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Exercise Set 10 covers CRR time discretization, recombining trees, risk-neutral probabilities, and log-return decompositions. We worked with standardized increments and the central limit theorem, and compared the real-world and risk-neutral continuous-time stock-price formulas.
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- Petters–Dong, Sec. 5.2.1, ...
In Lecture 14, we introduced the Black–Scholes–Merton model, including its marketplace assumptions, the money market account, and the stock-price model. For simplicity, we assumed zero dividends (students may also make this assumption). We discussed self-financing replicating portfolios and began deriving the BSM partial differential equation using Itô’s formula (the derivation will be completed on Thursday).
Material used:
- Petters–Dong, Sec. 8.1, The BSM Model, pp. 384–391.
- Formula sheet for Lecture 14
Exercise Set 9 is available for self-study, as there is no teaching on Thursday 29 October. It covers two-period European call and put pricing by direct risk-neutral expectation and backward induction, dividend-adjusted risk-neutral probabilities, and European call pricing using the CRR binomial sum.
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- Petters–Do...
In Lecture 13, we studied the continuous-time limit of CRR stock prices. We rewrote stock prices in terms of log returns and used the central limit theorem to obtain lognormal stock-price distributions. We compared the real-world and risk-neutral formulas, which have different drifts but the same volatility.
Material used:
- Petters–Dong, Sec. 5.3.2, The Continuous-Time Security Price Formula, pp. 241–245.
- Supporting material: Sec. 5.2.1, The Real-World CRR Tree, pp. 219–230; Sec. 5.2.2, The Risk-Neutral CRR Tree, pp. 230–236; Sec. 5.3.1, The Lindeberg Central Limit Theorem, pp. 237–242.
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In the eighth exercise session, we worked with paths and information nodes in the binomial model. We computed risk-neutral probabilities, checked the no-arbitrage condition, and priced digital and European call and put options using risk-neutral expectation and backward induction. We also constructed a replicating strategy for a digital option.
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In Lecture 12, we studied two-period European call pricing in the binomial model. We derived the price both by direct risk-neutral expectation over terminal paths and by backward induction, and connected these methods to the martingale pricing framework. We also introduced the Cox–Ross–Rubinstein (CRR) tree, its up and down factors, and its risk-neutral probability.
Material used:
- Petters–Dong, Sec. 8.5.3, Two-Period Binomial Pricing, pp. 417–420; Sec. 5.2.2, The Risk-Neutral CRR Tree, pp. 230–236.
- Pliska, Sec. 3.5, The Binomial Model, pp. 100–105.
- Formula sheet for Lecture 12
In the seventh exercise session, we worked with predictable trading strategies, self-financing rebalancing, and discounted gains. We computed local risk-neutral probabilities on a two-step tree and combined them into a martingale measure. We also checked the no-arbitrage criterion and constructed an arbitrage strategy when a local condition failed.
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The student representative is Daniel Junior Lifenya Fondo (juniorfl[at]math[dot]uio[dot]no).
Any comments or suggestions regarding the course (such as lectures, exercises, textbooks, etc.) can be shared with him. He will relay your feedback to me, or you may choose to speak with me directly.
In Lecture 11, we introduced the multiperiod binomial model, in which the stock price moves up or down at each step. We considered the full path tree and derived the risk-neutral up-probability from the martingale condition. We also discussed the no-arbitrage restriction and how the model leads to backward pricing of claims.
Material used:
- Pliska, Sec. 3.5, The Binomial Model, pp. 100–105.
- Formula sheet for Lecture 11
In the sixth exercise session, we worked with filtrations and information on finite trees. We computed conditional expectations node by node, used the pull-out and tower properties, and checked whether processes are martingales. We also connected conditional expectations to risk-neutral pricing on a two-period tree.
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In Lecture 10, we discussed predictable trading strategies, portfolio value and gains, self-financing, and discounted prices. We formulated the multiperiod no-arbitrage criterion using martingale measures and explained why discounted security prices are martingales under a risk-neutral measure. We also connected discounted portfolio values to gains for self-financing strategies. On Thursday, we will see how one-period risk-neutral probabilities can be combined along a tree.
Material used:
- Pliska, Sec. 3.1.3, Trading Strategies, pp. 80–81; Sec. 3.1.4, Value Processes and Gains Processes, pp. 81–82; Sec. 3.1.5, Self-Financing Trading Strategies, pp. 82–83; Sec. 3.1.6, Discounted Prices, pp. 83–84; and Sec. 3.4, Economic Considerations, pp. 92–99.
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The mandatory assignment is available on the mandatory assignment page. There you will find the assignment PDF, the Excel data workbook, and submission instructions.
Submit your solution as one PDF in Canvas by Thursday 15 October at 14:30.
Exercise Set 5 covers contingent claims, risk-neutral pricing, replicating strategies, attainability, and complete versus incomplete one-period markets. It also includes an example of arbitrage arising from a mispriced claim. There is no exercise session on Thursday 1 October, so this set is available for independent work.
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In Lecture 9, we moved from one-period to multiperiod models, where information is revealed gradually. We described this information using filtrations, calculated conditional expectations on a two-step tree, and introduced martingales. These ideas prepare us for risk-neutral pricing in multiperiod markets.
