Xinfu Chen Mathematical Finance II
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Mathematical Finance
Mathematical Finance 6.1I nterest and Effective Rates In this section, you will learn about various ways to solve simple and compound interest problems related to bank accounts and calculate the effective rate of interest. Upon completion you will be able to: • Apply the simple interest formula to various financial scenarios. • Apply the continuously compounded interest formula to various financial scenarios. • State the difference between simple interest and compound interest. • Use technology to solve compound interest problems, not involving continuously compound interest. • Compute the effective rate of interest, using technology when possible. • Compare multiple accounts using the effective rates of interest/effective annual yields. Working with Simple Interest It costs money to borrow money. The rent one pays for the use of money is called interest. The amount of money that is being borrowed or loaned is called the principal or present value. Interest, in its simplest form, is called simple interest and is paid only on the original amount borrowed. When the money is loaned out, the person who borrows the money generally pays a fixed rate of interest on the principal for the time period the money is kept. Although the interest rate is often specified for a year, annual percentage rate, it may be specified for a week, a month, or a quarter, etc. When a person pays back the money owed, they pay back the original amount borrowed plus the interest earned on the loan, which is called the accumulated amount or future value. Definition Simple interest is the interest that is paid only on the principal, and is given by I = Prt where, I = Interest earned or paid P = Present value or Principal r = Annual percentage rate (APR) changed to a decimal* t = Number of years* *The units of time for r and t must be the same. -
Math 581/Econ 673: Mathematical Finance
Math 581/Econ 673: Mathematical Finance This course is ideal for students who want a rigorous introduction to finance. The course covers the following fundamental topics in finance: the time value of money, portfolio theory, capital market theory, security price modeling, and financial derivatives. We shall dissect financial models by isolating their central assumptions and conceptual building blocks, showing rigorously how their gov- erning equations and relations are derived, and weighing critically their strengths and weaknesses. Prerequisites: The mathematical prerequisites are Math 212 (or 222), Math 221, and Math 230 (or 340) or consent of instructor. The course assumes no prior back- ground in finance. Assignments: assignments are team based. Grading: homework is 70% and the individual in-class project is 30%. The date, time, and location of the individual project will be given during the first week of classes. The project is mandatory; missing it is analogous to missing a final exam. Text: A. O. Petters and X. Dong, An Introduction to Mathematical Finance with Appli- cations (Springer, New York, 2016) The text will be allowed as a reference during the individual project. The following books are not required and may serve as supplements: - M. Capi´nski and T. Zastawniak, Mathematics for Finance (Springer, London, 2003) - J. Hull, Options, Futures, and Other Derivatives (Pearson Prentice Hall, Upper Saddle River, 2015) - R. McDonald, Derivative Markets, Second Edition (Addison-Wesley, Boston, 2006) - S. Roman, Introduction to the Mathematics of Finance (Springer, New York, 2004) - S. Ross, An Elementary Introduction to Mathematical Finance, Third Edition (Cambrige U. Press, Cambridge, 2011) - P. Wilmott, S. -
Chapter 4 — Complete Markets
Chapter 4 — Complete Markets The Basic Complete-Market Problem There are N primary assets each with price pi and payoff Xsi is state s. There are also a complete set of S pure state securities each of which pays $1 in its own state s and 0 otherwise. These pure state securities are often called Arrow-Debreu securities. Assuming there is no arbi- trage, the price of the pure state s security must be qs, the state price. In a complete market like this, investors will be indifferent between choosing a portfolio from the S state securities and all N + S securities. A portfolio holding n shares of primary assets and η1 shares of the Arrow-Debreu securities will provide consumption c = Xn + η1 in the states at time 1. A portfolio holding η2 ≡ Xn + η1 of just the Arrow-Debreu securities obviously gives the same consumption. The costs of the two portfolios are q′c and p′n + q′η1. From the definition of c and the no-arbitrage result that p′ = q′X, these are obviously the same. Of course in equilibrium the original primary assets must actually be held. However this can be done by financial intermediaries like mutual funds. They buy all the primary shares finan- cing the purchases by issuing the Arrow-Debreu securities. By issuing h = Xn Arrow-Debreu securities where n is the aggregate supply of the primary assets, the intermediary is exactly hedged, and their role can be ignored. So with no loss of generality, we can therefore assume that individual investors only purchases Arrow-Debreu securities ignoring the primary assets. -
Mathematics and Financial Economics Editor-In-Chief: Elyès Jouini, CEREMADE, Université Paris-Dauphine, Paris, France; [email protected]
