The CoxRossRubinstein Option Pricing Model


 Noreen O’Neal’
 3 years ago
 Views:
Transcription
1 Fnance 400 A. Penat  G. Pennacc Te CoxRossRubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te noarbtrage assumpton alone cannot determne an exact opton prce as a uncton o te underlyng asset prce. To do so, one needs to make addtonal assumptons regardng te dstrbuton o returns earned by te underlyng asset. Certan dstrbutonal assumptons can mply a complete market or te underlyng asset s rsk tat allows us to determne a unque opton prce. Te model n tese notes makes te assumpton tat te underlyng asset, ereater reerred to as a stock, takes on one o only two possble values eac perod. Wle ts may seem unrealstc, te assumpton leads to a ormula tat can accurately prce optons. Ts bnomal opton prcng tecnque s oten appled by Wall Street practtoners to numercally compute te prces o complex optons. Here, we start by consderng te prcng o a smple European opton wrtten on a nondvdendpayng stock. In addton to assumng te absence o arbtrage opportuntes, te bnomal model assumes tat te current stock prce, S, eter moves up, by a proporton u, or down, by a proporton d, eacperod. Teprobabltyoanupmovesq, so tat te probablty o a down move s 1 q. Ts can be llustrated as us wt probablty q S % & (1) ds wt probablty 1 q Denote as one plus te rskree nterest rate or te perod. Ts rskree return s assumed to be te constant over tme. To avod arbtrage between te stock and te rskree nvestment, we must ave d< <u. Let C equal te value o a European call opton wrtten on te stock and avng a strke prce o X. Atexpry,C = max[0,s T X]. Tus: One perod pror to expry: 1
2 C u max [0,uS X] wt probablty q C % & (2) C d max [0,dS X] wt probablty 1 q Wat s C one perod beore expry? Consder a portolo contanng sares o stock and $B o bonds. It as current value equal to S + B. Ten te value o ts portolo evolves over te perod as us + B wt probablty q S + B % & (3) ds + B wt probablty 1 q Wt two securtes (te bond and stock) and two states o te world (up or down), and B can be cosen to replcate te payo o te call opton: us + B = C u ds + B = C d (4a) (4b) Solvng or and B tat satsy tese two equatons, we ave = C u C d (u d) S (5a) B = uc d dc u (u d) (5b) Hence, a portolo o sares o stock and $B o bonds produces te same caslow as te call opton. Ts s possble because te market s complete. Tradng n te stock and bond produces payos tat span te two states. Now snce te portolo s return replcates tat o te opton, te absence o arbtrage mples 2
3 Example: IS = $50, u =2,d =.5, =1.25, X =$50,ten C = S + B (6) us = $100, ds=$25,c u =$50,C d =$0. Tereore: = 50 0 (2.5) 50 = 2 3 B = 0 25 (2.5) 1.25 = 40 3 so tat C = S + B = 2 3 (50) 40 3 = 60 3 = $20 I C< S + B, ten an arbtrage s to sort sell sares o stock, nvest $ B n bonds, and buy te call opton. Conversely, C> S + B, ten an arbtrage s to wrte te call opton, buy sares o stock, and borrow $ B. Te resultng opton prcng ormula as an nterestng mplcaton. It can be rewrtten as C = S + B = C u C d (u d) + uc d dc u (7) (u d) R d u d = max [0,uS X]+ u R u d max [0,dS X] wc does not depend on te probablty o an up or down move o te stock, q. Tus,gven S, nvestors wll agree on te noarbtrage value o te call opton even tey do not agree on q. Snce q determnes te stock s expected rate o return, uq + d(1 q) 1, ts does not need to be known or estmated n order to solve or te noarbtrage value o te opton, C. However, 3
