1 fixed income math and risk measure/basic ird 2008.12

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1

Fixed Income Math and Risk Measure/Basic IRD

2008.12

2

ContentsTopics Contents

Math Basic arithmetic for Fixed-income securities

Bond Bond Market OverviewYTM and PriceMacauley Duration/ Modified DurationImmunizationConvexityFinding YTM

Term structure of Interest Rates

Spot rate, Forward rate Term structure TheoryBootstrappingFitting the yield curve

Basic Interest Rate Derivatives(IRD)

Bond Futures: KTB FuturesFRAInterest Rate Swap

Cap and FloorSwaption

3

Math; Geometric series

• Geometric series– Initial value(scale factor);– Common ratio;– # of terms;

• [ex] Annuity, Bond price, Swaption,…

a

r

n1 nararaX

nn arararrX 10

naraXr 001

r

raX

n

1

1

4

Math; Matrix Operation

• Inverse Matrix: solving simultaneous equations

• Gaussian Elimination

• [ex] bootstrap, interpolation, FDM,…

bAx

bAx 1

6

5

43

21

2

1

x

x

6

5

5.05.1

12

10

01

2

1

x

x

5

Math; Differential

• 미분공식– 다항함수– 지수함수– 로그함수

• 곱함수의 미분

• 합성함수의 미분

• [ex] Duration, Convexity, Greek, …

1 nn nxdx

dfxxf

gfgffg

dx

dg

dg

df

dx

dfxgf

xx edx

dfexf

xdx

dfxxf

1ln

6

Math; Integral

• 적분공식– 다항함수 ;– 지수함수 ;– 로그함수 ;– 분수함수 ;

• 부분적분

• [ex] Nelson-Siegel Model, Option pricing,…

cxn

dxxfxxf nn 1

1

1

cedxxfexf xx

cxxxdxxfxxf lnln

cxdxxfx

xf ln1

dxxgxfxgxfdxxgxf

7

Math;Taylor Expansion

• We want to decompose(analyse) any function to “polynomial functions”

• How to find the coefficients– Above equation should hold when x=0

– Differentiating and inputting x=0 still hold equality

– and so on

• [ex] Numerical Method, Ito’s lemma,…

...2210 xaxaaxf

00 fa

01 fa

...000 2!2

1 xfxffxf

8

Taylor Expansion_cont.

• General case; expansion point ≠0 ...2

210 xaxaaxxf

00 axf

10 axf

220 axf

...2

1 2 xxfxxfxfxxf

9

Math; Numerical Methods

• Finding zeros;

• Numerical Methods

• [ex] YTM, Implied volatilities,…

Zero;

xfy

00 xf

[Bisection Method] [Newton Method]

10

Bonds/Market

채권시장

발행시장

유통시장 장내시장

장외시장

국채전문유통시장

일반채권거래시장

대고객상대매매시장

Inter Dealer Broker

사모발행

공모발행 직접발행

간접발행

매출발행

공모입찰발행

위탁모집

인수모집 잔액인수

총액인수

Conventional

Dutch

11

Bonds/Market

자료 :www.ksdabond.or.kr

12

Bonds/Market

자료 :www.ksdabond.or.kr

13

Bond/price

• Price or Present Value– In general;– Bond; ,– Excel formula;

CFDFPV

n

iiiCdP

1 itiy

d

1

1

14

Bond/Price

Example2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

A B C D E F G H I issue 97-12-27 1.calcuate "A"mat 00-12-27 date i cf df pvfreq 4 98-03-27 0 300 1 300cpn 12% 98-06-27 1 300 0.953516 286.0548yield 19.50% 98-09-27 2 300 0.909193 272.7579notional 10,000 98-12-27 3 300 0.86693 260.079trade 98-01-20 99-03-27 4 300 0.826632 247.9895daycount 30/360 99-06-27 5 300 0.788207 236.462

99-09-27 6 300 0.751568 225.4703price(함수) 8351.597 99-12-27 7 300 0.716632 214.9896

00-03-27 8 300 0.68332 204.99600-06-27 9 300 0.651557 195.46700-09-27 10 300 0.62127 186.380900-12-27 11 10300 0.592391 6101.625

8732.272let's try 98-03-01 2.caculate "B"

98-01-20 A set B setDSC 66 67 dirty priceE 90 90 8432.723 A setdf 0.965696 0.965186 8428.264 B set

