Frequently Asked Questions in Quantitative Finance E-Book

Paul Wilmott

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 698

Veröffentlichungsjahr: 2010

Beschreibung

Getting agreement between finance theory and finance practice is important like never before. In the last decade the derivatives business has grown to a staggering size, such that the outstanding notional of all contracts is now many multiples of the underlying world economy. No longer are derivatives for helping people control and manage their financial risks from other business and industries, no, it seems that the people are toiling away in the fields to keep the derivatives market afloat! (Apologies for the mixed metaphor!) If you work in derivatives, risk, development, trading, etc. you'd better know what you are doing, there's now a big responsibility on your shoulders. In this second edition of Frequently Asked Questions in Quantitative Finance I continue in my mission to pull quant finance up from the dumbed-down depths, and to drag it back down to earth from the super-sophisticated stratosphere. Readers of my work and blogs will know that I think both extremes are dangerous. Quant finance should inhabit the middle ground, the mathematics sweet spot, where the models are robust and understandable, and easy to mend. ...And that's what this book is about. This book contains important FAQs and answers that cover both theory and practice. There are sections on how to derive Black-Scholes (a dozen different ways!), the popular models, equations, formulae and probability distributions, critical essays, brainteasers, and the commonest quant mistakes. The quant mistakes section alone is worth trillions of dollars! I hope you enjoy this book, and that it shows you how interesting this important subject can be. And I hope you'll join me and others in this industry on the discussion forum on wilmott.com. See you there!" FAQQF2...including key models, important formulae, popular contracts, essays and opinions, a history of quantitative finance, sundry lists, the commonest mistakes in quant finance, brainteasers, plenty of straight-talking, the Modellers' Manifesto and lots more.

Leseprobe

Table of Contents

Title Page

Copyright Page

Dedication

Frequently Asked Questions

Preface to the Second Edition

Some more about the author

Preface to the First Edition

About the author

Chapter 1 - The Quantitative Finance Timeline

References and Further Reading

And Now a Brief Unofficial History!

References and Further Reading

Chapter 2 - FAQs

What are the Different Types of Mathematics Found in Quantitative Finance?

References and Further Reading

What is Arbitrage?

References and Further Reading

What is Put-Call Parity?

References and Further Reading

What is the Central Limit Theorem and What are its Implications for Finance?

References and Further Reading

How is Risk Defined in Mathematical Terms?

References and Further Reading

What is Value at Risk and How is it Used?

References and Further Reading

What is Extreme Value Theory?

References and Further Reading

What is CrashMetrics?

References and Further Reading

What is a Coherent Risk Measure and What are its Properties?

References and Further Reading

What is Modern Portfolio Theory?

References and Further Reading

What is the Capital Asset Pricing Model?

References and Further Reading

What is Arbitrage Pricing Theory?

References and Further Reading

What is Maximum Likelihood Estimation?

References and Further Reading

What is Cointegration?

References and Further Reading

What is the Kelly Criterion?

References and Further Reading

Why Hedge?

References and Further Reading

What is Marking to Market and How Does it Affect Risk Management in Derivatives Trading?

References and Further Reading

What is the Efficient Markets Hypothesis?

References and Further Reading

What are the Most Useful Performance Measures?

References and Further Reading

What is a Utility Function and How is it Used?

References and Further Reading

What is the Difference between a Quant and an Actuary?

References and Further Reading

What is a Wiener Process/Brownian Motion and What are its Uses in Finance?

References and Further Reading

What is Jensen’s Inequality and What is its Role in Finance?

References and Further Reading

What is Itô’s Lemma?

References and Further Reading

Why Does Risk-Neutral Valuation Work?

References and Further Reading

What is Girsanov’s Theorem, and Why is it Important in Finance?

References and Further Reading

What are the Greeks?

References and Further Reading

Why do Quants like Closed-Form Solutions?

References and Further Reading

What are the Forward and Backward Equations?

References and Further Reading

What is the Black-Scholes Equation?

References and Further Reading

Which Numerical Method should I Use and When?

References and Further Reading

What is Monte Carlo Simulation?

References and Further Reading

What is the Finite-Difference Method?

References and Further Reading

What is a Poisson Process and What are its Uses in Finance?

References and Further Reading

What is a Jump-Diffusion Model and How does it Affect Option Values?

References and Further Reading

What is Meant by ‘Complete’ and ‘Incomplete’ Markets?

References and Further Reading

Can I use Real Probabilities to Price Derivatives?

References and Further Reading

What is Volatility?

References and Further Reading

What is the Volatility Smile?

References and Further Reading

What is GARCH?

References and Further Reading

How Do I Dynamically Hedge?

References and Further Reading

What is Serial Autocorrelation and Does it Have a Role in Derivatives?

References and Further Reading

What is Dispersion Trading?

References and Further Reading

What is Bootstrapping using Discount Factors?

References and Further Reading

What is the LIBOR Market Model and its Principal Applications in Finance?

References and Further Reading

What is Meant by the ‘Value’ of a Contract?

References and Further Reading

What is Calibration?

References and Further Reading

What is Option Adjusted Spread?

References and Further Reading

What is the Market Price of Risk?

References and Further Reading

Can I Reverse Engineer a Partial Differential Equation to get at the Model and Contract?

References and Further Reading

What is the Difference Between the Equilibrium Approach and the No-Arbitrage ...

References and Further Reading

How Good is the Assumption of Normal Distributions for Financial Returns?

References and Further Reading

How Robust is the Black - Scholes Model?

References and Further Reading

Why is the Lognormal Distribution Important?

References and Further Reading

What are Copulas and How are they Used in Quantitative Finance?

References and Further Reading

What is Asymptotic Analysis and How is it Used in Financial Modelling?

References and Further Reading

What is a Free-Boundary Problem and What is the Optimal-Stopping Time for an ...

References and Further Reading

What are Low-Discrepancy Numbers?

References and Further Reading

What are the Bastard Greeks?

References and Further Reading

What are the Stupidest Things People have Said about Risk Neutrality?

References and Further Reading

What is the Best-Kept Secret in Quantitative Finance?

References and Further Reading

Chapter 3 - The Financial Modellers’ Manifesto

Preface

Manifesto

Chapter 4 - Essays

Science in Finance: Introduction

Science in Finance I Revisited: Supply and Demand, and Spoon Bending

Science in Finance II: ‘ ... ists’

Science in Finance IV: The Feedback Effect

Science in Finance VI: True Sensitivities, CDOs and Correlations

Science in Finance VII: Risk Management - What is the Point?

