Introduction to Quantitative Methods for Financial Markets

Frontmatter

1. Interest, Coupons and Yields

Abstract

ach of us has experience with paying or receiving interest. If you wish to purchase goods today despite having insufficient funds, you can, for example, borrow money from a bank. Your desired purchases could include a house, a car or consumption goods, and the borrowing could be in the form of a current account overdraft or a term loan.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

2. Financial Products

Abstract

In Chapter 1, bonds have been introduced as an important class of financial assets which is structurally similar to loans. The authorized issuer promises in the bond contract to make future payments according to a fixed schedule, up to some final time T (the term or maturity of the bond).

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

3. The No-Arbitrage Principle

Abstract

The term arbitrage is used for making risk-free profit by buying and selling financial assets in one’s own account. Let π _t be the value of a portfolio at times t ≥ 0, with π ₀ = 0.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

4. European and American Options

Abstract

We discussed in Chapter 2 that an option gives the buyer a particular right which can lead to financial upsides in the future, without including any obligations. Hence, there must be a positive price for obtaining this right, and we will now aim to determine this price.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

5. The Binomial Option Pricing Model

Abstract

The following chapters will be dedicated to the stochastic modeling of price movements of financial assets. Chapters 5 to 8 will focus on stocks, while Chapter 9 will deal with interest rates.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

6. The Black-Scholes Model

Abstract

In the last chapter we introduced a binomial model, which provided an intuitive way for pricing derivatives and finding replicating portfolios. However, the binomial model often oversimplifies the real world, so that in practice one would aim to choose a model setup that better describes reality. In this chapter we will discuss a continuous-time model which is broadly considered today the classical model of mathematical finance.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

7. The Black-Scholes Formula

Abstract

For the Black-Scholes model, as introduced in the last chapter, we can now derive the no-arbitrage price of a European-style option – the so-called Black-Scholes formula. In Section 7.1, we will discuss a direct approach to obtaining the Black-Scholes formula as the solution of a partial differential equation.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

8. Stock-Price Models

Abstract

In Chapter 7 it has been shown that the Black-Scholes model allows to derive explicit formulas for the prices of European call and put options. Having explicit pricing formulas is a great advantage; however, the Black-Scholes model has also been found to not fully explain market prices due to some of its assumptions and properties.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

9. Interest Rate Models

Abstract

So far we have assumed that interest rates are given either as constants or as deterministic functions of time. However, in reality interest rates show stochastic behavior (cf. Fig.​ 1.​2). While this often only plays a secondary role when dealing with stock derivatives, it is, of course, the core aspect when pricing interest rate derivatives. After a brief introduction to some of the most commonly traded interest rate products, this chapter will present a selection of popular interest rate models.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

10. Numerical Methods

Abstract

Numerical techniques prove particularly useful when explicit solution formulas in a certain model cannot be derived even for simple derivatives (e.g. in the Black-Karasinski model) or when the to-be-priced financial instrument has a complex structure so that analytical methods fail (e.g. if multiple cancelation rights exist).

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

11. Simulation Methods

Abstract

Many problems that arise in financial mathematics are structurally complex, so that one often cannot obtain explicit results (such as explicit pricing formulas for derivatives) or successfully apply numerical methods as outlined in Chapter 10. In such cases stochastic simulation can offer an efficient and powerful alternative for obtaining numerical estimates for specific quantities.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

12. Calibrating Models – Inverse Problems

Abstract

In the previous chapters we studied several model choices to describe stock price and interest rate dynamics. When using models to valuate derivatives or to obtain a hedging strategy, the used parameters will greatly impact the results. While there is broad agreement of how to model many problems in physics (such as the thermal conductivity of copper at room temperature), financial markets are fundamentally different. Many market participants have different views on the distributions of market variables, and market prices of liquid assets only represent an economic equilibrium resulting from those different views.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

13. Case Studies: Exotic Derivatives

Abstract

Today’s financial markets offer a wide range of complex financial products. In this chapter we will introduce several structured financial instruments and discuss ideas for their valuation. The exercises at the end of the chapter will then further illustrate the specific features of the presented instruments.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

14. Portfolio Optimization

Abstract

Insurers, banks, mutual funds, sovereign wealth funds and also individuals invest money in the financial markets in order to generate financial returns. Hereby the investor allocates capital to different investments, such as low-risk low-expected return investments (e.g. high quality government bonds, bank accounts) or higher-risk higher-expected return investments (e.g. stocks, real estate, commodities). It is one of the core problems in finance to provide decision making tools for the optimal (or: efficient) allocation of capital. Optimality in this context depends on the decision maker’s liquidity needs and risk aversion. This chapter will introduce the classical mean-variance optimization framework in a static one-period setup, and proceed to continuous-time portfolio optimization problems.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

15. Introduction to Credit Risk Models

Abstract

Lending money is one of the core businesses of banks. The income from this business line comes in the form of interest income and we will now discuss why different borrowers will be charged different interest costs in the same lending market.

Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

Backmatter