Quantitative Finance: Back to Basic Principles

Quantitative Finance

Back to Basic Principles

Adil Reghai

NATIXIS, France

©Adil Reghai 2015

Foreword I @ Cedric Dubois. 2015

Foreword II @ Eric Moulines. 2015

All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission.

No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6–10 Kirby Street, London EC1N 8TS.

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First published 2015 by

PALGRAVE MACMILLAN

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Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010.

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Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries

ISBN: 978–1–137–41449–6

This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin.

A catalogue record for this book is available from the British Library.

A catalog record for this book is available from the Library of Congress.

To my parents, my spouse Raja, and my daughters Rania, Soraya, Nesma & Amira

Contents

List of Figures

List of Tables

Foreword I

Cédric Dubois

Foreword II

Eric Moulines

Acknowledgments

1 Financial Modeling

Introduction

2 About Modeling

A Philosophy of modeling

B An example from physics and some applications in finance

3 From Black & Scholes to Smile Modeling

A Study of derivatives under the Black & Scholes model

Methodology

The search for convexity

Vanilla European option

Numerical application

Price scenarios

Delta gamma scenarios:

European binary option

Price Scenario

Delta and gamma scenarios

American binary option

Numerical application

Price scenario

Delta and gamma scenarios

Barrier option

Price scenario

Delta and gamma scenarios

Asian option

Numerical application

Price scenario

Delta and gamma scenarios

When is it possible to use Black & Scholes

B Study of classical Smile models

Black & Scholes model

Term structure Black & Scholes

Monte Carlo simulation

Terminal smile model

Replication approach (an almost model-free approach)

Monte Carlo simulation (direct approach)

Monte Carlo simulation (fast method)

Classic example

Separable local volatility

Term structure of parametric slices

Dupire/Derman & Kani local volatility model

Stochastic volatility model

C Models, advanced characteristics and statistical features

Local volatility model

Stochastic volatility model

4 What is the Fair Value in the Presence of the Smile?

A What is the value corresponding to the cost of hedge?

The Delta spot ladder for two barrier options

The vega volatility ladder

The vega spot ladder

Conclusion

5 Mono Underlying Risk Exploration

Dividends

Models: discrete dividends

Models: cash amount dividend model

Models: proportional dividend model

Models: mixed dividend model

Models: dividend toxicity index

Statistical observations on dividends

Interest rate modeling

Models: why do we need stochastic interest rates?

Models: simple hybrid model

Models: statistics and fair pricing

Forward skew modeling

The local volatility model is not enough

Local volatility calibration

Alpha stable process

Truncated alpha stable invariants

Local volatility truncated alpha stable process

6 A General Pricing Formula

7 Multi-Asset Case

A Study of derivatives under the multi-dimensional Black & Scholes

Methodology

PCA for PnL explanation

Eigenvalue decomposition for symmetric operators

Stochastic application

Profit and loss explanation

The source of the parameters

Basket option

Worst of option (wo: call)

Best of option (Bo: put)

Other options (Best of call and worst of put)

Model calibration using fixed-point algorithm

Model estimation using an envelope approach

Conclusion

8 Discounting and General Value Adjustment Techniques

Full and symmetric collateral agreement

Perfect collateralization

Applications

Repo market

Optimal posting of collateral

Partial collateralization

Asymmetric collateralization

9 Investment Algorithms

What is a good strategy?

A simple strategy

Reverse the time

Avoid this data when learning

Strategies are assets

Multi-asset strategy construction

Signal detection

Prediction model

Risk minimization

10 Building Monitoring Signals

A Fat-tail toxicity index

B Volatility signals

Nature of the returns

The dynamic of the returns

Signal definition

Asset and strategies cartography

Asset management

C Correlation signals

Simple basket model

Estimating correlation level

Implied correlation skew

Multi-dimensional stochastic volatility

Local correlation model

General Conclusion

Solutions

Bibliography

Index

List of Figures

List of Tables

Foreword I

The valuation of financial derivatives instruments, and to some extent the way they behave, rests on a numerous and complex set of mathematical models, grouped into what is called quantitative finance. Nowadays, it should be required that each and every one involved in financial markets has a good knowledge of quantitative finance. The problem is that the many books in this field are too theoretical, with an impressive degree of mathematical formalism, which needs a high degree of competence in mathematics and quantitative methods.

As the title suggests, from absolute basics to advanced trading techniques and P&L explanations, this book aims to explain both the theory and the practice of derivatives instruments valuation in clear and accessible terms. This is not a mathematical textbook, and long and difficult equations that are not understandable by the average person are avoided wherever possible.

Practitioners have lost faith in the ability of financial models to achieve their ultimate purpose, as those models are not at all precise in their application to the complex world of real financial markets. They need to question the hypotheses that are behind models and challenge them. The models themselves should be applied in practice only tentatively, with careful assessment of their limitations in each application and in their own validity domain, as these can vary significantly across time and place.

This is especially true after the global financial crisis. The financial world has changed a lot and witnesses a much faster pace of crisis. New regulations and their application in modeling have become a very important topic which is enforced through regulatory regimes, especially Basel III and fundamental review of the trading book for the banking industry.

This book nicely covers all these subjects from a pragmatic point of view. It shows that stochastic calculus alone is not enough for properly evaluating and hedging derivatives instruments. It insists on the importance of data analysis in parameters estimation and how this extra information can be helpful in the construction of the fair valuation and most importantly the right hedging strategy.

At first sight, this ambitious objective seems to be tough to achieve. As a matter of fact, Adil Reghai has done it and furthermore treated it in a very pedagogical way.

Finally, the reader should appreciate the overall aim of Adil’s book, allowing for useful comparisons – some valuation methods appearing to be more robust and trustworthy than others – and often warning against the lack of reliability of some quantitative models, due to the hypotheses on which they are built.

For all these reasons, this book is a must have for all practitioners and should be a great success.

Cédric Dubois

Global Head of Structured Equity and Funds Derivatives Trading Natixis SA London

Foreword II

Quantitative finance has been one of the most active research fields in the last 50 years. The initial push in mathematical finance was the ‘portfolio selection’ theory invented by H. Markowitz. This work was a first mathematical attempt towards trying to identify and understand the trade-offs between risks and returns that is the central problem in portfolio theory. The mathematical tools used to develop portfolio selection theory resulted in a rather elementary combination of the analysis of variance and multivariate linear regression. This model of assets price immediately leads to the optimization problem of choosing the portfolio with largest return subject to a given amount of risk (measured here rather simplistically as the variance of the portfolios, ignoring fat tails and non Gaussiannity). The work by H. Markowitz was considerably extended by W. Sharpe, who proposed using dimension reduction: instead of modeling the covariance of every pair of stocks, W. Sharpe proposed to identify only a few factors and to regress asset prices on these factors. For these pioneering works, H. Markowitz and W. Sharpe received 1990 Nobel prizes in economics, the first ever awarded to work in finance.

The work of Markowitz and Sharpe introduced mathematics into what was previously considered mainly as the ‘art’ of portfolio management. Since then, the mathematical sophistication of models for assets and markets increased quite rapidly. ‘One-period’ investment models were quickly replaced by