Table of Contents. Part I. Deterministic Models... 1

Transcription

1 Preface...xvii Part I. Deterministic Models... 1 Chapter 1. Introductory Elements to Financial Mathematics The object of traditional financial mathematics Financial supplies. Preference and indifference relations The subjective aspect of preferences Objective aspects of financial laws. The equivalence principle The dimensional viewpoint of financial quantities Chapter 2. Theory of Financial Laws Indifference relations and exchange laws for simple financial operations Two variable laws and exchange factors Derived quantities in the accumulation and discount laws Accumulation Discounting Decomposable financial lawas Weak and strong decomposability properties: equivalence relations Equivalence classes: characteristic properties of decomposable laws Uniform financial laws: mean evaluations Theory of uniform exchange laws An outline of associative averages Average duration and average maturity Average index of return: average rate Uniform decomposable financial laws: exponential regime... 39

2 vi Mathematical Finance Chapter 3. Uniform Regimes in Financial Practice Preliminary comments Equivalent rates and intensities The regime of simple delayed interest (SDI) The regime of rational discount (RD) The regime of simple discount (SD) The regime of simple advance interest (SAI) Comments on the SDI, RD, SD and SAI uniform regimes Exchange factors (EF) Corrective operations Initial averaged intensities and instantaneous intensity Average length in the linear law and their conjugates Average rates in linear law and their conjugated laws The compound interest regime Conversion of interests The regime of discretely compound interest (DCI) The regime of continuously compound interest (CCI) The regime of continuously comound discount (CCD) Complements and exercises on compound regimes Comparison of laws of different regimes Chapter 4. Financial Operations and their Evaluation: Decisional Criteria Calculation of capital values: fairness Retrospective and prospective reserve Usufruct and bare ownership in discrete and continuous cases Methods and models for financial decisions and choices Internal rate as return index Outline on GDCF and internal financial law Classifications and propert of financial projects Decisional criteria for financial projects Choice criteria for mutually exclusive financial projects Mixed projects: the TRM method Dicisional criteria on mixed projects Appendix: outline on numberical methods for the solution of equations General aspects The linear interpolation method Dichotomic method (or for successive divisions) Secants and tangents method Classical interation method...143

3 vii Chapter 5. Annuities-Certain and their Value at Fixed Rate General aspects Evaluation of constant installment annuities in the compound regime Temporary annual annuity Annual perpetuity Fractional and pluriannual annuities Inequalities between annuity values with different frequency: correction factors Evaluation of constant installment annuities according to linear laws The direct problem Use of correction factors Inverse problem Evaluation of varying installment annuities in the compound regime General case Specific cases: annual annuities in arithmetic progression Specific cases: fractional and pluriannual annuities in arithmetic progression Specific cases: annual annuity in geometric progression Specific cases: fractional and pluriannual annuity in geometric progression Evaluation of varying installment annuities according to linear laws General case Specific cases: annuities in arithmetic progression Specific cases: annuities in geometric progression Chapter 6. Loan Amortization and Funding Methods General features of loan amortization General loan amortization at fixed rate Gradual amortizatin with varying installments Particular case: delayed constant installment amortization Particular case: amortization with constant principal repayments Particular case: amortization with advance interests Particular case: American amortization Amortization in the continuous scheme Life amortization Periodic advance payments Periodic payments with delayed principal amounts Continuous payment flow Periodic funcing at fixed rate Delayed payments Advance payments Continuours payments...251

4 viii Mathematical Finance 6.5. Amortizations with adjustment of rates and values Amortizations with adjustable rate Amortizations with adjustment of the outstanding loan balance Valuation of reserves in unshared loans General aspects Makeham s formula Usufructs and bare ownership valuation for some amortization forms Leasing operation Ordinary leasing The monetary adjustment in leasing Amortizations of loans shared in securities An introduction on the securities Amortization from the viewpoint of the debtor Amortization from the point of view of the bondholder Drawing probabiity and mean life Adjustable rate bonds, indexed bonds and convertible bonds Rule variations in bond loans Valuation in shared loans Introduction Valuation of bonds with given maturity Valuation of drawing bonds Bond loan with varying rate or values adjusted in time Chapter 7. Exchanges and Prices on the Financial Market A reinterpretation of the financial quantities in a market and price logic: the perfect market The perfect market Bonds Spot contracts, price and rates. Yield rate Forward contracts, prices and rates The implicit structure of prices, rates and intensities Term structures Structures with discrete payments Structures with fractional periods Structures with flows in continuum Chapter 8. Annuities, Amortizations and Funding in the Case of Term Structures Capital value of annuities in the case of term structures Amortizations in the case of term structures Amortization with varying installments...337

5 ix Amortization with constant installments Amortization with constant principal repayments Life amortization Updating of valuations during amortization Funding in term structure environments Valuations referred to shared loans in term structure environments Financial flows by the issuer s and investors point of view Valuations of price and yield Chapter 9. Time and Variability Indicators, Classical Immunization Main time indicators Maturity and time to maturity Arithmetic mean maturity Average maturity Mean financial time length or duration Variability and dispersion indicators nd order duration Relative variation Elasticity Convexity and volatility convexity Approximated estimations of price fluctuation Rate risk and classical immunization An introductin to financial risk Preliminaries to classic immunization The optimal time of realization The meaning of classical immunization Single liability cover Multiple liability cover Part II. Stochastic Models Chapter 10. Basic Probabilistic Tools for Finance The sample space Probability space Random variables Expectation and independence Main distribution probabilities The binominal distribution The Poisson distribution The normal (or Laplace Gauss) distribution The log-normal distribution The negative exponential distribution...432

