Advanced finance theories

Contents
Preface
About the Author
Acknowledgements
Note for PhD Students
1 Utility Theory
	1.1 Risk Aversion and Certainty Equivalent
2 Pricing Kernel and Stochastic Discount Factor
	2.1 Arrow–Debreu State Prices
		2.1.1 The pricing kernel, φi
		2.1.2 Equilibrium model
	2.2 Cochrane Two-period Consumption Problem
		2.2.1 Stochastic discount factor
		2.2.2 Further notation
		2.2.3 Risk-free rate
		2.2.4 Risk corrections
		2.2.5 Idiosyncratic risk does not affect prices
	2.3 Expected Return-Beta Representation
3 Risk Measures
	3.1 One-period Portfolio Selection
	3.2 Rothschild and Stiglitz “Strict” Risk Aversion
		3.2.1 Efficient portfolio
		3.2.2 Portfolio analysis
	3.3 Merton’s Risk Measures
		3.3.1 Properties of Merton’s risk measure bp
		3.3.2 Relationship between bp and conditional expected return E[Zp|Ze]
		3.3.3 Discussion
	Exercises: Capital Market Theory, Risk Measures
4 Consumption and Portfolio Selection
	4.1 Basic Set-up
	4.2 One Risky and One Risk-Free Asset
		4.2.1 The Bellman equation
		4.2.2 Infinite time horizon
	4.3 Constant Relative Risk Aversion
		4.3.1 Solution for J
		4.3.2 Solution for C and w
		4.3.3 Economic interpretation
	4.4 Constant Absolute Risk Aversion
		4.4.1 Solve for J
		4.4.2 Solve for C* and w*
		4.4.3 Economic interpretation
	4.5 Hyperbolic Absolute Risk Aversion (HARA)
		4.5.1 Relationship with CRRA and CARA
		4.5.2 Portfolio choice
		4.5.3 Solution for J
		4.5.4 Solve for C* and w*
	4.6 Optimal Rules Under Finite Horizon
		4.6.1 CRRA with finite horizon
		4.6.2 CARA with finite horizon
	Exercises: Intertemporal Portfolio Section
5 Optimum Demand and Mutual Fund Theorem
	5.1 Asset Dynamics and the Budget Equation
	5.2 The Equation of Optimality
	5.3 Optimal Investment Weight and Special Cases
		5.3.1 No risk-free asset
		5.3.2 GBM and risk-free rate
		5.3.3 Economic interpretation
	5.4 Lognormality and Mutual Fund Theorem
		5.4.1 “Separation” or “mutual-fund” theorem
		5.4.2 Key assumptions and uniqueness
		5.4.3 Tobin–Markowitz separation theorem
	Exercises: Optimum Demand and Mutual Fund Separation
6 Mean–Variance Frontier
	6.1 Mean–Variance Frontier
		6.1.1 The Sharpe ratio
		6.1.2 Calculating the mean–variance frontier
		6.1.3 Decomposing the mean–variance frontier
		6.1.4 Spanning the frontier
		6.1.5 Hansen–Jagannathan bounds
7 Solving Black–Scholes with Fourier Transform
	7.1 Option Pricing with Fourier Transform
		7.1.1 Black–Scholes hedge portfolio
	7.2 Black–Scholes Fundamental PDE
		7.2.1 Fourier transform
		7.2.2 Solution through transform method
8 Capital Structure Theory
	8.1 Objective Function for the Firm
	8.2 Partial Equilibrium One-period Model
		8.2.1 Pricing kernel
		8.2.2 Probability-cum-utility function
		8.2.3 m assets
		8.2.4 Introducing the concept of dQ
		8.2.5 What is eητ?
	8.3 Payoff of Risky Debt
	8.4 Pricing Risky Debt
		8.4.1 Solving the FPDE
	8.5 Price of a Warrant
	8.6 Convertible Bond
		8.6.1 Reverse convertible
		8.6.2 Call option enhanced reverse convertible
		8.6.3 Policy implications
	8.7 Bankruptcy Cost and Tax Benefit
		8.7.1 Solution under time invariant
		8.7.2 Protected debt covenant
		8.7.3 Optimal capital structure
	8.8 Deposit Insurance
	Exercises: Capital Structure Theory
9 General Equilibrium
	9.1 Firms and Securities
	9.2 Individuals
	9.3 Aggregate Demand
	9.4 Market Portfolio
	9.5 Security Market Line
	9.6 Three-fund Separation
	9.7 Empirical Application of CAPM
	Exercises: General Equilibrium
10 Discontinuity in Continuous Time
	10.1 Counting and Marked Point Process
	10.2 Poisson Process
	10.3 Constant Jump Size
		10.3.1 Fundamental PDE with constant jump size
		10.3.2 Market price of jump risk
		10.3.3 European call price
		10.3.4 Immediate ruin
	10.4 Random Jump Size
		10.4.1 When J has a lognormal distribution
	10.5 Intertemporal Portfolio Selection with Jumps
		10.5.1 Portfolio selection
		10.5.2 Stock markets systemic and idiosyncratic risk
	Exercises: Discontinuity in Continuous Time
11 Spanning and Capital Market Theories
	11.1 Necessary Conditions for Non-trivial Spanning
	11.2 Efficient Portfolio and Spanning
	11.3 Market Portfolio Spanning and CAPM
	11.4 Arbitrage Pricing Theory (APT)
	11.5 Modigliani–Miller Hypothesis
	11.6 Comment on Spanning
	11.7 HARA
	Exercises: Spanning & Capital Market Theories
Bibliography
Calculus Notes
	Differentiation
	Integration
	Integral of Normally Distributed Variable
	Ito lemma
Index