ABCD springer.com 2nd Announcement and Call for Papers Mathematics and Financial Economics Editor-in-Chief: Elyès Jouini, CEREMADE, Université Paris-Dauphine, Paris, France; [email protected] New from Springer 1st issue in July 2007 NEW JOURNAL Submit your manuscript online springer.com Mathematics and Financial Economics In the last twenty years mathematical finance approach. When quantitative methods useful to has developed independently from economic economists are developed by mathematicians theory, and largely as a branch of probability and published in mathematical journals, they theory and stochastic analysis. This has led to often remain unknown and confined to a very important developments e.g. in asset pricing specific readership. More generally, there is a theory, and interest-rate modeling. need for bridges between these disciplines. This direction of research however can be The aim of this new journal is to reconcile these viewed as somewhat removed from real- two approaches and to provide the bridging world considerations and increasingly many links between mathematics, economics and academics in the field agree over the necessity finance. Typical areas of interest include of returning to foundational economic issues. foundational issues in asset pricing, financial Mainstream finance on the other hand has markets equilibrium, insurance models, port- often considered interesting economic folio management, quantitative risk manage- problems, but finance journals typically pay ment, intertemporal economics, uncertainty less -
QR43, Introduction to Investments Class Notes, Fall 2004 I. Prices, Portfolios, and Arbitrage
QR43, Introduction to Investments Class Notes, Fall 2004 I. Prices, Portfolios, and Arbitrage A. Introduction 1. What are investments? You pay now, get repaid later. But can you be sure how much you will be repaid? Time and uncertainty are essential. Real investments, such as machinery, require an input of re- sources today and deliver an output of resources later. Financial investments,suchasstocksandbonds,areclaimstotheoutput produced by real investments. For example: A company issues shares of stock to investors. The proceeds of the share issue are used to build a factory and to hire workers. The factory produces goods, and the sales proceeds give the company revenue or sales. After paying the workers and set- ting aside money to replace the machinery and buildings when they wear out (depreciation), the company is left with earnings.The company keeps some earnings to make new real investments (re- tained earnings), and pays the rest to shareholders in the form of dividends. Later, the company decides to expand production and needs to raisemoremoneytofinance its expansion. Rather than issue new shares, it decides to issue corporate bonds that promise fixed pay- ments to bondholders. The expanded production increases revenue. However, interest payments on the bonds, and the setting aside of 1 money to repay the principal (amortization), are subtracted from revenue. Thus earnings may increase or decrease depending on the success of the company’s expansion. Financial investments are also known as capital assets or finan- cial assets. The markets in which capital assets are traded are known as capital markets or financial markets. Capital assets are commonly divided into three broad categories: Fixed-income securities promise to make fixed payments in the • future. -
Expected Stock Returns and Volatility Kenneth R
University of Pennsylvania ScholarlyCommons Finance Papers Wharton Faculty Research 1987 Expected Stock Returns and Volatility Kenneth R. French G. William Schwert Robert F. Stambaugh University of Pennsylvania Follow this and additional works at: http://repository.upenn.edu/fnce_papers Part of the Finance Commons, and the Finance and Financial Management Commons Recommended Citation French, K. R., Schwert, G., & Stambaugh, R. F. (1987). Expected Stock Returns and Volatility. Journal of Financial Economics, 19 (1), 3-29. http://dx.doi.org/10.1016/0304-405X(87)90026-2 At the time of publication, author Robert F. Stambaugh was affiliated with the University of Chicago. Currently, he is a faculty member at the Wharton School at the University of Pennsylvania. This paper is posted at ScholarlyCommons. http://repository.upenn.edu/fnce_papers/363 For more information, please contact [email protected]. Expected Stock Returns and Volatility Abstract This paper examines the relation between stock returns and stock market volatility. We find ve idence that the expected market risk premium (the expected return on a stock portfolio minus the Treasury bill yield) is positively related to the predictable volatility of stock returns. There is also evidence that unexpected stock market returns are negatively related to the unexpected change in the volatility of stock returns. This negative relation provides indirect evidence of a positive relation between expected risk premiums and volatility. Disciplines Finance | Finance and Financial Management Comments At the time of publication, author Robert F. Stambaugh was affiliated with the University of Chicago. Currently, he is a faculty member at the Wharton School at the University of Pennsylvania. -
Estimating Value at Risk