Science in Finance IX: In Defence of Black, Scholes and Merton

Magicians and Mathematicians

Volatility Arbitrage

The Same Old Same Old

Results and Ideas: Two Classical Putdowns

It Is and It Isn’t

This is No Longer Funny

Frustration

Ponzi Schemes, Auditors, Regulators, Credit Ratings, and Other Scams

Economics Makes My Brain Hurt

Name and Shame in Our New Blame Game!

Chapter 5 - The Commonest Mistakes in Quantitative Finance: A Dozen Basic ...

Introduction

Quiz

Lesson 1: Lack of Diversification

Lesson 2: Supply and Demand

Lesson 3: Jensen’s Inequality Arbitrage

Lesson 4: Sensitivity to Parameters

Lesson 5: Correlation

Lesson 6: Reliance on Continuous Hedging (Arguments)

Lesson 7: Feedback

Lesson 8: Reliance on Closed-Form Solutions

Lesson 9: Valuation is Not Linear

Lesson 10: Calibration

Lesson 11: Too Much Precision

Lesson 12: Too Much Complexity

Bonus Lesson 13: The Binomial Method is Rubbish

Summary

References and Further Reading

Chapter 6 - The Most Popular Probability Distributions and Their Uses in Finance

References and Further Reading

Chapter 7 - Twelve Different Ways to Derive Black-Scholes

Hedging and the Partial Differential Equation

Martingales

Change of Numeraire

Local Time

Parameters as Variables

Continuous-Time Limit of the Binomial Model

CAPM

Utility Theory

Taylor Series

Mellin Transform

A Diffusion Equation

Black-Scholes for Accountants

Other Derivations

References and Further Reading

Chapter 8 - Models and Equations

Equity, Foreign Exchange and Commodities

Credit

References and Further Reading

Chapter 9 - The Black-Scholes Formulæ and the Greeks

Warning

Chapter 10 - Common Contracts

Things to Look Out For in Exotic Contracts

Chapter 11 - Popular Quant Books

Paul Wilmott Introduces Quantitative Finance, Second Edition by Paul Wilmott

Paul Wilmott on Quantitative Finance, Second Edition by Paul Wilmott

Advanced Modelling in Finance Using Excel and VBA by Mary Jackson and Mike Staunton

Option Valuation under Stochastic Volatility by Alan Lewis

The Concepts and Practice of Mathematical Finance by Mark Joshi

C++ Design Patterns and Derivatives Pricing by Mark Joshi

Heard on the Street by Timothy Crack

Monte Carlo Methods in Finance by Peter Jäckel

Credit Derivatives Pricing Models by Philipp Schönbucher

Principles of Financial Engineering by Salih Neftci

Options, Futures, and Other Derivatives by John Hull

The Complete Guide to Option Pricing Formulas by Espen Gaarder Haug

Chapter 12 - The Most Popular Search Words and Phrases on Wilmott.com

Chapter 13 - Brainteasers

The Questions

The Answers

Chapter 14 - Paul & Dominic’s Guide to Getting a Quant Job

Introduction

Writing a CV

Interviews

Appearance

What People Get Wrong

Index

This edition first published 2009

© 2009 Paul Wilmott

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Library of Congress Cataloging-in-Publication Data

Wilmott, Paul.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-74875-6

1. Finance—Mathematical models. 2. Investments—Mathematical models. 3. Options (Finance)—Mathematical models. I. Title.

HG4515.2.W55 2009

332.601’ 5118—dc22

2009027978

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

Typeset in 9/10.5 Cheltenham-Book by Laserwords Private Limited, Chennai, India

To my parents

Paul Wilmott has been called ‘the smartest of the quants, he may be the only smart quant’ (Portfolio magazine/Nassim Nicholas Taleb), ‘cult derivatives lecturer’ (Financial Times), ‘expert on quantitative easing,’ (Guardian)1, ‘the finance industry’s Mozart’ (Sunday Business) and ‘financial mathematics guru’ (BBC).