3.calcuate Accrual Interest clean priceAccrual 80 76.66667 8348.264 A set

8351.597 B set

15

Bonds/Excel functions정리 그밖에 채권 관련된 함수

함수명 설명

YIELD 정기적으로 이자를 지급하는 유가증권의 수익률

YIELDDISC 할인된 유가증권의 연수익률

YIELDMAT 만기 시 이자를 지급하는 유가증권의 연 수익률

COUPDAYSNC 결산일부터 다음 이자 지급일까지의 날짜 수

COUPDAYBS 이자 지급기간의 시작일부터 결산일까지의 날짜 수

COUPDAYS 결산일이 들어 있는 이자 지급 기간의 날짜 수

COUPNCD 결산일 다음 첫 번째 이자 지급일

COUPPCD 결산일 바로 전 이자 지급일

COUPNUM 결산일과 만기일 사이의 이자지급횟수 ( 정수로 반올림 ) 

참고 이자기간함수간의 관계COUPDAYS = COUPDAYSNC+COUPDAYBS = COUPNCD-COUPPCD

16

Bonds/Terminology

• When evaluation date ≠issue date(coupon date)?• Conventional price: Theoretical price

• Dirty price : Clean price– [1] Cash price, Invoice price

– [2] Quoted price

17

Bonds/note

• Conventional VS Theoretical

정리 관행적 계산법 vs 이론적 계산법(1) 관행적 계산법 : 2 단계 할인할 때 단리 할인법 적용(2) 이론적 계산법 : 2 단계 할인할 때 복리 할인법 적용

accrual

41

100

41

4100

41

11

11

N

N

kk

E

daysyldyld

rate

yldp

1pA

발행일 거래일 이자지급일 1 이자지급일 2 이자지급일 3 …… 만기일

1 단계 할인2 단계 할인

18

Bonds/Day count convention

• Calculating Interest Amount;– .

• Calculation End Date (vs) Payment date– Adjusting– No Adjusting

CountDay RateInterest Amount PrincipalAmountInterest

19

Bonds/Macaulay Duration

• History of bond sensitivity– Maturity– CF Weighted Average Term to Maturity– PV Weighted Average Term to Maturity

• Meaning of Macauley Duration– Investment Horizon → Immunization– Sensitivity ?

P

dctwtD ii

n

iii

n

iiMAC

11

DurationMacauley

20

A question?

80

90

100

110

120

130

140

0

0.25 0.5

0.75 1

1.25 1.5

1.75 2

2.25 2.5

2.75 3

3.25 3.5

3.75 4

4.25 4.5

4.75 5

time(year)

pric

e

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

4 5 6 7 8 9

10 11 12

A B today 08-12-04maturity 13-12-04coupon 5.00%frq 4basis 0yield 5.00%redemption 100price 100duration 4.454827

Macaulay Duration

시장수익률이 거래체결 직후에 9% 가 되었다고 하자 . 1 년 뒤 , ‘ 이자와 이자의 재투자 수익 및 당시 채권가격을 모두 합한 값 (TOTAL)’ 이 점 “ A” 로 표시되어 있다 .

21

A question_cont.

80

90

100

110

120

130

140

0

0.25 0.5

0.75 1

1.25 1.5

1.75 2

2.25 2.5

2.75 3

3.25 3.5

3.75 4

4.25 4.5

4.75 5

time(year)

pric

e

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

t

y

Total PL

22

Macaulay Duration

• Interpretation

– Check duration for Zero-coupon bond when YTM is defined as continuous one and discrete one!

• Application; Immunization– Case 1; buying a coupon-bond to hedge ‘single-payment

liability’– Case 2;– -

23

Bonds/Modified Duration

• What we want to know is…– What happens to my money if the yield moves up 1%p?

• (If we define YTM as discrete-compounding one) The Macaulay Duration does not give the answer

• Modified Duration

– % change in Price to % point change in YTM

y

y

P

P

y

y

P

P

A

BDmac

)1(

)1(

Py

Py

P

P

y

DD mac 1

1

24

Comparison

• Continuous-compounding YTM– Coupon bond

– Zero coupon bond

• Discrete-compounding YTM– Coupon bond

– Zero coupon bond

n

ii

yt CeP i

1

n

i

iyt

imac P

CetD

i

1

P

Cet

PD

n

ii

yti

dydP

i

1

mod

n

iit C

yP

i1 1

1

n

i

iyimac P

CtD

it

1

11

n

i

iyi P

CtD

it

1

11

mod

1

yDD mac

1

1mod

macDD mod

25

Bonds/Convexity

• If we hedged the bond’s duration, then what happens to the value of our portfolio when yield moves?

