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2016 | Buch

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Stochastic Analysis for Finance with Simulations

verfasst von: Geon Ho Choe

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is an introduction to stochastic analysis and quantitative finance; it includes both theoretical and computational methods. Topics covered are stochastic calculus, option pricing, optimal portfolio investment, and interest rate models. Also included are simulations of stochastic phenomena, numerical solutions of the Black–Scholes–Merton equation, Monte Carlo methods, and time series. Basic measure theory is used as a tool to describe probabilistic phenomena.
The level of familiarity with computer programming is kept to a minimum. To make the book accessible to a wider audience, some background mathematical facts are included in the first part of the book and also in the appendices. This work attempts to bridge the gap between mathematics and finance by using diagrams, graphs and simulations in addition to rigorous theoretical exposition. Simulations are not only used as the computational method in quantitative finance, but they can also facilitate an intuitive and deeper understanding of theoretical concepts.
Stochastic Analysis for Finance with Simulations is designed for readers who want to have a deeper understanding of the delicate theory of quantitative finance by doing computer simulations in addition to theoretical study. It will particularly appeal to advanced undergraduate and graduate students in mathematics and business, but not excluding practitioners in finance industry.

Inhaltsverzeichnis

Frontmatter

Introduction to Financial Mathematics

Frontmatter

Chapter 1. Fundamental Concepts

Abstract

A stock exchange is an organization of brokers and financial companies which has the purpose of providing the facilities for trade of company stocks and other financial instruments. Trade on an exchange is by members only. In Europe, stock exchanges are often called bourses. The trading of stocks on stock exchanges, physical or electronic, is called the stock market.

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Chapter 2. Financial Derivatives

Abstract

Financial assets are divided into two categories: The first group consists of primary assets including shares of a stock company, bonds issued by companies or governments, foreign currencies, and commodities such as crude oil, metal and agricultural products. The second group consists of financial contracts that promise some future payment of cash or future delivery of the primary assets contingent on an event in the future date. The event specified in the financial contract is usually defined in terms of the behavior of an asset belonging to the first category, and such an asset is called an underlying asset or simply an underlying. The financial assets belonging to the second category are called derivatives since their values are derived from the values of the underlying asset belonging to the first category. Securities are tradable financial instruments such as stocks and bonds.

Geon Ho Choe

Probability Theory

Frontmatter

Chapter 3. The Lebesgue Integral

Abstract

Given an abstract set $\Omega $, how do we measure the size of one of its subsets A? When $\Omega $ has finite or countably infinite elements, it is natural to count the number of elements in A. However, if A is an uncountable set, we need a rigorous and systematic method. In many interesting cases there is no logical way to measure the sizes of all the subsets of $\Omega $, however, we define the concept of size for sufficiently many subsets. Subsets whose sizes can be determined are called measurable subsets, and the collection of these measurable subsets is called a σ-algebra where the set operations such as union and intersection resemble operations on numbers such as addition and multiplication. After measures are introduced we define the Lebesgue integral.

Geon Ho Choe

Chapter 4. Basic Probability Theory

Abstract

When H. Lebesgue invented the Lebesgue integral, it was regarded as an abstract concept without applications. It was A.N. Kolmogorov [52] who first showed how to formulate rigorous axiomatic probability theory based on Lebesgue integration.

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Chapter 5. Conditional Expectation

Abstract

The concept of conditional expectation will be developed in three stages.

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Chapter 6. Stochastic Processes

Abstract

We collect and analyze sequential data from nature or society in the form of numerical sequences which are indexed by the passage of time, and try to predict what will happen next. Due to uncertainty of payoff in financial investment before maturity date, the theory of stochastic processes, a mathematical discipline which studies a sequence of random variables, has become the language of mathematical finance.

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Brownian Motion

Frontmatter

Chapter 7. Brownian Motion

Abstract

After the botanist Robert Brown discovered Brownian motion under a microscope in 1827, it was studied by Louis Bachelier in 1900 to study option price, and Albert Einstein did research on Brownian motion in 1905.

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Chapter 8. Girsanov’s Theorem

Abstract

Let $\{W_{t}\}_{t\geq 0}$ be a Brownian motion with respect to a probability measure $\mathbb{P}$. Take a constant $\theta$, and consider $X_{t} = W_{t} +\theta t$, $0 \leq t <\infty$, which is called a Brownian motion with drift. Our goal is to find a probability measure $\mathbb{Q}$ for which X _t, 0 ≤ t ≤ T, is a Brownian motion for some fixed T. We require an additional condition that $\mathbb{Q}$ is equivalent to $\mathbb{P}$, i.e., $\mathbb{P}(A) = 0$ if and only if $\mathbb{Q}(A) = 0$. In other words, an event occurs with positive $\mathbb{P}$-probability if and only if it happens with positive $\mathbb{Q}$-probability. Such a condition is important in financial applications since we have to deal with the same set of asset price movements even when we switch to a new probability measure. Igor Girsanov proved the existence of such a measure $\mathbb{Q}$. We will find first a necessary condition for the existence of an equivalent probability measure $\mathbb{Q}$ for which a Brownian motion with drift is a Brownian motion. Such a necessary condition will turn out to be crucial in defining $\mathbb{Q}$.

