The past thirty years have
seen explosive growth in the use of financial deritivatives along with the development of
an elegant mathematical theory for their valuation. The theory, based on stochastic
calculus, provides a conceptual framework for valuation as well as efficient computational
tools.
As derivative markets have
matured, much of the focus has shifted from the valuation of individual financial
instruments to the problems of hedging and risk management of large financial portfolios.
The challenge here is to understand the behaviour of a complex set of instruments that may
depend on hundreds or thousands of underlying risk factors. Financial Geometry will help
you to do so.
Mathematical yet accessible,
"Financial Geometry" provides intuitive geometric metaphors and powerful
computational machinery for describing the complex risks of the modern financial world.
Topics covered include:
- marking to market,
- valuation techniques and
models,
- risk-factor definition,
- sensitivity and scenario
analysis,
- interest rate calculations,
- hedge calculations,
- Value at Risk,
- risk-factor mapping,
- volatility curves and
surfaces,
- time effects.
'I applaud the author's use
of differential geometry to integrate the disparate approaches to hedging and risk
management. This approach is quite literally the shape of things to come!'
Dr. Peter Carr, Global Head of Quantitative Research, Bloomberg, and Director of the
Master's Program in Mathematical Finance, Courant Institute, NYU
'In Financial Geometry,
Kuruc provides a thoroughly consistent approach, within a clean and easily understandable
formalism. He does this by relating the problem to the field of differential geometry, to
which it is surprisingly well suited. Kuruc's writing style is exceptionally clear and
accessible, even pedagogical. Many examples are treated, which make application of his
approach to real-world problems straightforward.'
David Eliezer, Goldman Sachs
'Although the underlying
mathematics is quite sophisticated, the material is made accessible to a board audience.
Despite its sophistication, this is an eminently practical book that will prove useful to
anyone with an interest in financial derivatives or enterprise risk.'
James Lewis, Barclays Global Investors
'Alvin Kuruc's book is
unique in providing a unifying theoretical framework for the bewildering array of separate
concepts and approaches currently applied in practice to the hedging and risk management
of portfolios of financial instruments... with elegance and precision. The result is a
charming and very readable work that is bound to have a significant impact on students of
risk management - both researchers and practitioners.'
Professor Michael Dempster, Director, Centre for Financial Research; Professor of
