# Stock Profit Patterns

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 50 costs. Moreover, an actve future contract market for Farsta Shppng s avalable Buy Stock Sol Short Stock 1

2 Futures an Forwars Proft Patterns Sell Forwar contract(kort Buy Forwar contract (lang

3 Smple Optons Proft Patterns Buy call opton Sell call opton Buy put opton 3

4 Formulas Buy Sol Short Buy Sell Stock Stock call opton call opton B1-\$B\$3 \$B\$3-B1 (MAA(0,(B1-\$D\$3-\$D\$4 (-MAA(0,(B1-\$D\$3\$D\$4 Buy Sell Sell Buy put opton put opton Forwar contraforwar contr (MAA(0,(\$D\$3-B1-\$D\$5 (-MAA(0,(\$D\$3-B1\$D\$5 \$D\$6-B1 -\$D\$6B1 4

5 Propostons (arbtrage Restrctons for Optons No. 1 Assume that the present value of a rsk less securty pays off at tme T: e rt Proposton 1: Lower boun: C max [,0] rt S e 0 Example: Assume 80, S 0 83, T0.5 an r 10% C max [ ] e, Proof: (we bul on the conseraton of the CASH FLOW from a partcular strategy at Tme 0 at Tme T Buy one share of the stock Exercse f proftable Borrow the PV of the Opton Repay borrowe Funs Wrte a call on the stock 5

6 Propostons (arbtrage Restrctons for Optons No. 1 Toay Acton Buy the Stock Borrow PV( Wrte a call on the stock Total CF Cash Flow -S 0 e -rt C -S 0 e -rt C at Tme T S T < S T > S T S T (S T - S T -<0 0 A contract that has only non-postve pay-offs n the future must have a postve ntal cash flow: C rt S0 e > 0 or C > S0 The value of a call can n no case be less than zero: e rt C max [,0] rt S e 0 6

7 Propostons (arbtrage Restrctons for Optons No. Proposton : Early Exercse: Conser exercse t < T; then S t > 0. However, Proposton 1 says that the value of a call s at least r( T t Snce, S t e S t e r( T t > S t The opton holer s better off sellng the opton n the market than exercsng t. 7

8 Propostons (arbtrage Restrctons for Optons No. 3 P Strategy: max Acton Cash Flow Short the Stock S 0 Len PV( Toay [ 0, e S ] rt -e -rt Wrte a put on the stock P Total CF S 0 -e -rt P 0 at Tme T S T < S T > -S T -S T -(- S T S T <0 A contract that has only non-postve pay-offs n the future must have a postve ntal cash flow: P rt rt S0 e > 0 or P > e S 0 The value of a put can n no case be less than zero: P max [ 0, e S ] rt 0 8

9 Propostons (arbtrage Restrctons for Optons No. 4 C e rt P S 0 Toay Acton Buy the Call on the stock Buy a bon PV( Wrte a put on the stock Short on share of the stock Total CF Cash Flow -C -e -rt P S 0 -C -e -rt P S 0 at Tme T S T < S T > 0 S T - -(-S T 0 -S T -S T 0 0 A contract that has only zero pay-offs n the future must have a zero ntal cash flow: rt C e P S0 0 rt C e P S0 9

10 Propostons (Call Opton Prce Convexty No. 5 Exercse Prce 1 3 Opton Prce C 1 C C 3 1 Toay 3 then C C 1 3 < Proof: at Tme T Acton Cash Flow S T < 1 1 <S T < <S T < 3 S T > 3 Buy a Call on the stock -C 1 0 S T - S 1 T - 1 S T - 1 Buy a Call on the stock -C S T - 3 Wrte calls on the stock C 0 0 -(S T - -(S T - Total CF C -C 1 -C 3 0 S T - 1 >0-1 -S T 0 3 -S T >0 C1 C3 C C1 C3 < 0 C < C 10

