Introduction to Financial Derivatives

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Geraldine Allison
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1 Itroductio to Fiacial Derivatives Determiig Prices for Forwards ad Futures Week of October 1, 01 Where we are Last week: Itroductio to Iterest Rates, Future Value, Preset Value ad FRAs (Chapter 4, OFOD) This week: Revisit FRAs ad the we look at the Determiatio of Forward ad Futures Prices - Value (Chapter 5, OFOD) Next week: Iterest Rate Futures (Chapter 6, OFOD) Assigmet Assigmet For This Week (October 1 st ) Read: Hull Chapter 5 Problems (Due October 1 st ) Chapter 4: 5, 8, 9, 11, 1, 14, 16, ; 3 Chapter 4 (7e): 5, 8, 9, 11, 1, 14, 16, ; 7 Problems (Due October 8 th ) Chapter 5:, 4, 6, 7, 1, 16, 17, 0; 4 Chapter 5 (7e):, 4, 6, 7, 1, 16, 17, 0; For Next Week (October 8 th ) Read: Hull Chapter 6 Problems (Due October 8 th ) Chapter 5:, 4, 6, 7, 1, 16, 17, 0; 4 Chapter 5 (7e):, 4, 6, 7, 1, 16, 17, 0; 4 Problems (Due October 17 th ) Chapter 6: 4, 6, 9, 11, 14, 1; 6, 7 Chapter 6 (7e): 4, 6, 9, 11, 14, 1; 3, 4 Exams Midterm: October 1 th, 011 Fial: Thurs., Dec. 15 th, 9am 1oo 1.4 1

2 Pla for This Week Forward Rate Agreemet Look at FRAs (today, the:) Determiatio of Forward & Futures Prices Ivestmet Asset vs. Cosumptio Asset Forward Price for a Ivestmet Asset Whe the Ivestmet Asset has Icome/Divided Yield Value of Forward Cotracts Forward vs. Futures Pricig Forward ad Futures o Currecies Futures o Commodities Futures Details (carry, embedded optios) A forward rate agreemet (FRA) is a agreemet that a certai rate will apply to a specified pricipal, L, durig a certai future time period ( T 1 to T ) A FRA is equivalet to a agreemet where iterest at a predetermied rate, R K, is exchaged for iterest at the market rate, R M, If R K is received (paid), receiver (payer) value at T is L( R ( ) K RM)( T T1) L( RM RK)( T T1) FRAs are settled (w/payoff) at T 1 rather tha T : L( R R )( T T) L( R R )( T T) ( M K 1 ) 1 R ( T T) Futures Price vs. Expected Future Spot 1.5 M 1 M K M 1 1 R ( T T) Forward Rate Agreemet A FRA ca be preset valued by assumig that the forward iterest rate is certai to be realized A dealer who writes the FRA ca lock-i the forward rate to be paid or received at essetially o cost To receive, borrow to T 1 ad ivest to T Borrow $100 util T 1 ad ivest it to T T1 R 1 At T 1 you pay 100e T T R ad at T you receive 100e T Ad ormalizig, a deposit of 100 at T 1 returs (at T ): T RT T1 R T1 F T1, T T T1 100e 100e Hece, by defiitio of the forward rate, this rate o a ivestmet (to receive) has bee locked-i for the period T 1 to T Similarly, the dealer ca lock-i his borrowig cost from T 1 to T by borrowig util T ad ivestig to T Forward Rate Agreemet A FRA ca be preset valued by assumig that the forward iterest rate is certai to be realized Sice the dealer ca lock-i the forward rate, a cliet that wats to receive a fixed rate ca have the forward rate agreemet at the curret market forward rate at o cost (there is a fee charged by the dealer) The dealer just eters ito the trade o the last slide, ad o the cotract date ca deliver the cotract rate o the agreed upo pricipal to the cliet If the cliet wats a rate other tha the curret forward rate, he has to settle the differetial PV, up-frot this is how we establish the PV of the FRA for R K 1.8

