Lecture notes for STAT3006 / STATG017 Stochastic Methods in Finance 1
Department of Statistical Science, UCL 2011-2012
1 Financial Markets and Products 8
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1.1 Financialmarkets………………………… 8
1.2 Equities …………………………….. 8
1.3 Fixedincome(FI)………………………… 9
1.4 Currencies……………………………. 10
1.5 Commodities ………………………….. 11
1.6 Indices……………………………… 11
1.7 Furtherreading …………………………. 11
2 Time value of money 12
2.1 Compoundinterestandpresentvalue………………. 12 2.2 Compoundinterestwithnon-annualpayments . . . . . . . . . . . . . . 13 2.3 Presentvalue ………………………….. 15 2.4 GovernmentBondvaluation …………………… 15 2.5 Furtherreading …………………………. 17
3 Introduction to Derivatives 18
3.1 Forwards…………………………….. 19 3.2 Futures …………………………….. 21 3.3 Options …………………………….. 21 3.4 OptionPayoffs …………………………. 21 3.5 Hedging …………………………….. 24 3.6 Speculation …………………………… 24 3.7 Combiningderivatives ……………………… 25
4 Arbitrage and the pricing of forward contracts 28
4.1 Arbitrage ……………………………. 28
4.2 Example – Arbitrage opportunities in a forward contract . . . . . . . . 28
4.3 Pricing forward contracts for securities that provides no income . . . . 30
4.4 Valueofaforwardcontract …………………… 31
4.5 Forward contracts on a security that provides a known cash income . . 32
Stochastic Methods in Finance 1
4.6 Knowndividendyield………………………. 33 4.7 Forwardforeignexchangecontracts ……………….. 34
5 Pricing Options under the Binomial Model 35
5.1 Modelling the uncertainty of the underlying asset price . . . . . . . . . 35
5.2 Asimpleexample………………………… 35 5.2.1 Arbitrageopportunity ………………….. 37 5.2.2 No-arbitragepricing …………………… 38
5.3 One-stepbinomialtree ……………………… 39
5.4 Areplicatingportfolio ……………………… 40
5.5 Risk-neutralvaluation ……………………… 41
5.6 Appendix:Arisklessportfolio………………….. 42 5.6.1 Examplerevisited…………………….. 42 5.6.2 No-arbitragepricing …………………… 43 5.6.3 Risklessportfolio-generalresult …………….. 44
5.7 Furtherreading …………………………. 45
6 Applications of the Binomial Model 46
6.1 Thevalueofaforwardcontract …………………. 46 6.1.1 Areplicatingstrategy ………………….. 46 6.1.2 Risk-neutralvaluation ………………….. 47
6.2 AEuropeanputoption……………………… 49
6.3 Two-stepbinomialtrees …………………….. 50
6.4 Generalmethodforn-steptrees…………………. 51
6.5 PricingofAmericanoptions …………………… 52
7 Calculus refreshers 55
7.1 Taylorseries…………………………… 55 7.2 Chainruledifferentiation…………………….. 56 7.3 Partialdifferentiation………………………. 56 7.4 Linearordinarydifferentialequations ………………. 57
8 Continuous-time stochastic processes for stock prices 58
8.1 Randomwalk ………………………….. 59
8.2 OtherprocessesandMarkovproperty………………. 59
8.3 Takinglimitsoftherandomwalk ………………… 60
8.4 Brownianmotion(Wienerprocess) ……………….. 61
8.5 DefinitionofBrownianmotion………………….. 62
8.6 GeneralisedBrownianMotionprocess………………. 64
8.7 Itˆoprocess……………………………. 65
8.8 A process for stock prices: the geometric Brownian motion . . . . . . . 66
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8.9 Appendix: Brownian Motion as a limit of a discrete time random walk 68
9 Introduction to stochastic calculus and Itˆo’s lemma 70
9.1 Ordinaryandstochasticcalculus…………………. 70
9.2 Itˆo’sformula…………………………… 72
9.3 Examples ……………………………. 73
9.3.1 Derivation of the SDE of a generalised Brownian motion . . . . 73
9.3.2 Solution of the SDE of a geometric (exponential) Brownian motion 74
9.3.3 Logarithmofstockprices ………………… 75
9.3.4 Generictransformationofstockprices . . . . . . . . . . . . . . 75
9.3.5 Forwardprice ………………………. 76
9.4 Appendix:secondordereffectsinthelimit……………. 76
9.5 Furtherreading …………………………. 77
10 The Black-Scholes model 78
10.1Lognormalpropertyofstockprices ……………….. 78