Frequently Asked Questions

1. What are the different types of mathematics found in quantitative finance? 22

2. What is arbitrage? 27

3. What is put-call parity? 30

4. What is the Central Limit Theorem and what are its Implications for finance? 33

5. How is risk defined in mathematical terms? 38

6. What is value at risk and how is it used? 42

7. What is Extreme Value Theory? 46

8. What is CrashMetrics? 48

9. What is a coherent risk measure and what are its properties? 52

10. What is Modern Portfolio Theory? 55

11. What is the Capital Asset Pricing Model? 58

12. What is Arbitrage Pricing Theory? 62

13. What is Maximum Likelihood Estimation? 65

14. What is cointegration? 71

15. What is the Kelly criterion? 74

16. Why hedge? 77

17. What is marking to market and how does it affect risk management in derivatives trading? 83

18. What is the Efficient Markets Hypothesis? 87

19. What are the most useful performance measures? 90

20. What is a utility function and how is it used? 93

21. What is the difference between a quant and an actuary? 97

22. What is a Wiener process/Brownian motion and what are its uses in finance? 100

23. What is Jensen’s Inequality and what is its role in finance? 103

24. What is Itô’s lemma? 106

25. Why does risk-neutral valuation work? 109

26. What is Girsanov’s theorem, and why is it important in finance? 113

27. What are the greeks? 116

28. Why do quants like closed-form solutions? 122

29. What are the forward and backward equations? 125

30. What is the Black-Scholes equation? 129

31. Which numerical method should I use and when? 132

32. What is Monte Carlo simulation? 141

33. What is the finite-difference method? 145

34. What is a Poisson process and what are its uses in finance? 150

35. What is a jump-diffusion model and how does it affect option values? 152

36. What is meant by ‘complete’ and ‘incomplete’ markets? 155

37. Can I use real probabilities to price derivatives? 160

38. What is volatility? 162

39. What is the volatility smile? 167

40. What is GARCH? 174

41. How do I dynamically hedge? 179

42. What is serial autocorrelation and does it have a role in derivatives? 185

43. What is dispersion trading? 188

44. What is bootstrapping using discount factors? 191

45. What is the LIBOR market model and its principal applications in finance? 196

46. What is meant by the ‘value’ of a contract? 200

47. What is calibration? 203

48. What is Option Adjusted Spread? 206

49. What is the market price of risk? 208

50. Can I reverse engineer a partial differential equation to get at the model and contract? 212

51. What is the difference between the equilibrium approach and the no-arbitrage approach to modelling? 216

52. How good is the assumption of normal distributions for financial returns? 219

53. How robust is the Black-Scholes model? 223

54. Why is the lognormal distribution important? 226

55. What are copulas and how are they used in quantitative finance? 229

56. What is asymptotic analysis and how is it used in financial modelling? 233

57. What is a free-boundary problem and what is the optimal-stopping time for an American option? 236

58. What are low-discrepancy numbers? 240

59. What are the bastard greeks? 245

60. What are the stupidest things people have said about risk neutrality? 248

61. What is the best-kept secret in quantitative finance? 250

Preface to the Second Edition

The previous edition of this book was aimed to some extent at those people wanting to get their first job in quantitative finance and perhaps needed something to refresh their memories just as they were going into an interview. In part it was meant to be orthogonal to those interview-prep books that focus almost exclusively on the math. Math is important, but I think it’s the easy bit of this business. It’s where the math breaks down that I think is crucial, and so there’s a lot of that in both the previous edition and this one.

In one respect I couldn’t have been more wrong when I said that the last book was for newbies. Recent events have shown that some of those most in need of a refresher in the fundamentals are not necessarily the newbies, but sometimes also the ‘experienced’ quants, the ‘respected’ academics and the ‘genius’ Nobel laureates. I think this book has an additional purpose now if you are going for a job. Yes, take this book with you, but use it to interrogate the interviewer. If he can’t answer your questions then don’t accept a job from him, his bank might not be around for much longer!

This second edition is not an ‘updating.’ Nothing in the previous edition is out of date ... And that’s not a statement that many quant finance authors can make! One of the themes in my research and writing is that much of quantitative finance is too theoretical, it’s far too mathematical, and somewhere along the way common sense has got left behind. In 2000 I wrote that there needed to be a change in modelling methods if there was not to be a “mathematician-led market meltdown.” There wasn’t, so there was. In 2006 I narrowed this down to credit instruments and credit models. Sadly, money making got in the way of good modelling, and you no doubt know what ensued. But now more and more people are starting to appreciate the importance of getting the level of mathematics right, and this has to be a good thing for the industry. We ran a survey on wilmott.com, our famous “Name and Shame in Our New Blame Game!” in which we asked members to tell us which are the worst finance models. The results are published towards the end of this book so you’ll see that common sense and robustness of modelling are on their way back.

To the thankees in the first edition I would like to add Emanuel Derman for allowing me to republish the “Financial Modelers’ Manifesto” that we wrote together at the start of 2009.

Some more about the author

Paul Wilmott is still learning the guitar, after 36 years trying. He now knows six chords. His only hobby at which he has been successful is reading, always fiction. But even so, he has been stuck at half way through James Joyce’s Ulysses for a decade, and has never got beyond the first 10 pages of anything by Salman Rushdie. Paul divides his time between his home in London and airport lounges around the world, where he can often be found nursing a dry martini.

Preface to the First Edition

This book grew out of a suggestion by wilmott.com

Member ‘bayes’ for a Forum (as in ‘internet discussion group’) dedicated to gathering together answers to the most common quanty questions. We responded positively, as is our wont, and the Wilmott Quantitative Finance FAQs Project was born. This Forum may be found at www.wilmott.com/faq. (There anyone may read the FAQ answers, but to post a message you must be a member. Fortunately, this is entirely free!) The FAQs project is one of the many collaborations between Members of wilmott.com.

As well as being an ongoing online project, the FAQs have inspired the book you are holding. It includes FAQs and their answers and also sections on common models and formulæ, many different ways to derive the Black--Scholes model, the history of quantitative finance, a selection of brainteasers and a couple of sections for those who like lists (there are lists of the most popular quant books and search items on wilmott.com). Right at the end is an excerpt from Paul and Dominic’s Guide to Getting a Quant Job, this will be of interest to those of you seeking their first quant role.

FAQs in QF is not a shortcut to an in-depth knowledge of quantitative finance. There is no such shortcut. However, it will give you tips and tricks of the trade, and insight, to help you to do your job or to get you through initial job interviews. It will serve as an aide memoire to fundamental concepts (including why theory and practice diverge) and some of the basic Black--Scholes formulæ and greeks. The subject is forever evolving, and although the foundations are fairly robust and static there are always going to be new products and models. So, if there are questions you would like to see answered in future editions please drop me an email at [email protected].

I would like to thank all Members of the forum for their participation and in particular the following, more prolific, Members for their contributions to the online FAQs and Brainteasers: Aaron, adas, Alan, bayes, Cuchulainn, exotiq, HA, kr, mj, mrbadguy, N, Omar, reza, WaaghBakri and zerdna. Thanks also to DCFC for his advice concerning the book.

I am grateful to Caitlin Cornish, Emily Pears, Graham Russel, Jenny McCall, Sarah Stevens, Steve Smith, Tom Clark and Viv Wickham at John Wiley & Sons Ltd for their continued support, and to Dave Thompson for his entertaining cartoons.

I am also especially indebted to James Fahy for making the Forum happen and run smoothly. Mahalo and aloha to my ever-encouraging wife, Andrea.

About the author

Paul Wilmott is an author, researcher, consultant and trainer in quantitative finance. He owns wilmott.com, is the Editor in Chief of the bimonthly quant magazine Wilmott and is the Course Director for the Certificate in Quantitative Finance (cqf.com). He is the author of the student text Paul Wilmott Introduces Quantitative Finance, which covers classical quant finance from the ground up, and Paul Wilmott on Quantitative Finance, the three-volume research-level epic. Both are also published by John Wiley & Sons, Ltd.

Chapter 1

The Quantitative Finance Timeline

There follows a speedy, roller-coaster of a ride through the official history of quantitative finance, passing through both the highs and lows. Where possible I give dates, name names and refer to the original sources.2

1827 Brown The Scottish botanist, Robert Brown, gave his name to the random motion of small particles in a liquid. This idea of the random walk has permeated many scientific fields and is commonly used as the model mechanism behind a variety of unpredictable continuous-time processes. The lognormal random walk based on Brownian motion is the classical paradigm for the stock market. See Brown (1827).