Estimating Value at Risk Eric Marsden <[email protected]> Do you know how risky your bank is? Learning objectives 1 Understand measures of financial risk, including Value at Risk 2 Understand the impact of correlated risks 3 Know how to use copulas to sample from a multivariate probability distribution, including correlation The information presented here is pedagogical in nature and does not constitute investment advice! Methods used here can also be applied to model natural hazards 2 / 41 Warmup. Before reading this material, we suggest you consult the following associated slides: ▷ Modelling correlations using Python ▷ Statistical modelling with Python Available from risk-engineering.org 3 / 41 Risk in finance There are 1011 stars in the galaxy. That used to be a huge number. But it’s only a hundred billion. It’s less than the national deficit! We used to call them astronomical numbers. ‘‘ Now we should call them economical numbers. — Richard Feynman 4 / 41 Terminology in finance Names of some instruments used in finance: ▷ A bond issued by a company or a government is just a loan • bond buyer lends money to bond issuer • issuer will return money plus some interest when the bond matures ▷ A stock gives you (a small fraction of) ownership in a “listed company” • a stock has a price, and can be bought and sold on the stock market ▷ A future is a promise to do a transaction at a later date • refers to some “underlying” product which will be bought or sold at a later time • example: farmer can sell her crop before harvest, -
Risk, Return, and Diversification a Reading Prepared by Pamela Peterson Drake
Risk, return, and diversification A reading prepared by Pamela Peterson Drake O U T L I N E 1. Introduction 2. Diversification and risk 3. Modern portfolio theory 4. Asset pricing models 5. Summary 1. Introduction As managers, we rarely consider investing in only one project at one time. Small businesses and large corporations alike can be viewed as a collection of different investments, made at different points in time. We refer to a collection of investments as a portfolio. While we usually think of a portfolio as a collection of securities (stocks and bonds), we can also think of a business in much the same way -- a portfolios of assets such as buildings, inventories, trademarks, patents, et cetera. As managers, we are concerned about the overall risk of the company's portfolio of assets. Suppose you invested in two assets, Thing One and Thing Two, having the following returns over the next year: Asset Return Thing One 20% Thing Two 8% Suppose we invest equal amounts, say $10,000, in each asset for one year. At the end of the year we will have $10,000 (1 + 0.20) = $12,000 from Thing One and $10,000 (1 + 0.08) = $10,800 from Thing Two, or a total value of $22,800 from our original $20,000 investment. The return on our portfolio is therefore: ⎛⎞$22,800-20,000 Return = ⎜⎟= 14% ⎝⎠$20,000 If instead, we invested $5,000 in Thing One and $15,000 in Thing Two, the value of our investment at the end of the year would be: Value of investment =$5,000 (1 + 0.20) + 15,000 (1 + 0.08) = $6,000 + 16,200 = $22,200 and the return on our portfolio would be: ⎛⎞$22,200-20,000 Return = ⎜⎟= 11% ⎝⎠$20,000 which we can also write as: Risk, return, and diversification, a reading prepared by Pamela Peterson Drake 1 ⎡⎤⎛⎞$5,000 ⎡ ⎛⎞$15,000 ⎤ Return = ⎢⎥⎜⎟(0.2) +=⎢ ⎜⎟(0.08) ⎥ 11% ⎣⎦⎝⎠$20,000 ⎣ ⎝⎠$20,000 ⎦ As you can see more immediately by the second calculation, the return on our portfolio is the weighted average of the returns on the assets in the portfolio, where the weights are the proportion invested in each asset. -
A Securitization-Based Model of Shadow Banking with Surplus Extraction and Credit Risk Transfer
A Securitization-based Model of Shadow Banking with Surplus Extraction and Credit Risk Transfer Patrizio Morganti∗ August, 2017 Abstract The paper provides a theoretical model that supports the search for yield motive of shadow banking and the traditional risk transfer view of securitization, which is consistent with the factual background that had characterized the U.S. financial system before the recent crisis. The shadow banking system is indeed an important provider of high-yield asset-backed securi- ties via the underlying securitized credit intermediation process. Investors' sentiment on future macroeconomic conditions affects the reservation prices related to the demand for securitized assets: high-willing payer (\optimistic") investors are attracted to these investment opportu- nities and offer to intermediaries a rent extraction incentive. When the outside wealth is high enough that securitization occurs, asset-backed securities are used by intermediaries to extract the highest feasible surplus from optimistic investors and to offload credit risk. Shadow banking is pro-cyclical and securitization allows risks to be spread among market participants coherently with their risk attitude. Keywords: securitization, shadow banking, credit risk transfer, surplus extraction JEL classification: E44, G21, G23 1 Introduction During the last four decades we witnessed to fundamental changes in financial techniques and financial regulation that have gradually transformed the \originate-to-hold" banking model into a \originate-to-distribute" model based on a securitized credit intermediation process relying upon i) securitization techniques, ii) securities financing transactions, and iii) mutual funds industry.1 ∗Tuscia University in Viterbo, Department of Economics and Engineering. E-mail: [email protected]. I am most grateful to Giuseppe Garofalo for his continuous guidance and support. -