• Convexity– Definition;

– 듀레이션 헤지된 포트폴리오의 손익변화 측정에 사용– See the Taylor Expansion

Pdy

PdC

2

2

)(

26

Bonds/Numerical Convexity

21111 2111

Y

PPP

PY

PP

Y

PP

YP

dY

dP

dY

d

PdY

Pd

PC

112

2

Difference approximation

27

Bonds/Summary_Taylor expansion

• Taylor series expansion; general case

• Taylor series expansion of bond price

• The relationship

– (Modified) Duration: 1st Derivative;– Convexity: 2nd Derivative;

2

2

1yCyD

P

P

322

2

01 )(2

1yOy

dy

Pdy

dy

dPPP

y

2

2

1y

...2

1 2 xxfxxfxfxxf

28

Bonds/Summary_example

• Situation

– -• Duration effect;

• Convexity effect;

• Total effect;

– -

100

1y

100

5.21 E

000,10

5.0

2

1 2 y

D=2.5

C=8.5000,10

25.42 E

75.24525.425021 PEPEE

29

Bond/Finding YTM(1)

• Why numerical method?

• Bisection MethodMARKETPPE

Pfy 1 yfP ???

y

LL Ey ,

HH Ey ,

mm Ey ,

30

Bond/Finding YTM(2)

• Newton Method2 3 4 5 6 7 8 9

10 11 12 13 14 15

A B C D E F G H I J K L find YTM using Newton method =PRICE($B$4,$B$5,$B$6,E7,$B$8,$B$9,$B$10)bond spec Newton method =$B$11-F7settlement 08-07-01 target f= f(y)=p-p(y) =F8/(1+E8)

maturity 11-07-01 =I8*H8rate 5% y p(y) f p/(1+y) duration f' y(i+1)yield 6% unknown 0.0000% 115 -17.673 115 2.869565 330 5.3555% =E8-G8/J8redemption 100 5.3555% 99.03841 -1.71142 94.00406 2.858678 268.7274 5.9923%frequency 1 5.9923% 97.34714 -0.02015 91.84358 2.857363 262.4305 6.0000%basis 0 6.0000% 97.32699 -2.9E-06 91.81792 2.857347 262.3557 6.0000%price 97.32699 knownduration 2.857347 =DURATION($B$4,$B$6,$B$7,E8,$B$10,$B$11)P/(1+y) 91.81791dP/dy 262.3557

yield

yPPyf market

iy 1iy

31

Term structure of Interest rates

32

TS/Terminology & Notation

• Zero rate– Definition: or – Notation:

• Par yield– Definition: such that– Notation:

• (Implied) Forward rate (Forward Rate Agreement)– Notation:– see FRA for pricing →Relationship between spot rate and forward rate

• Discount factor– _

*y

TrP 0MARKET 11 TreP MARKET

nn

i

i

i yFycF *

1

* 11

),( Ttr

),( Tty

),,( 21 TTtf

33

Term structure of interest rates

• Definition– CIR(1985): the term structure of interest rates measures the

relationship among yields on default-free securities that differ only in their maturity

• A mapping of the time to– Price of a default-free zero coupon bond– Yield of a default-free zero coupon bond– Forward rates

• Equivalence of P-Y-F

– -

),( TtP

),( Ttr ),,( 21 TTtf

34

Equivalence of P-Y-F

TtP ,

t

Time Schedule;

T1 T2

tTTtPr

1

,

10

tT

TtPr

1

,ln

122

121

11

,

,,,

TTTtP

TtPTTtf

35

Term Structure of Interest rates

• Dynamics of Term structure of interest rates in the real world;

자료 : www.ksdabond.or.kr

36

Term structure of interest rates

• Which one to model(draw) yield curve?– Condition1: same credit

• [ex] CD(3M)- 산금 (1Y)- 삼성전자 (2Y)- 국고 (3Y)

– Condition2: same cashflow & no arbitrage cond’n• Yield of zero coupon bond;

• Zero price;

• Forward rate;

• [ex]

tsdd ts

0r

0f

37

TS/Issues

• A; Why Term structure has this/that shape?• B; Finding the current term structure of interest rates

– Fitting Yield Curve– To price/trade illiquid bonds

• C; Estimating the future term structure of interest rates– Economics/Econometrics– To trade bonds

• D; Modeling the dynamics of term structure of interest rates– Finance– To price/trade Fixed Income Derivatives

• In summaryt=0 t=T

B CD

A

38

Issue_A;Why yield curves differ?