4 we do need to know u and d, te sze o movements per perod, wc determne te stock s volatlty. Butnotetattecalloptonvalue, C, doesnotdrectly depend on nvestors atttudes toward rsk. It s a relatve (to te stock) prcng ormula. Note also tat we can rewrte C as [pc u +(1 p) C d ] (8) were p d u d. Snce 0 < p < 1, p as te propertes o a probablty. In act, ts pseudoprobablty p would equal te true probablty q nvestors were rskneutral, snce ten te expected return on te stock would equal : [uq + d (1 q)] S = S (9) or q = d u d = p. (10) Perapstssnotsurprsng,snceteexpresson [pc u +(1 p) C d ] does not depend on rskpreerences, and so t must be consstent wt all possble rsk preerences, ncludng rskneutralty. Next, consder te opton s value wt: Two perods pror to expraton: Testockprceprocesss 4
5 u 2 S us % & S % & dus (11) ds % & d 2 S so tat te opton prce process s C uu max 0,u 2 S X % C u & C % & C du max [0,duS X] (12) % C d & C dd max 0,d 2 S X Usng te results rom our analyss wen tere was only one perod to expry, we know tat C u = pc uu +(1 p) C du C d = pc du +(1 p) C dd (13a) (13b) Wt two perods to expry, te one perod to go caslows o C u and C d can be replcated once agan by te stock and bond portolo composed o = Cu C d (u d)s B = uc d dc u (u d) o bonds. Noarbtrage mples sares o stock and C = S + B = 1 [pc u +(1 p) C d ] (14) 5
6 Substtutng n or C u and C d,weave p 2 R 2 C uu +2p (1 p) C ud +(1 p) 2 C dd = 1 R 2 p 2 max 0,u 2 S X +2p (1 p)max[0,dus X]+(1 p) 2 max 0,d 2 S X Note tat C depends only current S, X, u, d,,andtetmeuntlmaturty,2perods. Repeatng ts analyss or tree, our, ve,..., n perods pror to expry, we always obtan (15) C = S + B = 1 [pc u +(1 p) C d ] By repeated substtuton or C u, C d, C uu, C ud, C dd, C uuu, etc., we obtan te ormula: n perods pror to expraton: R n µ j=0 X p j (1 p) n j max 0,u j d n j S (16) Ts ormula can be smpled by denng a as te mnmum number o upward jumps o S or t to exceed X. Tus a s te smallest nonnegatve nteger suc tat u a d n a S>X. Takng te natural logartm o bot sdes, a s te mnmum nteger >ln(x/sd n )/ln(u/d). Tereore or all j<a(te opton expres outote money), wle or all j>a(te opton expres ntemoney), max 0,u j d n j S X =0, (17a) Tus, te ormula or C can be rewrtten: R n max 0,u j d n j S X = u j d n j S X (17b) µ j=a Breakngtsupntotwoterms,weave: p j (1 p) n j u j d n j S X (18) 6
7 µ " C = S p j (1 p) n j u j d n j # R n (19) j=a µ XR n p j (1 p) n j j=a Te terms n brackets are complementary bnomal dstrbuton unctons, so tat we can wrte ts as were p 0 C = Sφ[a; n, p 0 ] XR n φ[a; n, p] (20) ³ u R p and φ[a; n, p] = te probablty tat te sum o n random varables wc equal 1 wt probablty p and 0 wt probablty 1 p wll be a. Tese ormulas mply tat C s te dscounted expected value o te call s termnal payo tat would occur n a rskneutral world. I we dene τ astetmeuntlmaturtyotecalloptonandσ 2 astevaranceperunt tme o te stock s rate o return (wc depends on u and d), ten by takng te lmt as te number o perods n, but te lengt o eac perod τ n 0, te CoxRossRubnsten bnomal opton prcng ormula becomes te wellknown BlackScolesMerton opton prcng ormula 1 were z ln µ S XR τ (σ τ) σ2 τ C = SN (z) XR τ N z σ τ (21) and N ( ) s tat standard normal dstrbuton uncton. 1 In te BlackScolesMerton ormula, s now te rskree return per unt tme rater tan te rskree return or eac perod. Te relatonsp between σ and u and d wll be dscussed sortly. 7
Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.
Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces
More informationStock Profit Patterns
Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationLevel Annuities with Payments Less Frequent than Each Interest Period
Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annutymmedate 2 Annutydue Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annutymmedate 2 Annutydue Symoblc approach
More informationA. Te densty matrx and densty operator In general, te manybody wave functon (q 1 ; :::; q 3N ; t) s far too large to calculate for a macroscopc syste
G25.2651: Statstcal Mecancs Notes for Lecture 13 I. PRINCIPLES OF QUANTUM STATISTICAL MECHANICS Te problem of quantum statstcal mecancs s te quantum mecancal treatment of an Npartcle system. Suppose te
More informationUsing Series to Analyze Financial Situations: Present Value
2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated
More informationInterest Rate Fundamentals
Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 738 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qngxn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationAddendum to: Importing SkillBiased Technology
Addendum to: Importng SkllBased Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our
More informationInterest Rate Futures
Interest Rate Futures Chapter 6 6.1 Day Count Conventons n the U.S. (Page 129) Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (n perod) 30/360 Actual/360 The day count conventon
More informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
More informationHedging InterestRate Risk with Duration
FIXEDINCOME SECURITIES Chapter 5 Hedgng InterestRate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cashflows Interest rate rsk Hedgng prncples DuratonBased Hedgng Technques Defnton of duraton
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationNPAR TESTS. OneSample ChiSquare Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationLongTerm Insurance Products and Volatility under the Solvency II Framework
LongTerm Insurance Products and Volatlty under te Solvency II Framework Korneel van den Broek AG Insurance  Employee Benets Boulevard Du Jardn Botanque 0 B1000 Brussels, Belgum korneel.vandenbroek@agnsurance.be
More information7.5. Present Value of an Annuity. Investigate
7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationThursday, December 10, 2009 Noon  1:50 pm Faraday 143
1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationImplied (risk neutral) probabilities, betting odds and prediction markets
Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT  We show that the well known euvalence between the "fundamental theorem of
More informationWorld currency options market efficiency
Arful Hoque (Australa) World optons market effcency Abstract The World Currency Optons (WCO) maket began tradng n July 2007 on the Phladelpha Stock Exchange (PHLX) wth the new features. These optons are
More information1. Math 210 Finite Mathematics
1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationSimple Interest Loans (Section 5.1) :
Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part
More informationThe Choice of Direct Dealing or Electronic Brokerage in Foreign Exchange Trading
The Choce of Drect Dealng or Electronc Brokerage n Foregn Exchange Tradng Mchael Melvn Arzona State Unversty & Ln Wen Unversty of Redlands MARKET PARTICIPANTS: Customers Endusers Multnatonal frms Central
More informationCourse outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy
Fnancal Tme Seres Analyss Patrck McSharry patrck@mcsharry.net www.mcsharry.net Trnty Term 2014 Mathematcal Insttute Unversty of Oxford Course outlne 1. Data analyss, probablty, correlatons, vsualsaton
More informationChapter 15 Debt and Taxes
hapter 15 Debt and Taxes 151. Pelamed Pharmaceutcals has EBIT of $325 mllon n 2006. In addton, Pelamed has nterest expenses of $125 mllon and a corporate tax rate of 40%. a. What s Pelamed s 2006 net
More informationCautiousness and Measuring An Investor s Tendency to Buy Options
Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, ArrowPratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets
More informationSection 2.3 Present Value of an Annuity; Amortization
Secton 2.3 Present Value of an Annuty; Amortzaton Prncpal Intal Value PV s the present value or present sum of the payments. PMT s the perodc payments. Gven r = 6% semannually, n order to wthdraw $1,000.00
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More informationPowerofTwo Policies for Single Warehouse MultiRetailer Inventory Systems with Order Frequency Discounts
Powerofwo Polces for Sngle Warehouse MultRetaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationEXAMPLE PROBLEMS SOLVED USING THE SHARP EL733A CALCULATOR
EXAMPLE PROBLEMS SOLVED USING THE SHARP EL733A CALCULATOR 8S CHAPTER 8 EXAMPLES EXAMPLE 8.4A THE INVESTMENT NEEDED TO REACH A PARTICULAR FUTURE VALUE What amount must you nvest now at 4% compoune monthly
More information10. (# 45, May 2001). At time t = 0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest
1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual e ectve
