# Deterministic and stochastic topics in computational finance

, Print BookEnglish, 2017Edition: View all formats and editionsPublisher: World Scientific, New Jersey, 2017

9789813203075, 9789813203082, 9813203072, 9813203080

Machine generated contents note: I.Introduction

1.Determinism or Stochasticity?

1.1.Determinism and Semi-determinism

1.2.How to Measure

1.3.An Uncertainty Principle

1.4.Market Completeness

1.5.Market Efficiency

1.6.Stopping Times

1.7.Martingales and Submartingales

1.8.Optional Stopping Theorem

1.9.Can Random be Deterministic?

1.10.Change of Time Scale

2.Calibration to the Market

2.1.Deterministic Regression

2.2.Stochastic Regression

2.3.Calibration

2.4.Alternatives to the Method of Least Squares

2.4.1.The Maximum Likelihood Method

2.4.2.The Maximum Entropy Method

2.4.3.The Kullback-Leibler Relative Entropy

2.4.4.The Cross Entropy

II.Interest Rates and Bonds

3.Modeling Stochastic Rates

3.1.Deterministic versus Stochastic Calculus

3.2.Langevin's Equation

3.3.Equilibrium Models

3.3.1.Rendleman and Bartter Model

3.3.2.Vasicek Model

3.3.3.Calibration of Vasicek's Model Note continued: 3.3.4.Cox-Ingersoll-Ross Model

3.4.No-arbitrage Models

3.4.1.Ho and Lee Model

3.4.2.Hull and White Model

3.5.Nonstationary Models

3.5.1.Black, Derman and Toy Model

4.Bonds, Forward Rates and Yield Curves

4.1.Bonds

4.2.Yield

4.3.Bootstrap Method

4.4.Forward Rates

4.5.Single-Factor HJM Models

4.5.1.Ho-Lee Model

4.5.2.Hull and White Model

4.5.3.Vasicek Model

4.6.Relation Formulas

4.7.A Simple Spot Rate Model

4.8.Bond Price for Ho-Lee Model

4.9.Bond Price for Vasicek's Model

4.10.Bond Price for CIR's Model

4.11.Mean Reverting Model with Jumps

4.12.A Model with Pure Jumps

III.Risk-Neutral Valuation Pricing

5.Modeling Stock Prices

5.1.Constant Drift and Volatility Model

5.2.Correlation of Two Stocks

5.3.When Does a Stock Hit a Given Barrier?

5.4.Probability to Hit a Barrier Before T

5.5.Multiple Barriers

5.6.Estimation of Parameters Note continued: 5.7.Time-dependent Drift and Volatility

5.8.Models for Stock Price Averages

5.9.Stock Prices with Rare Events

5.10.Dividend Paying Stocks

5.11.Currencies

6.Risk-Neutral Valuation

6.1.The Method of Risk-Neutral Valuation

6.2.The Superposition Principle and Applications

6.3.Packages

6.4.Strike Sensitivity

6.5.Volatility Sensitivity

6.6.Implied Volatility

6.7.Asian Forward Contracts

6.8.Asian Options

6.9.The dk Notations

6.10.All-or-nothing Look-back Options

6.11.Asset-or-nothing Look-back Options

6.12.Look-back Call Options

6.13.Forward Look-back Contracts

6.14.Immediate Rebate Options

6.15.Perpetual Look-back Options

6.16.Double Barrier Immediate Option

6.17.Pricing in a Rare Events Environment

6.17.1.Pricing a Call

6.17.2.Pricing a Forward Contract

6.18.Pricing with Pareto Distribution

7.Martingale Measures

7.1.Martingale Measures Note continued: 7.1.1.Is the stock price St a martingale?

7.1.2.Risk-neutral World and Martingale Measure

7.1.3.Finding the Risk-Neutral Measure

7.2.Risk-neutral World Density Functions

7.3.Self-financing Portfolios

7.4.The Sharpe Ratio

7.5.Risk-neutral Valuation for Derivatives

IV.PDE Approach

8.Black-Scholes Analysis

8.1.Heat Equation

8.2.What is a Portfolio?

8.3.Risk-less Portfolios

8.4.Black-Scholes Equation

8.5.Delta Hedging

8.6.Tradable Securities

8.7.Risk-less Investment Revised

8.8.Solving the Black-Scholes

8.9.Black-Scholes and Risk-neutral Valuation

8.10.Boundary Conditions

8.11.The Black-Scholes Operator

8.12.Hedging and Black-Scholes

8.12.1.Hedging Stocks

8.12.2.Hedging Derivatives

8.12.3.Hedging Bonds

8.12.4.Particular Cases

8.12.5.The Bond Formula

8.13.Interest Rate Swaps

8.13.1.Case of Deterministic Rates

8.13.2.A Black-Scholes Type Equation Note continued: 8.13.3.Solving the Equation

8.13.4.Some Particular Cases

8.13.5.Hedging with Swaps

9.Black-Scholes for Asian Derivatives

9.1.Weighted Averages

9.2.Setting up the Black-Scholes Equation

9.3.Weighted Average Strike Call Option

9.4.Boundary Conditions

9.5.Asian Forward Contracts

9.6.Put-Call Parity

9.7.Valuation of Arithmetic Asian Price Options

10.American Options

10.1.Perpetual American Options

10.1.1.Present Value of Barriers

10.1.2.Perpetual American Calls

10.1.3.Perpetual American Puts

10.2.Perpetual American Log Contract

10.3.Perpetual American Power Contract

10.4.Finitely Lived American Options

10.4.1.American Call

10.5.One-phase Stefan Problem

10.6.Free-Boundary of a Call

10.7.The Call as a Free Boundary Problem

10.7.1.Dynamics of the Free-Boundary

10.7.2.Local Analysis near Maturity

10.7.3.The Infinite Horizon Case

10.8.American Put Note continued: 11.4.1.Calls

11.4.2.Call Perpetuity

11.4.3.Analytic Approximation to a Call

12.GARCH Model

12.1.GARCH(1,1) Differential Model

12.2.Dynamics of the GARCH Model

12.3.The GARCH PDE

12.4.Simplifying the PDE

12.5.Solving the Equation

13.AR(1) Model

13.1.AR(1)-Differential Model

13.2.The PDE

14.Stochastic Return Models

14.1.Mean-reverting Ornstein-Uhlenbeck Process

14.2.A Continuous VAR Process

14.3.Probability Density of St

14.4.The PDE

14.5.Simplifying the PDE

14.6.Solving the PDE

14.7.All-or-Nothing Option

15.Hints and Solutions

Appendix

A.Useful Transforms

A.1.The Fourier Transform

A.2.The Laplace Transform

B.Probability Concepts

B.1.Events and Probability

B.2.Stochastic Processes

B.3.Expectation

B.4.Conditional Expectations

B.5.Martingales

B.6.Submartingales and their Properties

B.7.Jensen's Inequality

C.Elements of Stochastic Calculus Note continued: C.1.The Poisson Process

C.2.The Brownian Motion

C.3.Exponential Process

C.4.Ito's Lemma

C.5.Girsanov Theorem

C.6.Ito Integral

C.7.Brownian Motion with Drift

C.8.The Generator of an Ito Diffusion

D.Series and Equations

D.1.Confluent Hypergeometric Equation

D.2.Duhamel's Principle