Unformatted text preview: Advanced Fixed Income Analytics Backus & Zin April 1, 1999 Vasicek: The Fixed Income Benchmark
1. Prospectus
2. Models and their uses
3. Spot rates and their properties
4. Fundamental theorem of arbitragefree pricing
5. The Vasicek model:
Solution and properties
Parameter values calibration"
Hedging: Vasicek v. Duration
Where are the bodies buried?
6. Summary and nal thoughts Lecture 1 Advanced Fixed Income Analytics 12 1. Prospectus
Regard this course as an experiment: no one has ever taught
such advanced material at this level
What makes this possible?
unique approach minimizes technical demands
custom software does much of the work
steady pace: we refuse to get bogged down in details
great teaching!
Our approach features
discrete time much simpler than stochastic calculus
continuous states more accurate than trees"
emphasis on numbers a good model with bad parameters is
a bad model
handson experience work in progress, you be the judge
Software options:
Excel: the old reliable, but awkward for complex problems
Matlab: more e ort initially, but much more powerful
Our guarantees:
you won't understand everything, but you'll learn a lot
an honest attempt gets at least a B Advanced Fixed Income Analytics 13 2. Uses of Models
Valuation: What's this option worth?
Given prices of underlying assets, estimate the value of
related derivatives.
Hedging: How do I o set the risk in these DM swaptions?
Find combinations of assets that have relatively little
risk.
Replication: How can I synthetically reproduce the cash ows of
an American swaption?
Construct combinations of simple assets to approximate
the returns of a more complex asset.
Risk management: How bad can things get?
Estimate the statistical distribution of the oneday
return of a portfolio of assets.
Summary and assessment:
Di erent objectives demand di erent models. If we
thought we knew everything, the perfect model could
be used for all four purposes. Instead, models are
designed with strengths that suit their uses. They
invariably exhibit weaknesses in other uses. Thus
complex statistical models are helpful in quantifying
risk, but generally ignore the pricing of risk. Our
interest lies, instead, in simpler models in which the
pricing of risk is explicit. Advanced Fixed Income Analytics
3. Spot Rates and Other De nitions
The discount factor bn is the price of an n period zero: the
t
value at t of one dollar payable n periods in the future.
The spot rate ytn is related to the discount factor bn by
t
n
bn = e,nyt
t
n
yt = ,n, log bn
t
This is based on continuous compounding, an arbitrary
convention that helps us down the road.
1 The short rate is rt = yt .
1 The time interval is h years. For periods of one month,
h = 1=12 years.
To express spot rates as annual percentage rates:
Annual Percentage Rate = ytn 100=h
For monthly data, we multiply by 1200. 14 Advanced Fixed Income Analytics 15 4. Statistics Review
Data: we have observations yt for t = 1; : : :; T .
The sample mean is PT y =
The sample variance is y t=1 t T yt , y
T
s is the standard deviation. Some people divide by T , 1, an
minor issue we'll ignore. s =
2 PT 2 t=1 The sample autocorrelation an indicator of dynamics
PT
t yt , y yt, , y
Corr yt ; yt, =
PT
t yt , y
Values close to one indicate that changes tend to last 
they're persistent.
