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: 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: By Ito’s lemma,

Now, consider another asset price w/ terminal payoff as P*(T), within the same framework, we can write this dynamic process as

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: 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. 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: Given each x, we can use gaussian process to describe the above diffusion process.

Under the alternative hypothesis:

Under the null hypothesis: If no-arbitrage restriction holds, the expression below is true for any arbitrary asset: 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.

Hence, the likelihood function under the null: c(x) is a constant across all the assets F(.) is multivariate gaussian density function

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. 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 B i,m ={Y i, Y i+1, …,Y i+m-1 }’, the block consisting of m rows starting from Y i, and the r.v. m is drawn from geometric distribution (1-q) m-1 q 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.

Unsmooth Fama-BlissNelson-Siegel L(1) L(0) T*=L(1)-L(0) No. of replications Prob.>T*67%73% Mculloch-KwonSmooth Fama-Bliss L(1) L(0) T*=L(1)-L(0)25490 No. of replications 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.