• Expectation Hypothesis– Expectation on…

• Future short rates Unbiased Expectation Hypothesis

• Return-to-maturity RTM Expectation Hypothesis

• Yield-to-maturity YTM Expectation Hypothesis

• Short term yield Local Expectation Hypothesis

• Term Premium Theories– Liquidity Preference Theory; Hicks(1939)

– Preferred Habitat Theory; Modigliani and Sutch(1966)

– Market Segmentation Theory; Culbertson(1957)

39

Issue_B; Introduction

• What we want to do is…

• A; Bootstrapping– Coupon bond price Zero bond price ZBP

• B; (Statistical) Fitting– Coupon bond price Zero bond priceFunctional form

Functional form

b2

b1

40

Finding Yield Curve

• Bootstrapping and Interpolation– When prices (of interest rate) are quoted on equal-i

nterval-of-maturity base– [ex] Interest Rate Swap Market

• Fitting of yield curves– Piece-wise Spline– Parametric models– Nelson-Seigel function

41

TS/Bootstrapping

Tdddfdtd iiii

;,,,,0 121

42

TS/Bootstrapping

Example

행렬을 이용하는 방법

43

Fitting of yield curvesA brief history of yield curve fitting[4]• David Durand(1942); the pioneering efforts• Piecewise polynomial spline

– McCulloch(1971): Quadratic (polynomial) spline– McCulloch(1975): Cubic spline– Vasicek and Fong(1982): polynomial spline to exponential

transform of maturity

• Functional(parametric) models– Cohen, Kramer and Waugh(1966) and Fischer(1966): some

functional forms via ordinary least squares – Nelson and Siegel(1987): a parsimonious exponential mod

el

44

TS/Functional Forms

• Polynomial Model

• (Simplified) Vasicek & Fong(1982)

• Nelson-Siegel(1987)

55

44

33

2211),( aaaaaTtP

RRRRR eaeaeaeaeaaTtP 55

44

33

2210),(

5

1

1i

ia

tT

eCABAeTtP ),(

2cb

A aB c

C

45

Polynomial Model/Cubic Function

• Yield curve function– Model:

– No Arbitrage condition

• Bond price

• Find coefficients

nni tdctdctdcP 1211

33

2210 ttttd

m

MARKETii PP

1i

2 Min

0

t

d

032 s.t. 2321 tt

10 d

46

Polynomial Model/Cubic Function/Data

자료 : Bloomberg

47

Polynomial Model/Cubic Function

Example2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

A B C D E F G H I J K L M N fitting Yield Curve.xls coefficients

b0 1b1 -0.08896b2 0.04839b3 -0.01544

0.5 0.965686=PRICE($B$10,D14,E14,C14,100,2,0)/100 =cubicF(K13,$J$3,$J$4,$J$5,$J$6)

today 08-09-07 =SUMPRODUCT($I$12:$N$12,I14:N14)

Market Data market model t 0.980542 0.965686 0.953983 0.943987 0.923322 0.89211t yield maturity coupon price price cf 0.25 0.5 0.75 1 1.5 2

0.25 5.32 0.0532 08-12-07 0.0532 0.999825 1.006625 1.02660.5 5.37 0.0537 09-03-08 0.0537 0.999998 0.991615 0 1.02685

0.75 5.41 0.0541 09-06-07 0.0541 0.99991 1.006312 0.02705 0 1.027051 5.45 0.0545 09-09-07 0.0545 1 0.996025 0.02725 1.02725

1.5 5.53 0.0553 10-03-08 0.0553 0.999998 1.001654 0.02765 0.02765 1.027652 5.77 0.0577 10-09-07 0.0577 1 0.99958 0.02885 0.02885 0.02885 1.02885

