This course introduces students to derivatives, their use in financial markets and how they are priced and hedged in discrete time.
The course introduces the idea of derivative securities and why they exist, explaining the role of forward and option contracts in risk management. The concept of arbitrage free pricing (cash-and-carry pricing) is explained and developed into the fundamental theorem of asset pricing in discrete time. Pricing on the binomial tree (the CRR model) is explained, for both European and American style derivatives, in the context of the fundamental theorem and the relationship between the CRR model and the continuous time Black-Scholes-Merton formula discussed. The fundamental properties of option prices are given.
This course covers some of the material in Subject CT8 of the Institute/Faculty of Actuaries examinations and the is synoptic with Portfolio Theory and Asset Models (F79PA).
- Understand the main uses of derivatives in hedging, arbitrage and speculation.
- Compare and contrast the features of the main derivatives contracts.
- Demonstrate an awareness of the practical aspects involved in the trading of derivatives contracts.
- Calculate fair prices of forward contracts.
- Describe how the pricing of a forward contract on a commodity differs from an equity forward contract and show how to incorporate the convenience yield and storage costs.
- Describe the different types of commodities that can be traded and their associated derivatives contracts.
- Describe what is meant by an arbitrage-free market.
- Show how the replication argument breaks down when the trinomial model is used in place of the binomial model, and how the market can be completed by the inclusion of an additional risky asset.
- Demonstrate how the existence of a martingale measure proves the absence of arbitrage
- Define a discrete-time martingale.
- Understand the role of the risk-neutral probability measure as a computational tool.
- Describe the binomial (CRR) model.
- Understand the concepts of replication, hedging, and delta hedging in the context of the binomial model.
- Calculate prices of European and American call and put options using the binomial model
- Prove the convergence of the price of a European call or put option to the Black-Scholes formula.
- Understand put-call parity, gearing, and the dependence of option prices on underlying variables.
Learning Outcomes: Subject Mastery
At the end of studying this course, students will understand
- Forward contracts, over-the counter and exchange-traded derivatives, use in hedging.
- Commodities and dividends; convenience yield and storage costs.
- Options: basics, strategies and profit diagrams, European and American options, put-call parity.
- No-arbitrage pricing, the risk-neutral probability measure and incomplete markets.
- Pricing European-style derivative contracts using binary trees and the binomial model and American options using the binomial model.
- The binomial model for stock prices and the relationship between the Cox-Ross-Rubensein model and the Black-Scholes model.
The following books are recommended:
- Baxter, M. and Rennie, A. (1996), Financial calculus, Cambridge University Press;
- Pliska, S.R., (2000) Introduction to Mathematical Finance, Blackwell.
Students may also find the following books, which extend into continuous time model, useful:
- Hull, J., (2000) Options, futures and other derivative securities, Prentice Hall;
- Neftci, S.. (1998) An Introduction to the Mathematics of Financial Derivatives, Academic Press;
- Jarrow, R. & Turnbull, S. (2000) Derivative Securities, South-Western College Publishing.
2-hour end-of-semester exam (80%), continuous assessment (20%).
SCQF Level: 9.
Help: If you have any problems or questions regarding the course, you are encouraged to contact the lecturer
VISION: further information and course materials are available on VISION