1900 Bachelier Louis Bachelier was the first to quantify the concept of Brownian motion. He developed a mathematical theory for random walks, a theory rediscovered later by Einstein. He proposed a model for equity prices, a simple normal distribution, and built on it a model for pricing the almost unheard of options. His model contained many of the seeds for later work, but lay ‘dormant’ for many, many years. It is told that his thesis was not a great success and, naturally, Bachelier’s work was not appreciated in his lifetime. See Bachelier (1995).

1905 Einstein Albert Einstein proposed a scientific foundation for Brownian motion in 1905. He did some other clever stuff as well. See Stachel (1990).

1911 Richardson Most option models result in diffusion-type equations. And often these have to be solved numerically. The two main ways of doing this are Monte Carlo and finite differences (a sophisticated version of the binomial model).

The very first use of the finite-difference method, in which a differential equation is discretized into a difference equation, was by Lewis Fry Richardson in 1911, and used to solve the diffusion equation associated with weather forecasting. See Richardson (1922). Richardson later worked on the mathematics for the causes of war. During his work on the relationship between the probability of war and the length of common borders between countries he stumbled upon the concept of fractals, observing that the length of borders depended on the length of the ‘ruler.’ The fractal nature of turbulence was summed up in his poem “Big whorls have little whorls that feed on their velocity, and little whorls have smaller whorls and so on to viscosity.”

1923 Wiener Norbert Wiener developed a rigorous theory for Brownian motion, the mathematics of which was to become a necessary modelling device for quantitative finance decades later. The starting point for almost all financial models, the first equation written down in most technical papers, includes the Wiener process as the representation for randomness in asset prices. See Wiener (1923).

1950s Samuelson The 1970 Nobel Laureate in Economics, Paul Samuelson, was responsible for setting the tone for subsequent generations of economists. Samuelson ‘mathematized’ both macro and micro economics. He rediscovered Bachelier’s thesis and laid the foundations for later option pricing theories. His approach to derivative pricing was via expectations, real as opposed to the much later risk-neutral ones. See Samuelson (1955).

1951 Itô Where would we be without stochastic or Itô calculus? (Some people even think finance is only about Itô calculus.) Kiyosi Itô showed the relationship between a stochastic differential equation for some independent variable and the stochastic differential equation for a function of that variable. One of the starting points for classical derivatives theory is the lognormal stochastic differential equation for the evolution of an asset. Itô’s lemma tells us the stochastic differential equation for the value of an option on that asset.

In mathematical terms, if we have a Wiener process X with increments dX that are normally distributed with mean zero and variance dt, then the increment of a function F (X) is given by

This is a very loose definition of Itô’s lemma but will suffice. See Itô (1951).

1952 Markowitz Harry Markowitz was the first to propose a modern quantitative methodology for portfolio selection. This required knowledge of assets’ volatilities and the correlation between assets. The idea was extremely elegant, resulting in novel ideas such as ‘efficiency’ and ‘market portfolios.’ In this Modern Portfolio Theory, Markowitz showed that combinations of assets could have better properties than any individual assets. What did ‘better’ mean? Markowitz quantified a portfolio’s possible future performance in terms of its expected return and its standard deviation. The latter was to be interpreted as its risk. He showed how to optimize a portfolio to give the maximum expected return for a given level of risk. Such a portfolio was said to be ‘efficient.’ The work later won Markowitz a Nobel Prize for Economics but is problematic to use in practice because of the difficulty in measuring the parameters ‘volatility,’ and, especially, ‘correlation,’ and their instability.

1963 Sharpe, Lintner and Mossin William Sharpe of Stanford, John Lintner of Harvard and Norwegian economist Jan Mossin independently developed a simple model for pricing risky assets. This Capital Asset Pricing Model (CAPM) also reduced the number of parameters needed for portfolio selection from those needed by Markowitz’s Modern Portfolio Theory, making asset allocation theory more practical. See Sharpe, Alexander and Bailey (1999), Lintner (1965) and Mossin (1966).

1966 Fama Eugene Fama concluded that stock prices were unpredictable and coined the phrase ‘market efficiency.’ Although there are various forms of market efficiency, in a nutshell the idea is that stock market prices reflect all publicly available information, and that no person can gain an edge over another by fair means. See Fama (1966).

1960s Sobol’, Faure, Hammersley, Haselgrove and Halton ... Many people were associated with the definition and development of quasi random number theory or low-discrepancy sequence theory. The subject concerns the distribution of points in an arbitrary number of dimensions in order to cover the space as efficiently as possible, with as few points as possible (see Figure 1.1). The methodology is used in the evaluation of multiple integrals among other things. These ideas would find a use in finance almost three decades later. See Sobol’ (1967), Faure (1969), Hammersley & Handscomb (1964), Haselgrove (1961) and Halton (1960).

1968 Thorp Ed Thorp’s first claim to fame was that he figured out how to win at casino Blackjack, ideas that were put into practice by Thorp himself and written about in his best-selling Beat the Dealer, the “book that made Las Vegas change its rules.” His second claim to fame is that he invented and built, with Claude Shannon, the information theorist, the world’s first wearable computer. His third claim to fame is that he used the ‘correct’ formulæ for pricing options, formulæ that were rediscovered and originally published several years later by the next three people on our list. Thorp used these formulæ to make a fortune for himself and his clients in the first ever quantitative finance-based hedge fund. He proposed dynamic hedging as a way of removing more risk than static hedging. See Thorp (2002) for the story behind the discovery of the Black-Scholes formulæ.

Figure 1.1: They may not look like it, but these dots are distributed deterministically so as to have very useful properties.

1973 Black, Scholes and Merton Fischer Black, Myron Scholes and Robert Merton derived the Black-Scholes equation for options in the early seventies, publishing it in two separate papers in 1973 (Black & Scholes, 1973, and Merton, 1973). The date corresponded almost exactly with the trading of call options on the Chicago Board Options Exchange. Scholes and Merton won the Nobel Prize for Economics in 1997. Black had died in 1995.

The Black-Scholes partial differential equation for the value V of an option is then

1974 Merton, again In 1974 Robert Merton (Merton, 1974) introduced the idea of modelling the value of a company as a call option on its assets, with the company’s debt being related to the strike price and the maturity of the debt being the option’s expiration. Thus was born the structural approach to modelling risk of default, for if the option expired out of the money (i.e. assets had less value than the debt at maturity) then the firm would have to go bankrupt.