Function cubicF(t, b0, b1, b2, b3)cubicF = b0 + b1 * t + b2 * t ^ 2 + b3 * t ^ 3End Function

21 22 23 24 25 26 27 28 29 30 31

E F G H I J K L M N t_function

4.62E-05 =(F14-G14)^27.03E-05

4.1E-051.58E-052.74E-061.77E-07

sum 0.000176 =SUM(F22:F27)

48

Spline: Piece-wise polynomial

• More freedomMore accurate one

• Bond price;• Spline Condition;

• Find coefficients;

31,3

21,21,11,01 ttttd

32,3

22,22,12,02 ttttd

3,3

2,2,1,0 ttttd iiiii

m

MARKETii PP

1i

2 Min

T

Zero price

n

jjij

m

jjj

l

jjji tdctdctdcP

112

11

49

Polynomial Model/Nelson-Siegel

Instantaneous forward rate discount factor

50

Nelson-Siegel

• From instantaneous forward rates to discount factors; Relationship forward rates & DF

nn tPdtP 1

111

ttfn

netP

12 nn tPdtP

122

n

ttfn tPetP n

1122

ttftfn

nnetP

1

1

02

ttf

n

n

ii

etP

0t

nt

tdttf

n etP 0)(

2

0t 2nt 1nt nt

51

note

52

Understanding NS Curve

• Factor1; level

• Factor2; slope

• Factor3; curvature

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

A B C D E F G H I J alpha 0.5weight 1 1 1term factor1 factor2 factor3 sum

0.25 1 0.882497 0.220624 2.1031210.5 1 0.778801 0.3894 2.168201

0.75 1 0.687289 0.515467 2.2027561 1 0.606531 0.606531 2.213061

1.25 1 0.535261 0.669077 2.2043381.5 1 0.472367 0.70855 2.180916

1.75 1 0.416862 0.729509 2.1463712 1 0.367879 0.735759 2.103638

2.25 1 0.324652 0.730468 2.0551212.5 1 0.286505 0.716262 2.002767

2.75 1 0.25284 0.695309 1.9481483 1 0.22313 0.66939 1.892521

3.25 1 0.196912 0.639963 1.8368753.5 1 0.173774 0.608209 1.781983

3.75 1 0.153355 0.575081 1.728436

0

0.2

0.4

0.6

0.8

1

1.2

0.25 1

1.75 2.5

3.25 4

4.75 5.5

6.25 7

7.75

factor1

factor2

factor3

uu cuebeauf ,0

a

b

c

53

Basic Interest Rate Derivatives (BIRD)

54

Roadmap_Basic Interest Rate Derivatives

Theory on Forward;Forward on a Stock

Forward on a bond

Forward on an Interest Rate

KTB Futures

Forward Rate Agreement

Interest Rate Swap Cap/Floor

SwaptionTheory on Option;Black Formula

55

Bond Futures/ 주식선도

56

Bond Futures/ 채권 선도이표가 없는 경우

이표가 1회 있는 경우

57

Bond Futures/ 이자율 선도 (FRA)

• FRA Formula; zero rate version

58

FRA_cont.

• FRA Formula; discount factor version– RULE: Any investment with same (future) payoff should give same (pr

esent) price

– Formula;

• 3×6(3 by 6) FRA– Interest Rate for Contracting today, Investing on 3 months and Returni

ng on 6 months– Notation; – cf. “1y into 3y swaption”

1

1,,0

T

t

d

d

tTTtf

moth6,moth3,0f

t

T

0

거래시점 Fixing 시점 결제시점

59

Interest Rate Swap

60

Interest Rate Swap

• Terminology– Notional Amount

– Fixed Rate(Amount, Leg etc)

– Floating Rate(Amount, Leg etc)

– Payment Date

– Fixing Date

1st Fixing date(Reset Date)

Effective date Payment date

2nd Fixing date

CalculationPeriod Maturity date

Fixed Rate

Floating Rate

Fixed RatePayer

Floating RatePayer

Source: 2000 ISDA Definitions

61

IRS_confirmation

62

IRS Market

Source: www.kdb.co.kr

• Meaning of number– ‘kdb’ will receive(offer, sell) 3.92% for 3 years– ‘counterparty’ will pay(bid, buy) floating rate for 3 years

