Beskjeder
Today was the final exercise session of the semester, thereby brining MAT2700 to an end. Good luck with the exam!
Today I introduced the notion of a martingale measure, demonstrated by means of an example how to determine a martingale measure, and proved that in a multi-period market the absence of arbitrage opportunities is equivalent to the existence of a martingale measure.
Today I talked about martingales and Exercises 2.12, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6.
I will continue with the characterization of risk-netural probability measures on Thursday.
The "fasit" (solution) of the mandatory assignment will be made available as soon as everyone has has handed in their assignment (2. attempt), tentatively Wednesday next week.
Today's topic was conditional expectations of random variables, conditioned on events and algebras (all available information). I will continue with martingales and risk-netural probability measures on Tuesday.
NB! All the mandatory assignments will be handed out in lecture on Tuesday November 10. Those who do not get approved in their first attempt will get a new deadline on Tuesday November 17, kl.14.30, to make a second and final attempt.
Today I finished with Sections 3.1-3.3 (thereby covering topics like information model / filtration, adapted stock price model, predictable and self-financing trading strategies, total portfolio value / gains processes, discounted processes, return and dividend processes). On Thursday the topic is conditional expectations and Martingales.
Today I began lecturing from Chapter 3 on multiperiod markets. Most of the time was spent on describing the model for the flow of information, namely the filtration, which is a nested sequence of algebras containing the information available to the investor as time evolves. I will continue to lecture from Chapter 3 on Tuesday next week.
The topic today was portfolio optimization in incomplete markets. I outlined two approaches for solving such problems. The first one was a natural extension of "risk-netural probability" approach utilized several times before for complete markets, where the main new aspect amounted to a proper characterization of the set of attainable claims. The second approach was based on enlarging the market with enough "artificial" risky assets to make the resulting market complete, and then perform optimization in this enlarged market with the constraint that investment in the added artificial asset was not allowed, thereby replacing the original problem in a incomplete market with an optimization problem in a complete market with an additional constraint. To solve this constrained optimization problem one can effectively use the approach introduced last week for problems with short sales restrictions (Section 2.5).
I have decided to...
Today I revisited Section 2.5 (which is a difficult part of the book) by illustrating through several examples the "risk-neutral probability" approach for solving portfolio optimization problems with constraints (like no borrowing of money in the bank, no short-selling of stocks, etc).
Exercises (for Thursday): 2.8, 2.9, 2.10, 2.11.
You can find the obligatory assignment [here]
Today I lectured from Section 2.4. Two approaches were outlined for solving the mean-variance portfolio optimization problem. 1) Direct approach through a quadratic programming problem. 2) Risk-neutral probabilities approach / Lagrange multiplier method. Both approaches were illustrated on a concrete example from the book.
Tuesday next week, I will continue with the lectures (no exercises).
The obligatory assignment will be made available here on this website on Monday (00:00-->).
Today I continued with optimal portfolio problems (maximizing expected utility of terminal wealth), highlighting two approaches for solving such problems: 1. The direct approach in which optimization is done with respect to the trading strategies H and for which the first order conditions at a maximum provide the relevant (nonlinear) equations to determine the optimal H. 2. The in-direct (risk-neutral probability) approach in which the problem is decomposed into two separate optimization problems. In the first problem the optimal terminal wealth is determined. Given the optimal terminal wealth, we then in the second problem determine the trading strategy that generates this wealth. Continuing with optimization problems, I introduced the the so-called consumption-investment problem in which the goal is to maximize expected utility from consumption. Similarly, two approaches for solving this optimization problem were outlined.
Exercises (for Tuesday next week): 1.19, 2.1...
Today I discussed the notions of return, mean return, and risk-premium ("mean return - interest rate") of individual assets and portfolios. We showed that the Q-expectation of the return equals the interest rate r for any risk-netural probability measure Q. To quantify risk in a portfolio the concept of its beta was introduced. I showed that the ratio of the risk premium of an arbitrary portfolio and the risk premium of the "benchmark" portfolio (the state price density) is equal to the beta. Finally, I introduced the basic portfolio optimization problem of maximizing expected utility of terminal wealth, and proved that the existence of an optimal solution (trading strategy) to this portfolio problem implies the existence of a risk-netural probability measure (i.e., the absence of arbitrage). Moreover, such a measure comes out explicitly (in terms of the marginal utility of terminal wealth) thanks to the first order conditions at a maximum.
Today the topic was pricing of non-attainable contingent claims X (incomplete markets). Although there is no unique price for X in this case, it is possible to identify an interval [V-,V+] in which each value represents a fair (arbitrage) price. Here, V- is the supremum over Q-expected discounted payoffs Y/B1 with Y≤X being attainable and V+ is the infimum over Q-expected discounted payoffs Y/B1 with Y≥X being attainable. I showed how to reformulate these two problems as standard linear programming problems, which can be used to compute V-,V+ in concrete examples. Moreover, using duality, I showed that V- (V+) equals the infimum (supremum) of the Q-expected discounted payoff X/B1 with Q varying over the set of risk-neutral probability measures. This result was finally illustrated by an example.
Exercises (for Tuesday next week): I.12, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19.
The main new topic today was contingent claims and I derived the price (or the todays value) of an attainable contingent claim in terms of a risk-netural measure Q.
On Tuesday next week I will address the following exercises: 1.6, 1.7, 1.8, 1.9, 1.10, 1.11.
Finally, the student representative is Igor Pipkin [E-mail: igor.pipkin"@"umb.no without ""]
Today I introduced the notions of "arbitrage opportunity" and "risk neutral probability measure". I proved the important theorem stating that the absence of arbitrage opportunities is equivalent to the existence of a risk neutral probability measure. On Thursday I will continue lecturing from the book and also post some exercises to be discussed on Tuesday next week.
I started by recapitulating the basic quantities underlying a single-period market. I then introduced the notion of a linear pricing measure, collected some properties of such measures, and proved that the existence of linear pricing measures are equivalent to the absence of dominant trading strategies. As part of this, I spent some time on linear programming problems and recalled a few key facts from duality theory.
Exercises for Tuesday Sep 1: From the book, Exercises 1.1, 1.2, 1.3, 1.4, 1.5. Regarding Exercises 1.4 and 1.5, they involve the use of "arbitrage" and the "law of one price", which we have not yet covered in detail, so you have to read about it in the book. In addition to the above exercises, you should prove the following statement "the absence of dominant trading strategies implies the existence of a linear pricing measure" (the implication that we did not have time to complete in today's lecture).
Today I introduced the basic notation and definitions regarding single period markets, which were illustrated on a couple of simple examples. We discussed the numeraire (in particular the choice of the bank as the basic unit) and introduced the notion of a "dominant trading strategy", for which we proved several equivalent formulations.
Today I gave a short introduction to "financial derivatives", as a warm-up to what is coming. On Tuesday and Thursday next week, I will begin with the chapter on single-period markets from Pliska's book. There will be NO exercises next week.
Tuesday Sep 1, I will be out traveling and Giulia Di Nunno will step in for me. She will do exercises from the book (to be posted next week). The lecture on Thursday Sep 3 is cancelled (I am still out traveling).
First lecture will be thursday 20th of august(Exercises on tuesdays will start the following week). Welcome!