Credit risk became big, huge, in the 1990s. Theory and practice progressed at rapid speed during this period, urged on by some significant credit-led events, such as the Long Term Capital Management mess. One of the principals of LTCM was Merton who had worked on credit risk two decades earlier. Now the subject really took off, not just along the lines proposed by Merton but also using the Poisson process as the model for the random arrival of an event, such as bankruptcy or default. For a list of key research in this area see Schönbucher (2003).

1977 Boyle Phelim Boyle related the pricing of options to the simulation of random asset paths (Figure 1.2). He showed how to find the fair value of an option by generating lots of possible future paths for an asset and then looking at the average that the option had paid off. The future important role of Monte Carlo simulations in finance was assured. See Boyle (1977).

Figure 1.2: Simulations like this can be easily used to value derivatives.

1977 Vasicek So far quantitative finance hadn’t had much to say about pricing interest rate products. Some people were using equity option formulæ for pricing interest rate options, but a consistent framework for interest rates had not been developed. This was addressed by Vasicek. He started by modelling a short-term interest rate as a random walk and concluded that interest rate derivatives could be valued using equations similar to the Black-Scholes partial differential equation.

The bond pricing equation is a parabolic partial differential equation, similar to the Black-Scholes equation. See Vasicek (1977).

1979 Cox, Ross and Rubinstein Boyle had shown how to price options via simulations, an important and intuitively reasonable idea, but it was these three, John Cox, Stephen Ross and Mark Rubinstein, who gave option-pricing capability to the masses.

The Black-Scholes equation was derived using stochastic calculus and resulted in a partial differential equation. This was not likely to endear it to the thousands of students interested in a career in finance. At that time these were typically MBA students, not the mathematicians and physicists that are nowadays found on Wall Street. How could MBAs cope? An MBA was a necessary requirement for a prestigious career in finance, but an ability to count beans is not the same as an ability to understand mathematics. Fortunately Cox, Ross and Rubinstein were able to distil the fundamental concepts of option pricing into a simple algorithm requiring only addition, subtraction, multiplication and (twice) division. Even MBAs could now join in the fun. See Cox, Ross & Rubinstein (1979) and Figure 1.3.

1979-81 Harrison, Kreps and Pliska Until these three came onto the scene quantitative finance was the domain of either economists or applied mathematicians. Mike Harrison and David Kreps, in 1979, showed the relationship between option prices and advanced probability theory, originally in discrete time. Harrison and Stan Pliska in 1981 used the same ideas but in continuous time. From that moment until the mid 1990s applied mathematicians hardly got a look in. Theorem, proof everywhere you looked. See Harrison & Kreps (1979) and Harrison & Pliska (1981).

Figure 1.3: The branching structure of the binomial model.

1986 Ho and Lee One of the problems with the Vasicek framework for interest-rate derivative products was that it didn’t give very good prices for bonds, the simplest of fixed-income products. If the model couldn’t even get bond prices right, how could it hope to correctly value bond options? Thomas Ho and Sang-Bin Lee found a way around this, introducing the idea of yield-curve fitting or calibration. See Ho & Lee (1986).

1992 Heath, Jarrow and Morton Although Ho and Lee showed how to match theoretical and market prices for simple bonds, the methodology was rather cumbersome and not easily generalized. David Heath, Robert Jarrow and Andrew Morton (HJM) took a different approach. Instead of modelling just a short rate and deducing the whole yield curve, they modelled the random evolution of the whole yield curve. The initial yield curve, and hence the value of simple interest rate instruments, was an input to the model. The model cannot easily be expressed in differential equation terms and so relies on either Monte Carlo simulation or tree building. The work was well known via a working paper, but was finally published, and therefore made respectable in Heath, Jarrow & Morton (1992).

1990s Cheyette, Barrett, Moore and Wilmott When there are many underlyings, all following lognormal random walks, you can write down the value of any European non-path-dependent option as a multiple integral, one dimension for each asset. Valuing such options then becomes equivalent to calculating an integral. The usual methods for quadrature are very inefficient in high dimensions, but simulations can prove quite effective. Monte Carlo evaluation of integrals is based on the idea that an integral is just an average multiplied by a ‘volume.’ And since one way of estimating an average is by picking numbers at random we can value a multiple integral by picking integrand values at random and summing. With N function evaluations, taking a time of O(N) you can expect an accuracy of O(1/N1/2), independent of the number of dimensions. As mentioned above, breakthroughs in the 1960s on low-discrepancy sequences showed how clever, non-random, distributions could be used for an accuracy of O(1/N ), to leading order. (There is a weak dependence on the dimension.) In the early 1990s several groups of people were simultaneously working on valuation of multi-asset options. Their work was less of a breakthrough than a transfer of technology.

They used ideas from the field of number theory and applied them to finance. Nowadays, these low-discrepancy sequences are commonly used for option valuation whenever random numbers are needed. A few years after these researchers made their work public, a completely unrelated group at Columbia University successfully patented the work. See Oren Cheyette (1990) and John Barrett, Gerald Moore & Paul Wilmott (1992).

1994 Dupire, Rubinstein, Derman and Kani Another discovery was made independently and simultaneously by three groups of researchers in the subject of option pricing with deterministic volatility. One of the perceived problems with classical option pricing is that the assumption of constant volatility is inconsistent with market prices of exchange-traded instruments. A model is needed that can correctly price vanilla contracts, and then price exotic contracts consistently. The new methodology, which quickly became standard market practice, was to find the volatility as a function of underlying and time that when put into the Black-Scholes equation and solved, usually numerically, gave resulting option prices which matched market prices. This is what is known as an inverse problem: use the ‘answer’ to find the coefficients in the governing equation. On the plus side, this is not too difficult to do in theory. On the minus side, the practice is much harder, the sought volatility function depending very sensitively on the initial data. From a scientific point of view there is much to be said against the methodology. The resulting volatility structure never matches actual volatility, and even if exotics are priced consistently it is not clear how to best hedge exotics with vanillas in order to minimize any model error. Such concerns seem to carry little weight, since the method is so ubiquitous. As so often happens in finance, once a technique becomes popular it is hard to go against the majority. There is job safety in numbers. See Emanuel Derman & Iraj Kani (1994), Bruno Dupire (1994) and Mark Rubinstein (1994).