• Meaning of code(?)– Quarterly payment– Day count convention; A/365

63

IRS Pricing/Valuation

• Basic Principle;– When trading new deal, PL from this deal should b

e ‘fair(zero)’– For IRS, PV(payment) = PV(receiving)

[Note] Pricing (vs) ValuationPricing

[1]Find “Fixed Rate” or “Spread over Floating Rate” to make IRS’s PV zero[2]‘Give me swap price.’ ‘It’s 4.5%’

Valuation[1]Calculating PV of the given IRS[2]‘Evaluation the swap.’ ‘It’s 1,000,000(won).’

n

iii

n

ii dfdR

11

011

n

iii

n

ii dfdR or

TRfdfdRpn

iii

n

ii

;

11IRS

64

Meaning of the equation

n

ii

n

iii

d

dfR

1

1

n

iii

n

ii dfdR

11

n

iii fR

1

n

n

iiin

n

ii ddfddR

11

n

n

ii ddR

1

1

]pv[Annuity

leg floatingpv cashflow] pv[fixed

leg floatingpv

(Weighted) Average of IFR= STRIP (Portfolio) of FRA

Portfolio of StraightBond & FRN

Swap rate is ‘Par yield’

[Note] Review on Annuity & FRN

65

• pv[Annuity]– Derive formula for pv of annuity using geometric s

eries– -

• pv[FRN]

[Note] Review on Annuity & FRN

n

ii

mr

n

iidA

11 1

11

FRN

n

n

iii ddfp

1

11

i

ii d

df

n

nn

n

iii

n

ii

i

i

n

ii

i

i

d

dddddd

dddd

d

dd

dp

1

1

11

12110

11

1

1

1

1Coupon

pv[coupon]

n

iiidfp

1Coupon

66

Review on Bootstrapping;More considerations

n

n

iin

n R

dRd

1

11

1

Available prices Bootstrapping formula

1d 2d 3d 4d

Quarterly Grid point

Annual market quotes

CD=4.74

%4.41

%4.41 3214

dddd

365911 %74.41

1

d

2

122 1

1

R

dRd

3

2133 1

1

R

ddRd

Bootstrapping

Informationwe need

Informationwe have

[ ]

[ ]

67

Review on Bootstrapping;summary

68

Risk Measure• When we see risk as ‘YTM’

– BVP• BPV tells you how much money your position will gain or lose for 0.01% parallel m

ovement in the yield curve[2].– PVBP

• This measures, also called DV01, is the absolute value of the change in the price of a bond for a 1basis point change in yield[3]

– Dollar Duration• PVBP is equal to the effective dollar duration for a one-basis-point change in rates

([3]p.1260)• Dollar duration of swap([3]p.1306) =

Dollar duration of a fixed-rate bond −Dollar duration of a floating-rate bond– DV01=PVBP

• When we see risk as “FRA”– Delta0; FRA 에 대한 현금흐름의 변화가 스왑의 가치에 주는 효과– Delta1; FRA 에 대한 할인계수의 변화가 스왑의 가치에 주는 효과– Delta=Delta0+Delta1

69

Risk Measure_cont.

n

i

n

iiii

n

iiiiIRS pcdKfdP

1 11

Change in FRA

Change in Cashflow

Change in Discount Factor

Change in the Value of IRS

DELTA0

DELTA10 1 2 n

d1,f1,α1

iiiii fcfdp

i

j jji f

d1 1

1

ijj

j

i

i dff

d

1

ii

iii

ii

i cf

ddf

cf

p

곱함수의 미분

DELTA1 DELTA00

1

1

1

ii

iii

iiiii

ii

f

Kd

ddf

c

Properties (we assume the “pay-fix” swap)[1] Delta0 is always positive, but Delta1 depends on pi

[2] Delta is always positive. This implies the effect of Delta0 is dominant.