Material used:
- Pliska, Sec. 3.1, Model Specifications, Filtrations, and Stochastic Processes, pp. 72–80; Sec. 3.3, Conditional Expectation and Martingales, pp. 88–91.
- Formula sheet for Lecture 9
There will be no exercise session on Thursday 1 October.
In the fourth exercise session, we worked with risk-neutral probabilities in finite one-period models. We solved the risk-neutral pricing equations and used them to determine whether there were no, one, or many risk-neutral measures. We also used these equations as no-arbitrage tests.
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- Pliska, Sec. 1.3, R...
In Lecture 8, we introduced contingent claims in the one-period model. We defined attainability and replicating portfolios, and showed that the price of an attainable claim is determined by its replicating portfolio. Using risk-neutral probabilities, we derived the risk-neutral valuation principle for attainable claims and discussed why this price is independent of investor preferences. On Thursday, we will discuss completeness, i.e., the case in which every contingent claim is attainable.
Material used:
- Pliska, Sec. 1.4, Valuation of Contingent Claims, pp. 16–20; Sec. 1.5, Complete and Incomplete Markets, pp. 21–23.
- Formula sheet for Lecture 8
In the third exercise session, we worked with one-period securities-market models. We practised discounting by the bank account and computing portfolio values and ordinary and discounted gains. We identified arbitrage opportunities and used attainable payoff vectors to test the law of one price. We also discussed why a positive expected return alone does not constitute arbitrage.
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In Lecture 7, we introduced risk-neutral probability measures in the finite one-period model. We distinguished risk-neutral probabilities from real-world probabilities, and explained their role as pricing devices. We also worked through the linear equations defining risk-neutral probabilities, discussed the equivalence between no arbitrage and the existence of a risk-neutral probability measure, and considered how uniqueness relates intuitively to complete and incomplete markets.
Material used:
- Pliska, Sec. 1.3, Risk Neutral Probability Measures, pp. 11–15.
- Formula sheet for Lecture 7
In the second exercise session, we worked with the risk-free asset in the Markowitz framework. We used the capital market line to analyse lending and borrowing at the risk-free rate, and practised calculations involving the market portfolio and tangent portfolio. We also computed beta and expected return from CAPM, and considered CAPM-consistent pricing.
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In Lecture 6, we began the second part of the course, on arbitrage pricing, by introducing the finite single-period securities-market model. We defined the sample space and physical probability, the bank account and risky securities, trading strategies, portfolio values, and discounted prices. We then introduced dominant trading strategies. The law of one price and arbitrage opportunities, and why the absence of arbitrage is a basic pricing principle, will be discussed on Thursday.
Material used:
- Pliska, Sec. 1.1, Model Specifications, pp. 1–3; Sec. 1.2, Arbitrage and Other Economic Considerations, pp. 4–10.
- Formula sheet for Lecture 6
In the first exercise session, we worked with portfolio weights, expected portfolio return, portfolio variance, covariance, and diversification. We studied two-security portfolios and the Markowitz efficient frontier, including global minimum-variance portfolios, target returns, and the role of short selling. We also considered a three-security example using matrix notation.
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...In Lecture 5, we added a risk-free asset to the Markowitz framework. We introduced the capital market line and the market portfolio, and discussed the separation between the choice of risky portfolio and the allocation between the risk-free asset and the market portfolio. We also introduced beta and the capital asset pricing model, interpreting CAPM as a statement about compensation for systematic risk rather than total risk.
Material used:
- Petters–Dong, Sec. 4.1.1, The Capital Market Line (CML), pp. 153–157; Sec. 4.1.2, Expected Return and Risk of the Market Portfolio, pp. 157–158; Sec. 4.1.3, The Capital Asset Pricing Model (CAPM), pp. 158–163.
- Formula sheet for Lecture 5
In Lecture 4, we extended the portfolio framework from two assets to N securities using vector and matrix notation. We wrote portfolio return and variance in matrix form and derived the solution to the minimum-variance portfolio problem for a prescribed target mean return. We also discussed three perspectives for understanding why diversification can reduce risk.
Material used:
- Petters–Dong, Sec. 3.3, Efficient Frontier for N Securities with Short Selling, pp. 117–126
- Formula sheet for Lecture 4
In Lecture 3, we defined portfolio risk through variance and standard deviation. We introduced covariance and correlation as the key quantities governing diversification. We derived the mean and variance formulas for two-security portfolios, discussed the efficient frontier, including the roles of correlation and short selling, and derived the solution to Markowitz’s minimum-variance portfolio problem in the special case N = 2.
Material used:
- Petters–Dong, Sec. 3.1.6, Portfolio Risk, p. 96; Sec. 3.1.7, Risks and Covariances of the Portfolio’s Securities, pp. 96–99; Sec. 3.2, Two-Security Portfolio Theory, pp. 104–116
- Formula sheet for Lecture 3
We discussed the one-period assumption for mean–variance portfolio theory. We introduced portfolio weights, including negative weights for short selling, and explained the budget constraint. We then derived the expected portfolio return as a weighted sum of expected security returns. We also worked through the portfolio variance formula for N = 2.
Material used:
- Petters–Dong, Sec. 3.1.4, The One-Period Assumption, Weights, and Short Selling, pp. 88–93; Sec. 3.1.5, Expected Portfolio Return Rate, pp. 94–95
- Formula sheet for Lecture 2