6 x Mathematical Finance The multidimensional normal distribution Conditioning Stochastic processes Martingales Brownian motion Chapter 11. Markov Chains Definitions State classification Occupation times Absorption probabilities Asymptotic behavior Examples A management problem in an insurance company A case study in social insurance Chapter 12. Semi-Markov Processes Positive (J-X) processes Semi-Markov and extended semi-markov chains Primary properties Examples Markov renewal processes, semi-markov and associated counting processes Particular cases of MRP Renewal processes and Markov chains MRP of zero order Continuous Markov processes Markov renewal functions The Markov renewal equation Asymptotic behavior of an MRP Asymptotic behavior of SMP Irreducible case Non-irreducible case Non-homogenous Markov and semi-markov processes General definitions Chapter 13. Stochastic or Itô Calculus Problem of stochastic integration Stochastic integration of simple predictable processes and semi-martingales General definition of the stochastic integral...523

7 xi Itô s formula Quadratic variation of a semi-martingale Itô s formula Stochastic integral with standard Brownian motion as integrator process Case of predictable simple processes Extension to general integrand processes Stochastic differentiation Definition Examples Back to Itô s formula Stochastic differential of a product Itô s formula with time dependence Interpretation of Itô s formula Other extensions of Itô s formula Stochastic differential equations Existence and unicity general thorem Solution of stochastic differntial equations Diffusion processes Chapter 14. Option Theory Introduction The Cox, Ross, Rubinstein (CRR) or binomial model One-period model Multi-period model The Black-Scholes formula as the limit of the binomial model The lognormality of the underlying asset The Black-Scholes formula The Black-Scholes continuous time model The model The Solution of the Black-Scholes-Samuelson model Pricing the call with the Black-Scholes-Samuelson model Exercises on option pricing The Greek parameters Introduction Values of the Greek parameters Excercises The impact of dividend repartition Estimation of the volatility Historic method Implicit volatility method Black-Scholes on the market...587

8 xii Mathematical Finance Empirical studies Smile effect Exotic options Introduction Garman-Kohlhagen formula Greek parameters Theoretical models Binary or digital options Asset or nothing options The barrier options Lockback options Asiatic (or average) options Rainbow options The formula of Barone-Adesi and Whaley (1987): formula for American options Chapter 15. Markov and Semi-Markov Option Models The Janssen-Manca model The Markov extension of the one-period CRR model The multi-period discrete Markov chain model The multi-period discrete Markov chain limit model The extension of the Black-Scholes pricing formula with Markov environment: the Janssen-Manca formula The extension of the Black-Scholes pricing formula with a semi-markov environment: the Janssen-Manca-Volpe formula Introduction The Janssen-Manca-Çinlar model Call option pricing Stationary option pricing formula Markov and semi-markov option pricing models with arbitrage possibility Introduction The homogenous Markov model for the underlying asset Particular cases Numerical example for the Markov model The continuous time homogenous semi-markov model for the underlying asset Numerical example for the semi-markov model Conclusion...640

9 xiii Chapter 16. Interest Rate Stochastic Models Application to the Bond Pricing Problem The bond investments Introduction Yield curve Yield to maturity for a financial investment and for a bond Dynamic deterministic continuous time model for instantaneous interest rate Instantaneous interest rate Particular cases Yield curve associated with instantaneous interest rate Example of theoretical models Stochastic continuous time dynamic model for instantaneous interest rate The OUV stochastic model The CIR model (1985) The HJM model (1992) Zero-coupon pricing under the assumption of no arbitrage Stochastic dynamics of zero-coupons Application of the no arbitrage principle and risk premium Partial differential equatin for the structure of zero coupons Values of zero coupons without arbitrage opportunity for particular cases Values of a call on zero-coupon Option on bond with coupons A numerical example Appendix (solution of the OUV equation) Chapter 17. Portfolio Theory Quantitative portfolio management Notion of efficiency Exercises Markowitz theory for two assets Case of one risky asset and one non-risky asset Chapter 18. Value at Risk (VaR) Methods and Simulation VaR of one asset Introduction Definition of VaR for one asset Case of the normal distribution...705

10 xiv Mathematical Finance Example II: an internal model in case of the lognormal distribution Trajectory simulation Coherence and VaR extensions Risk measures General form of the VaR VaR extensions: TVaR and conditional VaR VaR of an asset portfolio VaR methodology General methods for VaR calculation VaR implementation VaR for a bond portfolio VaR for one plain vanilla option VaR and Monte Carlo simulation methods Introduction Case of one risk factor Case of several risk factors Monte Carlo simulation scheme for the VaR calculation of an asset portfolio Chapter 19. Credit Risk or Default Risk Introduction The Merton model Evaluation model of a risky debt Interpretation of Merton s result Spreads The Longstaff and Schwartz model (1995) Construction of a rating with Merton s model for the firm Rating construction Time dynamic evolution of a rating Discrete time semi-markov processes Purpose DTSMP definition Semi-Markov credit risk models NHSMP with backward conditioning time Examples Homogenous SMP application Non-homogenous downward example Non-homogenous downward backward example...784

11 xv Chapter 20. Markov and Semi-Markov Reward Processes and Stochastic Annuities Reward processes Homogenous and non-homogenous DTMRWP Homogenous and non-homogenous DTSMRWP The immediate cases The due cases MRWP and stochastic annuities Stochastic annuities Motorcar insurance application DTSMRWP and generalized stochastic annuities (GSA) Generalized stochastic annuities (GSA) GSA examples References Index...839