More informationThe Choice of Direct Dealing or Electronic Brokerage in Foreign Exchange Trading
The Choce of Drect Dealng or Electronc Brokerage n Foregn Exchange Tradng Mchael Melvn & Ln Wen Arzona State Unversty Introducton Electronc Brokerage n Foregn Exchange Start from a base of zero n 1992
More informationTime Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6  The Time Value of Money. The Time Value of Money
Ch. 6  The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21 Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationSimon Acomb NAG Financial Mathematics Day
1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Comovement between the ponts Postve
More informationOn the pricing of illiquid options with BlackScholes formula
7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠBTU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 On the prcng of llqud optons wth BlackScholes
More informationECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C White Emerson Process Management
ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C Whte Emerson Process Management Abstract Energy prces have exhbted sgnfcant volatlty n recent years. For example, natural gas prces
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Jord Galí: Monetary Polcy, Inflaton, and the Busness Cycle s publshed by Prnceton Unversty Press and copyrghted, 28, by Prnceton Unversty Press. All rghts reserved. No part of ths book
More informationIntrayear Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error
Intrayear Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationBANCO DE PORTUGAL Economics Research Department
BANCO DE PORUGAL Economcs Research Department he Estmaton of Rsk Premum Implct n Ol Prces Jorge Barros Luís WP 00 February 000 he analyses, opnons and fndngs of ths paper represent the vews of the author,
More informationA Model of Private Equity Fund Compensation
A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs
More informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
More informationESTIMATING THE MARKET VALUE OF FRANKING CREDITS: EMPIRICAL EVIDENCE FROM AUSTRALIA
ESTIMATING THE MARKET VALUE OF FRANKING CREDITS: EMPIRICAL EVIDENCE FROM AUSTRALIA Duc Vo Beauden Gellard Stefan Mero Economc Regulaton Authorty 469 Wellngton Street, Perth, WA 6000, Australa Phone: (08)
More informationStaff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall
SP 200502 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 148537801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent
More informationVasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio
Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of
More informationPricing MultiAsset Cross Currency Options
CIRJEF844 Prcng MultAsset Cross Currency Optons Kenchro Shraya Graduate School of Economcs, Unversty of Tokyo Akhko Takahash Unversty of Tokyo March 212; Revsed n September, October and November 212
More informationOPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004
OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected
More informationPricing index options in a multivariate Black & Scholes model
Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders AFI_1383 Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders Verson: October 2, 2013 1 Introducton In ths paper, we consder
More informationTime Value of Money Module
Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the
More informationStock market as a dynamic game with continuum of players 2
Stock market as a dynamc game wth contnuum of players Agneszka WsznewskaMatyszkel Insttute of Appled Mathematcs and Mechancs Warsaw Unversty emal: agnese@mmuw.edu.pl Abstract. Ths paper contans a gametheoretc
More information0.02t if 0 t 3 δ t = 0.045 if 3 < t
1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual effectve
More informationSection 5.3 Annuities, Future Value, and Sinking Funds
Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationThe Stock Market Game and the KellyNash Equilibrium
The Stock Market Game and the KellyNash Equlbrum Carlos AlósFerrer, Ana B. Ana Department of Economcs, Unversty of Venna. Hohenstaufengasse 9, A1010 Venna, Austra. July 2003 Abstract We formulate the
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationAkira Yanagisawa Leader Energy Demand, Supply and Forecast Analysis Group Energy Data and Modelling Center
Background of Surgng Ol Prces and Market Expectaton Seen n Optons Akra Yanagsawa Leader Energy Demand, Supply and Forecast Analyss Group Energy Data and Modellng Center Summary The crude ol prces (WTI
More informationFINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals
FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant
More informationOptimal Consumption and Investment with Transaction Costs and Multiple Risky Assets
THE JOURNAL OF FINANCE VOL. LIX, NO. 1 FEBRUARY 2004 Optmal Consumpton and Investment wth Transacton Costs and Multple Rsky Assets HONG LIU ABSTRACT We consder the optmal ntertemporal consumpton and nvestment
More informationA) 3.1 B) 3.3 C) 3.5 D) 3.7 E) 3.9 Solution.
