Asset Pricing in Discrete Time A Complete Markets Approach

by Poon, Ser-Huang; Stapleton, Richard C.

• ISBN13: 9780199271443
• ISBN10: 0199271445
• Format: Hardcover
• Copyright: 2005-04-07
• Publisher: Oxford University Press

Summary

Relying on the existence, in a complete market, of a pricing kernel, this book covers the pricing of assets, derivatives, and bonds in a discrete time, complete markets framework. It is primarily aimed at advanced Masters and PhD students in finance.-- Covers asset pricing in a single period model, deriving a simple complete market pricing model and using Stein's lemma to derive a version of the Capital Asset Pricing Model.-- Looks more deeply into some of the utility determinants of the pricing kernel, investigating in particular the effect of non-marketable background risks on the shape of the pricing kernel.-- Derives the prices of European-style contingent claims, in particular call options, in a one-period model; derives the Black-Scholes model assuming a lognormal distribution for the asset and a pricing kernel with constant elasticity, and emphasizes the idea of a risk-neutral valuationrelationship between the price of a contingent claim on an asset and the underlying asset price.-- Extends the analysis to contingent claims on assets with non-lognormal distributions and considers the pricing of claims when risk-neutral valuation relationships do not exist.-- Expands the treatment of asset pricing to a multi-period economy, deriving prices in a rational expectations equilibrium.-- Uses the rational expectations framework to analyse the pricing of forward and futures contracts on assets and derivatives.-- Analyses the pricing of bonds given stochastic interest rates, and then uses this methodology to model the drift of forward rates, and as a special case the drift of the forward London Interbank Offer Rate in the LIBOR Market Model.

Author Biography

Dick Stapleton, one of the most senior finance academics in Europe, has held senior posts at the Universities of Strathclyde, Lancaster, and Cambridge, and Manchester Business School. He is also a Professorial Fellow at the University of Melbourne, Australia. He has researched in many areas of finance including asset pricing and interest rate derivatives and has published extensively in all top ranking finance and economic journals. Ser-Huang Poon is known for her work in modelling and forecasting financial market volatility, and more recently the applications of extreme values theories in finance. She has published work on both areas in leading journals in finance and economics.

