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Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues

Book Description

The book aims to prioritise what needs mastering and presents the content in the most understandable, concise and pedagogical way illustrated by real market examples. Given the variety and the complexity of the materials the book covers, the author sorts through a vast array of topics in a subjective way, relying upon more than twenty years of experience as a market practitioner. The book only requires the reader to be knowledgeable in the basics of algebra and statistics.

The Mathematical formulae are only fully proven when the proof brings some useful insight. These formulae are translated from algebra into plain English to aid understanding as the vast majority of practitioners involved in the financial markets are not required to compute or calculate prices or sensitivities themselves as they have access to data providers. Thus, the intention of this book is for the practitioner to gain a deeper understanding of these calculations, both for a safety reason – it is better to understand what is behind the data we manipulate – and secondly being able to appreciate the magnitude of the prices we are confronted with and being able to draft a rough calculation, aside of the market data.

The author has avoided excessive formalism where possible. Formalism is securing the outputs of research, but may, in other circumstances, burden the understanding by non-mathematicians; an example of this case is in the chapter dedicated to the basis of stochastic calculus.

The book is divided into two parts:

- First, the deterministic world, starting from the yield curve building and related calculations (spot rates, forward rates, discrete versus continuous compounding, etc.), and continuing with spot instruments valuation (short term rates, bonds, currencies and stocks) and forward instruments valuation (forward forex, FRAs and variants, swaps & futures);

- Second, the probabilistic world, starting with the basis of stochastic calculus and the alternative approach of ARMA to GARCH, and continuing with derivative pricing: options, second generation options, volatility, credit derivatives;

- This second part is completed by a chapter dedicated to market performance & risk measures, and a chapter widening the scope of quantitative models beyond the Gaussian hypothesis and evidencing the potential troubles linked to derivative pricing models.

