1 fixed income math and risk measure/basic ird 2008.12
TRANSCRIPT
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
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
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
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
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
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
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
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
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
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
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
[ ]
[ ]
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
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)