ACTS 408 Instructor: Natala A. Humphreys SOLUTION TO HOMEWOR 4 Secton 7: Annutes whose payments follow a geometrc progresson. Secton 8: Annutes whose payments follow an arthmetc progresson. Problem Suppose
More informationProbability and Optimization Models for Racing
1 Probablty and Optmzaton Models for Racng Vctor S. Y. Lo Unversty of Brtsh Columba Fdelty Investments Dsclamer: Ths presentaton does not reflect the opnons of Fdelty Investments. The work here was completed
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationThe impact of hard discount control mechanism on the discount volatility of UK closedend funds
Investment Management and Fnancal Innovatons, Volume 10, Issue 3, 2013 Ahmed F. Salhn (Egypt) The mpact of hard dscount control mechansm on the dscount volatlty of UK closedend funds Abstract The mpact
More informationWhat is Advanced Corporate Finance? What is finance? What is Corporate Finance? Deciding how to optimally manage a firm s assets and liabilities.
Wat is? Spring 2008 Note: Slides are on te web Wat is finance? Deciding ow to optimally manage a firm s assets and liabilities. Managing te costs and benefits associated wit te timing of cas in and outflows
More informationThe Probability of Informed Trading and the Performance of Stock in an OrderDriven Market
AsaPacfc Journal of Fnancal Studes (2007) v36 n6 pp871896 The Probablty of Informed Tradng and the Performance of Stock n an OrderDrven Market Ta Ma * Natonal Sun YatSen Unversty, Tawan Mnghua Hseh
More informationDifferences of Opinion of Public Information and Speculative Trading in Stocks and Options
Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons H. Henry Cao Cheung Kong Graduate School of Busness CKGSB Hu OuYang Lehman Brothers and CKGSB We analyze the effects of
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationUnderwriting Risk. Glenn Meyers. Insurance Services Office, Inc.
Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether
More informationDB Global Short Maturity High Yield Bond Index
12 February 2015 DBIQ Index Gude DB Global Short Maturty Hgh Yeld Bond Index Summary The DB Global Short Maturty Hgh Yeld Bond Index ( Index ) tracks the performance of a selected basket of short term
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationIn our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is
Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns
More informationIn our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A
Amortzed loans: Suppose you borrow P dollars, e.g., P = 100, 000 for a house wth a 30 year mortgage wth an nterest rate of 8.25% (compounded monthly). In ths type of loan you make equal payments of A dollars
More informationA Master Time Value of Money Formula. Floyd Vest
A Master Tme Value of Money Formula Floyd Vest For Fnancal Functons on a calculator or computer, Master Tme Value of Money (TVM) Formulas are usually used for the Compound Interest Formula and for Annutes.
