Inference for stochastic volatility model using time change transformations

Kalogeropoulos, Konstantinos and Roberts, Gareth O. and Dellaportas, Petros (2007): Inference for stochastic volatility model using time change transformations.

Abstract

We address the problem of parameter estimation for diffusion driven stochastic volatility models through Markov chain Monte Carlo (MCMC). To avoid degeneracy issues we introduce an innovative reparametrisation defined through transformations that operate on the time scale of the diffusion. A novel MCMC scheme which overcomes the inherent difficulties of time change transformations is also presented. The algorithm is fast to implement and applies to models with stochastic volatility. The methodology is tested through simulation based experiments and illustrated on data consisting of US treasury bill rates.

• Item Type: MPRA Paper
• Language: English
• Keywords: Imputation, Markov chain Monte Carlo, Stochastic volatility
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C11 - Bayesian Analysis: General

References

\bibitem[A\"{i}t-Sahalia, 1996]{ait96} A\"{i}t-Sahalia, Y. (1996). \newblock Testing continuous-time models of the spot interest rate. \newblock {\em The Review of Financial Studies}, 9:385--426.

\bibitem[A\"{i}t-Sahalia, 2002]{ait02} A\"{i}t-Sahalia, Y. (2002). \newblock Maximum likelihood estimation of discretely sampled diffusions: a closed form approximation approach. \newblock {\em Econometrica}, 70:223--262.

\bibitem[A\"{i}t-Sahalia, 2005]{ait05} A\"{i}t-Sahalia, Y. (2005). \newblock Closed form likelihood expansions for multivariate diffusions. \newblock In preparation.

\bibitem[A\"{i}t-Sahalia and Kimmel, 2005]{ait:kim05} A\"{i}t-Sahalia, Y. and Kimmel, R. (2005). \newblock Maximum likelihood estimation for stochastic volatility models. \newblock To appear in the Journal of Financial Economics.

\bibitem[Andersen et~al., 2006]{and:bol:med02} Andersen, T., Bollerslev, T., and Meddahi, N. (2006). \newblock Analytic evaluation of volatility forecasts. \newblock Tech. rep., Northwestern University.

\bibitem[Andersen and Lund, 1998]{and:lun98} Andersen, T.~G. and Lund, J. (1998). \newblock Estimating continuous-time stochastic volatility models of the short term interest rate. \newblock {\em Journal of Econometrics}, 77:343--377.

\bibitem[Barndorff-Nielsen et~al., 2006]{bn:h:l:s06} Barndorff-Nielsen, O.~E., Hansen, P.~R., Lunde, A., and Shephard, N. (2006). \newblock Designing realized kernels to measure ex-post variation of equity prices in the presence of noise. \newblock Working paper.

\bibitem[Bergman et~al., 1996]{ber:gru:wie96} Bergman, Y., Grundy, B., and Wiener, Z. (1996). \newblock General properties of option prices. \newblock {\em Journal of Finance}, 51(5):1573--1610.

\bibitem[Beskos et~al., 2006]{bes:pap:rob:f06} Beskos, A., Papaspiliopoulos, O., Roberts, G., and Fearnhead, P. (2006). \newblock Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion). \newblock {\em Journal of the Royal Statistical Society: Series B (Statistical Methodology)}, 68(3):333--382.

\bibitem[Beskos and Roberts, 2005]{bes:rob05} Beskos, A. and Roberts, G.~O. (2005). \newblock Exact simulation of diffusions. \newblock {\em Ann. Appl. Probab.}, 15(4):2422--2444.

\bibitem[Bibby and Sorensen, 1995]{bib:sor95} Bibby, B. and Sorensen, M. (1995). \newblock Martingale estimating functions for discretely observed diffusion processes. \newblock {\em Bernoulli}, 1:17--39.

\bibitem[Black and Scholes, 1973]{bla:sch73} Black, F. and Scholes, M. (1973). \newblock The pricing of options and corporate liabilities. \newblock {\em Journal of Political Economy}, 81:637--654.

