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## Über dieses Buch

Yielding new insights into important market phenomena like asset price bubbles and trading constraints, this is the first textbook to present asset pricing theory using the martingale approach (and all of its extensions). Since the 1970s asset pricing theory has been studied, refined, and extended, and many different approaches can be used to present this material. Existing PhD–level books on this topic are aimed at either economics and business school students or mathematics students. While the first mostly ignore much of the research done in mathematical finance, the second emphasizes mathematical finance but does not focus on the topics of most relevance to economics and business school students. These topics are derivatives pricing and hedging (the Black–Scholes–Merton, the Heath–Jarrow–Morton, and the reduced-form credit risk models), multiple-factor models, characterizing systematic risk, portfolio optimization, market efficiency, and equilibrium (capital asset and consumption) pricing models. This book fills this gap, presenting the relevant topics from mathematical finance, but aimed at Economics and Business School students with strong mathematical backgrounds.

## Inhaltsverzeichnis

### Chapter 1. Stochastic Processes

This chapter presents the basics of stochastic processes needed to study asset pricing theory.

Robert A. Jarrow

### Chapter 2. The Fundamental Theorems

This chapter presents the three fundamental theorems of asset pricing. These theorems are the basis for pricing and hedging derivatives, understanding the risk return relations among assets including the notion of systematic risk, portfolio optimization, and equilibrium asset pricing.

Robert A. Jarrow

### Chapter 3. Asset Price Bubbles

An important recent development in the asset pricing literature is an understanding of asset price bubbles. This chapter discusses these new insights. They are motivated by the First and Third Fundamental Theorems, which show that NFLVR only implies the existence of a local martingale measure and not a martingale measure. Asset price bubbles clarify the economic meaning of this difference.

Robert A. Jarrow

### Chapter 4. Spanning Portfolios, Multiple-Factor Beta Models, and Systematic Risk

This chapter studies spanning portfolios, the multiple-factor beta model, and characterizes systematic risk. This is done for an incomplete market with asset prices that can have discontinuous sample paths. Multiple-factor beta models are used for active portfolio management and the determination of positive alphas. These models can be derived using only the Third Fundamental Theorem 2.5 of asset pricing. A special case of this chapter is Ross’s APT, which illustrates the notion of portfolio diversification.

Robert A. Jarrow

### Chapter 5. The Black–Scholes–Merton Model

This chapter presents the seminal Black–Scholes–Merton modelBlack–Scholes–Merton (BSM) model for pricing options. Since this chapter is a special case of the material contained in Sect. 2.7 in the fundamental theorems Chap. 2, the presentation will be brief.

Robert A. Jarrow

### Chapter 6. The Heath–Jarrow–Morton Model

This chapter presents the Heath–Jarrow–Morton modelHeath–Jarrow–Morton (HJM) (Heath et al, Econometrica 60(1):77–105, 1992) model for pricing interest rate derivatives. Given frictionless and competitive markets, and assuming a complete market, this is the most general arbitrage-free pricing model possible with a stochastic term structure of interest rates. This model, with appropriate modifications, can also be used to price derivatives whose values depend on a term structure of underlying assets, examples include exotic equity derivatives where the underlyings are call and put options, commodity options where the underlyings are futures prices, and credit derivatives where the underlyings are risky zero-coupon bond prices.

Robert A. Jarrow

### Chapter 7. Reduced Form Credit Risk Models

There are two models for studying credit risk. The first is called the structural approach. This model assumes that all of the assets of the firm trade, an unrealistic assumption. The second is called the reduced form model. This model assumes that only a subset of the firm’s liabilities trade, those that need to be priced and hedged. This is the model studied in this chapter.

Robert A. Jarrow

### Chapter 8. Incomplete Markets

This chapter studies the arbitrage-free pricing of derivatives in an incomplete marketsincomplete market satisfying NFLVR. This chapter is a modest generalization of the presentation contained in Pham (Continuous time stochastic control and optimization with financial applications. Springer, Berlin, 2009) to discontinuous risky asset price processes.

Robert A. Jarrow

### Chapter 9. Utility Functions

This chapter studies an investor’s utility functionsutility function. We start with a normalized market S,(Ft),ℙ $$\left (S,(\mathscr {F}_{t}),\mathbb {P}\right )$$ where the money market account’s (mma’s) value is B t  ≡ 1. For the chapters in Part I of this book, although unstated, we implicitly assumed that the trader’s beliefstrader’s beliefs were equivalent to the statistical probability measuretrader’s beliefsstatistical probability measure ℙ $$\mathbb {P}$$ , i.e. the trader’s beliefs and the statistical probability measure agree on zero probability events. For this part of the book, Part II, we let the probability measure ℙ $$\mathbb {P}$$ correspond to the trader’s beliefs. This should cause no confusion since we do not need additional notation for the statistical probability measure in this part of the book. We discuss differential beliefs in Sect. 9.7 below. In addition, consistent with this interpretation, we let the information filtration Ft $$\mathscr {F}_{t}$$ given above correspond to the trader’s information set. When we study the notion of an equilibrium in Part III of this book, we will introduce a distinction between the trader’s beliefs and the statistical probability measure, and a distinction between the trader’s information set and the market’s information set.

