backward induction
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In the first part of the lecture we wrap up the previous discussion of implied default probabilities, showing how to calculate them quickly by using the same duality trick we used to compute forward interest rates, and showing how to interpret them as spreads in the forward rates. The main part of the lecture focuses on the powerful tool of backward induction, once used in the early 1900s by the mathematician Zermelo to prove the existence...more
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In this lecture we move from present values to dynamic present values. If interest rates evolve along the forward curve, then the present value of the remaining cash flows of any instrument will evolve in a predictable trajectory. The fastest way to compute these is by backward induction. Dynamic present values help us understand the returns of various trading strategies, and how marking-to-market can prevent some subtle abuses of the syst...more
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This lecture is about optimal exercise strategies for callable bonds, which are bonds bundled with an option that allows the borrower to pay back the loan early, if she chooses. Using backward induction, we calculate the borrower's optimal strategy and the value of the option. As with the simple examples in the previous lecture, the option value turns out to be very large. The most important callable bond is the fixed rate amortizing mortg...more
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The class's examination of Nozick's minimal state has raised a number of important questions, most of which are rooted in his troublesome model of compensation. Nozick would respond with his threefold account of justice: (1) justice in acquisition, (2) justice in transfer, and (3) rectification of past injustices. Nozick brilliantly demonstrates that "liberty upsets patterns"--even though we can originally start off with any just distribut...more
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The Art of Illustration: Millais, the Pre-Raphaelites and the Idyllic School
Gresham College / Art & Architecture
The final and 'most indispensable' founding principle of the Pre-Raphaelites was: "to produce thoroughly good pictures and statues". Derided by the establishment (including Charles Dickens) for producing art which was ugly and backward, the work of John Everett Millais, Dante Gabriel Rossetti and William Holman Hunt have since come to take its place as perhaps the most important point of British art in the 19th Century. This lecture lo...more
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We first discuss Zermelo's theorem: that games like tic-tac-toe or chess have a solution. That is, either there is a way for player 1 to force a win, or there is a way for player 1 to force a tie, or there is a way for player 2 to force a win. The proof is by induction. Then we formally define and informally discuss both perfect information and strategies in such games. This allows us to find Nash equilibria in sequential games. But we fin...more
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