\bibitem[Chan et~al., 1992]{ckls92} Chan, K.~C., Karolyi, G.~A., Longstaff, F.~A., and Sanders, A.~B. (1992). \newblock An empirical comparison of alternative models of the short-term interest rate. \newblock {\em The Journal of Finance}, 47(3):1209--1227.

\bibitem[Chernov and Ghysels, 2000]{che:ghy00} Chernov, M. and Ghysels, E. (2000). \newblock A study towards a uinified approach to the joint estimation of objective and risk neutral measures for the purposes of options valuation. \newblock {\em Journal od Financial Economics}, 56:407--458.

\bibitem[Chib et~al., 2005]{ch:pit:she05} Chib, S., Pitt, M.~K., and Shephard, N. (2005). \newblock Likelihood based inference for diffusion models. \newblock Submitted.

\bibitem[Cox et~al., 1985]{cir85} Cox, J.~C., Ingersoll, J.~E., and Ross, S.~A. (1985). \newblock A theory of the term structure of interest rates. \newblock {\em Econometrica}, 53:385--407.

\bibitem[Derman and Kani, 1994]{der:kan94} Derman, E. and Kani, I. (1994). \newblock Riding on the smile. \newblock {\em Risk}, 7:32--39.

\bibitem[Dumas et~al., 1998]{dum:fle:wha98} Dumas, B., Fleming, J., and Whaley, R.~E. (1998). \newblock Implied volatility functions: Empirical tests. \newblock {\em Journal of Finance}, 53:2059--2106.

\bibitem[Dupire, 1994]{dup94} Dupire, B. (1994). \newblock Pricing with a smile. \newblock {\em Risk}, 7:18--20.

\bibitem[Durham, 2002]{dur02} Durham, G.~B. (2002). \newblock Likelihod based specification analysis of continuous time models of the short term interest rate. \newblock {\em Journal of Financial Economics}. \newblock Forthcoming.

\bibitem[Durham and Gallant, 2002]{dur:gal02} Durham, G.~B. and Gallant, A.~R. (2002). \newblock Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. \newblock {\em J. Bus. Econom. Statist.}, 20(3):297--316. \newblock With comments and a reply by the authors.

\bibitem[El~Karoui et~al., 1998]{el:jea:shr98} El~Karoui, N., Jeanblanc, M., and Shreve, S. (1998). \newblock Robustness of the {B}lack {S}choles formula. \newblock {\em Mathematical Finance}, 8(2):93--126.

\bibitem[Elerian et~al., 2001]{ele:ch:she01} Elerian, O.~S., Chib, S., and Shephard, N. (2001). \newblock Likelihood inference for discretely observed non-linear diffusions. \newblock {\em Econometrica}, 69:959--993.

\bibitem[Eraker, 2001]{era01} Eraker, B. (2001). \newblock Markov chain {M}onte {C}arlo analysis of diffusion models with application to finance. \newblock {\em J. Bus. Econom. Statist.}, 19(2):177--191.

\bibitem[F{\"o}llmer and Schweizer, 1991]{fo:sch91} F{\"o}llmer, H. and Schweizer, M. (1991). \newblock Hedging of contingent claims under incomplete information. \newblock In {\em Applied stochastic analysis (London, 1989)}, volume~5 of {\em Stochastics Monogr.}, pages 389--414. Gordon and Breach, New York.

\bibitem[Gallant and Long, 1997]{gal:lon97} Gallant, A.~R. and Long, J.~R. (1997). \newblock Estimating stochastic differential equations efficiently by minimum chi-squared. \newblock {\em Biometrika}, 84(1):125--141.

\bibitem[Gallant and Tauchen, 1996]{gal:tau96} Gallant, A.~R. and Tauchen, G. (1996). \newblock Which moments to match? \newblock {\em Econometric Theory}, 12(4):657--681.

\bibitem[Gallant and Tauchen, 1998]{gal:tau98} Gallant, A.~R. and Tauchen, G. (1998). \newblock Reprojecting partially observed systems with applications to interest rate diffusions. \newblock {\em Journal of American Statistical Association}, 93(441):10--24.

