[金融中的噪音]Asymptotic Chaos Expansions in Finance

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Springer Finance系列
In recent years, derivative trading and research has refocused on liquid instruments, and in particular on European options. Increased market turbulence,
unrelenting standardisation and stronger regulatory oversight altogether call for
robust and well-calibrated models of the static smile. Furthermore, many agents
(e.g. banks, hedge funds) now deploy sophisticated strategies, involving both
assets and options, to capture some type ofalphaor relative value. These algorithms demand accurate, non-arbitrable modelling of the joint dynamics of the
underlying and its implied volatility surface.
In principle, Stochastic Volatility (SV) model classes (such as SABR, Heston,
LSV or SV term structure frameworks) offer the most potential to fulfil these
objectives. Indeed they can reach the statics and represent the dynamics of the
smile in a rich, realistic and flexible fashion. In practice however, their lack of
tractability makes classical SV models difficult to manage. The primary cause is
that the derivation of the smile’s exact shape and dynamics from the model’s SDE
is rarely achievable in closed form, which leaves only numerical methods. This is
an issue not only for calibration, but also for computing and hedging the risk of
complex derivatives (especially Vega risk) and thus for model design and analysis.
The academic answer to these limitations of stochasticinstantaneous volatility
(SInsV) models has been twofold. The first tack has been to develop numerous
approximationmethods for the static smile of specific SInsV models, mostly using
small-time asymptotic techniques up to some low order. These methods exploit
eitheran analytic (i.e. PDE)ora probabilistic (i.e. SDE) approach, and include for
instance heat kernel and WKB expansions, singular perturbations, Malliavin calculus or saddlepoint approximations. Yet none of these approximation methods is
flexible enough to provide arbitrary precision across a wide range of SInsV
models, and neither do they address the dynamics of the smile. Therefore they
cannot adapt easily to rapidly changing and challenging market conditions.