More informationThe Shortterm and Longterm Market
A Presentaton on Market Effcences to Northfeld Informaton Servces Annual Conference he Shortterm and Longterm Market Effcences en Post Offce Square Boston, MA 0209 www.acadanasset.com Charles H. Wang,
More information10.2 Future Value and Present Value of an Ordinary Simple Annuity
348 Chapter 10 Annutes 10.2 Future Value and Present Value of an Ordnary Smple Annuty In compound nterest, 'n' s the number of compoundng perods durng the term. In an ordnary smple annuty, payments are
More informationMathematics of Finance
Mathematcs of Fnance 5 C H A P T E R CHAPTER OUTLINE 5.1 Smple Interest and Dscount 5.2 Compound Interest 5.3 Annutes, Future Value, and Snkng Funds 5.4 Annutes, Present Value, and Amortzaton CASE STUDY
More informationFinancial Mathemetics
Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,
More informationINTRODUCTION TO MERGERS AND ACQUISITIONS: FIRM DIVERSIFICATION
XV. INTODUCTION TO MEGES AND ACQUISITIONS: FIM DIVESIFICATION In the ntroducton to Secton VII, t was noted that frs can acqure assets by ether undertakng nternallygenerated new projects or by acqurng
More informationPragmatic Insurance Option Pricing
Paper to be presented at the XXXVth ASTIN Colloquum, Bergen, 6 9th June 004 Pragmatc Insurance Opton Prcng by Jon Holtan If P&C Insurance Company Ltd Oslo, Norway Emal: jon.holtan@f.no Telephone: +47960065
More informationLoss analysis of a life insurance company applying discretetime riskminimizing hedging strategies
Insurance: Mathematcs and Economcs 42 2008 1035 1049 www.elsever.com/locate/me Loss analyss of a lfe nsurance company applyng dscretetme rskmnmzng hedgng strateges An Chen Netspar, he Netherlands Department
More informationBeating the Odds: Arbitrage and Wining Strategies in the Football Betting Market
Beatng the Odds: Arbtrage and Wnng Strateges n the Football Bettng Market NIKOLAOS VLASTAKIS, GEORGE DOTSIS and RAPHAEL N. MARKELLOS* ABSTRACT We examne the potental for generatng postve returns from wagerng
More informationThe VIX Volatility Index
U.U.D.M. Project Report :7 he VIX Volatlty Index Mao Xn Examensarbete matematk, 3 hp Handledare och examnator: Macej lmek Maj Department of Mathematcs Uppsala Unversty Abstract. VIX plays a very mportant
More informationFinite Math Chapter 10: Study Guide and Solution to Problems
Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount
More informationTHE EFFECT OF PREPAYMENT PENALTIES ON THE PRICING OF SUBPRIME MORTGAGES
THE EFFECT OF PREPAYMENT PENALTIES ON THE PRICING OF SUBPRIME MORTGAGES Gregory Ellehausen, Fnancal Servces Research Program George Washngton Unversty Mchael E. Staten, Fnancal Servces Research Program
More informationTHE IMPLIED VOLATILITY OF ETF AND INDEX OPTIONS
The Internatonal Journal of Busness and Fnance Research Volume 5 Number 4 2011 THE IMPLIED VOLATILITY OF ETF AND INDEX OPTIONS Stoyu I. Ivanov, San Jose State Unversty Jeff Whtworth, Unversty of HoustonClear
More informationStress test for measuring insurance risks in nonlife insurance
PROMEMORIA Datum June 01 Fnansnspektonen Författare Bengt von Bahr, Younes Elonq and Erk Elvers Stress test for measurng nsurance rsks n nonlfe nsurance Summary Ths memo descrbes stress testng of nsurance
More informationTimeVarying Liquidity in Foreign Exchange
TmeVaryng Lqudty n Foregn Excange Martn D. D. Evans Rcard K. Lyons 26 October 200 Abstract Ts paper addresses weter currency trades ave greater prce mpact durng perods of rapd publc nformaton flow. Central
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationChapter 3 Research Method
Chapter 3 Research Method 3.1 Framework 3.1.1 Research Desgn A twophase study was desgned to explore the feasblty of RM n agng socetes from both supply and demand aspect. In the aspect of the borrowers,
More informationJournal of Corporate Finance
CORFIN0047; No of Pages Journal of Corporate Fnance xxx (00) xxx xxx Contents lsts avalable at ScenceDrect Journal of Corporate Fnance journal omepage: www.elsever.com/locate/jcorpfn Insttutonal tradng,
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationChapter 15: Debt and Taxes
Chapter 15: Debt and Taxes1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt
More informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 081 Institutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationConstruction Rules for Morningstar Canada Target Dividend Index SM
Constructon Rules for Mornngstar Canada Target Dvdend Index SM Mornngstar Methodology Paper October 2014 Verson 1.2 2014 Mornngstar, Inc. All rghts reserved. The nformaton n ths document s the property
More informationOn the Use of Bayesian Networks to Analyze Survey Data
On te Use of Bayesan Networks to Analyze Survey Data P. Sebastan 1 (1 and. Ramon ( (1 Department of atematcs and Statstcs, Unversty of assacusetts. ( Cldren's Hosptal Informatcs Program, Harvard Unversty
More information