Unformatted text preview: Advanced Fixed Income Analytics Backus & Zin April 1, 1999 Options 1: The Term Structure of Volatility
1. Approaches to valuation review and extension
2. The BlackScholes formula history and meaning
3. Option sensitivities
4. Term structure of volatility
5. Extended Vasicek
6. Options on zeros
7. Options on coupon bonds
8. Options on eurocurrency futures
9. Normals and lognormals
10. Summary and nal thoughts Lecture 2 Advanced Fixed Income Analytics 22 1. The Importance of BlackScholes
Elegant solution to option price once you get used to it
A purely academic innovation
Now the industry standard in most markets
Played critical role in growth of option markets
Together with related work by Samuelson and Merton, triggered
an explosion of work that laid the basis for modern nance in
universities and on Wall Street
New approach to valuation: replication, a precursor to the
kinds of arbitragefree models we are using
Constant interest rate: fortunately not essential or applications
to xed income would have died out rapidly Advanced Fixed Income Analytics
2. Two Approaches to Valuation Problem: value random future cash ows ct+n
Recursive valuation one period at a time:
In period t + n , 1, value is
pt+n,1 = Et+n,1 mt+n ct+n
In period t + n , 2, value is
pt+n,2 = Et+n,2 mt+n,1 pt+n,1
Etc.
In period t, value is
pt = Et mt+1 pt+1
Allatonce" valuation:
Apply multiperiod stochastic discount factor" Mt;t+n:
pt = Et Mt;t+nct+n
Same answer if Mt;t+n = mt+1mt+2 mt+n
Twoperiod zero: recursive approach
pt = Et mt+1pt+1
= Et mt+1Et+1 mt+2 1
= Et mt+1 mt+2 1 :
Twoperiod zero: allatonce" approach
pt = Et Mt;t+2 1 : 23 Advanced Fixed Income Analytics 24 3. Option Basics
A European call option is the right to buy an asset at a xed
price K at a xed date n periods in the future
S is the price of the underlying
K is the strike price
n is the maturity of the option
a call generates cash ow at t + n of
ct+n = St+n , K +
where x+ means maxx; 0 the positive part of x
The price C of a call satis es
h
i
Ctn = Et Mt;t+n St+n , K + :
A put option is a comparable right to sell an asset at price K :
ct+n = K , St+n+
A forward contract arranged at date t speci es the purchase at
t + n of an asset for price Ftn the forward price, xed at t
Prices of puts P and calls C are related by putcall parity :
Ctn , Ptn = bnFtn , K
t
This allows us to concentrate on calls, knowing we can convert
any answer we get to puts. Advanced Fixed Income Analytics 25 4. The BlackScholes Formula
Assume: log Mt;t+n; log St+n is bivariate normal
we say Mt;t+n; St+n is lognormal
Then BlackScholes formula:
Ctn = bnFtn d , bnK d , w
t
t bn
t
n
Ft
w2
d =
=
=
=
= nperiod discount factor
forward price of the underlying
cumulative normal distribution function
Var t log St+n
logFtn =K + w2=2
w Comments:
a little di erent from the original, but more useful for our
purposes
industry standard in most markets
similar formulas were derived by Merton dividendpaying
stocks, Black futures, and BigerHull and
GarmanKohlhagen currencies
one of the best parts: the only component of the formula
that's not observed in cash markets is volatility
we can turn this around: given call price, nd volatility
describes implicitly how changes in F; W; b a ect C Advanced Fixed Income Analytics 26 4. The BlackScholes Formula continued
Sample inputs streamlined notation:
F = 95:00
K = 97:50 slightly out of the money
maturity = 1 year
b = exp,:0500 = 0:9512
w = 17:30
Intermediate calculations:
d = ,0:0636
d = 0:4746
d , w = ,0:2366
d , w = 0:4065
Call price:
C = b F d , K d , w
= 0:9512 950:4746 , 97:50:4065
= 5:193
Put price for same strike use putcall parity:
P = C , bF , K
= 5:193 , 0:951295 , 97:5
= 7:571
we most often use this in reverse: convert a put to a call and
use the BlackScholes call formula. Advanced Fixed Income Analytics 27 5. BlackScholes: How We Got There
Step 1: Integral formulas
Setting: let x1 ; x2 = log X1; log X2 be bivariate normal with
mean and variance :
2 3
2 2
3
1 5 ; = 4 1 12 5
= 4
2
2
12 2
Think of x1 ; x2 = log Mt;t+n; log St+n.