\bibitem[Ghysels et~al., 1996]{ghy:har:ren96} Ghysels, E., Harvey, A., and Renault, E. (1996). \newblock Stochastic volatily, in. \newblock {\em Handbook of Statistics 14, Statistical Methods in Finance}. \newblock G.S. Maddala and C.R. Rao (eds), North Holland, Amsterdam.

\bibitem[Golightly and Wilkinson, 2006]{gol:wil06} Golightly, A. and Wilkinson, D. (2006). \newblock Bayesian sequential inference for nonlinear multivariate diffusions. \newblock {\em Statistics and Computing}, 16:323--338.

\bibitem[Golightly and Wilkinson, 2007]{gol:wil07} Golightly, A. and Wilkinson, D. (2007). \newblock Bayesian inference for nonlinear multivariate diffusions observed with error. \newblock {\em Computational Statistics and Data Analysis}. \newblock In press.

\bibitem[Gouri{\'e}roux et~al., 1993]{gou:mon:ren93} Gouri{\'e}roux, C., Monfort, A., and Renault, E. (1993). \newblock Indirect inference. \newblock {\em Journal of Applied Econometrics}, 8:S85--S118.

\bibitem[Harrison and Krepps, 1979]{har:kre79} Harrison, M. and Krepps, D. (1979). \newblock Martingales and arbitrage in multiperiod securities markets. \newblock {\em Journal of Economic Theory}, 20:381--408.

\bibitem[Harrison and Krepps, 1981]{har:pli81} Harrison, M. and Krepps, D. (1981). \newblock Martingales and stochastic integrals in the theory of continuous trading. \newblock {\em Stochastic Processes and Their Applications}, 11:215--260.

\bibitem[Henderson et~al., 2005]{hen:hob05} Henderson, V., Hobson, D., Howison, S., and Kluge, T. (2005). \newblock A comparison of $q-$ optimal option prices in a stochastic volatility model with correllation. \newblock {\em Review of Derivatives Research}, 8:5--25.

\bibitem[Heston, 1993]{hes93} Heston, S. (1993). \newblock A closed-form solution for options with stochastic volatility. with applications to bonds and currency options. \newblock {\em Review of Financial Studies}, 6:327--343.

\bibitem[Hobson, 1993]{hob98} Hobson, D. (1993). \newblock Volatility mis-specification, option pricing and super-replication. \newblock {\em Annals of Applied Probability}, 8(2):193--205.

\bibitem[Hull and White, 1987]{hul:whi87} Hull, J.~C. and White, A.~D. (1987). \newblock The pricing of options on assets with stochastic volatilities. \newblock {\em Journal of Finance}, 42(2):281--300.

\bibitem[Jacquier et~al., 1994]{jac:pol:ro94} Jacquier, E., Polson, N.~G., and Rossi, P. (1994). \newblock Bayesian analysis of stochastic volatility models. \newblock {\em Journal of Business and Economic Statistics}, 12:371--389.

\bibitem[Johannes et~al., 2006]{joh:pol:str06} Johannes, M., Polson, N., and Stroud, J. (2006). \newblock Optimal filtering of jump diffusions: Extracting latent state from option prices. \newblock In submision.

\bibitem[Jones, 1999]{jon99} Jones, C.~S. (1999). \newblock Bayesian estimation of continuous-time finance models. \newblock Unpublished paper, Simon School of Business, University of Rochester.

\bibitem[Jones, 2003]{jon03} Jones, C.~S. (2003). \newblock Nonlinear mean reversion in the short-term interest rate. \newblock {\em The review of financial studies}, 16:793--843.

\bibitem[Kalogeropoulos, 2007]{kal07} Kalogeropoulos, K. (2007). \newblock Likelihood based inference for a class of multidimensional diffusions with unobserved paths. \newblock {\em Journal of Statistical Planning and Inference}. \newblock To appear.

\bibitem[Kalogeropoulos et~al., 2007]{kal:del07} Kalogeropoulos, K., Dellaportas, P., and Roberts, G. (2007). \newblock Bayesian inference for correllated diffusions observed discretly in time. \newblock In submision.

