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CENTRE FOR ECONOMIC AND FINANCIAL STUDIES

FINANCIAL DERIVATIVES

2010-2011

SEMESTER TWO

COURSE COORDINATOR AND LECTURER: Dr Mario Cerrato

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Total number of teaching hours: 20 hours lectures, 5 hours computer labsCourse credits: 20Course code: MJES

Course aims and overview

The lectures provide a graduate-level analysis of different types of major option pricingmechanisms and advanced option hedging techniques encountered in financial markets. Thecourse focuses on pricing options within a Black, Scholes and Merton-type framework. MonteCarlo methods will be covered in detail. Topics will be covered from a practical and theoreticalaspect using examples from equity and interest rate markets.

Computer lab sessions provide an opportunity for students to analyze problems and to explorethe effectiveness of the different approaches to pricing and hedging presented in the lectures,using advanced Excel spreadsheet and MatLab 6 modeling.

Intended learning outcomes

By the end of this course, students should be able to:

understand the martingale valuation of price contingent claims and the Black andScholes model and its main model misspecifications; the concept of exotic derivativesand their valuation; and the main Gaussian term structure models, including Vasicek,Ho-Lee and Hull and White;

apply Monte Carlo simulation to price derivatives and other assets;

efficiently compute Greeks for a variety of options using Monte Carlo methods.

Assessment

Students are assessed on the basis of coursework (25%) and a final written examination (75%).Coursework consists of a computer project. The final examination takes the form of a two-hourpaper, with students being required to answer two questions from a choice of four.

Penalty for lateness

Penalties for late submission of coursework apply. Please refer to the MSc handbook, sectionIn-course assessment.

Lecture outline

Lecture 1: This lecture presents the martingale valuation approach to price contingent claimsand the Black and Scholes model and its main model misspecifications.

Security and trading strategies: The Fundamental Theorem of Asset Pricing.

Black and Scholes economy and the Black and Scholes model.

Computing Greeks.

Implied volatility and volatility smiles.

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Reading: Cerrato, 2008.

Lecture 2: This lecture presents the Cox, Ross and Rubinstein (1979) binomial model andshows how it can be used to price equity derivatives.

The Cox, Ross and Rubinstein (1979) binomial model.

Binomial extensions and applications to price derivatives.

Reading: Hull, J., 2006, Chapter 18; Baz, J. and Chacko, G., Chapter 2.

Lecture 3: This lecture introduces the main concept of exotic derivatives and describes some ofthe main exotic options and the main valuations approaches used for pricing.

Arithmetic and geometric average options.

Lookback options.

Cash or nothing options.

Barrier options.

Reading: Hull, J., 2006, Chapter 19; Baz, J. and Chacko, G., Chapter 2.6.

Lecture 4: This lecture introduces students to Monte Carlo simulation to price derivatives anddescribes ways of extending that methodology using variance reduction techniques. Practicalexamples on pricing a variety of exotic options will be provided.

Monte Carlo methods.

Implementing Monte Carlo methods using antithetic variates.

Implementing Monte Carlo methods using control variates.

Implementing Monte Carlo methods using importance sampling.

Reading: Cerrato, 2008; Boyle et al, 1997.

Lecture 5: This lecture builds on Lecture 4 and describes efficient ways of computing Greeksfor a variety of options using Monte Carlo methods.

Exact computation of Greeks using Monte Carlo methods.

Pathways methods.

Likelihood ratio methods.


Lecture 6: This lecture reviews the relevant literature on pricing American options focusing both

on semi-closed form solutions and on various computational methods used to estimate theexercise boundary.

American options.

The put`s critical exercise boundary.

The Barone Adesi and Wiley (1987) model.


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Lecture 7: This lecture describes ways of using Monte Carlo techniques to price Americanoptions.

A dynamic programming approach to pricing American options.

Monte Carlo applications to price American options.

Longstaff and Schwartz (2001) method.

Glasserman and Broadie (2002) method.

Reading: Cerrato, 2008; Longstaff and Schwartz, 2001; Glasserman and Broadie, 2002.

Lecture 8: This lecture introduces students to the main stochastic volatility models such asHeston (1999) model and compares the latter with local volatility models, with particularemphasis on model calibration.

The concept of stochastic volatility and local volatility.

Main issues for model calibration.

Heston economy and the Heston (1999) model.

Pricing American put options when volatility is stochastic.


Lecture 9: In this lecture students will be shown how to price and to compute Greeks efficientlyfor exotic options when volatility is stochastic. Practical examples will be provided.

Euler methods to solve stochastic differential equations.

Exact simulation of Greeks under stochastic volatility.

Computing Greeks for exotics options.


Lecture 10: This lecture introduces the main Gaussian term structure models.

The Vasicek model.

The Ho-Lee model.

The Hull and White model.

Reading: Cerrato, 2008; Baz, J. and Chacko, G., Chapter 3.

Books and other learning resources

Main reading list

Baz, J. and Chacko, G. (2004). Financial Derivatives, Cambridge University Press.Epps, T.W. (2002). Pricing Derivative Securities, World Scientific Publishing.Cerrato, M. (2008). The Mathematics of Derivative Securities, with MatLab Applications, Mimeo.Hull, J. (2006). Options, Futures, and Other Derivatives, Prentice Hall.

Additional learning resourcesBarone-Adesi, G. and Whaley, R.E. (1987). 'Efficient analytic approximation of American optionvalues', Journal of Finance, vol. 42, pp. 301-320.

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Boyle, P., Brodie, P.M. and Glasserman, P. (1997). Monte Carlo methods for security pricing,Journal of Economic Dynamics and Control, vol. 21, pp. 1267-1321.Broadie, M. and Kaya, O. (2004). 'Exact simulation of stochastic volatility and other affine jumpdiffusion processes', Columbia University, Graduate School of Business.Cox, J., Ross, C.S. and Rubinstein, M. (1979). 'Option pricing: a simplified approach', Journal ofFinancial Economics, vol. 7, pp. 229-263.

Glasserman, P. and Yu, B. (2004). 'Simulation for American options: regression now orregression later?', in (H. Niederreiter, ed.), Monte Carlo and Quasi Monte Carlo Methods 2002,pp. 213-226.Heston, S. (1999). 'A closed-form solution for options with stochastic volatility with applicationsto bond and currency options', Review of Financial Studies, vol.6, pp. 327-343.Longstaff, F.A. and Schwartz, E.S. (2001). 'Valuing American options by simulation: a simpleleast-squares approach', The Review of Financial Studies, vol. 14(1), pp. 113-147.

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