Testing Monotonicity of Pricing Kernels

Transcription

1 Yuri Golubev Wolfgang Härdle Roman Timofeev C.A.S.E. Center for Applied Statistics and Economics Humboldt-Universität zu Berlin

2 Motivation 2-2 Motivation An investor observes the evolution of a stock price X t in the past and forms his subjective opinion about the future evolution of the price dax year Figure 1: DAX, Daily observations.

3 Motivation 2-3 Basic Notions X t half year DAX returns An opinion on the future value S t of the stock at time t can be described by a density function p which is called subjective density, a.k.a. historical density or physical density. Examples: Black-Scholes model (Nobel prize 1997): log normal distribution GARCH model (Nobel prize 23, Engle): stochastic volatility non-parametric diffusion model (Ait-Sahalia 2)

4 Motivation 2-4 Basic Notions Besides the subjective density there is also a state-price density (SPD) q for the stock price implied by the market prices of options, a.k.a. risk-neutral density. The state-price density differs from the subjective density because it corresponds to replication strategies (martingale risk neutral measure). A person alone does not use in general a replication strategy but thinks in terms of his subjective density.

5 Motivation 2-5 Utility Function and Pricing Kernel The pricing kernel (PK) K: K(s) = q(s) p(s) is proportional to Marginal Rate of Substitution (MRS): β U (X T ) U (X t) = exp( rt ) K(X T ) The form of the utility function U( ) is defined by a pricing kernel

6 Motivation 2-6 Is There a Bump? 6 Pricing kernel for Black Scholes Model Pricing kernel for June 25, Classical Concave Utility Function Market Utility for EPK on June 25,

7 Motivation 2-7 Statistical Setup For increasing sequence of returns X (k) such that X (1) X (2),..., X (n) we test if there exists any interval I, J such that the sequence K (k) = q(x (k)) p(x (k) ), I k J is not monotone decreasing having only q(x t ) as a known density

8 Motivation 2-8 Outline 1. Motivation 2. Basic Idea 3. ML Test for a Fixed Interval 4. Multiple Testing 5. Multiple Testing on Blocks 6. Simulations and Results 7. Conclusion

9 Basic Idea 3-1 Test Idea 1. Reduce the test problem using Pyke s order statistics theorem 2. Construct a likelihood ratio test for a nested (reduced) model for a fixed interval 3. Use intersection of tests to construct a test independent of intervals 4. Find critical values via Monte Carlo simulations 5. Test monotonicity of pricing kernel K

10 Basic Idea 3-2 Order Statistics Consider U 1,..., U n be i.i.d with a uniform distribution on [, 1]. For the order statistics define uniform spacings S k as U (1) U (2),..., U (n) S 1 = U (1) and S k = U (k) U (k 1)

11 Basic Idea 3-3 Pyke s Theorem Theorem Let U 1,..., U n be i.i.d with a uniform distribution on [, 1] and e 1,..., e n be i.i.d. standard exponentially distributed random variables. Then for uniform spacings S k { } L {S k, 1 k n} = L ek n i=1 e, 1 k n k Using the fact that E(e k ) = 1 we obtain the following result n { U (k) U (k 1) } = n Sk e k (1)

12 Basic Idea 3-4 Simplification Let P(x) be the cdf associated with the (historical) density p(x): Using Taylor approximation we can write P(x) = x p(u)du P ( X (k+1) ) = P ( X(k) ) + P ( X (k) ) ( X(k+1) X (k) ) U (k+1) U (k) = P(X (k+1) ) P(X (k) ) p(x (k) ) (X (k+1) X (k) ) (2)

13 Basic Idea 3-5 Simplification Combining (1) and (2) we obtain the simplification: n q(x (k) ) (X (k+1) X (k) ) q(x (k) ) p(x (k) ) e k = K k e k (3) Taking n q(x (k) ) (X (k+1) X (k) ) = Z(X(k) ) = Z k our problem is reduced to the following Check monotonicity of K(X (k) ) = K k using Z k = K k e k

