Equilibrium Theory: A Simple Production Model

This is the first of several chapters dealing with the dynamic equilibrium theory. As an instructive first example we study a simple Cox–Ingersoll–Ross type of production model. The equilibrium concept is given a precise formulation and we derive the equilibrium short rate as well as the equilibrium stochastic discount factor. We also study the associated optimization problem for a central planner and prove that this is equivalent to the equilibrium problem.

Portfolio Optimization in Incomplete Markets

The object of this chapter is to give an overview of the dual approach to portfolio optimization in incomplete markets. The main result of this theory is that to every optimal investment problem there is a dual problem where we minimize a dual objective function over the class of martingale measures. For the case of a finite sample space we can present the full theory, but for the general case we only outline the proof. The theory is closely connected to convex duality theory and to the martingale approach to optimal consumption/investment discussed in Chapter 27.

Minimizing f-Divergence

The f-divergence between two measures can be viewed as a generalized “distance” between the measures. In order to find a unique martingale measure we can then choose the measure which minimizes the f-divergence to the objective measure. We derive the necessary theory of f-divergences and we present the corresponding dual utility maximization theory. We also study some examples, for example, minimal entropy measures.

Bonds and Interest Rates

In this chapter the reader is introduced to the basic concepts of interest rate theory. Starting with a market for zero coupon bonds we define the relevant interest rates such as the short rate, the spot rates, and the forward rates. There is an in-depth study of the relations between the dynamics of these rates, and we also discuss some more applied topics as fixed coupon bonds, floating rate bonds, yields, duration, and convexity.

Dividends

We extend the previously derived theory to include the case when the underlying assets are paying dividends. After a short discussion of discrete dividends we mainly study the case of continuous dividends. The theory is derived by reducing the dividend-paying model to an equivalent standard model with no dividends. For the case of a constant dividend yield we derive explicit option pricing formulas.

Multidimensional Models: Martingale Approach

In this chapter we study a very general multidimensional Wiener-driven model using the martingale approach. Using the Girsanov Theorem we derive the martingale equation which is used to find an equivalent martingale measure. We provide conditions for absence of arbitrage and completeness of the model, and we discuss hedging and pricing. For Markovian models we derive the relevant pricing PDE and we also provide an explicit representation formula for the stochastic discount factor. We discuss the relation between the market price of risk and the Girsanov kernel and finally we derive the Hansen–Jagannathan bounds for the Sharpe ratio.

A Primer on Incomplete Markets

We discuss market incompleteness within the relatively simple framework of a factor model. The corresponding pricing PDE is derived and we relate it to the market price of risk.

Arbitrage Pricing

The chapter starts with a detailed discussion of the bank account in discrete and continuous time. The Black–Scholes model is then introduced, and using the principle of no arbitrage we study the problem of pricing an arbitrary financial derivative within this model. Using the classical delta hedging approach we derive the Black–Scholes PDE for the pricing problem and using Feynman–Kač we also derive the corresponding risk neutral valuation formula and discuss the connection to martingale measures. Some concrete examples are studied in detail and the Black–Scholes formula is derived. We also discuss forward and futures contracts, and we derive the Black-76 futures option formula. We finally discuss the concepts and roles of historic and implied volatility.

Utility Indifference Pricing and Other Topics

In this chapter we discuss two methods of pricing in incomplete markets, based on utility functions. This theory comes in the shape of a global and a local version. Both versions are discussed, and for the local version we connect to the theory of stochastic discount factors and equilibrium theory.

Optimal Stopping Theory and American Options

In this chapter we present the dynamic programming approach to optimal stopping problems. We start by presenting the discrete time theory, deriving the relevant Bellman equation. We present the Snell envelope and prove the Snell Envelope Theorem. For Markovian models we explore the connection to alpha-excessive functions. The continuous time theory is presented by deriving the free boundary value problem connected to the stopping problem, and we also derive the associated system of variational inequalities. American options are discussed in some detail.