3 FRA Valuatio Formulas PV of FRA where a fixed rate R K will be received o a pricipal L betwee times T 1 ad T is RT L( RK RF )( T T1 ) e PV of FRA where a fixed rate is paid is RT L( RF RK )( T T1 ) e R F is the forward rate for the period ad R is the zero rate for maturity T What compoudig frequecies are used i these formulas for R K, R F, ad R? R K & R F are defied by the period of the forward T T T 1 R is the cotiuously compouded rate 1.9 Duratio Bod Duratio is the (value-weighted) average time to receipt of cash Duratio, D, of a bod that provides cash flow c i at time t i is yt i c i e D t i i 1 B yti with price B cie ad (cotiuously compouded) yield y i 1 Sice a small chage i yield, y, leads to a chage i price as B db dy y ad yt B i So B y citie ad D y i 1 B The percetage chage i price is egatively related to a chage i yield. db dy i 1 c t e yti i i 1.10 Duratio Covexity Whe the yield y is expressed with compoudig m times per year BD y D B B y 1 y m y m 1 / The expressio D D* 1 y m defies D* ad it is referred to as Modified Duratio Duratio measures are importat for risk maagemet B D * y B Which exhibits the well kow characteristic for bods that yield ad price move i opposite directio Note that for cotiuous compoudig, duratio does t have to be modified The covexity of a bod is defied as 1 B C B y so that i 1 c t e yti i i B 1 D y C( y) B A useful result to show the limitatio of duratio hedgig as yield chages become larger the effect of the oliearity of the price-yield relatioship B 1.1 3

4 The Ed for Last Week Pla for This Week Questios? Now for this week 1.13 Determiatio of Forward & Futures Prices Ivestmet Asset vs. Cosumptio Asset Short Sellig & Arbitrage Forward Price for a Ivestmet Asset Whe the Ivestmet Asset has Icome/Divided Yield Value of Forward Cotracts Forward vs. Futures Pricig Forward ad Futures o Currecies Futures o Commodities Futures Details (carry, embedded optios) Futures Price vs. Expected Future Spot 1.14 Cosumptio vs. Ivestmet Assets Short Sellig Ivestmet assets are assets held by sigificat umbers of people purely for ivestmet purposes (Examples: gold, silver, stocks, bods) Cosumptio assets are assets held primarily for cosumptio (Examples: copper, oil, hogs, OJ) Ivestmet assets readily admit arbitrage ad therefore permit ratioal pricig of futures ad forwards from spot prices Cosumptio assets are slightly more ivolved Short sellig ivolves sellig securities you do ot ow Your broker borrows the securities from aother cliet ad sells them i the market i the usual way Margi accout provides protectio

Short Sellig (cotiued) At some stage you must buy the securities back so they ca be replaced i the accout of the cliet You must pay divideds ad other beefits the ower of the securities receives

5 Short Sellig (cotiued) At some stage you must buy the securities back so they ca be replaced i the accout of the cliet You must pay divideds ad other beefits the ower of the securities receives Notatio for Valuig Futures ad Forward Cotracts S 0 : Spot price today F 0 : Futures or forward price today T: Time util delivery date r: Risk-free iterest rate for maturity T (lets thik r ( T) 0 ) Whe Iterest Rates are Measured with Cotiuous Compoudig Arbitrage Example F 0 = S 0 e rt This equatio relates the forward price ad the spot price for ay ivestmet asset that provides o icome ad has o storage costs The relatioship is govered by the term, T, to the forward date ad the time value of moey, expressed through the term spot rate,

6 Whe a Ivestmet Asset Provides a Kow Dollar Icome Arbitrage Example F 0 = (S 0 I )e rt Where I is the preset value (at t = t 0 ) of the icome durig life of forward cotract Whe a Ivestmet Asset Provides a Kow Yield F 0 = S 0 e (r q )T Where q is the average yield, durig the life of the cotract (expressed with cotiuous compoudig), provided by the asset 1.3 Valuig a Forward Cotract Suppose that K is delivery price i a forward cotract ad F 0 is forward price that would apply to the cotract today (at t 0 ) The value of a log forward cotract, ƒ, o a asset w/o icome ad with delivery price K is rt f ( F0 K) e rt rt rt ( Se 0 Ke ) S0 Ke Similarly, for kow icome ad divided yield q f S0 I Ke rt ad qt rt f Se 0 Ke 1.4 6