11 Propostons (Put Opton Prce Convexty No then P < P 1 P 3 11

12 Propostons (Dven Payng Stock No. 7 We assume that stock pays ven (D at t < T. C max Toay Tme t Acton Cash Flow [,0] rt rt S De e 0 at Tme T S T < S T > Buy the stock -S 0 D S T S T Borrow PV (D De -rt -D Borrow PV ( e -rt - - Wrte Call C 0 -(S T - Total CF De -rt e -rt C-S 0 0 S T -<0 0 D e e C S > 0 0 rt rt Ths result C>0 > C max [,0] rt rt S De e 0 1

13 The Bnomal Opton Prcng Moel Fn the combnaton of the bons an stocks that exactly replcates (tracks the call optons payoff. 13

14 The Bnomal Opton Prcng Moel Two equatons an two unknowns: 55 A 1.06 B A 1.06 B 0 The secon equaton B A: The frst equaton: 55A * A A B B

15 15 Styrng og Fonsmeglng The Bnomal Opton Prcng Moel: State Prces Thnk about the market etermnng a prce q u for NOK 1 n the up state of the worl an a prce q for NOK 1 n the own state of the worl. Hence, for the stock 1 (1 (1 (1 (1 q u q S S q u S q u u An for the bon 1 (1 1 ( q q u Solves to gve: ( (1 ( (1 u u q u q u

16 16 Styrng og Fonsmeglng The Bnomal Opton Prcng Moel: State Prces Hence The put-call party gves: [ ] [ ],0 (1,0 1 ( S max q u S max q C u S 0 e C P T Consequently: [ ] [ ],0 (1 (1,0 (1 (1 n n u n n u u S max q q n P u S max q q n C

17 Prcng Amercan Optons Usng the Bnomal Prcng Moel The Bnomal Prcng Moel can prce Amercan Optons. Example: S50, 50, an 6%. q u 10% an q 3% Stock Prce Amercan Put payoffs Date 0 Date 1 Date Date 0 Date 1 Date {Max( ,0} 55?? ?? 0.00 {Max( ,0} 48.5?? {Max( ,0} 17

18 Prcng Amercan Optons Usng the Bnomal Prcng Moel At ate 1, the holer of an Amercan put can choose whether to hol the put or to exercse t. Hence, Put value Value of put f exercse at ate max qu Put payoff at uu q Put payoff at u state u max( Su max qu Put payoff at uu q Put payoff at u A smlar functon hols for the put value n state at ate. At ate 0, a smlar value functon recurs. 18

19 Prcng Amercan Optons Usng the Bnomal Prcng Moel The value tree for the Amercan put: Max(max( S (1 u,0, q putpayoff q putpayoff u uu u Max(max( S (1 u,0, q putpayoff q putpayoff u u Max(max( S,0, q putvalue q putvalue u u 19

20 The Lognormal Dstrbuton How o stock prces look lke? 0

21 Stock Prces: 5 Propertes (Black & Scholes 1. Wggly Lnes. Lnes that are contnuous, wth no obvous jumps 3. Lnes that are always postve an never cross zero, no matter how low they get 4. That at a gven pont n tme, the average over all plausble lnes s greater than the ntal prce of the stock. The farther out we go, the hgher ths average becomes. 5. That the stanar evaton over all plausble lnes s greater the farther out we go. 1

22 Lognormal Prce Dstrbutons an Geometrc Dffusons Let us enote by S t the prce at tme t of a share of stock. The lognormal strbuton assumes that the natural logarthm og one plus the return from holng a share of stock between tme t an t t s normally strbute wth mean µ an stanar evaton σ. Let us enote the uncertan return over the nterval t wth ~. Hence, S t t S t e [ ~ r t ] t In the lognormal strbuton the rate of return over a short tme pero t s normally strbute wth mean µ t an varance σ t. Hence, St S t [ t Z t ] t exp µ σ where Z s a stanar normal varable (mean0, varance1. ~ r t r t

