ECO423: Syllabus Jørgen Haug Jan. 2022 Tentative Syllabus “H #” refers to chapter # in the 11th edition of Hull’s textbook. Topics are listed in the order they are (most likely) covered. There are no set dates for the topics, as we will adjust the pace to the needs and preferences of the class. As the topics vary in extensiveness we will spend varying amounts of time on each one of them. If you are new to derivatives or the binomial model, you should prepare before the first class, by browsing through the chapters listed for topic 1. (1) Intuition from the binomial model for pricing through replication, and risk-adjusted probabilities (H 13, plus 1, 10, 11 if you’re new to derivatives) Learning outcome: The purpose is to give simple economic intuition for some technical, seemingly abstract concepts that we meet in the course. You are not expected to master the binomial model in this course—only the key economic intuition we go through. (2) Modeling asset prices as diffusions (H 14, 15) Learning outcome: You should know the definitions and basic properties of the standard, generalized, and geometric Wiener processes, and diffusions. You should be able to give relevant economic interpretations of a diffusion’s drift and dispersion terms. You should be able to work out basic ‘statistical’ properties of diffusions. (3) Project 1: Verify distributional properties of stock price model, in Excel or R Learning outcome: You should be able to simulate numerical observations that are consistent with the analytical models—by implementing them in a spreadsheet or in R—and to understand how the former relates to the latter. You should understand basic statistical properties of our stock price model. You should also attain a ‘hands-on’ feeling for the standard Wiener process, and be able to simulate it. (4) Modeling derivatives’ price changes: Itô’s Lemma (H 14.6, Appendix) Learning outcome: You should be able to compute the total derivative of arbitrary functions of time and a diffusion: compute df (t, Yt ) where f (t, y) is a real function, t is time and Yt is a diffusion. You are not expected to know the differentiation rule by heart, but to be able to apply the formula. (5) Derivatives prices as replicating portfolios, and the role of Partial Differential Equations (PDE) (H 15) Learning outcome: You should be able to construct a self-financing, replicating portfolio, and explain the economic role of ‘self-financing,’ ‘replicating,’ and ‘no arbitrage.’ You must know how to use a replicating portfolio to derive a PDE (Partial Differential Equation) that the price of all derivatives must satisfy. Finally, you should at the end of the course understand how aspects of the model for the underlying asset affects the structure of the PDE. (6) Example: The Black-Scholes-Merton (BSM) call option formula as the (Feynman-Kac) solution to the Fundamental PDE (H 15) Learning outcome: You should be able to use the Feynman-Kac formula to solve the Fundamental PDE for the price of the derivative. You are not expected to remember the formula. 1 (7) Project 2: Compare BSM value with value of replicating portfolio, in Excel or R Learning outcome: You should be able to construct a replicating portfolio when asset prices follow diffusions, and understand its role in determining the price of derivative assets. You should also understand the role of the expected rate of return (priced/systematic risk) of the underlying asset in determining the price of the derivative asset. (8) Monte Carlo simulation (MC) for European derivatives (H 14.3, 21.6–7) Learning outcome: You should be able to simulate the value of European derivatives on assets that follow geometric Wiener processes. You should be able to use simple Euler schemes to simulate discrete approximations of generic diffusions. (9) Lessons from derivatives ‘mishaps’; Long Term Capital Management, Einar Aas, OptionSellers.com, . . . (H 37, YouTube video(s), news paper articles) Learning outcome: You should know the key reasons for why some individuals and institutions have experienced fatal financial losses due to derivatives trading. (10) Derivatives price as risk-adjusted expected payoff discounted at the riskless rate: The equivalent martingale measure approach (H 28) Learning outcome: You should know what an equivalent martingale measure (EMM) is, and the meaning/role of ‘equivalent’, ‘martingale’, and ‘measure’. You must know when EMMs exists, and how they help simplify the estimation of present values. (11) Rule for risk-adjusting probabilities: Girsanov’s Theorem (H 28) Learning outcome: You should be able to adjust the physical probabilities for systematic risk (change measure) by modifying the standard Wiener process. You should be able to compute analytic expressions for present values in simple cases. You should moreover understand the economic intuition for why one common risk-adjustment works for all derivative cash flows that depend on the same underlying asset (provided we have time to cover this insight). (12) Project 3: MC estimation of BSM call price using the equivalent martingale measure/risk-adjusted Wiener process, in Excel or R Learning outcome: You should be able to numerically estimate