Formula 1:
Z1
E X1 = 2 1,1 ,1 ex e,x , =2 dx1
2
= exp1 + 1 =2
1 1 1 2 2
1 Formula 2:
Z Z
,1 jj,1=2 1 1 ex e ,x,
E X1jX2 K = 2
,1 log K
= E X1d;
with
d = 2 , log K + 12
1 ,1 x,=2 dx2dx1 2 Formula 3:
Z Z
,1 jj,1=2 1 1 ex +x e ,x,
E X1X2jX2 K = 2
,1 log K
= E X1X2d;
with
2
2 , log K + 12 + 2
d=
1 2 2 ,1 x,=2 dx2dx1 Advanced Fixed Income Analytics 28 5. BlackScholes: How We Got There continued Step 2: BlackScholes
Call prices satisfy
Ctn = Et Mt;t+nSt+njSt+n K , KEt Mt;t+njSt+n K
A forward contract arranged at date t speci es the purchase at
t + n of an asset for a price Ftn set at t
Forward price Ftn satis es
0 = Et Mt;t+n St+n , Ftn
The usual pricing relation with current payment of zero.
Solution includes use Formula 1:
2
2
bnFtn = Et Mt;t+nSt+n = exp1 + 2 + 1 + 2 + 2 12 =2
t
2
bn = Et Mt;t+n = exp1 + 1 =2
t
2
Ftn = Et Mt;t+nSt+n =Et Mt;t+n = exp2 + 12 + 2 =2:
Term 1 use Formula 3:
Et Mt;t+nSt+njSt+n K = bnFtn d
t
with
n
2
2
d = 2 , log K + 12 + 2 = logFt =K + 2 =2 :
2 2 Term 2 use Formula 2:
Et Mt;t+njSt+n K K = bnK d0 ;
t
with
n
2
0 = 2 , log K + 12 = logFt =K , 2 =2 = d ,
d
2 2 2 Advanced Fixed Income Analytics 29 5. BlackScholes: How We Got There continued
Comments
Clever substitution:
volatility w is the only thing not observable directly
from other markets
the features of M matter only via their e ect on b and F
Role of lognormality:
assumption is that log of underlying  whatever that
might be is normal
prices of many assets exhibit more frequent large
changes than this implies kurtosis, skewness
Is model appropriate?
it's a good starting point
and the industry standard in most markets
a useful benchmark even when not strictly appropriate
mathematically Advanced Fixed Income Analytics 210 6. Implied Volatility
Annual units
it's standard practice to measure maturity in years
if periods are h years, then maturity is N = hn
volatility v below in annual units is:
Nv2 = w2 v = w=N 1=2 v is often multiplied by 100 and reported as a percentage
BlackScholes formula becomes and this will be our standard:
CtN = bN FtN d , bN K d , N 1=2v
t
t
N
2
d = logFt =K=2+ Nv =2
N1 v Implied volatility: given a call price, nd the value of v that
satis es the formula
since the price is increasing in v, we can generally nd a
solution numerically
limits: the minimum call price is bN FtN , K + v = 0, the
t
N F N for v = 1
maximum is bt t
example continued: if call price is 6.00, v = 19:54 Advanced Fixed Income Analytics 211 6. Implied Volatility continued
Numerical methods bene t from good rst guess
If solution fails, check limits
BrennerSubrahmanyam approximations for options
atthemoney forward F = K :
1=2
N 1=2v = 2bF C
N 1=2v bF
C = 21=2
Example continued
Let K = 95 atthemoney forward
With v = 17:30 same as before, call price is C = 6:229
BrennerSubrahmanyam approximation:
2 1=2
v = 0: 6:229
951295
= 17:28!