Robert A. Jarrow

### Chapter 10. Complete Markets (Utility over Terminal Wealth)

This chapter studies an individual’s portfolio optimization problem. In this complete marketsutility optimizationoptimization, the solution differs depending on whether the market is complete or incomplete. This chapter investigates the optimization problem in a complete markets setting, and the next chapter analyzes incomplete markets.

Robert A. Jarrow

### Chapter 11. Incomplete Markets (Utility over Terminal Wealth)

This chapter studies the investor’s portfolio optimization problem in an incomplete marketsutility optimizationincomplete market. The solution in this chapter parallels the solution for the complete market setting in the portfolio optimization Chap. 1.

Robert A. Jarrow

### Chapter 12. Incomplete Markets (Utility over Intermediate Consumption and Terminal Wealth)

This chapter studies the investor’s optimization problem in an incomplete market where the investor has a utility function defined over both terminal wealth and intermediate consumptionintermediate consumption. The presentation parallels the portfolio optimization problem studied in Chap. 11. This chapter is based on Jarrow.

Robert A. Jarrow

### Chapter 13. Equilibrium

This chapter presents the description of an economy, the definition of an economic equilibrium, and some necessary conditions implied by the existence of an economic equilibrium.

Robert A. Jarrow

### Chapter 14. A Representative Trader Economy

To characterize the equilibrium and to facilitate existence proofs, this chapter introduces the notion of a representative trader. A representative trader is a hypothetical individual whose trades, in a sense to be made precise below, reflect the aggregate trades of all individuals in the economy. A representative trader is defined by her beliefs, utility function, and endowments.

Robert A. Jarrow

### Chapter 15. Characterizing the Equilibrium

Assuming that an equilibrium exists, this chapter characterizes the economic equilibrium. The key result in this chapter is a characterization of the equilibrium supermartingale deflator as a function of the economy’s primitives: beliefs, preferences, and endowments. Indeed, using a representative trader economy equilibrium that reflects the equilibrium in the original economy, an equilibrium supermartingale deflator is characterized as a function of the representative trader’s (aggregate) utility function and aggregate market wealth. Finally, this chapter derives the intertemporal capital asset pricing model (ICAPM) and the consumption capital asset pricing model (CCAPM) as special cases of this characterization.

Robert A. Jarrow

### Chapter 16. Market Informational Efficiency

This chapter studies market informational efficiency which is a key concept used in financial economics, introduced by Fama in the early 1970s. A rigorous definition of an efficient market is provided which is contrasted with the intuitive definition. It is shown that this rigorous definition requires only the existence, and not the characterization of an economic equilibrium. Such a rigorous definition allows new insights into the testing of an informationally efficient market, which are discussed as well.

Robert A. Jarrow

### Chapter 17. Epilogue (The Static CAPM)

This chapter studies the static CAPMstatic CAPM for two reasons. First, because it is of historical interest. Second, because it highlights the advances and insights obtained from the dynamic models studied in this book. This chapter provides a new derivation of the static CAPM that uses the martingale approach.

Robert A. Jarrow

### Chapter 18. The Trading Constrained Market

This chapter introduces trading constraints to the markets studied in Parts I–III of this book. Most of the structure extends in a straightforward fashion.

Robert A. Jarrow

### Chapter 19. Arbitrage Pricing Theory

This chapter studies the modifications needed due to the introduction of trading constraints in the arbitrage pricing theory of the fundamental theorems Chap. 2. Most, but not all of the three fundamental theorems of asset pricing extend with trading constraints.

Robert A. Jarrow

### Chapter 20. The Auxiliary Markets

This chapter studies how to transform a trading constrained market into an “equivalent” market which is incomplete, but without trading constraints. The transformed market is called the auxiliary marketsauxiliary unconstrained market. The goal of this transformation is to use the theorems from an incomplete but unconstrained market to understand trading constrained markets.

Robert A. Jarrow

### Chapter 21. Super- and Sub-replication

This chapter studies super- and sub-replication in a trading constrained market. In a trading constrained market, not all derivatives can be synthetically constructed using constrained admissible s.f.t.s.’s. To price derivatives, we can obtain upper and lower bounds using super- and sub-replication. This is analogous to super- and sub-replication in an incomplete but unconstrained market, see the super- and sub-replication Chap. 8.

Robert A. Jarrow

### Chapter 22. Portfolio Optimization

This chapter studies a trader’s portfolio optimization problem under trading constraints.

Robert A. Jarrow

### Chapter 23. Equilibrium

This section studies equilibriumtrading constrained economyequilibrium in a trading constrained economy.

Robert A. Jarrow

### Backmatter

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