10.2 The Black-Scholes-Merton differential equation . . . . . . . . . . . . . . 79 10.2.1 AssumptionsoftheBlack-Scholesmodel . . . . . . . . . . . . . 79 10.2.2 Boundaryconditions …………………… 82
10.3 Black-Scholes formulas for the pricing of vanilla options . . . . . . . . . 83
10.4 BSMPDEusingarisklessportfolioapproach . . . . . . . . . . . . . . 85
10.5 SimpleextensionstotheBlack-Scholesmodel . . . . . . . . . . . . . . 86
10.6Furtherreading …………………………. 86
11 Hedging and the Greeks 88
11.1TheGreeks …………………………… 88 11.2DeltaHedging………………………….. 88 11.3Gammaandgammaneutralportfolios ……………… 90 11.4Furtherreading …………………………. 92
12 Volatility 93
12.1 Estimatingvolatilityfromhistoricaldata . . . . . . . . . . . . . . . . . 93 12.2ImpliedVolatility………………………… 94 12.3Volatilitysmiles…………………………. 94 12.4Furtherreading …………………………. 98
13 Risk-neutral pricing in continuous time 99
13.1Therisk-neutralprocess …………………….. 99 13.2Risk-neutralpricing……………………….. 100 13.2.1 Example:Europeancalloption ……………… 100 13.3Ausefullog-normaldistributionresult ……………… 102
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13.4 The risk-neutral Monte Carlo approach to derivative pricing . . . . . . 102 13.4.1 Simulating Geometric Brownian motions . . . . . . . . . . . . . 103 13.4.2 Generatingrandomvariables……………….. 103
13.5 Appendix – Risk neutral pricing and the Black-Scholes equation . . . . 104
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STOCHASTIC METHODS IN FINANCE 1 2011–12
Lectures: Term 1, Tue 5–7pm, starting 4 October. No lecture in reading week.
In Course Assessment: In class assessment on Week 5; Tuesday 1 Novem-
Course guidance notes
This introductory note provides some guidance around the course.
Aims of course
To introduce mathematical and statistical concepts and tools used in the finance in- dustry, in particular stochastic models and techniques used for derivative pricing.
Prerequisites
No prior knowledge of finance is assumed, though an interest in the subject is a distinct advantage.
A course covering probability and distribution theory, and a foundation in calculus and differential equations is required. For example you should be comfortable with ordinary differential equations (in particular be familiar with linear and exponential growth equations), Taylor expansions, partial differentiation and standard integration techniques such as integration by parts and substitution. You should also be comfort- able with conditional probabilities, Markov processes, probability spaces, probability density functions and normal distributions.
Stochastic Methods in Finance 1
Course content
1. Intro to financial products, markets and derivatives 2. Time value of money
3. Arbitrage pricing
4. The Binomial pricing model
5. Brownian motion and continuous time modelling of assets 6. Stochastic calculus
7. The Black-Scholes framework
8. Risk-neutral pricing
Lecture notes
The printed notes should be used as a guide to some of the key topics in the course, and are aimed at providing written copies of some of the working in the lectures so that students can concentrate on the subject without needing to copy too many lines of algebra. At the end of each set of printed lecture notes a selection of further reading is provided, which will give a guide to the sections of the course-related texts in which the lecture topic can be found, as well as a selection of more advanced reading for the interested student.
Note that the printed lecture notes do not contain everything that is outlined in the lectures, and so should not be used as a substitute for lecture attendance. The full set of course material consists of material presented in the lectures, combined with the handouts and the ideas and techniques developed through exercise sheets and the two workshops.
