No-Arbitrage Testing with Single Factor

1. No-Arbitrage Testing with Single Factor Presented by Meg Cheng

2. 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.

3. 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,

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

5. 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.

6. 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)

7. 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.

8. Under the alternative hypothesis:

9. 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.

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

11. 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

12. 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:

13. 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. • Have all the residuals estimated from each asset into a matrix • by columns and denote it by Y (NxP). • Let i be i.i.d. random variable generated from Uniform • Distribution U(N). • 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).

14. 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. • Put Y* back to the Euler euqation to get the new LHS. • 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.

19. 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.