Limits of Bayesian decision related quantities of binomial asset price models

Wolfgang Stummer; Wei Lao

Kybernetika (2012)

  • Volume: 48, Issue: 4, page 750-767
  • ISSN: 0023-5954

Abstract

top
We study Bayesian decision making based on observations X n , t : t ∈ { 0 , T n , 2 T n , ... , n T n } ( T > 0 , n ∈ ℕ ) of the discrete-time price dynamics of a financial asset, when the hypothesis a special n -period binomial model and the alternative is a different n -period binomial model. As the observation gaps tend to zero (i. e. n → ∞ ), we obtain the limits of the corresponding Bayes risk as well as of the related Hellinger integrals and power divergences. Furthermore, we also give an example for the “non-commutativity” between Bayesian statistical and optimal investment decisions.

How to cite

top

Stummer, Wolfgang, and Lao, Wei. "Limits of Bayesian decision related quantities of binomial asset price models." Kybernetika 48.4 (2012): 750-767. <http://eudml.org/doc/246426>.

@article{Stummer2012,
abstract = {We study Bayesian decision making based on observations $\left(X_\{n,t\} : t\in \lbrace 0,\frac\{T\}\{n\},2\frac\{T\}\{n\},\ldots ,n\frac\{T\}\{n\}\rbrace \right)$ ($T>0, n\in \mathbb \{N\}$) of the discrete-time price dynamics of a financial asset, when the hypothesis a special $n$-period binomial model and the alternative is a different $n$-period binomial model. As the observation gaps tend to zero (i. e. $n \rightarrow \infty $), we obtain the limits of the corresponding Bayes risk as well as of the related Hellinger integrals and power divergences. Furthermore, we also give an example for the “non-commutativity” between Bayesian statistical and optimal investment decisions.},
author = {Stummer, Wolfgang, Lao, Wei},
journal = {Kybernetika},
keywords = {Bayesian decisions; power divergences; Cox-Ross-Rubinstein binomial asset price models; power divergences; Bayesian decisions; Cox-Ross-Rubinstein binomial asset price models},
language = {eng},
number = {4},
pages = {750-767},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Limits of Bayesian decision related quantities of binomial asset price models},
url = {http://eudml.org/doc/246426},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Stummer, Wolfgang
AU - Lao, Wei
TI - Limits of Bayesian decision related quantities of binomial asset price models
JO - Kybernetika
PY - 2012
PB - Institute of Information Theory and Automation AS CR
VL - 48
IS - 4
SP - 750
EP - 767
AB - We study Bayesian decision making based on observations $\left(X_{n,t} : t\in \lbrace 0,\frac{T}{n},2\frac{T}{n},\ldots ,n\frac{T}{n}\rbrace \right)$ ($T>0, n\in \mathbb {N}$) of the discrete-time price dynamics of a financial asset, when the hypothesis a special $n$-period binomial model and the alternative is a different $n$-period binomial model. As the observation gaps tend to zero (i. e. $n \rightarrow \infty $), we obtain the limits of the corresponding Bayes risk as well as of the related Hellinger integrals and power divergences. Furthermore, we also give an example for the “non-commutativity” between Bayesian statistical and optimal investment decisions.
LA - eng
KW - Bayesian decisions; power divergences; Cox-Ross-Rubinstein binomial asset price models; power divergences; Bayesian decisions; Cox-Ross-Rubinstein binomial asset price models
UR - http://eudml.org/doc/246426
ER -