23 Lognormal Prce Smulaton; VBA Proceure Sub prcepathsmulaton( Range("starttme" Tme Applcaton.screenupatng False N Range("runs".Value mean Range("mean" sgma Range("sgma" elta_t 1 / ( * N ReDm prce(0 To * N As Double prce(0 Range("ntal_prce" For Inex 1 To N start: Statc ran1, ran, S1, S, 1, ran1 * Rn - 1 ran * Rn - 1 S1 ran1 ^ ran ^ If S1 > 1 Then GoTo start S Sqr(- * Log(S1 / S1 1 ran1 * S ran * S prce( * Inex - 1 prce( * Inex - * Exp(mean * elta_t _ sgma * Sqr(elta_t * 1 prce( * Inex prce( * Inex - 1 * Exp(mean * elta_t _ sgma * Sqr(elta_t * Next Inex For Inex 0 To * N Range("output".Cells(Inex 1, prce(inex Next Inex Range("stoptme" Tme Range("elapse" Range("stoptme" - Range("starttme" Range("elapse".NumberFormat "hh:mm:ss" En Sub 3

24 Calculatng Parameters for Lognormal Dstrbuton (Stock Prces 1. Mean S t t E ln( E S σ t. Varance St Var ln( S t lnearty n tme. Estmaton: µ [ µ t Z t ] Var [ µ t σ Z t ] σ t t St Mean ln( S t t t t, σ St Var ln( S t t 4

25 Calculatng Parameters for Lognormal Dstrbuton (Stock Prces Return: Mean Annual Returns n * Mean Peroc Return where n s the number of peros n one year (aly > n 5 Stanar evaton: σ Annual Returns Sqrt(n * σ Peroc Return where n s the number of peros n one year (aly > n Sqrt(5 5

26 The Black & Scholes Moel; Call Opton. C S N rt ( e N( 1 where 1 ln( S / ( r σ σ T / T 1 σ T Where C enotes the prce of a call, S s the prce of the unerlyng nstrument, s the exercse prce of the call, T s the tme to exercse an r s the rsk free nterest rate. 6

27 where P The Black & Scholes Moel; Put Opton. Applyng the put-call party relaton a put wth the same exercse ate T, exercse prce : P C S e -rt. Hence, 1 S ln( S 1 rt N( 1 e N( / σ ( r σ σ T T / T Where P enotes the prce of a put, S s the prce of the unerlyng nstrument, s the exercse prce of the call, T s the tme to exercse an r s the rsk free nterest rate. 7

28 The Black & Scholes Moel; VBA-Functons. Call Opton: Put Opton: Functon CallOpton(Stock, Exercse, Tme, Interest, sgma CallOpton Stock * Applcaton.NormSDst(One(Stock, Exercse, _ Tme, Interest, sgma - Exercse * Exp(-Tme * Interest * _ Applcaton.NormSDst(One(Stock, Exercse, Tme, Interest, sgma _ - sgma * Sqr(Tme En Functon Functon PutOpton(Stock, Exercse, Tme, Interest, sgma PutOpton CallOpton(Stock, Exercse, Tme, Interest, sgma _ Exercse * Exp(-Interest * Tme - Stock En Functon 8

29 The Black & Scholes Moel; VBA-Imple Volatlty. Imple Volatlty: Functon CallVolatlty(Stock, Exercse, Tme, Interest, Target Hgh Low 0 Do Whle (Hgh - Low > If CallOpton(Stock, Exercse, Tme, Interest, (Hgh Low / > _ Target Then Hgh (Hgh Low / Else: Low (Hgh Low / En If Loop CallVolatlty (Hgh Low / En Functon 9

