No-Arbitrage Testing with Single Factor
Presented by Meg Cheng
Motivation
• No-arbitrage condition is one of the most popular
assumptions in the area of asset pricing.
• If changes in the price of the asset are driven by some
underlying factors, the excess return should be consist of all
the prices of the associated underlying factors risks.
• To be free from any model specification, nonparametric
estimation method is adopted to recover the embedded
information directly from the data.
Bond Pricing Theory
Suppose the economy can be driven by a state variable X,
defined in a stochastic differential equation:
dX (t )   x ( x)dt   x ( x)dW
Consider the dynamics of an asset price at current time t
of a claim to terminal payoff P(T) at some future date T
as follows:
dP(t )   ( x)dt   ( x)dW
By Ito’s lemma,
1
 ( x)  Pt  Px  x  Pxx x2
2
 ( x)  Px x
Now, consider another asset price w/ terminal payoff as P*(T),
within the same framework, we can write this dynamic process
as
dP*   * ( X )dt   * ( X )dW
1 * 2
*
*
*
 ()  Pt  Px  x  Pxx x
2
 * ()  Px* x
If we want to make a portfolio Z to hedge against the factor
risk with these two assets, then portfolio Z can be represented
as follows:
Z  P   * P *
dZ  dP   *dP *
 [   *  * ]dt  [   * * ]dW
where  and * are portfolio weights on each asset
Let

*
 
*
,  
*

 * 
Then portfolio Z becomes a risk free asset.
Hence, the drift term of dZ should be equal to risk-free rate r.
i.e.
 *    *
r
*
 
  r *  r


  ( x)
*


Notice that if the underlying market is arbitrage free, this
relationship holds for any arbitrary asset. This addresses
our null hypothesis.
i.e.
1(x)= 2(x)= …=P(x)
Hypothesis testing
We choose short rate as the factor, and use the three-month
yield to maturity as the proxy for the short rate.
Suppose we have P different assets to be estimated, and each one
of them follows:
dPi
 Xdt  ai ( x)dt  bi ( x)dW (t ) i  1, 2,..., P
Pi
ai (.) and bi (.) are local constants.
Given each x, we can use gaussian process to describe the above
diffusion process.
Under the alternative hypothesis:
L1 (a( x), b( x))   log( f i [ai ( x), bi ( x)]) K h ( X  x)
n
i
(a, b)  arg max L1 (a( x), b( x))
a ,b
where
a( x)  ai ( x) i  1,2,..., P
b( x)  {bi ( x) i  1,2,..., P}
f i () is Gaussian density function
K h () is kernel function
Under the null hypothesis:
If no-arbitrage restriction holds, the expression below is true
for any arbitrary asset:
  r *  r

 ...   ( x)
*


So that given each x, the risk premium of each asset should be
proportional to its diffusion term with a constant term across all
assets.
dPi
 Xdt  [c( x)bi( x)]dt  bi( x)dW , i  1, 2,..., P
Pi
Hence, the likelihood function under the null:
L0 (b( x), c( x))   F (b( x), c( x)) K h ( X  x)
n
(b, c)  arg max L0 (b( x), c( x))
c(x) is a constant across all the assets
F(.) is multivariate gaussian density function
b( x)  {bi( x), i  1,2,..., P}
Data
• We use weekly values for the annualized zero-coupon yields
with six different maturities (0.5, 1, 2, 3, 5, 10 years).
• Generally speaking, almost each bond/security comes w/
coupons or dividends, except treasury bills.
• Since there is no generally accepted “best” practice for extracting
zero coupon prices from coupon bonds, we construct our data by
four methods: 1.
Smoothed Fama-Bliss
2.
Unsmoothed Fama-Bliss
3.
McCulloch-Kwon
4.
Nelson-Siegel
•To test our null hypothesis, we propose to use empirical
Likelihood Ratio (LR) test, since we’ve already constructed
likelihood both under the alternative and the null.
•We interpolate all the estimates associated with the chosen grids
to compute the likelihood at each observation.
•To get LR test statistics distribution under the null hypothesis,
We adopt stationary bootstrap method proposed by Romano
(1994).
The procedures are described as follows:
1. We use first order Euler approximation to fit the model
under the null, i.e.
dPi
 [ X  cˆ( X )bˆi( X )]  bˆi( X )ˆti i  1,2,..., P
Pi
Since we don’t literally have maximum likelihood estimated
on every data point, there still exists some dependence in time
in the residuals extracted from the above.
2. Have all the residuals estimated from each asset into a matrix
by columns and denote it by Y (NxP).
3. Let i be i.i.d. random variable generated from Uniform
Distribution U(N).
4. Generate Bi,m={Yi, Yi+1, …,Yi+m-1}’ , the block consisting of
m rows starting from Yi, and the r.v. m is drawn from geometric
distribution (1-q)m-1q for m=1,2,…N. where qR(0,1).
5. Repeat step 3 and 4, stack each block matrix end to end,
till the number of columns and rows of the newly generated Y*
are equal to Y.
6. Put Y* back to the Euler euqation to get the new LHS.
7. Implement local MLE both under the null and the alternative.
8. Replicate step3~step7 for sufficiently enough time (around
1,000), then the statistics distribution will then be constructed.
Nelson-Siegel
3
3
2
2
1
1
lamda
lamda
Unsmooth Fama-Bliss
0
-1
-2
-1
0
5
10
15
short rate %
Mculloch-Kwon
-2
20
6
0
5
10
15
short rate %
Smooth Fama-Bliss
20
4
4
2
2
lamda
lamda
0
0
0
-2
-2
-4
-4
0
5
10
15
short rate %
20
0
5
10
15
short rate %
20
0.8
autocovariance
autocovariance
1
0.5
0
-0.5
0
100
200
lags
300
0.4
0.2
0
-0.2
400
0
100
200
lags
300
400
0
100
200
lags
300
400
0.8
autocovariance
1.5
autocovariance
0.6
1
0.5
0
-0.5
0.6
0.4
0.2
0
-0.2
0
100
200
lags
300
400
Nelson-Siegel
10
50
8
40
frequency
frequency
Unsmooth Fama-Bliss
6
4
2
0
30
20
10
0
500
statistics
Mculloch-Kwon
0
1000
60
0
500
statistics
Smooth Fama-Bliss
1000
50
40
frequency
frequency
40
20
30
20
10
0
0
200
400
600
statistics
800
0
0
500
1000
statistics
1500
Unsmooth Fama-Bliss
Nelson-Siegel
L(1)
-8345
-8162
L(0)
-8474
-8318
T*=L(1)-L(0)
129
156
No. of replications
100
500
Prob.>T*
67%
73%
Mculloch-Kwon
Smooth Fama-Bliss
L(1)
-6035
-8138
L(0)
-6289
-8228
T*=L(1)-L(0)
254
90
No. of replications
600
500
Prob.>T*
33%
69%
Conclusion:
So far, based on our result, the hypothesis of
no-arbitrage condition tested with six different
yields to maturities is not rejected.
Put in another way, the no-arbitrage restriction
may still holds in U.S. Treasury Bill and Bond market.