Front cover image for Deterministic and stochastic topics in computational finance

Deterministic and stochastic topics in computational finance

Ovidiu Calin (Author)
What distinguishes this book from other texts on mathematical finance is the use of both probabilistic and PDEs tools to price derivatives for both constant and stochastic volatility models, by which the reader has the advantage of computing explicitly a large number of prices for European, American and Asian derivatives. The book presents continuous time models for financial markets, starting from classical models such as Black Scholes and evolving towards the most popular models today such as Heston and VAR. A key feature of the textbook is the large number of exercises, mostly solved, which are designed to help the reader to understand the material. The book is based on the author's lectures on topics on computational finance for senior and graduate students, delivered in USA (Princeton University and EMU), Taiwan and Kuwait. The prerequisites are an introductory course in stochastic calculus, as well as the usual calculus sequence. The book is addressed to undergraduate and graduate students in Masters of Finance programs as well as to those who wish to become more efficient in their practical applications.-- Provided by Publisher
Print Book, English, 2017
World Scientific, New Jersey, 2017
xix, 461 pages ; 26 cm
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
2.4.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
4.3.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
6.Risk-Neutral Valuation
6.1.The Method of Risk-Neutral Valuation
6.2.The Superposition Principle and Applications
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
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.4.Conditional Expectations
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