Derivatives and Risk Management

• Forwards and futures
• Stock options: general properties and specific models (the binomial model, the Black-Scholes-Merton model and alternative
• odels)
• Hedging and risk management with options
• Risk-neutral pricing and absence of arbitrage
• Bonds, yield curves, and interest rate derivatives
• Models of the term structure of interest rates
• Interest rate risk management
• Numerical methods: trees, finite differences, Monte Carlo simulation
• Stochastic (Itô) calculus 

Demonstrate knowledge about the course’s focus areas enabling the student to

• Describe and compare the payoffs and practical applications of forwards and futures.
• Explain how forward prices and futures prices can be computed,
• Describe the payoffs and practical applications of European and American call and put options on stocks.
• Describe when early exercise of American options can be optimal.
• Explain how stock prices are modelled in the binomial and Black-Scholes-Merton models.
• Explain how replicating portfolios and option prices can be computed in the binomial and Black-Scholes-Merton models.
• Criticise and reflect over the crucial assumptions in the binomial and Black-Scholes-Merton models
• Describe implied volatility smiles and provide potential explanations of such smiles.
• Describe alternative models for stock option pricing and compare them to the Black-Scholes-Merton model.
• Describe the so-called Greek letters in relation to option pricing and explain how delta, theta, and gamma are related
• Explain the concept of arbitrage and why the absence of arbitrage implies that a derivative asset can be priced by computing a conditional expectation of the properly discounted payoff under a risk-adjusted probability measure or, in some cases, by solving a specific partial differential equation.
• Explain basic bond market terminology
• Describe the payoffs, practical applications, and general pricing results of interest rate forwards and futures, Eurodollar-futures, bond options, caps, floors, collars, swaps, and swaptions
• Explain and criticize selected continuous-time models of the term structure of interest rates
• Explain how interest rate risk can be measured and managed using popular continuous-time models of the term structure of interest rates.
• Explain how selected numerical methods (binomial/trinomial trees, finite difference solutions of partial differential equations, Monte Carlo simulation) can be applied to the pricing of derivatives in selected models and reflect over the applicability of the different methods.  

• Derive forward and futures prices, and identify when and how the price of a futures contract differs from the price of an otherwise identical forward contract
• Derive the put-call parity
• Apply the binomial and Black-Scholes-Merton models for stock option pricing.
• Derive pricing formulas and replicating portfolios in the binomial and Black-Scholes-Merton model.
• Derive the fundamental partial differential equation and solve it in the Black-Scholes-Merton model and in similar frameworks.
• Compute implied volatility smiles
• Apply alternative models for stock option pricing and compare them to the Black-Scholes-Merton model.
• Apply the Greek letters to hedging and risk management.
• Compute, and relate bond prices, bond yields, and the term structure of interest rates.
• Extract yield curves from bond prices
• Derive general pricing results of interest rate forwards and futures, Eurodollar-futures, bond options, caps, floors, collars, swaps, and swaptions, and derive relations between these derivative securities.
• Derive the fundamental partial differential equation and pricing formulas for selected interest rate derivatives within selected continuous-time models of the term structure of interest rates.
• Compare selected continuous-time models of the term structure of interest rates.
• Compute dynamic interest rate risk measures and apply them to interest rate risk management.
• Compare dynamic interest rate risk measures to the classical Macaulay and Fisher-Weil risk measures.
• Implement selected numerical methods (binomial/trinomial trees, finite difference solutions of partial differential equations, Monte Carlo simulation) and apply them to the pricing of derivatives in selected models.
• Apply Itô calculus. 

• Identify theoretical or practical applications on which the knowledge and skills obtained above can be applied independently.
• Apply the knowledge and skills obtained above in an interdisciplinary application (e.g., within asset pricing, corporate finance, econometrics, or accounting).
• Use the above knowledge and skills to participate in team work so that the student obtains competences in collaboration and communication. 

• Chapters from Hull, J.C.: "Options, Futures, and Other Derivatives", Prentice Hall, newest edition, or similar material.
• Chapters from Munk, C.: "Fixed Income Modelling", Oxford University Press, newest edition.
• Articles and lecture notes.