1996 Avellaneda and Parás Marco Avellaneda and Antonio Parás were, together with Arnon Levy and Terry Lyons, the creators of the uncertain volatility model for option pricing. It was a great breakthrough for the rigorous, scientific side of finance theory, but the best was yet to come. This model, and many that succeeded it, was nonlinear. Nonlinearity in an option pricing model means that the value of a portfolio of contracts is not necessarily the same as the sum of the values of its constituent parts. An option will have a different value depending on what else is in the portfolio with it, and an exotic will have a different value depending on what it is statically hedged with. Avellaneda and Parás defined an exotic option’s value as the highest possible marginal value for that contract when hedged with any or all available exchange-traded contracts. The result was that the method of option pricing also came with its own technique for static hedging with other options. Prior to their work the only result of an option pricing model was its value and its delta, only dynamic hedging was theoretically necessary. With this new concept, theory became a major step closer to practice. Another result of this technique was that the theoretical price of an exchange-traded option exactly matched its market price. The convoluted calibration of volatility surface models was redundant. See Avellaneda & Parás (1996).

1997 Brace, Gatarek and Musiela Although the HJM interest rate model had addressed the main problem with stochastic spot rate models, and others of that ilk, it still had two major drawbacks. It required the existence of a spot rate and it assumed a continuous distribution of forward rates. Alan Brace, Dariusz Gatarek & Marek Musiela (1997) got around both of those difficulties by introducing a model which only relied on a discrete set of rates - ones that actually are traded. As with the HJM model the initial data are the forward rates so that bond prices are calibrated automatically. One specifies a number of random factors, their volatilities and correlations between them, and the requirement of no arbitrage then determines the risk-neutral drifts. Although B, G and M have their names associated with this idea many others worked on it simultaneously.

2000 Li As already mentioned, the 1990s saw an explosion in the number of credit instruments available, and also in the growth of derivatives with multiple underlyings. It’s not a great step to imagine contracts depending on the default of many underlyings. Examples of these are the once ubiquitous Collateralized Debt Obligations (CDOs). But to price such complicated instruments requires a model for the interaction of many companies during the process of default. A probabilistic approach based on copulas was proposed by David Li (2000). The copula approach allows one to join together (hence the word ‘copula’) default models for individual companies in isolation to make a model for the probabilities of their joint default. The idea was adopted universally as a practical solution to a complicated problem. However with the recent financial crisis the concept has come in for a lot of criticism.

2002 Hagan, Kumar, Lesniewski and Woodward There has always been a need for models that are both fast and match traded prices well. The interest-rate model of Pat Hagan, Deep Kumar, Andrew Lesniewski and Diana Woodward (2002), which has come to be called the SABR (stochastic, α, β, ρ) model, is a model for a forward rate and its volatility, both of which are stochastic. This model is made tractable by exploiting an asymptotic approximation to the governing equation that is highly accurate in practice. The asymptotic analysis simplifies a problem that would otherwise have to be solved numerically. Although asymptotic analysis has been used in financial problems before, for example in modelling transaction costs, this was the first time it really entered mainstream quantitative finance.

August 2007 quantitative finance in disrepute In early August 2007 several hedge funds using quantitative strategies experienced losses on such a scale as to bring the field of quantitative finance into disrepute. From then, and through 2008, trading of complex derivative products in obscene amounts using simplistic mathematical models almost brought the global financial market to its knees: Lend to the less-than-totally-creditworthy for home purchase, repackage these mortgages for selling on from one bank to another, at each stage adding complexity, combine with overoptimistic ratings given to these products by the ratings agencies, with a dash of moral hazard thrown in, base it all on a crunchy base of a morally corrupt compensation scheme, and you have the recipe for the biggest financial collapse in decades. Out of this many people became very, very rich, while in many cases the man in the street lost his life savings. And financial modelling is what made this seem all so simple and safe.

References and Further Reading

Avellaneda, M & Buff, R 1997 Combinatorial implications of nonlinear uncertain volatility models: the case of barrier options. Courant Institute, NYU

Avellaneda, M & Parás, A 1994 Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Applied Mathematical Finance1 165-194

Avellaneda, M & Parás, A 1996 Managing the volatility risk of derivative securities: the Lagrangian volatility model. Applied Mathematical Finance3 21-53

Avellaneda, M, Lévy, A & Parás, A 1995 Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance2 73-88

Bachelier, L 1995 Théorie de la Spéculation. Jacques Gabay

Barrett, JW, Moore, G & Wilmott, P 1992 Inelegant efficiency. Risk magazine 5 (9) 82-84

Black, F & Scholes, M 1973 The pricing of options and corporate liabilities. Journal of Political Economy81 637-659

Boyle, P 1977 Options: a Monte Carlo approach. Journal of Financial Economics4 323-338

Brace, A, Gatarek, D & Musiela, M 1997 The market model of interest rate dynamics. Mathematical Finance7 127-154

Brown, R 1827 A Brief Account of Microscopical Observations. London

Cheyette, O 1990 Pricing options on multiple assets. Advances in Futures and Options Research4 68-91

Cox, JC, Ross, S & Rubinstein M 1979 Option pricing: a simplified approach. Journal of Financial Economics7 229-263

Derman, E & Kani, I 1994 Riding on a smile. Risk magazine 7 (2) 32-39 (February)

Derman, E, Ergener, D & Kani, I 1997 Static options replication. In Frontiers in Derivatives (eds Konishi, A & Dattatreya, RE) Irwin

Dupire, B 1993 Pricing and hedging with smiles. Proc AFFI Conf, La Baule June 1993

Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18-20 (January)

Fama, E 1965 The behaviour of stock prices. Journal of Business38 34-105

Faure, H 1969 Résultat voisin d’un théoréme de Landau sur le nombre de points d’un réseau dans une hypersphere. C.R. Acad. Sci. Paris Sér. A269 383-386

Hagan, P, Kumar, D, Lesniewski, A & Woodward, D 2002 Managing smile risk. Wilmott magazine, September

Halton, JH 1960 On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Num. Maths.2 84-90

Hammersley, JM & Handscomb, DC 1964 Monte Carlo Methods. Methuen, London

Harrison, JM & Kreps, D 1979 Martingales and arbitrage in multi-period securities markets. Journal of Economic Theory20 381-408

Harrison, JM & Pliska, SR 1981 Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications11 215-260

Haselgrove, CB 1961 A method for numerical integration. Mathematics of Computation15 323-337

Heath, D, Jarrow, R & Morton, A 1992 bond pricing and the term structure of interest rates: a new methodology. Econometrica60 77-105

Ho, T & Lee, S 1986 Term structure movements and pricing interest rate contingent claims. Journal of Finance42 1129-1142