1 =2 1 =1 2 Higher moments for future reference only:
PT
j for j 2
t yt , y
mj m T
skewness = m =
m
kurtosis = m , 3
=1 3 2 3 2 4 2 2 Skewness measures asymmetry. Kurtosis summarizes the
probability of extreme events. Both are zero for normal random
variables. Advanced Fixed Income Analytics 16 5. Properties of Spot Rates
US Treasury spot rates, 197095, monthly data
Maturity Mean St Dev Skewness Kurtosis Autocorr 1 month
3 months
6 months
9 months
12 months
24 months
36 months
48 months
60 months
84 months
120 months 6.683
7.039
7.297
7.441
7.544
7.819
8.008
8.148
8.253
8.398
8.529 2.699
2.776
2.769
2.721
2.667
2.491
2.370
2.284
2.221
2.138
2.069 1.073
1.007
0.957
0.924
0.903
0.887
0.913
0.944
0.971
1.008
1.035 1.256
0.974
0.821
0.733
0.669
0.484
0.378
0.321
0.288
0.251
0.217 Note
Means: increase with maturity picture coming
Standard deviations
Autocorrelations: close to one very persistent! 0.959
0.971
0.971
0.970
0.970
0.973
0.976
0.977
0.978
0.979
0.981 Advanced Fixed Income Analytics 17 5. Properties of Spot Rates continued
Spreads Over Short Rate: ytn , yt
1 Maturity Mean St Dev Skewness Kurtosis Autocorr 3 months
6 months
9 months
12 months
24 months
36 months
48 months
60 months
84 months
120 months 0.357
0.614
0.758
0.861
1.136
1.325
1.465
1.570
1.715
1.846 Skip for now 0.350
0.507
0.601
0.683
0.929
1.095
1.213
1.300
1.420
1.526 1.811
1.325
1.117
0.863
0.017
0.424
0.599
0.677
0.728
0.735 5.642
4.538
4.647
4.177
2.197
1.368
0.983
0.765
0.532
0.385 0.403
0.527
0.618
0.680
0.792
0.833
0.855
0.868
0.882
0.892 Advanced Fixed Income Analytics 18 5. Properties of Spot Rates continued
OneMonth Changes: ytn , ytn,
1 Maturity Mean St Dev Skewness Kurtosis Autocorr 1 month
3 months
6 months
9 months
12 months
24 months
36 months
48 months
60 months
84 months
120 months 0.009
0.010
0.010
0.009
0.009
0.009
0.009
0.009
0.008
0.008
0.007 Skip for now 0.764
0.662
0.656
0.648
0.634
0.558
0.499
0.461
0.434
0.401
0.374 1.043
1.747
1.352
1.088
0.926
0.583
0.424
0.345
0.300
0.256
0.220 9.939
11.158
10.691
10.514
10.251
7.902
5.829
4.598
3.845
3.000
2.337 0.003
0.146
0.150
0.153
0.154
0.157
0.157
0.152
0.143
0.120
0.094 Advanced Fixed Income Analytics 19 6. The Fundamental Theorem
An arbitrage opportunity is a combination of long and short
positions that costs nothing but has a positive return a free
lunch, in other words.
We say prices are arbitragefree if there are no arbitrage
opportunities. Statement of faith in markets, good
approximation but not perfect.
Notation: let Rt be the gross return between dates t and t + 1.
For example, the return on an nperiod zero is Rt = bn, =bn.
t
t
+1 +1 1
+1 Theorem. In any arbitragefree setting, there is a positive random variable m a pricing kernel satisfying 1 = Et mt Rjt
for all assets j . Et means the expectation as of time t when the
trade is done. Ignore the t's for now, they're a distraction.
+1 +1 Risk premiums are governed by covariance with m:
Let Rt = 1=bt be the riskfree oneperiod return
Excess returns are xjt = Rjt , Rt .
Risk premiums are
j j Cov t mt ; xt
Et xt = , E m
t t
Comment: this is a lot like the CAPM, with m playing the
role of the market return.
1
+1 1 +1 +1 1
+1 +1 +1 Needed: a good m +1 +1 Advanced Fixed Income Analytics 110 7. Vasicek: The Model
Goal: a useful description of the behavior of m
The archetype: the Vasicek model
, log mt = =2 + zt + "t
zt = 1 , ' + 'zt + "t
where " is normal with mean zero and variance one and
0 ' 1. The twopart structure is typical.
+1 2 +1 +1 +1 The pricing kernel m controls risk:
log guarantees positive m if m is positive log m can be
anything, and vice versa
If = 0, m discounts at rate z :
mt = e,zt
+1 controls risk recall: risk premiums are proportional to
covariance with m
=2 is largely irrelevant you'll see
2 Advanced Fixed Income Analytics 111 7. Vasicek: The Model continued
The state variable" z will turn out to be the short rate. Its
properties include:
Over one period, the conditional mean and variance are:
Etzt = 1 , ' + 'zt
Var t zt
=
Stop if this isn't clear.
Over two periods, z looks like this:
zt = 1 , ' + 'zt + "t
= 1 , ' + ' zt + "t + '"t
The conditional mean and variance are:
Etzt = 1 , ' + ' zt
Var t zt
= 1 + '
Over n periods:
n,
X j
n
n
zt n = 1 , ' + ' zt +
' "t n,j
+1 2 +1 +2 +1 2 +2 2 +2 2 +2 2 +2 +1 2 2 1 + + j =0 The conditional mean and variance are:
Etzt n = 1 , 'n + 'nzt
Var t zt n =
1 + ' + + ' n,
As n grows, the mean and variance approach
Ez = Var z =
1 + ' + ' + = 1 , '
We refer to these as the unconditional mean and variance,
the theoretical analogs of the mean and variance we might
estimate from a sample time series of z .