30 The Bang of the Bucks wth Optons. Suppose that you are convnce that a gven stock wll go up n a very short pero of tme. You want to buy calls on the stock that have a maxmum bang for the buck - that s, you want the percentage proft on your opton nvestment to be maxmal. Apply the B&S formula: Buy calls wth shortest possble maturty Buy calls that are most hghly out of the money (.e. wth the hghest exercse prce possble Implement! 30

31 Valung Optons on More Complex Assets The Forwar Prce Verson of the B&S moel: One we know how to calculate the forwar prce of an unerlyng asset, t s possble to etermne the B&S value for a European call on the unerlyng asset. Remember: sf S 0 (1 where s F s the forwar prce of an unerlyng asset. Hence, r T C e rt [ s N( N( F 1 ] 1 ln( S / ( r σ σ T / T 1 σ T 31

32 Valung Optons on More Complex Assets The U.S. Dollar rsk-free rate s assume to be 6% per year an all σ s are assume to be 5% per year. E want to apply the forwar prce verson of B&S to value European call optons one year from now: a 1 DM at a strke prce of \$0.50. The Spot Exchange rate s \$0.40/DM an the one year rsk free rate s 8% Germany. b A 30-year bon wth an 8% sem-annual coupon at a strke prce of \$100. The bon s currently sellng at a full prce of \$10 (nclue accrue nterest per \$100 of face value an has two scheule coupons of \$4 before opton expraton, to be pa sx months an one year form now. c The S&P500 contract wth a strke prce of \$850. The S&P500 has a current prce of \$800 an a present value of next year s vens of \$0. A barrel of Ol wth a strke prce of \$0. A barrel of ol currently sells for \$18. The present value of next year storage costs s \$1 an the present value of the convenence yel havng a barrel of ol avalable over the next year (e.g., f there are long gas lnes ue to an ol embargo an eplorable government polcy s \$1. 3

33 Forwar prces: a b c Valung Optons on More Complex Assets (\$ /1.08 \$0.396 \$ \$ \$4 \$ (\$ 800 \$ \$86.80 (\$ 18 \$1 \$ \$19.08 Forwar Prce verson of B&S moel: \$0.396 N( \$0.5 N( ln(0.396 / 0.5 C 1 where, \$ N( \$100 N( ln( / \$86.80 N( \$850 N( ln(86.80 / 850 C 1 where, \$19.08 N( \$0 N( ln(19.08/ 0 C 1 where, a 1 >C\$0.01 b C 1 where, >C\$9.39 c >C\$68. >C\$

### EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A 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

### The Cox-Ross-Rubinstein Option Pricing Model

Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten 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 no-arbtrage

### 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

### = i δ δ s n and PV = a n = 1 v n = 1 e nδ

Exam 2 s Th March 19 You are allowe 7 sheets of notes an a calculator 41) An mportant fact about smple nterest s that for smple nterest A(t) = K[1+t], the amount of nterest earne each year s constant =

### Section 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

### Hedging Interest-Rate Risk with Duration

FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

### Interest 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

### AS 2553a Mathematics of finance

AS 2553a Mathematcs of fnance Formula sheet November 29, 2010 Ths ocument contans some of the most frequently use formulae that are scusse n the course As a general rule, stuents are responsble for all

### Pricing 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

### Lecture 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 annuty-mmedate, and ts present value Study annuty-due, and

### Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes-1 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

### Simple 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

### Using 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

### 10.2 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

### Interest 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

### Multiple discount and forward curves

Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

### benefit 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

### Answer: 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 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

### An 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

### Time 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

### CURRENCY OPTION PRICING II

Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility Black-Scholes Moel for Currency Options Properties of the BS Moel Option Sensitivity

### Recurrence. 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.

### Level Annuities with Payments Less Frequent than Each Interest Period

Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

### The Application of Fractional Brownian Motion in Option Pricing

Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

### Lecture 14: Implementing CAPM

Lecture 14: Implementng CAPM Queston: So, how do I apply the CAPM? Current readng: Brealey and Myers, Chapter 9 Reader, Chapter 15 M. Spegel and R. Stanton, 2000 1 Key Results So Far All nvestors should