Itô, K 1951 On stochastic differential equations. Memoirs of the American Mathematical Society4 1-51

Li, DX 2000 On default correlation: a copula function approach. RiskMetrics Group

Lintner, J 1965 Security prices, risk, and maximal gains from diversification. Journal of Finance20 587-615

Markowitz, H 1959 Portfolio Selection: Efficient Diversification of Investment . John Wiley & Sons Ltd (www.wiley.com)

Merton, RC 1973 Theory of rational option pricing. Bell Journal of Economics and Management Science4 141-183

Merton, RC 1974 On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance29 449-470

Merton, RC 1992 Continuous-Time Finance. Blackwell

Mossin, J 1966 Equilibrium in a capital asset market. Econometrica34 768-783

Niederreiter, H 1992 Random Number Generation and Quasi-Monte Carlo Methods. SIAM

Ninomiya, S & Tezuka, S 1996 Toward real-time pricing of complex financial derivatives. Applied Mathematical Finance3 1-20

Paskov 1996 New methodologies for valuing derivatives. In Mathematics of Derivative Securities (eds Pliska, SR & Dempster, M)

Paskov, SH & Traub, JF 1995 Faster valuation of financial derivatives. Journal of Portfolio Management Fall 113-120

Richardson, LF 1922 Weather Prediction by Numerical Process. Cambridge University Press

Rubinstein, M 1994 Implied binomial trees. Journal of Finance69 771-818

Samuelson, P 1955 Brownian motion in the stock market. Unpublished

Schönbucher, PJ 2003 Credit Derivatives Pricing Models. John Wiley & Sons Ltd

Sharpe, WF, Alexander, GJ & Bailey, JV 1999 Investments. Prentice-Hall

Sloan, IH & Walsh, L 1990 A computer search of rank two lattice rules for multidimensional quadrature. Mathematics of Computation54 281-302

Sobol’, IM 1967 On the distribution of points in cube and the approximate evaluation of integrals. USSR Computational Mathematics and Mathematical Physics7 86-112

Stachel, J (ed.) 1990 The Collected Papers of Albert Einstein. Princeton University Press

Thorp, EO 1962 Beat the Dealer. Vintage

Thorp, EO 2002 Wilmott magazine, various papers

Thorp, EO & Kassouf, S 1967 Beat the Market. Random House

Traub, JF & Wozniakowski, H 1994 Breaking intractability. Scientific American January 102-107

Vasicek, OA 1977 An equilibrium characterization of the term structure. Journal of Financial Economics5 177-188

Wiener, N 1923 Differential space. Journal of Mathematics and Physics58 131-174

And Now a Brief Unofficial History!

Espen Gaarder Haug, as well as being an option trader, author, lecturer, researcher, gardener, soldier, and collector of option-pricing formulæ, is also a historian of derivatives theory. In his excellent book Derivatives: Model on Models (John Wiley and Sons Ltd, 2007) he gives the ‘alternative’ history of derivatives, a history often ignored for various reasons. He also keeps us updated on his findings via his blog http://www.wilmott.com/blogs/collector. Here are a few of the many interesting facts Espen has unearthed.

1688 de la Vega Possibly a reference to put-call parity. But then possibly not. De la Vega’s language is not particularly precise.

1900s Higgins and Nelson They appear to have some grasp of delta hedging and put-call parity.

1908 Bronzin Publishes a book that includes option formulæ, and seems to be using risk neutrality. But the work is rapidly forgotten!

1915 Mitchell, 1926 Oliver and 1927 Mills They all described the high-peak/fat-tails in empirical price data.

1956 Kruizenga and 1961 Reinach They definitely describe put-call parity. Reinach explains ‘conversion,’ which is what we know as put-call parity, he also understands that it does not necessarily apply for American options.

1962 Mandelbrot In this year Benoit Mandelbrot wrote his famous paper on the distribution of cotton price returns, observing their fat tails.

1970 Arnold Bernhard & Co They describe market-neutral delta hedging of convertible bonds and warrants. And show how to numerically find an approximation to the delta.

For more details about the underground history of derivatives see Espen’s excellent book (2007).

References and Further Reading

Haug, EG 2007 Derivatives: Models on Models, John Wiley & Sons, Ltd.

Mandelbrot, B & Hudson, R 2004 The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward. Profile Books

Chapter 2

FAQs

What are the Different Types of Mathematics Found in Quantitative Finance?

Short answer

The fields of mathematics most used in quantitative finance are those of probability theory and differential equations. And, of course, numerical methods are usually needed for producing numbers.

Example

The classical model for option pricing can be written as a partial differential equation. But the same model also has a probabilistic interpretation in terms of expectations.

Long answer

The real-world subject of quantitative finance uses tools from many branches of mathematics. And financial modelling can be approached in a variety of different ways. For some strange reason the advocates of different branches of mathematics get quite emotional when discussing the merits and demerits of their methodologies and those of their ‘opponents.’ Is this a territorial thing? What are the pros and cons of martingales and differential equations? What is all this fuss, and will it end in tears before bedtime?

Here’s a list of the various approaches to modelling and a selection of useful tools. The distinction between a ‘modelling approach’ and a ‘tool’ will start to become clear.

Modelling approaches:

• Probabilistic

• Deterministic

• Discrete: difference equations

• Continuous: differential equations

Useful tools:

• Simulations

• Discretization methods

• Approximations

• Asymptotic analysis

• Series solutions

• Green’s functions

While these are not exactly arbitrary lists, they are certainly open to some criticism or addition. Let’s first take a look at the modelling approaches.

Probabilistic One of the main assumptions about the financial markets, at least as far as quantitative finance goes, is that asset prices are random. We tend to think of describing financial variables as following some random path, with parameters describing the growth of the asset and its degree of randomness. We effectively model the asset path via a specified rate of growth, on average, and its deviation from that average. This approach to modelling has had the greatest impact over the last 30 years, leading to the explosive growth of the derivatives markets.

Deterministic The idea behind this approach is that our model will tell us everything about the future. Given enough data, and a big enough brain, we can write down some equations or an algorithm for predicting the future. Interestingly, the subjects of dynamical systems and chaos fall into this category. And, as you know, chaotic systems show such sensitivity to initial conditions that predictability is in practice impossible. This is the ‘butterfly effect,’ that a butterfly flapping its wings in Brazil will ‘cause’ rainfall over Manchester. (And what doesn’t!) A topic popular in the early 1990s, this has not lived up to its promises in the financial world.