+ 2 + 2 2 2 4 2 1 2 2 Advanced Fixed Income Analytics 112 8. Vasicek: The Solution
Discount factors are loglinear functions of z :
, log bn = An + Bn zt
t
This relatively simple structure is one of the most useful
features of the model. All we need to do is nd the coe cients
An; Bn.
The fundamental theorem gives us
An = An + =2 + Bn 1 , ' , + Bn =2
= An + Bn1 , ' , Bn , Bn =2
Bn = 1 + Bn'
2 +1 2 2 +1 For future reference, note that
Bn = 1 + ' + + 'n, = 1 , 'n=1 , '
1 Starting point: bt = 1 a dollar today costs a dollar, so
log bt = ,A , B zt = 0 A = B = 0
0 0 0 0 0 0 These formulas are easily computed in a spreadsheet: starting
with A = B = 0, we compute An ; Bn from An; Bn.
0 0 +1 +1 Needed: values for the parameters ; ; ';  coming soon. Advanced Fixed Income Analytics 113 9. Vasicek: How We Got There
Step 1: Fundamental theorem applied to zeros
Theorem says returns R satisfy
1 = Et mt Rt
+1 +1 Returns on zeros are Rn = bn =bn
t
t
t
ratio of sale price to purchase price
+1
+1 +1 +1 Theorem becomes bn = Et mt bn
t
t
An n + 1period zero is a claim to an nperiod zero one period
in the future.
+1 +1 +1 Advanced Fixed Income Analytics 114 9. Vasicek: How We Got There continued
Step 2: Show that z is the short rate
Since bt = 1, the short rate satis es:
e,yt = bt = Et mt
Stop now if you have any questions about this.
0 1 1 +1 Fact: if log x is normal with mean a and variance b, then
E x = ea + b=2
log E x = a + b=2
In words: the log of the mean a + b=2 isn't quite the mean of
the log a. There's an extra term due to the variance b.
Since " is normal, log mt is normal, too:
log mt = , =2 , zt , "t
Its conditional mean and variance are
Et log mt = , =2 , zt
Var t log mt =
+1 2 +1 +1 2 +1
+1 2 Apply the fact: Et mt = e, =2 , zt + =2 = e,zt ;
2 +1 Result: z is the short rate. 2 Advanced Fixed Income Analytics 115 9. Vasicek: How We Got There continued Step 3: Solution for an arbitrary maturity
Guess that bond prices are loglinear functions of z :
, log bn = An + Bn zt
t
We know this works for n = 1: A = 0 and B = 1.
Given a solution for n, nd n + 1:
bn = Et mt bn
t
t
Work on righthandside:
log mt = , =2 , zt , "t
log bn = ,An , Bn zt
t
= ,An , Bn 1 , ' + 'zt + "t
The sum is
log mt bn = , =2 , An , Bn1 , ' , 1 + Bn'zt
t
, + Bn "t :
The conditional mean and variance are
Et log mt bn = , =2 , An , Bn1 , ' , 1 + Bn 'zt
t
n
Var t log mt bt = + Bn
Apply the fact:
log bn = , =2 , An , Bn 1 , ' , 1 + Bn 'zt
t
+ + Bn =2
= ,An , Bn zt:
Recursions for computing An; Bn one maturity at a time:
An = An + =2 + Bn 1 , ' , + Bn =2
Bn = 1 + Bn'
1 +1 +1 2 +1 1 +1 +1 +1 +1 +1 +1 2 +1 +1 +1
+1 2 +1
+1 +1 2 2 2 +1 +1 +1 2 +1 2 Advanced Fixed Income Analytics 116 10. Vasicek: Choosing Parameters
Premise: a model with bad parameters is a bad model
Question: How do we choose them ; ; '; ?
Answer for now: reproduce broad featues of spot rates see
table several pages above
Mean and variance of spot rates:
ytn = n, An + Bn zt
E yn = n, An + Bn
Bn ! Var z = Bn !
n
Var y =
n
n 1,'
n n
Corr yt ; yt, = '
The last follows since linear functions of z inherit z 's
autocorrelation.