present values by combining the EMM technique with simulations. (13) Risk management: The Greeks (H 19) Learning outcome: You should be able to compute the Greeks, and understand their role in risk management. You should also understand their limitations; understand the economic interpretation, and limitations, of comparative statics. (14) Risk management: Value at Risk and Expected Shortfall (H 22, Basak and Shapiro (2001)) Learning outcome: You should know and understand the definitions of Value at Risk and Expected Shortfall, and be able to compute them based on simulated samples. You should moreover know their strengths and limitations. (15) Simulation methods for American derivatives: Approximating conditional expectation by Least Squares regression on Monte Carlo simulated observations (LSMC) (Longstaff and Schwartz (2001); H 27.8) Learning outcome: You should have an intuitive understanding of the LSMC method; how a regression equation can be used to a approximate conditional expectation, and the economic role of the latter. You should also understand the intuition behind the major implementation choices behind the method. (16) Project 4: LSMC simulation of American/Bermudian put option, in Excel or R Learning outcome: You should understand and be able to implement the LSMC method, to estimate the present value of simple American derivatives. 2 (17) Real options (Notes, H 35, Schwartz and Moon (2000)) Learning outcome: You should know the ideal conditions for using option pricing methods, and when they are not suitable. You should be able to model real investment projects using diffusions, and apply PDE or EMM methods to estimate their present values. You should in particular be able to make the appropriate adjustments for systematic risk when the underlying risk factors are not traded. You should also understand the role and meaning of convenience yields. (18) Modelling commodities prices (H 35, Gibson and Schwartz (1990); Schwartz (1997)) Learning outcome: You should be able to express the basic diffusion models for commodities, know their economic interpretation, and understand the basics of their time-series counterparts. (19) If time: Volatility; historical, implied, stochastic Learning outcome: You should be able to compute ‘classic’ BSM-based implied volatility, and understand its basic properties. You should also be able to implement the model-free approach to implied volatility, and have an intuitive understanding of its construction. You should be able to recognize and give economic interpretation of a stochastic volatility model and be able to do MC with GARCH and estimate its parameters (in R). (20) If time: Seasonality Learning outcome: You should understand when seasonality in the underlying asset can and cannot affect the price of a derivative. You should be able to implement diffusion models with seasonality, and provide appropriate economic interpreations of the models. Guest lecture TBA Literature • H: Hull’s “Options, Futures, and other Derivatives,” 11th edition (essentially the same as earlier versions, but possible differences in chapter references) • Class notes, distributed on Canvas Projects • Work in groups of 1–3 students. Individual solutions ok, but collaboration strongly encouraged as it is known to enhance learning! • Have to hand in report and self-assessment of one project. • Report no more than one page; brief and to the point. • Write about what you believe are the important insights from the project. • There is a clear division of labor between projects and exam. Projects are not primarily intended as preparation for the exam: – projects ensure you can implement in practice ideas and methods covered in class. For some of you they moreover play an important role to de-mystify seemingly abstract concepts. – exam to test your mastery and level of insight into the principles behind the ideas/methods covered in class. 3 Use of Software • Projects must be solved in either a spreadsheet (Excel) or R • Lecture notes includes R code – shows what goes on “behind” figures, simulations etc – an aid to help you on the way to learn R, if you don’t already know it? • Software implementation is not part of the learning outcome: You will not be evaluated on this on the exam – some level of mastery big plus on your CV. . . Course Approval Course approval is given for satisfactory participation in all Projects. Exam You must finish all four projects to get course approval, which is required to take the exam. The projects are assessed by self-evaluation. You must in addition hand in a report for one project (to be announced on Canvas), and your assessment of your performance on this project. The exam is an individual four hour digital home exam. The exam is scheduled for Monday May 23, 2022. Related courses • Econometrics. Any course in Time Series is useful companion course; where parameter values come from. Very useful for writing a thesis on derivatives! • FIE425 Derivatives and Risk Management. Covers more basic and wider set of topics. Simpler, less powerful methods (the binomial model). • ECO421 Asset Pricing. The economic principles behind pricing/finding present values. Low precision wrt. applications. 4