Comment: this is more accuracy than is justi ed by the
underlying data, but it illustrates the quality of the
approximation. Advanced Fixed Income Analytics
7. Sensitivities 212 E ect of small changes in F :
@C
delta" = @F
= bd
inthemoney options more sensitive than outofthemoney
E ect of small changes in v:
vega" = @C
@v
= bF 0 dN 1=2
more sensitive at d = 0  roughly at the money
E ect of small changes in r use b = e,rN :
rho" = @C
@r
= ,NC
Comments:
standard approach to quantifying risk in options
based on BlackScholes: other models imply di erent
sensitivities
volatility critical: dealers tend to be short options
customers buy puts and calls, so they are typically short
volatility" even if they are delta hedged"
xed income poses basic problem with the approach: hard
to disentangle say delta and rho , since both re ect
changes in interest rates
need to tie all of the components to underlying
variables  a z , so to speak Advanced Fixed Income Analytics 213 8. Volatility Term Structures
In the theoretical environment that led to BlackScholes,
volatility v is the same for options of di erent maturities and
strikes
In xed income markets, volatility v varies across several
dimensions:
maturity of the option N
tenor" of underlying asset
strike price K
Volatility term structures for March 19, 1999
interest rate caps:
Maturity yrs Volatility
1
12.50
2
15.50
3
16.55
5
17.25
7
17.15
10
16.55
swaption volatilities :
Maturity
1 mo
6 mos
1 yr
5 yrs 1 yrs
10.00
13.00
15.65
16.40 2 yrs
11.75
14.00
15.55
15.85 Tenor
5 yrs 10 yrs
13.88 13.88
14.75 14.60
15.20 14.90
15.10 14.20 20 yrs
11.30
11.80
11.90
10.55 Advanced Fixed Income Analytics 214 9. Extended Vasicek Model
Model generates BlackScholes option prices for zeros
Extended Vasicek model translated from Hull and White:
model consists of
, log mt+1 = 2=2 + zt + "t+1
zt+1 = 1 , 't + 'zt + "t+1
bond prices satisfy
log bn = ,Ant , Bnzt
t
Bn = 1 + ' + '2 + + 'n,1
details:
ts structure of BlackScholes formula:
log Mt;t+n; log bt+n are bivariate normal for all
maturities n and tenors
extended" refers to the t" subscripts on and An
allows us to reproduce current spot rates exactly
Forward prices:
a forward contract locks in a price F on an n + period
zero in n periods
price now bn times F must equal current price:
bnF = bn+
for completeness, we should probably write F as F n;
Key issue: what are the volatilities? Advanced Fixed Income Analytics 215 10. Volatility Term Structures in Vasicek How does volatility vary in theory! with maturity and tenor?
Volatility of bond prices:
log bt+n = ,A , B zt+n
Var t log bt+n = B 2Var tzt+n
E ect of maturity:
Var tzt+n =
= 2 1 + '2 + '4 + + '2n,1 0 2n 1
A
1 , '2 2 @1 , ' If 0 ' 1, this grows more slowly than n
E ect of tenor: how B varies with
Annualized perperiod volatility is
0 10
1
Var tzt+n = B 2 @ 2 A @ 1 , '2n A
vn; = B 2 hn
hn 1 , '2
Comments:
note that volatility has two dimensions: it's a matrix, like
the swaption volatility matrix we'll see shortly
e ects of mean reversion apparent through
maturity: v declines with n if 0 ' 1
constant if ' = 1
tenor: v increases with , but the rate of increase is
greater if ' is closer to one
model required  not optional  for B ; Var tzt+n Advanced Fixed Income Analytics 216 10. Volatility Term Structures in Vasicek continued
Volatility term structures for 6month and 5year zeros
5
4.5 Volatility (Annual Percentage) 4
3.5
5−year zeros 3
2.5
2
1.5 6−month zeros 1
0.5
0
0 20 40
60
80
Maturity of Option in Months Less mean reversion ' closer to one . . .
