Contemporary Quantitative Finance Essays in Honour of Eckhard Platen 1st Edition by Carl Chiarella, Alexander Novikov 3642034780 9783642034787

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ISBN 10: 3642034780

ISBN 13: 9783642034787

Author: Carl Chiarella, Alexander Novikov

This volume contains a collection of papers dedicated to Professor Eckhard Platen to celebrate his 60th birthday, which occurred in 2009. The contributions have been written by a number of his colleagues and co-authors. All papers have been – viewed and presented as keynote talks at the international conference “Quantitative Methods in Finance” (QMF) in Sydney in December 2009. The QMF Conference Series was initiated by Eckhard Platen in 1993 when he was at the Australian – tional University (ANU) in Canberra. Since joining UTS in 1997 the conference came to be organised on a much larger scale and has grown to become a signi?cant international event in quantitative ?nance. Professor Platen has held the Chair of Quantitative Finance at the University of Technology, Sydney (UTS) jointly in the Faculties of Business and Science since 1997. Prior to this appointment, he was the Founding Head of the Centre for Fin- cial Mathematics at the Institute of Advanced Studies at ANU, a position to which he was appointed in 1994. Eckhard completed a PhD in Mathematics at the Technical University in Dresden in 1975 and in 1985 obtained his Doctor of Science degree (Habilitation degree in the German system) from the Academy of Sciences in Berlin where he headed the Stochastics group at the Weierstrass Institute.

Contemporary Quantitative Finance Essays in Honour of Eckhard Platen 1st Table of contents:

Introduction

Preliminaries

Change of Variables

The Model

Numéraire and Log-Optimality Properties

Relative Arbitrage

The Föllmer “Exit Measure”

The Functionally-Generated Portfolio

Induced Drifts

Conditioning

Numéraire and Log-Optimality Properties of pi (·)

Relative Entropy

Stochastic Control

Stochastic Game

Conclusion

References

Finitely Additive Probabilities and the Fundamental Theorem of Asset Pricing

Introduction

Arbitrages of the First Kind and Weakly Equivalent Local Martingale Measures

General Probabilistic Remarks

The Market and Investing

Arbitrages of the First Kind

Weakly Equivalent Local Martingale Measures

Local Probabilities Weakly Equivalent to P

Density Processes

Local Martingales

Weakly Equivalent Local Martingale Measures

The Main Result

The FTAP of Delbaen and Schachermayer

Proving the FTAP

NFLVR and the Supermartingale Property of Wealth Processes Under a WELMM

The Case of Continuous-Path Semimartingales

References

M6-On Minimal Market Models and Minimal Martingale Measures

Introduction

General Financial Market Models

Continuous Financial Market Models

Minimal Market Models

References

The Economic Plausibility of Strict Local Martingales in Financial Modelling

Introduction

An Overview of the Two Models

The Stock Price Bubble

The Bond Price Bubble

References

A Remarkable sigma-finite Measure Associated with Last Passage Times and Penalisation Problems

Notation

Introduction

A New Kind of Augmentation of Filtrations Consistent with the Problem of Extension of Measures

Understanding the Problem

The N-usual Augmentation

Extension of Measures and the N-usual Augmentation

A Universal sigma-finite Measure Q

The Class (Sigma)

A Special Case Related to Financial Modeling

The General Case

Further Properties of Q and Some Remarkable Associated Martingales, and Penalisation Results

References

Pricing Without Equivalent Martingale Measures Under Complete and Incomplete Observation

Introduction

The Model

Relative Arbitrage and Absence of Martingale Measures

Pricing Under Absence of an Equivalent Martingale Measure (the Case of Complete Observation)

The “Fernholz-Karatzas Approach”

The “Benchmark Approach” of Platen

Equivalence of the Two Pricing Approaches

Pricing Under Absence of an Equivalent Martingale Measure (the Case of Incomplete Observation)

The Model

Corresponding Complete Observation Model

Price Dynamics

GOP Dynamics

GOP Strategy and Growth Rate Under Complete and Incomplete Observation

Expected Utility Maximization Under Complete and Incomplete Observation

Pricing According to the “Fernholz-Karatzas Approach”

First Variant

Second Variant

Pricing According Platen’s “Benchmark Approach”

Comments on the Equivalence of the Two Approaches in the Case of Incomplete Observations

