# Coursebooks

## Derivatives

Hugonnier Julien

English

#### Summary

The objective of this course is to provide a detailed coverage of the standard models for the valuation and hedging of derivatives products such as European options, American options, forward contracts, futures contract and exotic options.

#### Content

Part I: Discrete-time models

• Introduction to derivatives products and markets
• Model free results and static models
• Multiperiod discrete-time models
• American options and applications
• Convergence

Part II: Continuous-time models

• Arbitrage, valuation and hedging in continuous-time
• The Black-Scholes model
• Foreign exchange products
• American derivatives
• Implied, local, and stochastic volatility
• Exotic options

#### Keywords

Derivatives, options, arbitrage valuation, hedging

#### Learning Prerequisites

##### Required courses

• Introduction to finance
• Stochastic calculus

• Econometrics

##### Important concepts to start the course

To follow this course students need to have taken an introduction to finance, and must possess solid foundations in probability theory and stochastic calculus.

#### Learning Outcomes

By the end of the course, the student must be able to:
• Describe the principal types of derivatives contracts including forwards, futures and options and compare their basic usages for hedging or speculation
• Describe and analyse the most common types of options strategies such as spreads, straddles, collars, and covered calls or puts.
• Formulate the no-arbitrage principle and illustrate its basic application in a model-free setting: cash and carry relations for different types of underlying securities with or without dividends, put/call parity, arbitrage bounds on option prices, early exercise of American options.
• Discuss the main characteristics of a general discrete time model with finitely many states of nature, multiple securities and possibly stochastic interest rates.
• Work out / Determine whether a given discrete-time model with finitely many states of nature is arbitrage free and has complete markets; Relate these properties to the existence and uniqueness of an equivalent martingale measure.
• Discuss and apply risk-neutral valuation to price and hedge derivatives of either European or American type in the context of a given discrete time model with finitely many states and complete markets.
• Construct and implement a binomial model to price and hedge both plain vanilla derivatives of European or American type as well as any exotic derivative.
• Describe the main assumptions of the Black-Scholes model and its limitations, derive the valuation partial differential equation and the Black-Scholes-Merton formula for the price of standard European options.
• Discuss the main option Greeks and use them appropriately for risk management and financial engineering purposes in the context of the Black-Scholes model or its extensions to futures contracts and foreign exchange.
• Work out / Determine whether a general Brownian-driven model of financial markets admits an equivalent martingale measure, relate the uniqueness of this probability measure to market completeness, and derive the risk-neutral dynamics of traded securities prices and relevant state variables.
• Derive the partial differential equation satisfied by the price of a European derivative in a given Markovian model, and use it with appropriate boundary conditions to price options in specific models.
• Formulate the valuation of American options as a free boundary problem for the valuation PDE in the context of the Black-Scholes model, derive and discuss exact solutions for the infinite horizon case and the Barone-Addesi-Whaley approximation for the finite horizon case.

#### Transversal skills

• Plan and carry out activities in a way which makes optimal use of available time and other resources.
• Evaluate one's own performance in the team, receive and respond appropriately to feedback.
• Continue to work through difficulties or initial failure to find optimal solutions.
• Use both general and domain specific IT resources and tools

#### Teaching methods

Lectures and exercise sessions

#### Expected student activities

• Participate in weekly lectures
• Participate in weekly exercise sessions
• Solve and turn in the weekly homework assignment (20%)
• Write a midterm exam (30%) and a final exam (50%)

#### Assessment methods

• 50% Homework assignments
• 50% Final examination

#### Supervision

 Office hours Yes Assistants Yes Forum Yes

#### Resources

##### Bibliography

• K. Back, A course in derivative securities, Springer Verlag, New York, 2005.
• N. Bingham and R. Kiesel, Risk neutral valuation, Springer Verlag, New York, 2004.
• J. Hull, Options, futures and other derivatives, Prentice Hall.
• D. Lamberton & B. Lapeyre, Introduction to stochastic calculus applied to finance, Second edition, Chapman and Hall, 2008.
• S. Shreve, Stochastic Calculus for Finance I and II, Springer Verlag, New York, 2004.
• T. Bjork, Arbitrage theopry in continuous-time, 2nd Edition, Oxford University Press, New York, 2004

#### Prerequisite for

• Advanced topics in financial econometrics
• Credit risk
• Fixed income analysis
• Real options and financial structuring

### Reference week

MoTuWeThFr
8-9
9-10 EXTRANEF126CM1221
10-11
11-12
12-13
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14-15
15-16
16-17
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20-21
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Lecture
Exercise, TP
Project, other

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• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German