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Unformatted text preview: PDE for Finance Notes, Spring 2011 Section 9 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use only in connection with the NYU course PDE for Finance, G63.2706. Prepared in 2003, minor updates made in 2011. Scope of the final exam: As previously announced, our exam is Monday May 9, at the usual time and place: 5:10-7pm, WWH 517. You may bring two sheets of notes (8 . 5 11, both sides, any font). The preparation such notes is an excellent study tool. The exam covers the material in Sections 1-8 of the lecture notes, and Homeworks 1-6. Exception : only the first 3 pages of Section 8 are potential exam material (in particular: there will be no questions requiring use of the Fourier Transform, and none about the discussion at the end of Section 8 of hedging and the risk-neutral process. This Section 9 will not be on the exam, nor will the material we cover next week. A good exam question can be answered with very little calculation, provided you understand the relevant ideas. Most of the exam questions will be similar to (parts of) homework problems or examples discussed in the notes. The website of my 2003 PDE for Finance lecture notes includes the exam I gave then. ********************* The martingale method for dynamic portfolio optimization . Sections 5-7 were devoted to stochastic control. We discussed the value function and the principle of dynamic programming . In the discrete-time setting dynamic programming gives an iterative scheme for finding the value function; in the continuous-time setting it leads to the Hamilton-Jacobi- Bellman PDE. Stochastic control is a powerful technique for optimal decision-making in the presence of uncertainty. In particular it places few restrictions on the sources of randomness, and it does not require special hypotheses such as market completeness. In the continuous-time setting, a key application (due to Merton, around 1970) is dynamic portfolio optimization . We examined two versions of this problem: one optimizing the utility of consumption (Section 5), the other optimizing the utility of final-time wealth (Homework 5). This section introduces an alternative approach to dynamic portfolio optimization. It is much more recent the main papers were by Cox & Huang; Karatzas, Lehoczky, & Shreve; and Pliska, all in the mid-80s. A very clear, rather elementary account is given in R. Korn and E. Korn, Option Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics (American Mathematical Society, 2001). My discussion is a simplified (i.e. watered-down) version of the one in Korn & Korn. This alternative approach is called the martingale method, for reasons that will become clear presently. It is closely linked to the modern understanding of option pricing via the discounted risk-neutral expected value. (Therefore this Section, unlike the rest of the course, requires some familiarity with continous-time finance.) The method is much less general than stochastic control; in particular, it requires that the market be complete. When itthan stochastic control; in particular, it requires that the market be complete....
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.
- Fall '11