References

top
  1. A. K. Bera, Y. Bilias, 10.1016/S0304-4076(01)00113-0, J. Econometrics 107 (2002), 51-86. Zbl1088.62505MR1889952DOI10.1016/S0304-4076(01)00113-0
  2. A. Berlinet, I. Vajda, 10.1080/02331880903573385, Statistics 45 (2011), 479-495. Zbl1229.62035MR2832180DOI10.1080/02331880903573385
  3. M. Broniatowski, I. Vajda, Several applications of divergence criteria in continuous families., To appear in Kybernetika (2012). See also Research Report No. 2257, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Prague 2009; moreover, see arXiv:0911.0937v1 [math.ST]. 
  4. J. C. Cox, S. A. Ross, M. Rubinstein, 10.1016/0304-405X(79)90015-1, J. Finan. Econ. 7 (1979), 229-263. Zbl1131.91333DOI10.1016/0304-405X(79)90015-1
  5. N. Cressie, T. R. C. Read, Multinomial goodness-of-fit tests., J. Roy. Stat. Soc. Ser. B Stat. Methodol. 46 (1984), 440-464. Zbl0571.62017MR0790631
  6. I. Csiszár, F. Matúš, 10.1007/s00440-007-0084-z, Probab. Theory Related Fields 141 (2008), 213-246. Zbl1133.62039MR2372970DOI10.1007/s00440-007-0084-z
  7. I. Csiszár, P. C. Shields, 10.1561/0100000004, Found. Trends Commun. Inform. Theory 1 (2004), 4, 417-528. Zbl1157.62300DOI10.1561/0100000004
  8. I. V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures., Theory Probab. Appl. 5 (1960), 285-301. Zbl0100.34004MR0133152
  9. A. Golan, 10.1016/S0304-4076(01)00110-5, J. Econometrics 107 (2002), 1-15. Zbl1088.62519MR1889949DOI10.1016/S0304-4076(01)00110-5
  10. A. Gretton, L. Györfi, Consistent nonparametric tests of independence., J. Mach. Learn. Res. 11 (2010), 1391-1423. Zbl1242.62033MR2645456
  11. P. Harremoes, I. Vajda, 10.1109/TIT.2007.911155, IEEE Trans. Inform. Theory 54 (2008), 321-331. MR2446756DOI10.1109/TIT.2007.911155
  12. P. Harremoes, I. Vajda, On Bahadur efficiency of power divergence statistics., Preprint arXiv:1002.1493v1 [math.ST] (2010). 
  13. T. Hobza, L. Pardo, D. Morales, 10.1016/j.stamet.2009.03.001, Statist. Methodol. 6 (2009), 424-436. MR2751084DOI10.1016/j.stamet.2009.03.001
  14. F. Liese, K.-J. Miescke, Statistical Decision Theory., Springer-Verlag, New York 2008. Zbl1154.62008MR2421720
  15. F. Liese, D. Morales, I. Vajda, 10.1109/TIT.2006.885495, IEEE Trans. Inform. Theory 52 (2006), 5599-5606. MR2300722DOI10.1109/TIT.2006.885495
  16. F. Liese, I. Vajda, Convex Statistical Distances., Teubner, Leipzig 1987. Zbl0656.62004MR0926905
  17. F. Liese, I. Vajda, 10.1109/TIT.2006.881731, IEEE Trans. Inform. Theory 52 (2006), 4394-4412. MR2300826DOI10.1109/TIT.2006.881731
  18. E. Maasoumi, 10.1080/07474939308800260, Econometrics Rev. 12 (1993), 2, 137-181. Zbl0769.62003MR1222574DOI10.1080/07474939308800260
  19. D. Morales, I. Vajda, Generalized information criteria for optimal Bayes decisions., Research Report No. 2274, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Prague 2010. 
  20. D. B. Nelson, K. Ramaswamy, 10.1093/rfs/3.3.393, Rev. Financ. Stud. 3 (1990), 393-430. DOI10.1093/rfs/3.3.393
  21. L. Pardo, Statistical Inference Based on Divergence Measures., Chapman & Hall, Boca Raton 2005. Zbl1118.62008MR2183173
  22. M. C. Pardo, 10.1007/s00184-009-0275-y, Metrika 73 (2011), 231-253. Zbl1206.62131MR2769272DOI10.1007/s00184-009-0275-y
  23. T. R. C. Read, N. A. C. Cressie, Goodness-of-Fit Statistics for Discrete Multivariate Data., Springer-Verlag, New York 1988. Zbl0663.62065MR0955054
  24. H. Strasser, Mathematical Theory of Statistics., De Gruyter, Berlin 1985. Zbl0594.62017MR0812467
  25. W. Stummer, Exponentials, Diffusions, Finance, Entropy and Information., Shaker, Aachen 2004. Zbl1140.91013
  26. W. Stummer, I. Vajda, 10.1016/j.jeconom.2005.10.001, J. Econometrics 137 (2007), 441-471. MR2354952DOI10.1016/j.jeconom.2005.10.001
  27. W. Stummer, I. Vajda, 10.1080/02331880902986919, Statistics 44 (2010), 169-187. MR2674416DOI10.1080/02331880902986919
  28. I. Vajda, E.C. van der Meulen, Goodness-of-Fit criteria based on observations quantized by hypothetical and empirical percentiles., In: Handbook of Fitting Statistical Distributions with R (Z.A. Katrian, E.J. Dudewicz eds.), Chapman & Hall / CRC, 2010, pp. 917-994. 
  29. I. Vajda, J. Zvárová, On generalized entropies, Bayesian decisions and statistical diversity., Kybernetika 43 (2007), 675-696. Zbl1143.94006MR2376331

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.