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Chapter 9. The Reflection Principle of Brownian Motion

Abstract

We investigate the reflection properties of Brownian motion. The results in this chapter will be used for the pricing of barrier options in Sect. 18.2 For the sake of simplicity of exposition we consider only one barrier problems.

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Itô Calculus

Frontmatter

Chapter 10. The Itô Integral

Abstract

We define the Itô integral of a stochastic process and investigate its properties. To define a Riemann–Stieltjes type integral $\int _{0}^{T}f(t)\,\mathrm{d}\alpha (t)$ using a function $\alpha: [0,T] \rightarrow \mathbb{R}$ as an integrator, we need the condition that the variation of α is bounded. (For the definition of variation, see Sect. A.3.) However, a sample path of a Brownian motion is of unbounded variation since the growth rate of δ W is approximately equal to $\sqrt{\delta t}$, which is very large compared with δ t as $\delta t \downarrow 0$. Therefore a Brownian sample path cannot be used as an integrator in a definition of a Riemann–Stieltjes type integral. K. Itô’s idea is to take a suitable average over all possible Brownian paths. This idea will be explained gradually since it requires a considerable amount of preparation. For an elementary introduction to the Itô integral, see [82, 102].

Geon Ho Choe

Chapter 11. The Itô Formula

Abstract

The Itô formula, or the Itô lemma, is the most frequently used fundamental fact in stochastic calculus. It approximates a function of time and Brownian motion in a style similar to Taylor series expansion except that the closeness of approximation is measured in terms of probabilistic distribution of the increment in Brownian motion.

Geon Ho Choe

Chapter 12. Stochastic Differential Equations

Abstract

Let $\mathbf{W} = (W^{1},\ldots,W^{m})$ be an m-dimensional Brownian motion, and let

$$\displaystyle{\boldsymbol{\sigma }= (\sigma _{ij})_{1\leq i\leq d,1\leq j\leq m}: [0,\infty ) \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d} \times \mathbb{R}^{m}}$$

and

$$\displaystyle{\boldsymbol{\mu }= (\mu ^{1},\ldots,\mu ^{d}): [0,\infty ) \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}}$$

be continuous functions where $\boldsymbol{\sigma }$ is regarded as a d × m matrix.

Geon Ho Choe

Chapter 13. The Feynman–Kac Theorem

Abstract

As an alternative method for the derivation of the Schrödinger differential equation in quantum mechanics, the path integral approach was introduced in the 1960s by the physicist Richard Feynman. Along a similar line a certain type of partial differential equation can be solved using the expectation over the sample paths of a stochastic process.

Geon Ho Choe

Option Pricing Methods

Frontmatter

Chapter 14. The Binomial Tree Method for Option Pricing

Abstract

Not long after the partial differential equation approach was developed for option pricing by Black, Scholes [6] and Merton [65] in 1973, the binomial tree method was introduced by Cox, Ross and Rubinstein [23] in 1979, which is a discrete time model and much easier to understand and implement in practice.

Geon Ho Choe

Chapter 15. The Black–Scholes–Merton Differential Equation

Abstract

The simultaneous publications of Black and Scholes [6] and Merton [65] in 1973 mark the beginning of the theory of option pricing. Using the theory of stochastic calculus, they derived the so-called Black–Scholes–Merton differential equation. It is essentially a heat equation with the direction of time reversed.

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Chapter 16. The Martingale Method

Abstract

In this chapter we introduce two proofs of the option pricing formula given by (16.2) by applying martingale theory. The first method is based on hedging of a portfolio process, and the second on replication of the payoff at expiry date T.

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Examples of Option Pricing

Frontmatter

Chapter 17. Pricing of Vanilla Options

Abstract

The Black–Scholes–Merton formula for a European call option is derived in Sect. 16.3 using the martingale method. In this chapter we present more examples of pricing of vanilla options. For more information consult [9] and [29].

Geon Ho Choe

Chapter 18. Pricing of Exotic Options

Abstract

Options with nonstandard features are called exotic options. In this chapter we introduce exotic options such as Asian options and barrier options. For an encyclopedic collection of option pricing formulas consult [36].

Geon Ho Choe

Chapter 19. American Options

Abstract

An option that can be exercised early, i.e., before or on the expiry date, is called an American option while a European option can be exercised only on the expiry date. Such an early exercise feature makes the pricing of American options harder. In the last section we introduce a very practical algorithm called the least squares Monte Carlo method, which is based on regression to estimate the continuation values from simulated sample paths.