14 MLT for a Fixed Interval 4-1 Testing Hypothesis for a Fixed Interval [I,J] Let A(I, J) be set of all positive decreasing sequences A(I, J) = {a k : a k a k+1, I k J} Hypothesis H : K A(I, J) and pricing kernel K(X k ) is monotone decreasing function Hypothesis H 1 : K is an arbitrary function

15 MLT for a Fixed Interval 4-2 Likelihood Ratio Test for a Fixed Interval [I,J] The monotonicity LRT is defined by the function: { } maxk A(I,J) {p(z, K)} φ(z) = 1 H α (I, J) max K {p(z, K)} where H α (I, J) is the critical value with significance level α. With h α (I, J) = log H α (I, J): { φ(z) = 1 log max } K A(I,J) {p(z, K)} h α (I, J) (4) max K {p(z, K)} H is rejected if φ(z) 1 and accepted if φ(z) = 1

16 MLT for a Fixed Interval 4-3 Computation of ML function Recall from equation (3) that Z k = K k e k. The likelihood function is: J Z k J log {p(z, K)} = log(k k ) (5) K k which gives us analytically: k=i k=i max K log {p(z, K)} = (J I ) J k=i log(z k)

17 MLT for a Fixed Interval 4-4 Computation of ML function Computation of max K A(I,J) log {p(z, K)} is performed with Newton-Raphson and projection on the decreasing sequence A(I, J). Search for the maximum likelihood over all possible monotone decreasing sequences by iterative optimization via the Newton Raphson algorithm. Isotonic regression over Z k parameters since Z k gives us max K log {p(z, K)}: max K A(I,J) log {p(z, K)} = J k=i Z k f iso (Z k ) J k=i log {f iso(z k )}

18 MLT for a Fixed Interval 4-5 Isotonic Regression and Newton Raphson Algorithm Initial Zs Optimal Isotonic Regression Figure 2: Isotonic Regression combined with Newton Rapshon algorithm

19 Multiple Testing 5-1 Likelihood Ratio Test for Multiple Intervals Construct a test which does not depend on intervals Typically solved with the help of tests intersection The hypothesis H of monotone decreasing function is refuted if φ(z) 1 at least on one of the intervals [I, J]. Look for a minimal critical surface h(i, J) such that: { P {min log max } } K A(I,J) p(z, r) h α (I, J) = α. I,J max K p(z, K)

20 Multiple Testing 5-2 Multiple Testing via Monte-Carlo Simulations Solution is extremely difficult and therefore Monte Carlo simulations are used to find a reasonable surface Generate the worst non-increasing case of the sequence K (k) as a constant: K (1) = K (2) =... = K (n) = 1 Then using the result that Z k = K k e k we generate Z k exp(1) as an i.i.d standard exponential random variable

21 Multiple Testing 5-3 Test Statistics Define ξ(i, J) as the log-lr with simulated Z k : ξ(i, J) = log max K A(I,J) p(z,k) max K p(z,k) Define mean M(I, J) and variance V 2 (I, J) of ξ(i, J): M(I, J) = E ξ(i, J) V 2 (I, J) = E [ ξ 2 (I, J) E ξ(i, J) ] 2 Parameters M(I, J) and V (I, J) are calculated by simulations.

22 Multiple Testing 5-4 Critical Surface The critical value t α is calculated via Monte-Carlo simulations: } P {min {ξ(i, J) M(I, J) + t αv (I, J)} = α (6) I,J Equation (6) gives us a corresponding critical surface h α (I, J) h α (I, J) = M(I, J) t α V (I, J)

23 Multiple Testing on Blocks 6-1 Introduction of Block Parameter Block parameter improves the power of the test by allowing the function to be monotone decreasing on interval of size b Instead of calculating ML for each point, we calculate it for averages of each block Block size b is a trade-off between the variance and the shift of test statistics distribution Test procedure successfully captures the shift of the cdf due to smaller variance