7 Forward vs. Futures Prices Forward & futures prices usually assumed equal Whe iterest rates are ucertai they are, i theory, slightly differet: A strog positive correlatio betwee iterest rates ad the asset price implies that the futures price should be slightly higher tha the forward price A strog egative correlatio implies the reverse: futures price should be slightly lower tha forward price The differece is usually small eough to be igored for short dated futures Stock Idex Ca be viewed as a ivestmet asset payig a divided yield The futures price ad spot price relatioship is therefore F 0 = S 0 e (r q )T where q is the average divided yield o the portfolio represeted by the idex durig life of cotract The textbook will assume they are the same Stock Idex Idex Arbitrage For the formula to be true it is importat that the idex represet a ivestmet asset I other words, chages i the idex must correspod to chages i the value of a tradable portfolio The Nikkei idex viewed as a dollar umber does ot represet a ivestmet asset (See Busiess Sapshot 5.3, page 113) If S is value of Nikkei 5; the CME futures is 5xS We ca oly ivest i a PF worth 5xQxS (Q:$valueY) 1.7 Whe F 0 > S 0 e (r-q)t a arbitrageur buys the stocks uderlyig the idex ad sells futures Whe F 0 < S 0 e (r-q)t a arbitrageur buys futures ad shorts or sells the stocks uderlyig the idex 1.8 7

8 Idex Arbitrage Futures ad Forwards o Currecies Idex arbitrage ivolves simultaeous trades i futures ad may differet stocks Very ofte a computer is used to geerate the trades Occasioally (e.g., o Black Moday) simultaeous trades are ot possible ad the theoretical o-arbitrage relatioship betwee F 0 ad S 0 does ot hold A foreig currecy is aalogous to a security providig a divided yield The cotiuous divided yield is the foreig risk-free iterest rate It follows that if r f is the foreig risk-free iterest rate ( F S e r r f ) T Why the Relatio Must Be True Futures o Cosumptio Assets r T 1000e f uits of foreig currecy at time T r T 1000F f 0 e dollars at time T 1000 uits of foreig currecy at time zero 1000S 0 dollars at time zero rt 1000S 0 e dollars at time T F 0 S 0 e (r+u )T where u is the storage cost per uit time as a percet of the asset value. Alteratively, F 0 (S 0 +U )e rt where U is the preset value of the storage costs. Storage costs act as egative icome We might fid the iequality surprisig, but ot so for cosumptio commodities vs. ivestmet commodities (gold, silver): the iequality may hold!

9 The Cost of Carry The cost of carry, c, is the storage cost plus the iterest costs less the icome eared c=r for a o-divided payig stock c=r-q for a idex w/yield q (c=r-r f ) c=r-q+u w/storage rate u For a ivestmet asset F 0 = S 0 e ct For a cosumptio asset F 0 S 0 e ct The coveiece yield o the cosumptio asset, y, is defied so that F 0 = S 0 e (c y )T If c>y beefit of holdig < tha carry; short: deliver early Ear the iterest o moey received If c<y coveiece is better tha carry; short: deliver later 1.33 Futures Prices & Expected Future Spot Prices Suppose k is the expected retur required by ivestors o a asset We ca ivest F 0 e r T at the risk-free rate ad eter ito a log futures cotract so that there is a cash iflow of S T at maturity This shows that, with the required retur k rt kt ( Fe 0 ) E( ST ) e or ( r k) T F E( S ) e 0 T 1.34 Futures Prices & Future Spot Prices (cotiued) If the asset has o systematic risk, the k = r ad F 0 is a ubiased estimate of S T positive systematic risk (positively correlated with stocks), the k > r ad F 0 < E (S T ) ormal backwardatio egative systematic risk (egatively correlated with stocks), the k < r ad F 0 > E (S T ) cotago

Binomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge

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