1
1 2 2 2 2 1 Parameter values:
Choose to match the mean short rate: = 6:683 = 0:005569
1200 Choose ' to match autocorrelation of short rate:
' = 0:959
Choose to match variance of short rate:
2:699 !
= 0:0006374
1 , ' = 1200
What about ?
2 2 2 Advanced Fixed Income Analytics 117 10. Vasicek: Choosing Parameters continued
controls risk premiums on long bonds
Mean spot rates in model and data for di erent choices of :
9 Mean Yield (Annual Percentage) 8.5 8 lambda < 0 7.5 7
lambda = 0
6.5 6 5.5 lambda > 0 5
0 20 40 60
Maturity in Months 80 100 120 Result: = ,0:1308 matches mean 10year rate
Why? Need negative value to get right sign on risk premium
Problem: miss the curvature Advanced Fixed Income Analytics 118 11. Hedging
Durationbased hedging of a 5year zero
Sensitivity to yield changes:
n
bn = e,ny n
bn ,ne,ny yn = ,nbnyn
n is the duration, measured in months.
Arrange positions in nmonth zero that o set risk in 60month:
Portfolio consists of
v = x b + xnbn
60 60 Change in value:
v = x b + xnbn
,x 60b y , xn nbnyn
60 60 60 60 60 Hedge ratio results from setting v = 0 the de nition of a
hedge and y = yn the assumption underlying
duration:
xnbn = , 60
Hedge Ratio = x b
n
The usual: the ratio of the durations.
60 60 60 Advanced Fixed Income Analytics 119 11. Hedging continued
Vasicekbased hedging of a 5year zero
Sensitivity to z changes:
bn = e,An , Bnz
bn ,Bn bnz
Arrange positions in nmonth zero that o set risk in 60month:
Portfolio consists again of
v = x b + xnbn
60 60 Change in value:
v = x b + xnbn
,x B b z , xnBnbnz
60 60 60 60 60 Hedge ratio results from setting v = 0 the de nition of a
hedge:
xnbn = , B
Hedge Ratio = x b
B
60 60 60 Comparison:
Recall: Bn = 1 + ' + + 'n, n 1 When ' = 1, Bn = n and Vasicek and durationbased
hedge ratios are the same
When 0 ' 1, Bn n: Vasicek attributes relatively less
risk to long zeros than duration Advanced Fixed Income Analytics 120 11. Hedging continued
Thought experiment: If the world obeyed the Vasicek model,
how far wrong could you go by using duration to hedge?
Buy $100 of 5year zeros
Construct duration and Vasicek hedges with 24month zeros
Compute pro t loss for given changes in z
3 Profit/Loss on Hedged Position 2 1
Vasicek−based hedge
0 −1
Duration−based hedge −2 −3
−2 −1.5 −1 −0.5
0
0.5
Change in z (x 1200) 1 1.5 2 Advanced Fixed Income Analytics 121 11. Hedging continued
Comments on the thought experiment
Duration hedging: you're overhedged
Vasicek hedging: not exactly zero due to convexity
Is this reasonable? Qualitatively yes: less volatility for long
rates than duration assumes see table for monthly changes Advanced Fixed Income Analytics
12. Where are the Bodies Buried?
Strengths of the Vasicek model:
relatively simple and transparent
ts general features of bond prices
Blacklike option prices coming soon
Weaknesses:
insu cient curvature in mean spot rates
crude approximation to current spot rates
volatility is constant
persistence same for spot rates and spreads
allows negative interest rates 122 Advanced Fixed Income Analytics 123 Summary 1. Modern pricing theory is based on a pricing kernel, which plays
a role analogous to the market return in the CAPM
2. Models contain
descriptions of interest rate dynamics ; ; '
assessment of risk
3. The Vasicek model
an archetype: many other models have similar structures
strengths include simplicity, Blacklike option formulas
weaknesses include its single factor, constant volatility
4. Coming up:
derivatives pricing
comparisons with other models
5. Vote on these topics:
convexity adjustment for eurocurrency futures
Hull and White model
HeathJarrowMorton approach
volatility smiles for options
stochastic volatility
twofactor models
American options
CMT CMS swaps
Currency and commodity derivatives ...
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 Spring '15
 Backus
 Standard Deviation, Variance, Mathematical finance, spot rates

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