Money and Prices Under Uncertainty1
Money and prices under uncertainty1 Tomoyuki Nakajima2 Herakles Polemarchakis3 Review of Economic Studies, 72, 223-246 (2005) 4 1We had very helpful conversations with Jayasri Dutta, Ronel Elul, John Geanakoplos, Demetre Tsomokos and Michael Woodford; the editor and referees made valuable comments and suggestions. The work of Polemarchakis would not have been possible without earlier joint work with Gaetano Bloise and Jacques Dr`eze. 2Department of Economics, Brown University; tomoyuki [email protected] 3Department of Economics, Brown University; herakles [email protected] 4Earlier versions appeared as \Money and prices under uncertainty," Working Paper No. 01- 32, Department of Economics, Brown University, September, 2001 and \Monetary equilibria with monopolistic competition and sticky prices," Working Paper No. 02-25, Department of Economics, Brown University, 2002 Abstract We study whether monetary economies display nominal indeterminacy: equivalently, whether monetary policy determines the path of prices under uncertainty. In a simple, stochastic, cash-in-advance economy, we find that indeterminacy arises and is characterized by the initial price level and a probability measure associated with state-contingent nom- inal bonds: equivalently, monetary policy determines an average, but not the distribution of inflation across realizations of uncertainty. The result does not derive from the stability of the deterministic steady state and is not affected essentially by price stickiness. Nom- inal indeterminacy may affect real allocations in cases we identify. Our characterization applies to stochastic monetary models in general, and it permits a unified treatment of the determinants of paths of inflation. Key words: monetary policy; uncertainty; indeterminacy; fiscal policy; price rigidities. JEL classification numbers: D50; D52; E31; E40; E50. -
Intermediation Markups and Monetary Policy Passthrough ∗
Intermediation Markups and Monetary Policy Passthrough ∗ Semyon Malamudyand Andreas Schrimpfz This version: December 12, 2016 Abstract We introduce intermediation frictions into the classical monetary model with fully flexible prices. Trade in financial assets happens through intermediaries who bargain over a full set of state-contingent claims with their customers. Monetary policy is redistributive and affects intermediaries' ability to extract rents; this opens up a new channel for transmission of monetary shocks into rates in the wider economy, which may be labelled the markup channel of monetary policy. Passthrough efficiency depends crucially on the anticipated sensitivity of future monetary policy to future stock market returns (the \Central Bank Put"). The strength of this put determines the room for maneuver of monetary policy: when it is strong, monetary policy is destabilizing and may lead to market tantrums where deteriorating risk premia, illiquidity and markups mutually reinforce each other; when the put is too strong, passthrough becomes fully inefficient and a surprise easing even begets a rise in real rates. Keywords: Monetary Policy, Stock Returns, Intermediation, Market Frictions JEL Classification Numbers: G12, E52, E40, E44 ∗We thank Viral Acharya, Markus Brunnermeier, Egemen Eren, Itay Goldstein, Piero Gottardi, Michel Habib, Enisse Kharroubi, Arvind Krishnamurthy, Giovanni Lombardo, Matteo Maggiori, Jean-Charles Rochet, Hyun Song Shin, and Michael Weber, as well as seminar participants at the BIS and the University of Zurich for helpful comments. Semyon Malamud acknowledges the financial support of the Swiss National Science Foundation and the Swiss Finance Institute. Parts of this paper were written when Malamud visited BIS as a research fellow. -
The Cross-Section of Volatility and Expected Returns
THE JOURNAL OF FINANCE • VOL. LXI, NO. 1 • FEBRUARY 2006 The Cross-Section of Volatility and Expected Returns ANDREW ANG, ROBERT J. HODRICK, YUHANG XING, and XIAOYAN ZHANG∗ ABSTRACT We examine the pricing of aggregate volatility risk in the cross-section of stock returns. Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns. Stocks with high idiosyncratic volatility relative to the Fama and French (1993, Journal of Financial Economics 25, 2349) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility. IT IS WELL KNOWN THAT THE VOLATILITY OF STOCK RETURNS varies over time. While con- siderable research has examined the time-series relation between the volatility of the market and the expected return on the market (see, among others, Camp- bell and Hentschel (1992) and Glosten, Jagannathan, and Runkle (1993)), the question of how aggregate volatility affects the cross-section of expected stock returns has received less attention. Time-varying market volatility induces changes in the investment opportunity set by changing the expectation of fu- ture market returns, or by changing the risk-return trade-off. If the volatility of the market return is a systematic risk factor, the arbitrage pricing theory or a factor model predicts that aggregate volatility should also be priced in the cross-section of stocks.