### Hull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5

Binomial Moel Hull, Chapter 11 + ections 17.1 an 17.2 Aitional reference: John Cox an Mark Rubinstein, Options Markets, Chapter 5 1. One-Perio Binomial Moel Creating synthetic options (replicating options)

### Vasicek 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

### 7.5. 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

### On the pricing of illiquid options with Black-Scholes formula

7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 On the prcng of llqud optons wth Black-Scholes

### A 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

### Mathematics of Finance

5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty;Amortzaton Chapter 5 Revew Extended Applcaton:Tme, Money, and Polynomals Buyng a car

### Simon 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 Co-movement between the ponts Postve

### Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

### 1. 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

### DEGREES OF EQUIVALENCE IN A KEY COMPARISON 1 Thang H. L., Nguyen D. D. Vietnam Metrology Institute, Address: 8 Hoang Quoc Viet, Hanoi, Vietnam

DEGREES OF EQUIVALECE I A EY COMPARISO Thang H. L., guyen D. D. Vetnam Metrology Insttute, Aress: 8 Hoang Quoc Vet, Hano, Vetnam Abstract: In an nterlaboratory key comparson, a ata analyss proceure for

### Section 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

### Cautiousness 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, Arrow-Pratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets

### Pricing Multi-Asset Cross Currency Options

CIRJE-F-844 Prcng Mult-Asset 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

### On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

### 8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value

8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest \$000 at

### Thursday, 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

### Joe 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

### THE 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

### Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared

### n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

### Interest Rate Forwards and Swaps

Interest Rate Forwards and Swaps Forward rate agreement (FRA) mxn FRA = agreement that fxes desgnated nterest rate coverng a perod of (n-m) months, startng n m months: Example: Depostor wants to fx rate

### On the computation of the capital multiplier in the Fortis Credit Economic Capital model

On the computaton of the captal multpler n the Forts Cret Economc Captal moel Jan Dhaene 1, Steven Vuffel 2, Marc Goovaerts 1, Ruben Oleslagers 3 Robert Koch 3 Abstract One of the key parameters n the

### Finite 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

### In 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

### Financial 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,

### ECONOMICS 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

### A) 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

### Section 2.2 Future Value of an Annuity

Secton 2.2 Future Value of an Annuty Annuty s any sequence of equal perodc payments. Depost s equal payment each nterval There are two basc types of annutes. An annuty due requres that the frst payment

### Chapter 21: Options and Corporate Finance

Chapter 21: Options and Corporate Finance 21.1 a. An option is a contract which gives its owner the right to buy or sell an underlying asset at a fixed price on or before a given date. b. Exercise is the

### THE 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

### What is Candidate Sampling

What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

### Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance

Applcaton of Quas Monte Carlo methods and Global Senstvty Analyss n fnance Serge Kucherenko, Nlay Shah Imperal College London, UK skucherenko@mperalacuk Daro Czraky Barclays Captal DaroCzraky@barclayscaptalcom

### Lecture 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

### 1 De nitions and Censoring

De ntons and Censorng. Survval Analyss We begn by consderng smple analyses but we wll lead up to and take a look at regresson on explanatory factors., as n lnear regresson part A. The mportant d erence

### Mathematics 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

### Option pricing. Vinod Kothari

Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate

### Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the

Bond futures INRODUCION Bond futures contracts are futures contracts that allow nvestor to buy n the future a theoretcal government notonal bond at a gven prce at a specfc date n a gven quantty. Compared

### In 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

### A 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.

### Communication Networks II Contents

8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

### Mathematical Option Pricing

Mark H.A.Davs Mathematcal Opton Prcng MSc Course n Mathematcs and Fnance Imperal College London 11 January 26 Department of Mathematcs Imperal College London South Kensngton Campus London SW7 2AZ Contents

### ADVA FINAN QUAN ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS VAULT GUIDE TO. Customized for: Jason (jason.barquero@cgu.edu) 2002 Vault Inc.