Table of Contents

  1. Cover
  2. Series
  3. Title Page
  4. Copyright
  5. Dedication
  6. Foreword
  7. Main Notations
  8. Introduction
  9. Part I: The Deterministic Environment
    1. Chapter 1: Prior to the yield curve: spot and forward rates
      1. 1.1 INTEREST RATES, PRESENT AND FUTURE VALUES, INTEREST COMPOUNDING
      2. 1.2 DISCOUNT FACTORS
      3. 1.3 CONTINUOUS COMPOUNDING AND CONTINUOUS RATES
      4. 1.4 FORWARD RATES
      5. 1.5 THE NO ARBITRAGE CONDITION
      6. FURTHER READING
    2. Chapter 2: The term structure or yield curve
      1. 2.1 INTRODUCTION TO THE YIELD CURVE
      2. 2.2 THE YIELD CURVE COMPONENTS
      3. 2.3 BUILDING A YIELD CURVE: METHODOLOGY
      4. 2.4 AN EXAMPLE OF YIELD CURVE POINTS DETERMINATION
      5. 2.5 INTERPOLATIONS ON A YIELD CURVE
      6. FURTHER READING
    3. Chapter 3: Spot instruments
      1. 3.1 SHORT-TERM RATES
      2. 3.2 BONDS
      3. 3.3 CURRENCIES
      4. FURTHER READING
    4. Chapter 4: Equities and stock indexes
      1. 4.1 STOCKS VALUATION
      2. 4.2 STOCK INDEXES
      3. 4.3 THE PORTFOLIO THEORY
      4. FURTHER READING
    5. Chapter 5: Forward instruments
      1. 5.1 THE FORWARD FOREIGN EXCHANGE
      2. 5.2 FRAs
      3. 5.3 OTHER FORWARD CONTRACTS
      4. 5.4 CONTRACTS FOR DIFFERENCE (CFD)
      5. FURTHER READING
    6. Chapter 6: Swaps
      1. 6.1 DEFINITIONS AND FIRST EXAMPLES
      2. 6.2 PRIOR TO AN IRS SWAP PRICING METHOD
      3. 6.3 PRICING OF AN IRS SWAP
      4. 6.4 (RE)VALUATION OF AN IRS SWAP
      5. 6.5 THE SWAP (RATES) MARKET
      6. 6.6 PRICING OF A CRS SWAP
      7. 6.7 PRICING OF SECOND-GENERATION SWAPS
      8. FURTHER READING
    7. Chapter 7: Futures
      1. 7.1 INTRODUCTION TO FUTURES
      2. 7.2 FUTURES PRICING
      3. 7.3 FUTURES ON EQUITIES AND STOCK INDEXES
      4. 7.4 FUTURES ON SHORT-TERM INTEREST RATES
      5. 7.5 FUTURES ON BONDS
      6. 7.6 FUTURES ON CURRENCIES
      7. 7.7 FUTURES ON (NON-FINANCIAL) COMMODITIES
      8. FURTHER READING
  10. Part II: The Probabilistic Environment
    1. Chapter 8: The basis of stochastic calculus
      1. 8.1 STOCHASTIC PROCESSES
      2. 8.2 THE STANDARD WIENER PROCESS, OR BROWNIAN MOTION
      3. 8.3 THE GENERAL WIENER PROCESS
      4. 8.4 THE ITÔ PROCESS
      5. 8.5 APPLICATION OF THE GENERAL WIENER PROCESS
      6. 8.6 THE ITÔ LEMMA
      7. 8.7 APPLICATION OF THE ITô LEMMA
      8. 8.8 NOTION OF RISK NEUTRAL PROBABILITY
      9. 8.9 NOTION OF MARTINGALE
      10. ANNEX 8.1: PROOFS OF THE PROPERTIES OF dZ(t)
      11. ANNEX 8.2: PROOF OF THE ITÔ LEMMA
      12. FURTHER READING
    2. Chapter 9: Other financial models: from ARMA to the GARCH family
      1. 9.1 THE AUTOREGRESSIVE (AR) PROCESS
      2. 9.2 THE MOVING AVERAGE (MA) PROCESS
      3. 9.3 THE AUTOREGRESSION MOVING AVERAGE (ARMA) PROCESS
      4. 9.4 THE AUTOREGRESSIVE INTEGRATED MOVING AVERAGE (ARIMA) PROCESS
      5. 9.5 THE ARCH PROCESS
      6. 9.6 THE GARCH PROCESS
      7. 9.7 VARIANTS OF (G)ARCH PROCESSES
      8. 9.8 THE MIDAS PROCESS
      9. FURTHER READING
    3. Chapter 10: Option pricing in general
      1. 10.1 INTRODUCTION TO OPTION PRICING
      2. 10.2 THE BLACK–SCHOLES FORMULA
      3. 10.3 FINITE DIFFERENCE METHODS: THE COX–ROSS–RUBINSTEIN (CRR) OPTION PRICING MODEL
      4. 10.4 MONTE CARLO SIMULATIONS
      5. 10.5 OPTION PRICING SENSITIVITIES
      6. FURTHER READING
    4. Chapter 11: Options on specific underlyings and exotic options
      1. 11.1 CURRENCY OPTIONS
      2. 11.2 OPTIONS ON BONDS
      3. 11.3 OPTIONS ON INTEREST RATES
      4. 11.4 EXCHANGE OPTIONS
      5. 11.5 BASKET OPTIONS
      6. 11.6 BERMUDAN OPTIONS
      7. 11.7 OPTIONS ON NON-FINANCIAL UNDERLYINGS
      8. 11.8 SECOND-GENERATION OPTIONS, OR EXOTICS
      9. FURTHER READING
    5. Chapter 12: Volatility and volatility derivatives
      1. 12.1 PRACTICAL ISSUES ABOUT THE VOLATILITY
      2. 12.2 MODELING THE VOLATILITY
      3. 12.3 REALIZED VOLATILITY MODELS
      4. 12.4 MODELING THE CORRELATION
      5. 12.5 VOLATILITY AND VARIANCE SWAPS
      6. FURTHER READING
    6. Chapter 13: Credit derivatives
      1. 13.1 INTRODUCTION TO CREDIT DERIVATIVES
      2. 13.2 VALUATION OF CREDIT DERIVATIVES
      3. 13.3 CONCLUSION
      4. FURTHER READING
    7. Chapter 14: Market performance and risk measures
      1. 14.1 RETURN AND RISK MEASURES
      2. 14.2 VaR OR VALUE-AT-RISK
      3. FURTHER READING
    8. Chapter 15: Beyond the Gaussian hypothesis: potential troubles with derivatives valuation
      1. 15.1 ALTERNATIVES TO THE GAUSSIAN HYPOTHESIS
      2. 15.2 POTENTIAL TROUBLES WITH DERIVATIVES VALUATION
      3. FURTHER READING
  11. Bibliography
  12. Index