Pricing in (In)Complete Markets: Structural Analysis and Applications

ISBN-13: 9783540208174 / Angielski / Miękka / 2004 / 122 str.

Angelika Esser

Opis

In this book, the authors investigate structural aspects of no arbitrage pricing of contingent claims and applications of the general pricing theory in the context of incomplete markets. A quasi-closed form pricing equation in terms of artificial probabilities is derived for arbitrary payoff structures. Moreover, a comparison between continuous and discrete models is presented, highlighting the major similarities and key differences. As applications, two sources of market incompleteness are considered, namely stochastic volatility and stochastic liquidity. Firstly, the general theory discussed before is applied to the pricing of power options in a stochastic volatility model. Secondly, the issue of liquidity risk is considered by focusing on the aspect of how asset price dynamics are affected by the trading strategy of a large investor.

Spis treści

1 Motivation and Overview.- 2 Pricing by Change of Measure and Numeraire.- 2.1 Introduction.- 2.2 Model Setup.- 2.3 Equivalent Measures.- 2.3.1 Radon-Nikodym Derivative.- 2.3.2 Martingale Measures.- 2.3.3 Change of Numeraire.- 2.4 Derivation of a General Pricing Equation.- 2.5 Is Every Equivalent Measure a Martingale Measure?.- 2.5.1 Complete Market.- 2.5.2 Incomplete Market.- 2.5.3 Review of the Pricing Equation.- 2.6 Conclusion.- 3 Comparison of Discrete and Continuous Models.- 3.1 Introduction.- 3.2 Dynamics of the Underlying Processes.- 3.2.1 Diffusion Model.- 3.2.2 Discrete Model.- 3.3 Model-Specific Change of Measure.- 3.3.1 Diffusion Model.- 3.3.2 Discrete Model.- 3.4 Normalized Price Processes.- 3.4.1 Discounted Price Processes and Risk-Neutral Measure.- 3.4.1.1 Diffusion Model.- 3.4.1.2 Discrete Model.- 3.4.2 Price Processes Normalized by a Risky Basis Asset.- 3.4.2.1 Diffusion Model.- 3.4.2.2 Discrete Model.- 3.4.3 Price Processes Normalized by a Portfolio.- 3.4.3.1 Diffusion Model.- 3.4.3.2 Discrete Model.- 3.5 Examples.- 3.5.1 Complete Market with Two Basis Assets in the Discrete Setup.- 3.5.2 Binomial Tree.- 3.5.3 Two Correlated Assets.- 3.5.4 Stochastic Volatility Setup.- 3.6 Conclusion.- 4 Valuation of Power Options.- 4.1 Introduction.- 4.2 General Pricing Equation.- 4.2.1 Power Option.- 4.2.2 Powered Option.- 4.2.3 Capped Power Option.- 4.3 Examples.- 4.3.1 Black-Scholes Model.- 4.3.1.1 Pricing Equation.- 4.3.1.2 Numeraire Portfolio.- 4.3.2 Stochastic Volatility Models.- 4.3.2.1 Attainable Payoffs.- 4.3.2.2 Quasi-Closed Form Pricing Equation.- 4.4 Conclusion.- 5 Modeling Feedback Effects Using Stochastic Liquidity.- 5.1 Introduction.- 5.2 The Liquidity Framework.- 5.2.1 Constant Liquidity.- 5.2.2 Stochastic Liquidity.- 5.2.2.1 The Model.- 5.2.2.2 Stock Price Dynamics with Feedback Effects.- 5.2.2.3 Risk-Neutral Dynamics.- 5.3 Examples.- 5.3.1 Numerical Analysis of the Effective Stock Price Dynamics for Two Trading Strategies.- 5.3.1.1 Typical Feedback Strategies.- 5.3.1.2 Parameter Specifications for the Sample Paths.- 5.3.1.3 Positive Feedback Strategy.- 5.3.1.4 Contrarian Feedback Strategy.- 5.3.2 Liquidity Insurance.- 5.3.2.1 Specification and Pricing of the Contract.- 5.3.2.2 Alternative Scenario.- 5.4 Conclusion.- 6 Summary and Outlook.- A Power Options in Stochastic Volatility Models.- A.1 Calculations of the Characteristic Functions.- A.2 Ornstein-Uhlenbeck Process for Volatility.- References.- Abbreviations.- List of Symbols.- List of Figures.- List of Tables.