ADVA FINAN QUAN 00 Vault Inc. VAULT GUIDE TO ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS Copyrght 00 by Vault Inc. All rghts reserved. All nformaton n ths book s subject to change wthout notce. Vault

### Section 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

### World 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

### BERNSTEIN POLYNOMIALS

On-Lne 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

### Options: Valuation and (No) Arbitrage

Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial

### INVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMA-HDR NETWORKS

21 22 September 2007, BULGARIA 119 Proceedngs of the Internatonal Conference on Informaton Technologes (InfoTech-2007) 21 st 22 nd September 2007, Bulgara vol. 2 INVESTIGATION OF VEHICULAR USERS FAIRNESS

### Akira 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

### Extending Probabilistic Dynamic Epistemic Logic

Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

### Pragmatic 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

### The covariance is the two variable analog to the variance. The formula for the covariance between two variables is

Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.

### The 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

### Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies

Insurance: Mathematcs and Economcs 42 2008 1035 1049 www.elsever.com/locate/me Loss analyss of a lfe nsurance company applyng dscrete-tme rsk-mnmzng hedgng strateges An Chen Netspar, he Netherlands Department

### An Analysis of Pricing Methods for Baskets Options

An Analyss of Prcng Methods for Baskets Optons Martn Krekel, Johan de Kock, Ralf Korn, Tn-Kwa Man Fraunhofer ITWM, Department of Fnancal Mathematcs, 67653 Kaserslautern, Germany, emal: krekel@twm.fhg.de

### Portfolio Performance Manipulation and Manipulation-Proof Performance Measures

Portfolo Performance Manpulaton and Manpulaton-Proof Performance Measures Wllam Goetzmann, Jonathan Ingersoll, Matthew Spegel, Ivo Welch March 5, 006 Yale School of Management, PO Box 0800, New Haven,

### The 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 End-users Multnatonal frms Central

### Hedging with Futures and Options: Supplementary Material. Global Financial Management

Hedging with Futures and Options: Supplementary Material Global Financial Management Fuqua School of Business Duke University 1 Hedging Stock Market Risk: S&P500 Futures Contract A futures contract on

### The Short-term and Long-term Market

A Presentaton on Market Effcences to Northfeld Informaton Servces Annual Conference he Short-term and Long-term Market Effcences en Post Offce Square Boston, MA 0209 www.acadan-asset.com Charles H. Wang,

### EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

### Question 2: What is the variance and standard deviation of a dataset?

Queston 2: What s the varance and standard devaton of a dataset? The varance of the data uses all of the data to compute a measure of the spread n the data. The varance may be computed for a sample of

### Lecture 21 Options Pricing

Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Put-call

### 1 Example 1: Axis-aligned rectangles

COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

### Guide to the Volatility Indices of Deutsche Börse

Volatlty Indces of Deutsche Börse Verson.4 Volatlty Indces of Deutschen Börse Page General Informaton In order to ensure the hghest qualty of each of ts ndces, Deutsche Börse AG exercses the greatest care

### Texas Instruments 30Xa Calculator

Teas Instruments 30Xa Calculator Keystrokes for the TI-30Xa are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the tet, check

### FINANCIAL 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

### Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration

CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put:

### Mathematics of Finance

CHAPTER 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty; Amortzaton Revew Exercses Extended Applcaton: Tme, Money, and Polynomals Buyng

### The 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

### Texas Instruments 30X IIS Calculator

Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the

### Options Pricing. This is sometimes referred to as the intrinsic value of the option.

Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the Put-Call Parity Relationship. I. Preliminary Material Recall the payoff

### Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.

Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When

### greatest common divisor

4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

### An Overview of Financial Mathematics

An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take