Finance and Professorial, Judge Institute of Management, University of Cambridge
Contents
Foreword
Preface
Acknowledgements
Prologue
PART I: Foundations
1 Marking to market
1.1 Mark to market
1.2 On-market and off-market instruments
1.2.1 Types of financial markets
1.2.2 Key instrument distinctions
1.2.3 On-market instruments
1.2.4 Off-market instruments
1.3 Units of account
1.3.1 Accounting currencies
1.3.2 Currency conversion
1.4 Accounting verses risk management
1.4.1 Reporting frequency
1.4.2 Past verses future orientation
1.4.3 Probabilistic modelling requirements
1.4.4 Liquidity and funding risks
1.4.5 Specific accounting requirements
1.4.6 Modelling uncertainty
1.5 Notes and references
2 Valuation techniques
2.1 Asset-value notation
2.2 Future cash flows
2.2.1 Cash rates
2.2.2 Interpolation
2.3 Forward contracts
2.4 Interest-rate swaps
2.4.1 Par swaps
2.4.2 Off-market swaps
2.5 Calls and puts
2.5.1 Option-pricing theory
2.5.2 Black-Scholes model
2.5.3 Black-Scholes volatility surface
2.6 General options
2.7 Notes and references
3 Introduction to risk
management
3.1 Market-state space and
risk factors
3.1.1 Market-state space
3.1.2 Risk factors
3.1.3 Coordinate system
3.1.4 Valuation functions
3.1.5 Example: zero-coupon valuation
3.1.6 Alternative coordination systems
3.1.7 Observables
3.1.8 Types of risk factors
3.2 Risk-management techniques
3.2.1 Mark-to market
3.2.2 Sensitivity analysis
3.2.3 Scenario analysis
3.2.4 Value-at-risk analysis
3.2.5 What are we missing?
3.3 The devil is in the details
3.4 Notes and references
4 Classical bond geometry
4.1 Bonds
4.2 Risk factors for bonds
4.2.1 Price
4.2.2 Yield to maturity
4.2.3 Arbitrage restrictions
4.2.4 Coordinate maps
4.3 Valuation functions
4.4 Reconciling risk factors
4.5 Sensitivities
4.5.1 Differentiation of univariate functions
4.5.2 Univariate chain rule
4.5.3 First-order sensitivities
4.5.4 Second-order sensitivities
4.6 Tangent vectors and differentials
4.6.1 Path derivatives
4.6.2 Tangent vectors
4.6.3 Differentials
4.7 Reconciling sensitivities
4.7.1 Tangent vetors
4.7.2 Differentials
4.8 Classical sensitivities
4.8.1 Ten-year equivalents
4.8.2 Modified duration
4.8.3 Macaulay duration
4.8.4 Dollar duration
4.8.5 Convexity
4.9 Commensurability
4.10 Notes and references
5 Modern bond geometry
5.1 Time value of money
5.1.1 Absolute and relative dates
5.1.2 Nominal and actual terms
5.1.3 Credit quality
5.2 Risk factors
5.2.1 Zero-coupon discount factors
5.2.2 Log discount factors
5.2.3 Zero-coupon discount rates
5.2.4 Forward discount factors
5.2.5 Forward rates
5.2.6 Par rates
5.2.7 Market risk factors
5.3 Valuation functions
5.4 Sensitivities
5.4.1 Differentiation of multivariate functions
5.4.2 Multivariate chain rule
5.4.3 Sensitivities to discount factors
5.4.4 Sensitivities to log discount factors
5.4.5 Sensitivities to zero rates
5.4.6 Sensitivities to forward discount factors
5.4.7 Sensitivities to forward rates
5.4.8 Sensitivities to par rates
5.4.9 "Market" sensitivities
5.5 Tangent vectors and differentials
5.5.1 Tangent and vectors
5.5.2 Differentials
5.6 Reconciling sensitivities
5.6.1 Tangent vectors
5.6.2 Differentials
5.7 Notes and references
PART II
Asset values
6 Interpolation
6.1 Introduction
6.2 Risk factors
6.2.1 Entire term structure
6.2.2 Key term structure
6.3 Valuation functions
6.3.1 Entire term structure
6.3.2 Key term structure
6.4 Sensitivities
6.4.1 Entire term structure
6.4.2 Key term structure
6.5 Perturbations
6.5.1 Entire term structure
6.5.2 Key term structure
6.6 Differentials
6.7 Bucketing
6.8 Notes and references
7 Interest-rate hedging
7.1 Introduction to hedging
7.2 Delta hedging
7.2.1 Basic theory
7.2.2 Duration hedging
7.2.3 Cash-flow hedging
7.2.4 Curve hedging
7.3 Subspace hedging
7.4 Weighted hedging
7.5 Notes and references
8 Foreign-currency geometry
8.1 Foreign exchange
8.1.1 Market description
8.1.2 Systematic description
8.1.3 No-arbitrage relations
8.2 Risk factors
8.2.1 Exchange rates
8.2.2 Log foreign-exchange rates
8.3 Valuation functions
8.4 Sensitivities
8.4.1 Exchange rates
8.4.2 Log exchange rates
8.5 Reconciling sensitivities
8.5.1 Perturbations
8.5.2 Differentials
8.6 Change of base currency
8.6.1 Asset values
8.6.2 Log asset values
8.7 Notes and references
9 Demand-deposit geometry
9.1 Introduction
9.2 Risk factors
9.2.1 Asset-value manifold
9.2.2 Asset values