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Portfolio Management

Frontmatter

Chapter 20. The Capital Asset Pricing Model

Abstract

How can we measure the performance of mutual funds and their investment risk? What is the use of a market index such as S&P 500? The portfolio theory can provide us with the answers. This chapter presents the Capital Asset Pricing Model (CAPM) , which deals with an efficient portfolio management. For a historical introduction see [4].

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Chapter 21. Dynamic Programming

Abstract

We investigate continuous time models for combined problems of optimal portfolio selection and consumption. An optimal strategy maximizes a given utility, and the solution depends on what to choose as a utility function. Under the assumption that a part of wealth is consumed, we find an explicit solution for optimal consumption and investment under certain assumptions. Utility increases as more wealth is spent, however, less is reinvested and the capability for future consumption decreases, therefore the total utility over the whole period under consideration may decrease. See [9, 63, 64, 83] for more information.

Geon Ho Choe

Interest Rate Models

Frontmatter

Chapter 22. Bond Pricing

Abstract

In this chapter we derive a fundamental pricing equation for bond pricing under a general assumption on interest rate movements. For a comprehensive introduction to bonds, consult [100].

Geon Ho Choe

Chapter 23. Interest Rate Models

Abstract

While the assumption that the interest rate is constant produces reasonable estimates in option pricing, the same assumption would produce less reliable results in pricing bonds and interest rate derivatives. One of the reasons is that the bonds usually have much longer maturity.

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Chapter 24. Numeraires

Abstract

A numeraire is a reference asset against which all other assets are evaluated. For example, the concept of time value of money is equivalent to discounting assets using the risk-free bond as a numeraire. Sometimes, a suitable choice of a numeraire makes the computation of option prices easier.

Geon Ho Choe

Computational Methods

Frontmatter

Chapter 25. Numerical Estimation of Volatility

Abstract

Volatility is the most important parameter in the geometric Brownian motion model for asset price movement for option pricing. All other parameters such as asset price, strike price, time to expiry and the risk-free interest rate can be observed in the financial market.

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Chapter 26. Time Series

Abstract

A time series is a sequence of collected data over a time period. Theoretically, we usually assume that a given sequence is infinitely long into the future and sometimes also into the past, and it is regarded as an observed random sample realized from a sequence of random variables. Time series models are used to analyze historical data and to forecast future movement of market variables. Since financial data is collected at discrete time points, sometimes it is appropriate to use recursive difference equations rather than differential equations which are defined for continuous time. Throughout the chapter we consider only the discrete time models. A process is stationary if all of its statistical properties are invariant in time, and a process is weakly stationary if its mean, variance and covariance are invariant in time.

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Chapter 27. Random Numbers

Abstract

The Monte Carlo method was invented by John von Neumann and Stanislaw Ulam. It is based on probabilistic ideas and can solve many problems at least approximately. The method is powerful in option pricing which will be investigated in Chap. 28.We need random numbers to apply the method. For efficiency, we need a good random number generator.

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Chapter 28. The Monte Carlo Method for Option Pricing Monte Carlo method

Abstract

Option price is expressed as an expectation of a random variable representing a payoff. Thus we generate sufficiently many asset price paths using random number generators, and evaluate the average of the payoff. In this chapter we introduce efficient ways to apply the Monte Carlo method. The key idea is variance reduction, which increases the precision of estimates for a given sample size by reducing the sample variance in the application of the Central Limit Theorem. The smaller the variance is, the narrower the confidence interval becomes, for a given confidence level and a fixed sample size.

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Chapter 29. Numerical Solution of the Black–Scholes–Merton Equation

Abstract

The price of a European call option is given by the Black–Scholes–Merton partial differential equation with the payoff function $(x - K)^{+}$ as the final condition. However, for a more general option with an arbitrary payoff function there is no simple formula for option price, and we have to resort to numerical methods studied in this Chapter. For further information, the reader is referred to [38, 92, 98].

Geon Ho Choe

Chapter 30. Numerical Solution of Stochastic Differential Equations

Abstract

Stochastic differential equations (SDEs) including the geometric Brownian motion are widely used in natural sciences and engineering. In finance they are used to model movements of risky asset prices and interest rates. The solutions of SDEs are of a different character compared with the solutions of classical ordinary and partial differential equations in the sense that the solutions of SDEs are stochastic processes. Thus it is a nontrivial matter to measure the efficiency of a given algorithm for finding numerical solutions.

Geon Ho Choe

Backmatter

Titel: Stochastic Analysis for Finance with Simulations
verfasst von: Geon Ho Choe
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-25589-7
Print ISBN: 978-3-319-25587-3
DOI: https://doi.org/10.1007/978-3-319-25589-7