Table of Contents

Preface	vii
1 Asset Prices in a Single-period Model	1	(18)
1.1 Initial Setup and Key Assumptions	1	(2)
1.2 Properties of the State Price, qi	3	(1)
1.3 A Simplification of the State Space	3	(1)
1.4 The Pricing Kernel, øi	4	(3)
1.5 The Capital Asset Pricing Model	7	(2)
1.6 The Arbitrage Pricing Theory	9	(1)
1.7 Risk Aversion and the Pricing Kernel in an Equilibrium Model	10	(2)
1.8 Examples	12	(1)
1.8.1 Case 1: Risk Neutrality	12	(1)
1.8.2 Case 2: Utility is Quadratic	12	(1)
1.9 A Note on the Equivalent Martingale Measure	13	(1)
1.10 A Note on the Asset Specific Pricing Kernel, ψ(χj)	14	(1)
1.11 Conclusions	15	(4)
2 Risk Aversion, Background Risk, and the Pricing Kernel	19	(20)
2.1 Risk Aversion and Declining Marginal Utility of Wealth	19	(2)
2.2 Absolute Risk Aversion	21	(9)
2.2.1 Relative Risk Aversion	22	(3)
2.2.2 The Elasticity of the Pricing Kernel	25	(1)
2.2.3 Prudence	26	(4)
2.3 Background Risk and the Pricing Kernel	30	(5)
2.3.1 Consumption Optimisation Under Background Risk	32	(1)
2.3.2 The Precautionary Premium and the Shape of the Pricing Kernel	33	(2)
2.4 Conclusion	35	(1)
2.5 Appendix: Properties of the Precautionary Premium	35	(4)
3 Option Pricing in a Single-period Model	39	(18)
3.1 The General Case	39	(1)
3.2 An example: Quadratic Utility and Joint-normal Distribution for χi and χm	40	(1)
3.3 Option Valuation When χj is Lognormal	41	(6)
3.3.1 Notation for the Lognormal Case	42	(1)
3.3.2 The Asset-Specific Pricing Kernel	42	(1)
3.3.3 The Risk-adjusted PDF	43	(3)
3.3.4 The Forward Price of the Underlying Asset under Lognormality	46	(1)
3.3.5 The Lognormal RNVR	46	(1)
3.4 The Black-Scholes Price of a European Call Option	47	(4)
3.4.1 Some Applications of the General Black-Scholes Formula	49	(2)
3.5 The Black-Scholes Model and the Elasticity of the Pricing Kernel	51	(1)
3.6 Sufficient Conditions for ψ(χj) to have Constant Elasticity	52	(1)
3.7 Conclusion	53	(1)
3.8 Appendix: The Normal Distribution	54	(3)
4 Valuation of Contingent Claims: Extensions	57	(20)
4.1 Sufficient Conditions for Constant Elasticity	58	(3)
4.1.1 Asset Price Follows a Geometric Brownian Motion	58	(1)
4.1.2 Lognormal Wealth and Power Utility	59	(2)
4.2 RNVR on Non-Lognormal Prices	61	(4)
4.2.1 The Transformed Normal Distribution	61	(1)
4.2.2 The Asset-specific Pricing Kernel	62	(1)
4.2.3 The Price of Contingent Claims	63	(1)
4.2.4 Pricing Options on a Normally Distributed Asset Price	63	(2)
4.3 A Generalisation of the RNVR: Missing Parameters in the Option Pricing Function	65	(3)
4.4 Contingent Claim Pricing given Non-constant Elasticity of the Pricing Kernel	68	(6)
4.4.1 Bounds on Option Prices	73	(1)
4.5 Conclusions	74	(1)
4.6 Appendix	74	(3)
4.6.1 The Mean of a Truncated Normal Variable	74	(3)
5 Multi-period Asset Pricing	77	(20)
5.1 Basic Setup	77	(1)
5.2 A Complete Market: The Multi-period Case	78	(2)
5.3 Pricing Multi-period Cash Flows	80	(6)
5.3.1 The Time-State Preference Approach	80	(2)
5.3.2 The Rational Expectations Model	82	(2)
5.3.3 The Relationship between the Time-State Preference and the Rational Expectations Equilibria Prices	84	(1)
5.3.4 The Relationship between the Pricing Kernels when Interest Rates are Non-stochastic	85	(1)
5.4 Multi-Period Valuation Equilibrium: Joint-Normal Cash Flows	86	(3)
5.5 Time-State Preference: Pricing Kernels in a Multi-period Equilibrium	89	(3)
5.6 Marginal Utility of Consumption and Wealth in a Normal Distribution and Exponential Utility Model	92	(2)
5.7 Conclusions	94	(3)
6 Forward and Futures Prices of Contingent Claims	97	(16)
6.1 Forward and Futures Cash Flows	97	(1)
6.2 No-arbitrage Relationships	98	(2)
6.3 Forward and Futures Prices in a Rational Expectations Model	100	(4)
6.4 Futures and Forward Prices given Lognormal Variables	104	(3)
6.5 Futures Rates and Forward Rates in a Normal Interest-rate Model	107	(1)
6.6 Futures and Forward Prices of European-style Contingent Claims	108	(2)
6.7 Conclusions and Further Reading	110	(3)
7 Bond Pricing, Interest-rate Processes, and the LIBOR Market Model	113	(18)
7.1 Bond Pricing under Rational Expectations	113	(4)
7.1.1 Bond Forward Prices	115	(1)
7.1.2 Some Further Implications of Forward Parity and Rational Expectations	116	(1)
7.2 The Drift of Forward Rates	117	(7)
7.2.1 FRA Pricing and the Drift of the Forward Rate: One-period Case	119	(1)
7.2.2 FRA Pricing and the Drift of the Forward Rate: Two-period Case	120	(2)
7.2.3 The Drift of the Forward Rate under Lognormality	122	(2)
7.3 An Application of the Forward Rate Drift: The LIBOR Market Model	124	(2)
7.4 Conclusions	126	(1)
7.5 Appendix	127	(4)
Appendix: Stein's Lemma	131	(4)
Bibliography	135	(4)
Index	139