One-Step Binomial Model: Discussion of Outcomes .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 233 3.11 A Brief Excursus on the Degree of Luck and Skill in Games . . . . . . . 246 3.12 Fair Price of Derivatives in the Binomial Model on Underlying Assets with Payouts/Costs . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 250 3.13 The Two-step Binomial Model.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 251 3.14 Derivative Valuation in the Two-Step Binomial Model: Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 255 3.15 Hedging and Arbitrage in the Two-Step Binomial Model . . . . . . . . . . . 257 3.16 Numerical Example of How to Value and Hedge Derivatives and Execute Arbitrage Trades in a Two-Step Binomial Model .. . . . . 258 3.17 Derivative Valuation in the N-Step Binomial Model.. . . . . . . . . . . . . . . . 262 3.18 Comments on the Valuation of Derivatives in the N-Step Binomial Model and an Example . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 268 3.19 Derivative Valuation in the N-Step Binomial Model on Underlying Assets with Payouts or Costs . . . . . . . .. . . . . . . . . . . . . . . . . . . . 271 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 272 4 The Wiener Stock Price Model and the Basic Principles of Black-Scholes Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 273 4.1 Basic Tools for Analysing Real Stock Prices: Trend, Volatility, Distribution of Returns, Skewness, and Kurtosis . . . . . . . . . 274 4.2 Basic Tools for Analysing Real Stock Prices: Covariances and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 292 4.3 Basic Tools for Analysing Real Stock Prices: Autocorrelations of Stock Returns . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 304 4.4 What Does Mathematical Modeling Mean? What Is a Stock Price Model? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 306 4.5 The Wiener Stock Price Model . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 313 4.6 Simulation of Stock Prices in the Wiener Model.. . . . . . . . . . . . . . . . . . . . 316 4.7 Simulation of Two Correlated Stock Prices . . . . . .. . . . . . . . . . . . . . . . . . . . 319 4.8 Simulation of Several Correlated Stock Prices . . .. . . . . . . . . . . . . . . . . . . . 324 4.9 Simulation of a Wiener Model for Given Initial and Final Values: The Brownian Bridge .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 329 4.10 Expectations, Variances, and Probability Distributions of Stock Prices in the Wiener Model . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 332 4.11 Approximation of the Wiener Model Through Binomial Models: Preliminary Remarks . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 338 4.12 Approximation of the Wiener Model Through Binomial Models: Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 338 4.13 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 340 4.14 Approximation of the Wiener Model Through Binomial Models: The Proof (Sketch of Proof) .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 345 xviii Contents 4.15 The Brownian Motion, Motivation, and Definition . . . . . . . . . . . . . . . . . . 347 4.16 The Brownian Motion: Basic Properties . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 359 4.17 The Wiener Model as a Geometric Brownian Motion and the Brownian Motion with Drift . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 371 4.18 The Black-Scholes Formula in the Wiener Model . . . . . . . . . . . . . . . . . . . 371 4.19 The Fair Price of a European Call Option and a European Put Option in the Wiener Model . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 379 4.20 A (Very) Short History of the Black-Scholes Formula . . . . . . . . . . . . . . 382 4.21 Perfect Hedging in the Black-Scholes Model . . . .. . . . . . . . . . . . . . . . . . . . 383 4.22 Another Example of the Application of the Black-Scholes Formula and of Perfect Hedging as Well as Its Implementation in Discrete Hedging . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 387 4.23 Discretely Approximated Perfect Hedging for European Derivatives, Especially for European Call and Put Options . . . . . . . . . 393 4.24 Detailed Discussion of the Black-Scholes Formula for European Call Options I (Dependence on S and t, Intrinsic Value, Fair Value) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 397 4.25 Detailed Discussion of the Black-Scholes Formula for European Call Options II (Dependence on Volatility) . . . . . . . . . . . . . . . 406 4.26 Detailed Discussion of the Black-Scholes Formula for European Call Options III (Dependence on Risk-Free Interest Rate) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 410 4.27 Some Brief Remarks on the Use of the Black-Scholes Formula and Its Parameters r and σ . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 413 4.28 Program and Test: Valuation of Derivatives by Approximation Using an N-Step Binomial Model Where Volatility Is Correlated with the Price of the Underlying Asset . . . . . . . . . . . . . . . . . . . 416 4.29 Break-Even for Call-Only Strategies . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 419 4.30 Analysis of the Black-Scholes Price of Put Options . . . . . . . . . . . . . . . . . 426 4.31 Break-Even for Put-Only Strategies . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 433 4.32 Analysis of the Price Paths of a Few Other Basic Option Strategies: Short Iron Butterfly. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 439 4.33 Analysis of the Price Curves of a Few Other Basic Option Strategies: Naked Short Butterfly .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 445 4.34 Excursus: Brief Remark on the “Asymmetry of Call and Put Prices” .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 449 4.35 Analysis of the Price Curves of a Few Other Basic Option Strategies: Simple Time Spreads .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 456 4.36 The Greeks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 461 4.37 The Greeks for Call Options and Put Options .. . .. . . . . . . . . . . . . . . . . . . . 463 4.38 Graphical Illustration of the Greeks of Call Options.. . . . . . . . . . . . . . . . 464 4.39 Graphical Illustration of the Greeks of Put Options.. . . . . . . . . . . . . . . . . 475 4.40 Delta and Gamma: Analysis of a Put Bull Spread . . . . . . . . . . . . . . . . . . . 479 4.41 Test Simulations of Exit Strategies for Bull Put Spread Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 496 Contents xix 4.42 Delta/Gamma Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 501 4.43 Delta/Gamma Hedging: A Realistic Example.. . .. . . . . . . . . . . . . . . . . . . . 504 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 516

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