Hansjoerg Albrecher

The Binomial Option Pricing Model

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.

Introduction to Quantitative Methods for Financial Markets

I Interest Rates.- II Financial Products.- III The No-Arbitrage Principle.- IV European and American Options.- The Binomial Option Pricing Model.- VI The Black-Scholes Model.- VII The Black-Scholes Formula.- VIII Stock-Price Models.- IX Interest Rate Models and the Valuation of Interest Rate Derivatives.- X Numerical Tools.- XI Simulation Methods.- XII Calibrating Models - Inverse Problems.- XIII Case Studies: Exotic Derivatives.- XIV Portfolio-Optimization.- XV Introduction to Credit Risk Models.

Interest, Coupons and Yields

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.

European and American Options

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.

Case Studies: Exotic Derivatives

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.

The No-Arbitrage Principle

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.

Interest Rate Models

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.

Introduction to Credit Risk Models

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.

Calibrating Models – Inverse Problems

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.

The Black-Scholes Model

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.