24 Multiple Testing on Blocks 6-2 Influence of Block Parameter 1.9 Initial and block tests distribution functions test distribution without block test distribution after block 1.9 Influence of block parameter b on cdf variance and shift b = 1 b = 5 b = cdf.5.4 Shift of cdf after increase of slope cdf Figure 3: Resulting Distributions after change of slope Figure 4: Influence of the block on shift and variance

25 Multiple Testing on Blocks 6-3 Influence of Block Parameter For small size block distribution is shifted, but variance is also big For large blocks the distribution function is less shifted but has smaller variance Block size b = 1 corresponds to no block We test for different b in order to increase the power of the test

26 Multiple Testing on Blocks 6-4 Test with Block Parameter Test statistics ξ(i, J, b) is obtained as an average for each block, where m is a number of blocks: m ( max log {p(z, K)} = m j b k=j b b+1 log Z ) k K b H hypothesis is refuted when monotonicity is rejected at least on one of the intervals I, J with any block size b: { min ξ(i, J, b) M(I, J, b) + tα,b V (I, J, b) } (7) (I,J,b) j=1

27 Multiple Testing on Blocks 6-5 Test Summary 1. Compute Z(X (k) ) = n q(x (k) ) {X (k+1) X (k) } 2. Compute test statistics ξ(i, J, b) = log max K A(I,J) p(z,k) max K p(z,k) = max K A(I,J) log {p(z, K)} max K log {p(z, K)} 3. Take decision: if { ξ(i, J, b) M(I, J, b) + tα,b V (I, J, b) } min I,J,b then K( ) is a non-monotone decreasing function

28 Applications and Simulations 7-1 Real Z k 9 Calculated Z(k) values for June 3th Zs Dax returns Figure 5: Calculated Z k and optimized isotonic regression over DAX returns data in 22

29 Applications and Simulations 7-2 Simulated surfaces M(I, J, b) and V (I, J, b) M(I,J) surface for n=255 V(I,J) surface for n= M(I,J) 6 8 V(I,J) I J I J Figure 6: Simulated surface M(I, J, b) with Monte-Carlo method for n = 255 and b = 1 Figure 7: Simulated surface V (I, J, b) with Monte-Carlo method for n = 255 and b = 1

30 Applications and Simulations 7-3 Simulated t α,b 7 5% and 1% critival values for different b parameters 5% crit. values 1% crit. values Crit. Value b Figure 8: Distribution of critical values t.5,b and t.1,b over block parameter b

31 Applications and Simulations 7-4 Test surface ξ(i, J, b) M(I, J, b) + t α V (I, J, b) Testing surface ξ(i,j) M(I,J)+ t.5 V and Zero surface Testing surface ξ(i,j) M(I,J)+ t.5 V and Zero surface 5 5 ξ(i,j M(I,J) + t α V) ξ(i,j M(I,J) + t α V) I J I J Figure 9: Testing surfaces for year 22, α = 5% and b = 5 Figure 1: Testing surfaces for year 24, α = 5% and b = 5

32 Conclusion 8-1 Obtained results for EPK in 2, 22 and 24 Sign. level/year of analysis % Significance level min I,J,b Accepted H H H 1 H 1% Significance level min I,J,b Accepted H 1 H 1 H 1 H Table 1: Summary of results on monotonicity of EPK in 2, 22 and 24.

33 Conclusion 8-2 Conclusion We constructed a test which allows to verify monotonicity of PKs Strictly decreasing PKs correspond to concave utility functions and vice versa Introduction of block parameter significantly improved the power of the test We found the evidence of non-monotone EPKs for DAX in 2 and 22 Test findings suggest that PK structure may change over time

34 Conclusion 8-3 References Ait-Sahalia, Y. and Lo, A. (2) Nonparametric risk management and implied risk aversion, Journal of Econometrics 94(1-2), Giacomini, E., Handel, M. and Härdle, W. (26) Time Dependent Relative Risk Aversion, proceedings, 9th Karlsruhe Econometrics Workshop 26. Jackwerth, J. C. (22) Recovering Risk Aversion from Option Prices and Realized Returns, Review of Financial Studies, 13, Pyke, R. (1965) Spacings, J.R. Statistical Society, B 27,