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1 Introduction ........................................ 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Stochastic Volatility Market Models . . . . . . . . . . . . . . . 2
1.2.2 Asymptotic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Asymptotic Chaos Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Outline and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 General Spirit and Edited Material . . . . . . . . . . . . . . . . . . . . . 16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Part I Single Underlying
2 Volatility Dynamics for a Single Underlying: Foundations....... 23
2.1 Framework and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.1 Market and Underlyings. . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.2 Vanilla Options Market and Sliding
Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.3 The Two Stochastic Volatility Model Frameworks . . . . . 34
2.1.4 The Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2 Derivation of the Zero-Drift Conditions . . . . . . . . . . . . . . . . . . 42
2.2.1 The Main Zero-Drift Condition. . . . . . . . . . . . . . . . . . . 42
2.2.2 The Immediate Zero Drift Conditions . . . . . . . . . . . . . . 47
2.2.3 The IATM Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.4 Synthesis and Overture . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3 Recovering the Instantaneous Volatility: The First Layer . . . . . . 53
2.3.1 Computing the Dynamics ofσt................... 53
2.3.2 Interpretation and Comments . . . . . . . . . . . . . . . . . . . . 57
xv
2.4 Generating the Implied Volatility: The First Layer . . . . . . . . . . 61
2.4.1 Computing the Immediate ATM Differentials. . . . . . . . . 62
2.4.2 Interpretation and Comments . . . . . . . . . . . . . . . . . . . . 65
2.5 Illustrations and Applications . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.5.1 An Overview of Possible Applications. . . . . . . . . . . . . . 77
2.5.2 Illustration: Qualitative Analysis
of a Classical SV Model Class . . . . . . . . . . . . . . . . . . . 87
2.5.3 Second Illustration: Smile-Specification
of SInsV Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.6 Conclusion and Overture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3 Volatility Dynamics for a Single Underlying:
Advanced Methods.................................... 117
3.1 Higher-Order Expansions: Methodology and Automation . . . . . . 118
3.1.1 Tools and Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.1.2 Computing the First Column
of the Differentiation Matrix . . . . . . . . . . . . . . . . . . . . 123
3.1.3 Computing Subsequent Columns
of the Differentiation Matrix . . . . . . . . . . . . . . . . . . . . 126
3.2 Higher-Order Expansions: Illustration and Interpretation . . . . . . 135
3.2.1 Justification and Outline . . . . . . . . . . . . . . . . . . . . . . . 135
3.2.2 Interpretation of the Results . . . . . . . . . . . . . . . . . . . . . 138
3.2.3 Illustration of the Maturity Effect . . . . . . . . . . . . . . . . . 142
3.3 Framework Extensions and Generalisation . . . . . . . . . . . . . . . . 146
3.3.1 Building Blocks and Available Extensions . . . . . . . . . . . 146
3.3.2 An Important Example: The Normal
BaselineviaItsZDC.......................... 156
3.3.3 The Generic Baseline Transfer . . . . . . . . . . . . . . . . . . . 164
3.4 Multi-dimensional Extensions, or the Limitations
of Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.4.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
3.4.2 Derivation of the Zero-Drift Conditions . . . . . . . . . . . . . 173
3.4.3 Recovering the Instantaneous Volatility:
The First Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.4.4 Generating the Implied Volatility: The First Layer . . . . . 180
3.5 Illustration of the Vectorial Framework: TheBasket Case..... 185
3.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
3.5.2 Framework and Objectives . . . . . . . . . . . . . . . . . . . . . . 190
3.5.3 The Coefficient Basket . . . . . . . . . . . . . . . . . . . . . . . . 192
3.5.4 The Asset Basket in the General Case . . . . . . . . . . . . . . 194
3.5.5 The Asset Basket Specialised to Fixed Weights . . . . . . . 202
3.5.6 Interpretation and Applications . . . . . . . . . . . . . . . . . . . 204
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
xvi Contents
4 Practical Applications and Testing........................ 211
4.1 General Considerations on Practical Applications . . . . . . . . . . . 212
4.2 Application to the Generic SABR Class . . . . . . . . . . . . . . . . . . 218
4.2.1 Presentation of the Model . . . . . . . . . . . . . . . . . . . . . . 218
4.2.2 Coefficients of the Chaos Dynamics . . . . . . . . . . . . . . . 219
4.2.3 Mapping the Model and the Smile . . . . . . . . . . . . . . . . 226
4.3 Application to the CEV-SABR Model . . . . . . . . . . . . . . . . . . . 230
4.3.1 Presentation of the Model . . . . . . . . . . . . . . . . . . . . . . 231
4.3.2 Coefficients of the Chaos Dynamics . . . . . . . . . . . . . . . 232
4.3.3 Mapping the Model and the Smile Shape. . . . . . . . . . . . 236
4.3.4 Compatibility with Hagan et al. . . . . . . . . . . . . . . . . . . 240
4.4 Application to the FL-SV Class (Exercise) . . . . . . . . . . . . . . . . 242
4.4.1 Presentation of the Model . . . . . . . . . . . . . . . . . . . . . . 243
4.4.2 Derivation Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
4.5 Numerical Implementation and Testing . . . . . . . . . . . . . . . . . . 245
4.5.1 Testing Environment and Rationale. . . . . . . . . . . . . . . . 245
4.5.2 Tests Data and Results . . . . . . . . . . . . . . . . . . . . . . . . 250
4.5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Part II Term Structures
5 Volatility Dynamics in a Term Structure ................... 273
5.1 Framework and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
5.1.1 Numeraires, Underlyings and Options . . . . . . . . . . . . . . 274
5.1.2 Absolute and Sliding Implied Volatilities. . . . . . . . . . . . 276
5.1.3 The Two Stochastic Volatility Models . . . . . . . . . . . . . . 277
5.1.4 The Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
5.2 Derivation of the Zero-Drift Conditions . . . . . . . . . . . . . . . . . . 282
5.2.1 The Main Zero-Drift Condition. . . . . . . . . . . . . . . . . . . 283
5.2.2 The Immediate Zero Drift Condition . . . . . . . . . . . . . . . 292
5.2.3 The IATM Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
5.3 Recovering the Instantaneous Volatility . . . . . . . . . . . . . . . . . . 296
5.3.1 Establishing the Main Result . . . . . . . . . . . . . . . . . . . . 296
5.3.2 Interpretation and Comments . . . . . . . . . . . . . . . . . . . . 304
5.4 Generating the SIV Surface: The First Layer . . . . . . . . . . . . . . 308
5.4.1 Computing the Differentials . . . . . . . . . . . . . . . . . . . . . 309
5.4.2 Interpretation and Comments . . . . . . . . . . . . . . . . . . . . 317
5.5 Extensions, Further Questions and Conclusion . . . . . . . . . . . . . 320

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Asymptotic Chaos Expansions in Finance

[金融中的噪音]Asymptotic Chaos Expansions in Finance