Exercise sheets
The exercise sheets are an integral part of the learning on the course. These should be attempted on a weekly basis. As well as providing opportunities for consolidating learning from the lectures, these exercises will also be used as a learning tool themselves to explore the concepts in the course. Therefore a student that does not complete all the exercises will not have achieved the full set of learning outcomes for the course.
These exercise sheets will not be marked and so do not need to be handed in.
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The course builds quickly on material learnt previously. You should therefore plan each week to review your lecture notes, review the previous week’s exercise solutions, as well as attempt the current week’s exercises before the next lecture. It is strongly recommended that you attempt the exercises each week as subsequent lectures usually build on these exercises.
The two workshops also form an integral part of the learning for the course. They will provide an opportunity to further develop your understanding of concepts introduced in the lectures, and also introduce some new techniques. They are usually timetabled for Friday afternoons – see the Statistics timetable for the specific dates and times for this course.
In course assessments
The in course assessment (ICA) will be a closed book classroom test during the lecture of the 5th week of term.
Office hours
Office hours for students requiring help with course material will be after each lecture, arranged by prior appointment only. Please indicate the area of the course and prob- lem that you require help on when arranging this.
Reading List
The course is self sufficient, but this is a list of both related texts and some wider reading. There is now a vast range of books on mathematical finance and derivative pricing, so this is necessarily just a small selection with the aim of providing a good range of approaches taken.
Additional books are also referenced at the end of some of the lecture notes in the reading list sections where they are particularly relevant to the lecture topic.
Main Related Texts
– . Hull (2005) Options Futures and Other Derivative Securities. Now in the 8th edition, .
– & (1996) Financial Calculus. Cambridge University
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– (2001) Introduces Quantitative Finance Wiley
Further introductory texts in the area
These books provides good introduction to the main topics covered in the course, each presenting the material with a different perspective.
– (2000) An Introduction to the Mathematics of Financial Derivatives. Academic Press.
Excellent coverage of the mathematics needed for pricing derivatives, explained in an easy to follow way. Includes a good introductory presentation of stochastic calculus.
– (2002), A course in Financial Calculus. Cambridge University Press
A nice introduction to the principals underlying derivative pricing, with the emphasis on laying foundations of understanding. Takes a similar approach to the Baxter and Rennie book, but at a slightly simpler level.
– Wilmott, Howison, and Dewynne (1995), The Mathematics of Financial Deriva- tives. Cambridge University Press.
This book presents derivative pricing from a physicist’s perspective, and so focuses on solving partial differential equations, a less common perspective in more recent books. Worth a read for a good explanation of the p.d.e.s relevant to finance, and particularly for those with a background in applied maths, but not essential for this course.
– ̈ork (1998) Arbitrage Theory in Continuous Time. Oxford University Press.
Good coverage of the key topics. Could be used as an excellent introductory text if you are comfortable with a more mathematical presentation. Provides some good intuition.
– (2004), An Introduction to Financial Option Valuation. Cam- bridge University Press
A simple presentation of the basics of options pricing, this book also provides a guide to computational implementation of the results, with guides to Matlab code at the end of each chapter. Useful for those who want to work on their own to implement and extend the results we see in the course.
– – Probability theory in finance – a mathematical guide to the Black- Scholes formula. American Mathematical Society, Graduate Series in Mathematics.
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A good introduction to the the use of martingales in no-arbitrage pricing. Provides a more mathematically rigorous treatment of topics covered in this course, but also takes the time to introduce the motivation for and intuition of the results presented, and proceeds at a slow pace. Chatty style of writing.
More Advanced Texts
– (2004) Volatility and Correlation; Second Edition; Wiley.
– (2004) – Stochastic Calculus for Finance – 1 Discrete-Time models; 2 Continuous-Time models. Springer.
A rigorous, mathematical approach, and goes into much more depth of the underlying probability theory results used in martingale pricing. Two volumes.
– and – Martingale Methods in Financial Mod- elling (Springer)
In-depth coverage of a wide range of models and products, using a rigorous, probability based approach.