References

Existence and Non-uniqueness of Solutions for BSDE

Notation

A Convergence Result

Pathwise Subquadratic Drivers

The Relation with Minimal Elements

An Example

References

Comparison Theorems for Finite State Backward Stochastic Differential Equations

Introduction

A Mathematical Setting

Some Lemmas

Linear BSDEs

A Scalar Comparison Theorem

General Comparison Theorems

F-Expectations

Applications to Risk Measures

References

Results on Numerics for FBSDE with Drivers of Quadratic Growth

Introduction

Preliminaries

Pricing and Hedging with Correlated Assets

Smoothness and Path Regularity Results

Smoothness Results

Regularity and Bounds for the Solution Process

A Truncation Procedure

The Exponential Transformation Method

Back to the Pricing Problem

Some Results on BMO Martingales

Basics of Malliavin’s Calculus

Some Results on SDE

Path Regularity for Lipschitz FBSDE

References

Variance Swap Portfolio Theory

Introduction

Autocorrelation and the Hardy Littlewood Gauss Transform

NonGaussian Dependence Model

Variance Swap Fixed Leg Pricing

Optimization Criterion and Portfolio Design

Results on Portfolio Design

Conclusion

References

Stochastic Partial Differential Equations and Portfolio Choice

Introduction

The Investment Model

The Backward Formulation of the Portfolio Choice Problem and the Associated SPDE

Examples: Markovian Stochastic Factor Models

Single Stochastic Factor Models

The CRRA Case: uT(x) = 1gamma xgamma, 0<gamma<1, gamma0

The Logarithmic Case: uT(x) =lnx, x>0

Multi-stochastic Factor Models

The Forward Formulation of the Portfolio Choice Problem and the Associated SPDE

Stochastic Optimization and Forward Investment Performance Process

Examples

The Case of Zero Volatility: a(x,t) 0

Single Stochastic Factor Models

Cases of Non-zero Volatility

The Market View Case: a(x,t) = U(x,t) phit

The Benchmark Case: a(x,t) = xUx (x,t) deltat

The Combined Market View/Benchmark Case: a (x,t) = -xUx (x,t) deltat + U (x,t) phit

Single Stochastic Factor Models

References

Issuers’ Commitments Would Add More Value than Any Rating Scheme Could Ever Do

Introduction

Rating Schemes

Floor

Conclusion

References

Pricing and Hedging of CDOs: A Top Down Approach

Introduction

(T,x)-Bond Dynamics

(T,x)-Forward Rates

Relation to the Top-Down Model in Lipton-Shelton 13.lipshe09

Single Tranche CDOs (STCDOs)

Variance-Minimizing Hedging

Deterministic Risk Free Rates

Affine Term Structure

Proof of Proposition 1

Conclusion and Outlook

References

Constructing Random Times with Given Survival Processes and Applications to Valuation of Credit Deri

Introduction

Filtering Example

Azéma Supermartingale

Conditional Distributions

Preliminary Results

Properties of (P,G)-Martingales

Girsanov’s Theorem

Construction Through a Change of Measure

Case of the Brownian Filtration

Applications to Valuation of Credit Derivatives

Defaultable Zero-Coupon Bonds

Credit Default Swaps

References

Representation of American Option Prices Under Heston Stochastic Volatility Dynamics Using Integral

Introduction

Problem Statement – the Heston Model

Finding the Density Function Using Integral Transforms

Solution for the American Call Option

Conclusion

Deriving the Jamshidian Formulation Under Stochastic Volatility

Proof of Proposition 1

Proof of Proposition 3

Proof of Proposition 4

Proof of Proposition 5 – The Inverse Laplace Transform

Proof of Proposition 8 – The European Option Price

Proof of Proposition 9

The Evaluation of Some Complex Integral Terms Occurring in Appendices 6 and 7

References

Buy Low and Sell High

Introduction

Preliminaries

Buying Problems

Selling Problems

Optimal Buying and Selling Strategies

Buying Strategies

Selling Problems

Concluding Remarks

Appendix: Proofs

Some Transformations

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3

References

Continuity Theorems in Boundary Crossing Problems for Diffusion Processes

Introduction

Approximation Rates in the Brownian Motion Case

Approximation Rates for General Diffusions

Differentiability of the Boundary Crossing Probabilities

References

Binomial Models for Interest Rates

The Methodology

The Weak Solution in Continuous Time

The Approximation of the Weak Solution

The Black and Scholes Model

One Factor Interest Rate Models

The Hull and White One-factor Model

A Black-Karasinsky Type Model

Cox, Ingersoll and Ross Model

Two Factor Interest Rate Models

The Brennan-Schwartz Two-factor Model

The Hull-White Two-factor Model

Conclusions

Appendix

References

Lognormal Forward Market Model (LFM) Volatility Function Approximation

Introduction

Generic Forward Rates

Base Definitions

Generic Forward Rate Definition

Interpolation by Day-Count Fractions

Caplet Volatility

Ito Calculus

Step 1

Step 2: Derivation of nuX

Step 3: Derivation of nuY

Step 4

Non-standard Forward SDE

Volatility Approximation

Dead Rate Volatility

Caplet Volatility

LFM Convexity Adjustment

Swaption Volatility Calculation

Definition

Approximate Swaption Volatility Formula

Testing the Approximate Formulae

Assumptions and Descriptions for Performance Tests

Conclusion

Appendix

Quotient SDE

Modified Black’s Formula

Step 1

Step 2

References

Maximum Likelihood Estimation for Integrated Diffusion Processes

Introduction

Model and Data

The Likelihood Function and the EM-Algorithm

Likelihood with Full Diffusion Observation

EM Algorithm

Conditional Diffusion Process Simulation

The Ornstein-Uhlenbeck Process: A Simulation Study

The Likelihood and the EM-algorithm

A Simulation Study

The CIR Process and a Stochastic Volatility Model

The Likelihood and the EM-algorithm

A Simulation Study

Concluding Remarks

References

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