reduces rate of decline with maturity
increases volatility of long zeros 100 120 Advanced Fixed Income Analytics 217 11. Options on Coupon Bonds
A coupon bond has known cash ows on xed dates
label set of dates by J = fn1; n2; n3; : : :g
eg, with a monthly time interval and semiannual payments,
a 5year bond would have payments at
J = f6; 12; 18; 24; 30; 36; 42; 48; 54; 60g
label cash ows by cj for each j 2 J
Bond price is pt = X
j 2J cj bn
t j If prices of zeros are lognormal, price of a coupon bond is not
log of a sum not equal to sum of logs
BlackScholes not strictly appropriate
Nevertheless: treat BlackScholes as
a convenient reporting tool eg, its implied volatility
an approximation to the exact formula Advanced Fixed Income Analytics 218 12. Options on Eurodollar Futures
Money market futures:
similar contracts on eurodollars CME, short sterling
LIFFE, euros LIFFE, euroyen TIFFE, etc
maturities out 1015 years
price F related to yield" Y by F = 100 , Y
settles at Y = 3m LIBOR
Call options on these contracts are
claims to future cash ow 90
F , K + 360 equivalent to put options on the yield
90
90
100 , Y , K + 360 = 100 , K , Y + 360
90
= KY , Y + 360
Similarly: put options are equivalent to calls on the yield
90
90
K , F + 360 = 100 , K , 100 , Y + 360
90
= Y , KY + 360 Advanced Fixed Income Analytics 219 12. Options on Eurodollar Futures continued
Option quotes on Mar 16 for options on the June contract:
F = 94:955
N = 0:25 3 months
b = 0:9874 roughly 5 for 3 months
call price mid of bad ask for K = 95:00: C = 0:0425
put price mid of bad ask for K = 95:00: P = 0:0925
Yieldbased implied volatilities streamlined notation:
C = b100 , F d , b100 , K d , N 1=2v
log 100 , F =100 , K + Nv2=2
d =
N 1=2v
This is more work than nding volatility for the futures
directly, but connects futures to related OTC markets for rates.
put is equivalent to a call on the yield:
volatility solution to formula is v = 0:0687
call is equivalent to a put on the yield:
putcall parity gives comparable call price as
CY = PY + b 100 , F , 100 , K
= 0:0425 + :98740:045
= 0:0869
volatility is v = 0:0629
Comment: do this slowly and carefully! Advanced Fixed Income Analytics 220 12. Options on Eurodollar Futures continued
Current term structure of yield volatilities:
Contract Maturity Futures Price
Jun 99
0.25
94.955
Sep 99
0.50
94.860
Dec 99
0.75
94.550
Mar 00
1.00
94.640
Jun 00
1.25
94.550
Sep 00
1.50
94.475 Strike Put Price Volatility
95.00 0.0925
0.0687
94.75 0.1075
0.1087
94.50 0.2500
0.1502
94.75 0.3650
0.1541
94.50 0.3500
0.1628
94.50 0.5700
0.1766 Strikes are closest to atthemoney. Discount factors are
bN = e,N r , with r = 0:05.
Comments:
contracts are for xed tenor but di erent maturities
good source of yield volatilities standard model input
suggests that markets are more complex than Vasicek
standard pattern shows declines in volatility at long
maturities, like Vasicek, but increases at short end Advanced Fixed Income Analytics 221 13. On Normals and LogNormals We have made two di erent assumptions about interest rates
rates are normal: built into Vasicek logs of prices are
normal, interest rates are approx normal
rates are lognormal: our analysis of eurodollar futures is
based on lognormal yields; caps and swaptions are
generally treated the same way 0.25 Probability Density Function 0.2 log−normal 0.15 0.1 normal 0.05 0
0 5
10
3−Month LIBOR in 12 Months 15 This apparently technical issue can have a large e ect on option
prices, especially for options far out of the money Advanced Fixed Income Analytics 222 Summary
1. The BlackScholes formula is both a theoretical benchmark for
option pricing and an industry standard for quoting prices.
2. BlackScholes can be viewed as a theoretical framework for
thinking about options a model, a convention for quoting
prices a formula, or both.
3. In xed income markets, volatility varies across both the
maturity of the option and the tenor" of the underlying
instrument.
4. In Vasicek, the relation between volatility across maturities and
tenors is controlled largely by the mean reversion parameter.
5. Market prices suggest two issues that deserve further attention:
the di erence between normal and lognormal interest rates
the shape of the volatility term structure ...
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 Spring '15
 Backus
 Normal Distribution, Volatility, Mathematical finance

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