DE

LT

A

70

Interest Rate Options

71

Black formula

• Black Formula– The Black formula can be used to price European

options when the underlying security is a forward or futures contract with initial price F [5]

21 dKNdFNec rT

T

TKFd

2ln 2

1

Tdd 12

12 dFNdKNep rT

KFEec QrT

F

K

The payoff

72

Caplet/Floorlet

• Terminology & Notation

• Caplet: call option on FRA

Expiry date(fixing date)

Effective date

Payment Date

Trade date

FRA(CD)

PL

K

73

Caplet pricing

• Caplet pricing formula– If we assume that FRA is log-normally distributed

on expiry date, following Black Formula gives the price of the caplet

Expiry date(fixing date)

Effective date

Payment DateTrade date

00 t T2t1t

74

Caplet pricing

21

~~

1dKNdNFe

FNc rT

12 tt

T

Td K

F

2

~

1

2

ln

Tdd 12

KFF

N

1

1payoff• The payoff;

• Pricing formula

KFE

FNeKF

FNEec QrTQrT

1

1

1

1

75

Cap pricing

• Caplet: option on FRA

• Cap: a series of Caplets– Different expiries– Same strike and tenor

– Pricing Formula;

Expiry date(fixing date)

Effective date

Payment Date

Trade date

Trade date

TenorTermAccrual Period

n

iiTpp

1CAPLETCAP

76

Swaption

• Option on swap;– If we exercise the option on expiry, we can enter a new IRS

pre-defined.• Payer swaption

– Pre-defined IRS is that the option owner will pay the fixed rate to the option writer.

– Call on swap rate• Receiver swaption

– Pre-defined IRS is that the option owner will receive the fixed rate from the option writer.

– Put on swap rate• The fixed rate is “strike”

77

Swaption; pricing

• Call on swap rate

• Put on “swap value”• Swaption pricing formula

– If we assume that forward IRS rate is log-normally distributed on expiry date, following Black Formula gives the price of the Swaption

PL

R

KRAKR

KR

n

ii

mR

n

ii

mR

1

1

1

1

1

1payoff

K

78

Swapton; pricing_cont.

• Meaning of “call on swap rate” and “put on swap value”– There exist a map from R to PV; and– The mapping is “reciprocal”– So, if we draw the payoff in terms of PV, it’s “leftside right”

PL

K R

PV(IRS)

PV(IRS)

PL

Analogy; Option on bondIf we have a option to buy abond one month later, of whichcoupon is 5%, it is totally sameto say “I have a option to buy thebond at 5%” and “I have a optionto buy the bond at par.But the payoff graph is different!

79

Swaption; pricing_cont.

• The payoff, again;

• Swaption price;

n

ii

mR

KR1 1

1payoff

RA nmR

1

11

T

Td K

R

2

1

2

ln

Tdd 12

KREeKREep Q

n

ii

mR

rTn

ii

mR

QrT

11 1

1

1

1

21 dKNdRNAep rT

Geometric Series[1]Initial value:

[2]Common Ratio:

[3]# of terms:

mR11

mR11

n

80

Markets

• Actually, Two instruments’ prices are quoted on one panel– Cap/Floor (upper table)– Swaption (lower table)

• Meaning of numbers– A: 2year ATM vol(volatility) for Cap/Floor– B: 1y into 3y ATM vol. for swaption

81

Application

• Put-call parity– Buying a call option and selling a put option with same

maturity and strike, gives the same payoff as the forward contract.

– Draw the payoff on expiry– We can hedge a option with forward!

• Cap-floor parity– “Buying a cap and selling a floor” with same maturity and

strike, gives the same payoff as the IRS– Draw a payoff of one of any fixing date!– We can hedge a cap/floor with IRS!

82

Application_cont.

• For the interbank-option deal, Market Convention is “the Delta Exchange”– [ex] Delta Exchange for FX option

• If we buy 10mio ATM Call option,• Delta Exchange is selling “5mio forward”

– In this case, Market Convention means• Quoted prices are for the deal that includes the delta exchange• If we don’t want Delta Exchange, the price should go unfavorable to us

• Delta Exchange for BIRD– We’ve just done a ATM cap with N=100mio, T=2y, then what is the D

elta Exchange for this deal?– We’ve just done a ATM ATM 2y into 3y payer swaption with N=100

mio, the what is the Delta Exchange for this deal?

83

References

• [1] 박건엽 , 엑셀 VBA 를 이용한 금융공학실습 , 서울경제경영

• [2]http://www.barbicanconsulting.co.uk/quickguides/bpv

• [3]Fabozzi, the Handbook of Fixed Income Securities(6/e), McGraw-Hill

• [4] C.Nelson, A.Siegel, Parsimonious modeling of yield curves, J. Business(1987)

• [5] E. Haug, the Complete Guide to Option Pricing Formulas, McGraw-Hill(1997)

84

Thank you!

[email protected]