– ochrane – Asset Pricing – Excellent book that presents general frameworks for the range of pricing techniques, with an aim of unifying asset pricing techniques and ”clarifying, relating and simplifying the set of tools we have all learned in a hogdepodge manner”. Hence goes beyond no-arbitrage derivative pricing covered in this course to include other types of pricing models (including CAPM etc), and on the way develops an awareness of the links between theoretical finance and macro-economics. Also in- cludes interesting chapters on the theory of statistical estimation. Aimed at graduate level, so I suggest reading after completing some introductory courses in asset pricing.
– , Introduction to Mathematical Finance: Discrete Time Models ( )
More of an introductory text, but I have put in this section as it looks more widely at models beyond derivative pricing. Provides a rigorous study of use of risk-neutral probability measures, takes a mathematical approach, but is still accessible if you are comfortable with calculus and elementary probability theory, and prepared to put some work in.
– & Vladimir Vovk (2001) Probability and Finance: It’s Only a Game! Wiley
Interesting book that presents a new framework for looking at probability and finance.
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Tough going without a strong background in probability theory.
Background and references in finance
These books provide guides to financial concepts and jargon, and may be useful as references as the course progresses, or for students with no prior finance knowledge.
– (1995) How to read the financial pages. 4th edition, Century Busi- ness.
– , & (1997) Dictionary of finance and bank- ing. Oxford University Press.
– , (2003) Dictionary of Finance and Investment Terms 6th edition, Barron’s.
A selection of other finance related books
– and (2007), How I became a quant – insights from 25 of Wall Street’s elite (Wiley)
Twenty five different industry professionals contribute a chapter each, and therefore this book provides an eclectic mix of views on a range of areas and industries in which quantitative techniques are used. Also highlights some of the challenges of developing modelling techniques that are effective in a business environment, as well as a number of personal views on career steps and challenges. Worth bearing in mind that you don’t have to empathise with all of the authors (or even any) to have a rewarding and successful career in finance – there is a huge range of different working cultures in industry.
An excellent read for those interested in using their stats, maths, economic or pro- gramming skills in the finance industry. An even better read for those of you who have these skills and are not currently interested. And probably essential reading if there is anyone who thinks they know they only want to research hybrid quantos stoch. vol. models for one of 4 select IBs/develop equity stat-arb trading algorithms on their favourite platform… etc.
– (2000) – When genius failed – the risk and fall of Long Term Capital Management
Well told story of the potential financial crises resulting from hedge fund LTCM’s po- sitions in 1998, and the response of regulators and investment banks. A good case
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study in the importance of stress testing underlying model assumptions.
– – Liars Poker
Classic book telling the story of the world of bond trading and investment banking in the eighties. Still relevant today.
– Sorkin (2009) Too big to fail – Inside the battle to save Wall Street Another well put together telling of a financial crisis, this time the story of the fall of Lehmans in 2008, based on interviews with many of the key players in banking and regulation at the time.
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Financial Markets and Products
In this lecture we provide a basic introduction to the financial markets that will provide the background to the problems in finance that we will study in the course. We also look at the types of risks that financial institutions operating in these financial markets are exposed to.
1.1 Financial markets
The financial markets are institutions and procedures that facilitate transactions in all types of financial securities (claim on future income).
We can have organised security or stock exchanges, or over-the-counter markets (for very specific requests).
Why do we have financial markets?
• In order to transfer efficiently funds from economic units who have them to people who can use them
• To reallocate risk and to manage risk
1.2 Equities
• Equity (or stock or share) is the ownership of a small piece of a company (claim on the earnings and on the assets of the company).
• The shareholders are the people who own the company, and have a say in the running of the business by the directors.
• Most companies give out lump sums every 6 months or a year, which are called dividends.
Stochastic Methods in Finance 1
• The shares of large companies are traded in regulated stock exchanges.
Later in the course what we will want to study (and model) is the stock price. The price of a share is determined by the market, and depends on the demand and supply of shares in the market.
1.3 Fixed income (FI)
FI securities are financial contracts between two counterparties where a fixed exchange of cash flows is agreed (which depends on the interest rate).
Interest rate is the cost of borrowing or the price paid for the rental of funds and is usually expressed as % per year.
Broadly, there are two types of interest:
• fixed interest rat
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