A Course in Financial Calculus 1st Edition by Alison Etheridge 0521813859 9780521813853

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ISBN 10: 0521813859

ISBN 13: 9780521813853

Author: Alison Etheridge

Finance provides a dramatic example of the successful application of advanced mathematical techniques to the practical problem of pricing financial derivatives. This self-contained 2002 text is designed for first courses in financial calculus aimed at students with a good background in mathematics. Key concepts such as martingales and change of measure are introduced in the discrete time framework, allowing an accessible account of Brownian motion and stochastic calculus: proofs in the continuous-time world follow naturally. The Black-Scholes pricing formula is first derived in the simplest financial context. The second half of the book is then devoted to increasing the financial sophistication of the models and instruments. The final chapter introduces more advanced topics including stock price models with jumps, and stochastic volatility. A valuable feature is the large number of exercises and examples, designed to test technique and illustrate how the methods and concepts can be applied to realistic financial questions.

A Course in Financial Calculus 1st Table of contents:

1 Single period models

Summary

1.1 Some de.nitions from .nance

Derivatives

The pricing problem

Payoffs

Packages

1.2 Pricing a forward

Expectation pricing

The risk-free rate

Arbitrage pricing

1.3 The one-step binary model

Pricing a European call

Pricing formula for European options

1.4 A ternary model

Bigger models

1.5 A characterisation of no arbitrage

A market with N assets

Arbitrage pricing

1.6 The risk-neutral probability measure

State prices and probability

Expectation recovered

Risk-neutral pricing

Complete markets

The main results so far

Trading in two different markets

Exercises

2 Binomial trees and discrete parameter martingales

Summary

2.1 The multiperiod binary model

The stock

The cash bond

Replicating portfolios

Backwards induction on the tree

Binomial trees

Path probabilities

2.2 American options

Calls on nondividendpaying stock

Put on nondividendpaying stock

2.3 Discrete parameter martingales and Markov processes

Random variables

Stochastic processes

Conditional expectation

The martingale property

The Markov property

Examples

New martingales from old

Discrete stochastic integrals

The Fundamental Theorem of Asset Pricing

2.4 Some important martingale theorems

Stopping times

Optional stopping

A convergence theorem

Compensation

American options and supermartingales

2.5 The Binomial Representation Theorem

From martingale representation to replicating portfolio

2.6 Overture to continuous models

Model with constant stock growth and noise

Under the martingale measure

Pricing a call option

Exercises

3 Brownian motion

Summary

3.1 Definition of the process

A characterisation of simple random walks

Rescaling random walks

Definition of Brownian motion

Behaviour of Brownian motion

3.2 Levy’s construction of Brownian motion

A polygonal approximation

Convergence to Brownian motion

3.3 The reflection principle and scaling

Stopping times

The reflection principle

Hitting a sloping line

Transformation and scaling of Brownian motion

3.4 Martingales in continuous time

Filtrations

Martingales

Optional stopping

Brownian hitting time distribution

Dominated Convergence Theorem

Exercises

4 Stochastic calculus

Summary

4.1 Stock prices are not differentiable

Quantifying roughness

Bounded variation and arbitrage

4.2 Stochastic integration

A differential equation for the stock price

Quadratic variation

Integrating Brownian motion against itself

Defining the integral

Integrating simple functions

Construction of the Ito integral

Other integrators

4.3 Ito’s formula

The stochastic chain rule

Geometric Brownian motion

Ito’s formula for geometric Brownian motion

Levy’s characterisation of Brownian motion

Stochastic differential equations

Solving stochastic differential equations

4.4 Integration by parts and a stochastic Fubini Theorem

Covariation

A stochastic Fubini Theorem

4.5 The Girsanov Theorem

Changing probability on a binomial tree

Change of measure in the continuous world

4.6 The Brownian Martingale Representation Theorem

4.7 Why geometric Brownian motion?

4.8 The Feynman–Kac representation

Solving pde’s probabilistically

Kolmogorov equations

Exercises

5 The Black–Scholes model

Summary

5.1 The basic Black–Scholes model

Self- financing strategies

A strategy for pricing

An equivalent martingale measure

The Fundamental Theorem of Asset Pricing

5.2 Black–Scholes price and hedge for European options

Pricing calls and puts

Hedging calls and puts

5.3 Foreign exchange

The Sterling investor

Change of numeraire

5.4 Dividends

Continuous payments

Periodic dividends

5.5 Bonds

5.6 Market price of risk

Martingales and tradables

Tradables and the market price of risk

Exercises

6 Different payoffs

Summary

6.1 European options with discontinuous payoffs

Digitals and pin risk

6.2 Multistage options

General strategy

Compound options

6.3 Lookbacks and barriers

Joint distribution of the stock price and its minimum

An expression for the price

Probability or pde?

6.4 Asian options

6.5 American options

The discrete case

Continuous time

An explicit solution

Exercises

7 Bigger models

Summary

7.1 General stock model

The model

A martingale measure

Second step to replication

Replicating a claim

The generalised Black–Scholes equation

7.2 Multiple stock models

Correlated security prices

Multifactor Ito formula

Integration by parts

Change of measure

A martingale measure

Replicating the claim

The multidimensional Black–Scholes equation

Numeraires

Quantos

Pricing a quanto forward contract

7.3 Asset prices with jumps

A Poisson process of jumps

Compensation

Poisson exponential martingales

Change of measure

Market price of risk

Multiple noises

7.4 Model error

Implied volatility

Hedging error

Stochastic volatility and implied volatility

Exercises

Bibliography

Background reading:

Supplementary textbooks:

Further topics in financial mathematics:

Brownian motion, martingales and stochastic calculus:

Additional references from the text:

Notation

Financial instruments and the Black–Scholes model

General probability

Martingales and other stochastic processes

Miscellaneous

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