\bibitem[Kim et~al., 1998]{kim:she:ch98} Kim, S., Shephard, N., and Chib, S. (1998). \newblock Bayesian analysis of stochastic volatility models. \newblock {\em Review of Economic Studies}, 65:361--393.

\bibitem[Kloeden and Platen, 1995]{klo:pla95} Kloeden, P. and Platen, E. (1995). \newblock {\em Numerical Solutions of Stochastic Differential Equations}. \newblock New York: Springer.

\bibitem[Merton, 1995]{mer73} Merton, B. (1995). \newblock The theory of rational option pricing. \newblock {\em Bell Journal of Economics and Management Science}, 4:141--183.

\bibitem[Oksendal, 2000]{oks00} Oksendal, B. (2000). \newblock {\em Stochastic differential equations}. \newblock Springer, 5th edition.

\bibitem[Papaspiliopoulos and Roberts, 2005]{pap:rob05} Papaspiliopoulos, O. and Roberts, G. (2005). \newblock Retrospective {MCMC} for {D}irichlet process hierarchical models. \newblock Submitted.

\bibitem[Pedersen, 1995]{ped95} Pedersen, A.~R. (1995). \newblock A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. \newblock {\em Scand. J. Statist.}, 22(1):55--71.

\bibitem[Pitt and Shephard, 1999]{pit:she99} Pitt, M.~K. and Shephard, N. (1999). \newblock Filtering via simulation: Auxilliary particle filter. \newblock {\em Journal of American Statistical Association}, 446:590--599.

\bibitem[Roberts and Stramer, 2001]{rob:str01} Roberts, G. and Stramer, O. (2001). \newblock On inference for partial observed nonlinear diffusion models using the metropolis-hastings algorithm. \newblock {\em Biometrika}, 88(3):603--621.

\bibitem[Roberts and Stramer, 2004]{rob:str04} Roberts, G. and Stramer, O. (2004). \newblock On bayesian analysis of non-linear continuous time autoregressive models. \newblock Submitted.

\bibitem[Rogers and Williams, 1994]{rog:wil94} Rogers, L. C.~G. and Williams, D. (1994). \newblock {\em Diffusions, Markov processes and martingales, 2, Ito calculus}. \newblock Wiley, Chicester.

\bibitem[Romano and Touzi, 1997]{rom:tou97} Romano, M. and Touzi, N. (1997). \newblock Contingent claims and market completeness in a stochastic volatility model. \newblock {\em Math. Finance}, 7(4):399--412.

\bibitem[Rubinstein, 1995]{rub95} Rubinstein, B. (1995). \newblock As simple as one, two, three. \newblock {\em Risk}, 8:44--47.

\bibitem[Schweizer, 1999]{sch99} Schweizer, M. (1999). \newblock A minimality property of the minimal martingale measure. \newblock {\em Statistics \& Probability Letters}, 42(1):27--31.

\bibitem[S{\o}rensen, 2004]{sor04} S{\o}rensen, H. (2004). \newblock Parametric inference for diffusion processes observed at discrete points in time: a survey. \newblock {\em International Statistical Review}, 72(3):337--354.

\bibitem[Stein and Stein, 1991]{ste:ste91} Stein, E.~M. and Stein, J.~C. (1991). \newblock Stock proce distributions with stochastic volatility: an analytic approach. \newblock {\em Review of Financial Studies}, 4(4):727--752.

\bibitem[Tanner and Wong, 1987]{tan:won87} Tanner, M.~A. and Wong, W.~H. (1987). \newblock The calculation of posterior distributions by data augmentation. \newblock {\em Journal of the American Statistical Association}, 82(398):528--540.

\bibitem[Whaley, 1993]{wha93} Whaley, R.~E. (1993). \newblock Derivatives on market volatility: Hedging tools long overdue. \newblock {\em Journal of Derivatives}, 1:71--83.

\bibitem[Whaley, 2000]{wha00} Whaley, R.~E. (2000). \newblock The investor fear gauge. \newblock {\em Journal of portfolio management}, 26:12--17.