Discrete/Continuous Whether probabilistic or deterministic, the eventual model you write down can be discrete or continuous. Discrete means that asset prices and/or time can only be incremented in finite chunks, whether a dollar or a cent, a year or a day. Continuous means that no such lower increment exists. The mathematics of continuous processes is often easier than that of discrete ones. But then when it comes to number crunching you have in any case to turn a continuous model into a discrete one.

In discrete models we end up with difference equations. An example of this is the binomial model for option pricing. Time progresses in finite amounts, the time step. In continuous models we end up with differential equations. The equivalent of the binomial model in discrete space is the Black-Scholes model, which has continuous asset price and continuous time. Whether binomial or Black-Scholes, both of these models come from the probabilistic assumptions about the financial world.

Now let’s take a look at some of the tools available.

Simulations If the financial world is random then we can experiment with the future by running simulations. For example, an asset price may be represented by its average growth and its risk, so let’s simulate what could happen in the future to this random asset. If we were to take such an approach we would want to run many, many simulations. There’d be little point in running just the one; we’d like to see a range of possible future scenarios.

Simulations can also be used for non-probabilistic problems. Just because of the similarities between mathematical equations, a model derived in a deterministic framework may have a probabilistic interpretation.

Discretization methods The complement to simulation methods, and there are many types of these. The best known are the finite-difference methods which are discretizations of continuous models such as Black-Scholes.

Depending on the problem you are solving, and unless it’s very simple, you will probably go down the simulation or finite-difference routes for your number crunching.

Approximations In modelling we aim to come up with a solution representing something meaningful and useful, such as an option price. Unless the model is really simple, we may not be able to solve it easily. This is where approximations come in. A complicated model may have approximate solutions. And these approximate solutions might be good enough for our purposes.

Asymptotic analysis This is an incredibly useful technique, used in most branches of applicable mathematics, but until recently almost unknown in finance. The idea is simple: find approximate solutions to a complicated problem by exploiting parameters or variables that are either large or small, or special in some way. For example, there are simple approximations for vanilla option values close to expiry.

Series solutions If your equation is linear (and they almost all are in quantitative finance) then you might be able to solve a particular problem by adding together the solutions of other problems. Series solutions are when you decompose the solution into a (potentially infinite) sum of simple functions, such as sines and cosines, or a power series. This is the case, for example, with barrier options having two barriers, one below the current asset price and the other above.

Green’s functions This is a very special technique that only works in certain situations. The idea is that solutions to some difficult problems can be built up from solutions to special cases of a similar problem.

References and Further Reading

Joshi, M 2003 The Concepts and Practice of Mathematical Finance. Cambridge University Press.

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd.

Wilmott, P 2007 Paul Wilmott Introduces Quantitative Finance, second edition. John Wiley & Sons Ltd.

What is Arbitrage?

Short answer

Arbitrage is making a sure profit in excess of the risk-free rate of return. In the language of quantitative finance we can say that an arbitrage opportunity is a portfolio of zero value today which is of positive value in the future with positive probability, and of negative value in the future with zero probability.

The assumption that there are no arbitrage opportunities in the market is fundamental to classical finance theory. This idea is popularly known as ‘there’s no such thing as a free lunch.’

Example

An at-the-money European call option with a strike of $100 and an expiration of six months is worth $8. A European put with the same strike and expiration is worth $6. There are no dividends on the stock and a six-month zero-coupon bond with a principal of $100 is worth $97.

Long answer

The principle of no arbitrage is one of the foundations of classical finance theory. In derivatives theory it is assumed during the derivation of the binomial model option-pricing algorithm and in the Black-Scholes model. In these cases it is rather more complicated than the simple example given above. In the above example we set up a portfolio that gave us an immediate profit, and that portfolio did not have to be touched until expiration. This is a case of a static arbitrage. Another special feature of the above example is that it does not rely on any assumptions about how the stock price behaves. So the example is that of model-independent arbitrage. However, when deriving the famous option-pricing models we rely on a dynamic strategy, called delta hedging, in which a portfolio consisting of an option and stock is constantly adjusted by purchase or sale of stock in a very specific manner.

Now we can see that there are several types of arbitrage that we can think of. Here is a list and description of the most important.

• A static arbitrage is an arbitrage that does not require rebalancing of positions

• A dynamic arbitrage is an arbitrage that requires trading instruments in the future, generally contingent on market states

• A statistical arbitrage is not an arbitrage but simply a likely profit in excess of the risk-free return (perhaps even suitably adjusted for risk taken) as predicted by past statistics

• Model-independent arbitrage is an arbitrage which does not depend on any mathematical model of financial instruments to work. For example, an exploitable violation of put-call parity or a violation of the relationship between spot and forward prices, or between bonds and swaps

• Model-dependent arbitrage does require a model. For example, options mispriced because of incorrect volatility estimate. To profit from the arbitrage you need to delta hedge, and that requires a model

Not all apparent arbitrage opportunities can be exploited in practice. If you see such an opportunity in quoted prices on a screen in front of you then you are likely to find that when you try to take advantage of them they just evaporate. Here are several reasons for this.

• Quoted prices are wrong or not tradeable

• Option and stock prices were not quoted synchronously

• There is a bid-offer spread you have not accounted for

• Your model is wrong, or there is a risk factor you have not accounted for

References and Further Reading

Merton, RC 1973 Theory of rational option pricing. Bell Journal of Economics and Management Science4 141-183

Wilmott, P 2007 Paul Wilmott Introduces Quantitative Finance, second edition. John Wiley & Sons Ltd

What is Put-Call Parity?

Short answer

Example Stock price is $98, a European call option struck at $100 with an expiration of nine months has a value of $9.07. The nine-month, continuously compounded, interest rate is 4.5%. What is the value of the put option with the same strike and expiration?

The put must therefore be worth $7.75.

Long answer

This is put-call parity.

Another way of interpreting put-call parity is in terms of implied volatility. Calls and puts with the same strike and expiration must have the same implied volatility.

The beauty of put-call parity is that it is a model-independent relationship. To value a call on its own we need a model for the stock price, in particular its volatility. The same is true for valuing a put. But to value a portfolio consisting of a long call and a short put (or vice versa), no model is needed. Such model-independent relationships are few and far between in finance. The relationship between forward and spot prices is one, and the relationships between bonds and swaps is another.