9.2.3 Relation to foreign-exchange rates
9.3 Valuations functions
9.3.1 Abstract valuation space
9.3.2 Valuation of demand deposits
9.3.3 Relation to foreign-exchange rates
9.4 Sensitivities
9.5 Relationship to foreign-exchange sensitivities
9.5.1 Perturbations
9.5.2 Differentials
9.5.3 Application
9.6 Notes and references
10 Asset-value geometry
10.1 Asset values
10.2 Risk factors
10.2.1 Absolute asset values
10.2.2 Relative asset values
10.3 Valuation functions
10.3.1 Abstract asset values
10.3.2 Absolute asset values
10.3.3 Relative asset values
10.3.4 Foreign-exchange rates and discount factors
10.4 Sensitivities
10.4.1 Abstract asset values
10.4.2 Absolute asset values
10.4.3 Log absolute asset values
10.4.4 Relative asset values
10.4.5 Log relative asset values
10.5 Implementation
10.5.1 Bucketing
10.5.2 Implementation by asset value and discount-factor perturbations
10.6 Notes and references
PART III
Metrics
11 Value at risk
11.1 Value-at-risk
methodologies
11.2 Probabilistic model
11.3 Geometrical interpretation
11.3.1 Length and angle
11.3.2 Change of coordinates
11.4 Change of base currency
11.5 Benchmarking
11.6 Notes and references
12 Risk-factor mapping
12.1 Natural and modelled
risk factors
12.2 Perturbations and differentials
12.3 Covariance structure
12.4 Mapping term structures
12.4.1 Asset series
12.4.2 Factor-name association
12.4.3 Factor substitution and scaling
12.5 Mapping stocks
12.5.1 Beta
12.5.2 Native currency
12.6 Notes and references
13 Risk contributions
13.1 Basic theory
13.2 Variance/covariance results
13.3 Limiting cases
13.3.1 Uncorrelated instruments
13.3.2 Perfectly correlated instruments
13.4 Notes and references
14 Variance/covariance
hedging
14.1 Motivation
14.2 Implementation
14.3 Application
14.4 Notes and references
PART IV
Implied vol
15 Option geometry
15.1 Stock options
15.2 Risk factors
15.2.1 Market prices
15.2.2 Black-Scholes model
15.2.3 Bachelier model
15.2.4 Model inconsistency
15.3 Valuation functions
15.3.1 Market-price coordinate system
15.3.2 Black-Scholes coordinate system
15.3.3 Bachelier coordinate system
15.4 Sensitivities
15.4.1 Market-price coordinate system
15.4.2 Black-Scholes coordinate system
15.4.3 Bachelier coordinate system
15.5 Reconciling sensitivities
15.5.1 Perturbations
15.5.2 Differentials
15.6 Valuing other instruments
15.7 Profit and loss attributions
15.8 Notes and references
16 Volatility curves
16.1 Market-state space
16.2 Risk factors
16.2.1 Call prices
16.2.2 Black-Scholes volatility curve
16.2.3 Bull-option prices
16.2.4 Arrow-Debreu prices
16.3 Valuation functions
16.3.1 Arrow-Debreu coordinate system
16.3.2 Bull-option coordinate system
16.3.3 Call-price coordinate system
16.3.4 Black-Scholes coordinate system
16.4 Sensitivities
16.4.1 Functional derivatives
16.4.2 Call-price coordinate system
16.4.3 Bull-option coordinate system
16.4.4 Arrow-Debreu coordinate system
16.4.5 Black-Scholes coordinate system
16.5 Reconciling sensitivities
16.6 Notes and references
17 Volatility surfaces
17.1 Market-state space
17.2 Risk factors
17.2.1 Call prices
17.2.2 Black-Scholes volatility surfce
17.2.3 Local volatility surface
17.2.4 Arrow-Debreu price surface
17.3 Valuation functions
17.3.1 Arrow-Debreu coordinate system
17.3.2 Call-price coordinate system
17.3.3 Black-Scholes coordinate system
17.3.4 Local-volatility coordinate system
17.4 Sensitivities
17.4.1 Functional derivatives
17.4.2 Call-prices
17.4.3 Arrow-Debreu prices
17.4.4 Black-Scholes coordinate system
17.5 Reconciling sensitivities
17.6 Notes and references
18 Valuation models
18.1 Market-state spaces
18.2 Risk factors
18.2.1 Risk-neutral models
18.2.2 Parameterized valuation models
18.3 Valuation functions
18.3.1 Risk-neutral valuation models
18.3.2 Parameterized valuation models
18.3.3 Calibration procedures
18.4 Sensitivities
18.4.1 Parameterized model sensitivities
18.4.2 European option prices
18.5 Notes and references
Epilogue
Appendix A
The dimension of time
A.1 Time concepts
A.2 Choice of time horizon
A.2.1 Present valuation functions
A.2.2 Future valuation functions
A.2.3 Practical considerations
A.3 Notes and references
Appendix B
Black-Scholes and Bachelier calculations
B.1 Valuation functions
B.1.1 Black-Scholes model
B.1.2 Bachelier model
B.2 Sensitivities
B.2.1 Black-Scholes model
B.2.2 Bachelier model
Appendix C
Index of notation
References
Index
381 pages
