ó áp7]c @sÙdZddlmZddlZddlmZmZddlj Z de fd„ƒYZ de fd„ƒYZ d e fd „ƒYZd e fd „ƒYZd e fd„ƒYZde fd„ƒYZdefd„ƒYZde fd„ƒYZde fd„ƒYZedkrÕdZdZgZdekre ƒZejddeƒZer­e jƒe jedjƒZe jedj dƒddƒZe j!dƒnd „Z"ej#e"d!d"deƒZ$er e jƒe je$djƒZe je$dddƒZe j!d#ƒnej%j&e$dej'd$e$dd%ƒej(ƒZ)e*e)ƒed&d'd(d)d*d+ƒZ+e+j,d!d,dd,ƒZ-er;e jƒe je-jƒZe je-j dƒddƒZe j!d-ƒe jƒe jej.e-ƒjƒZe jej.e-j dƒƒddƒZe j!d.ƒned&dd(d/d*dƒZ/e/j,d!d,dd,ƒZ0er¾e jƒe je0jƒZe je0j dƒddƒZe j!d0ƒned&dd(dd1d+d*d2ƒZ1e1j,ƒZ2e1j3ddej4j5d3dFƒƒZ6e1j3dej7dd5d5d2ƒdƒe1j8dd5ƒe*e1j8dd5d6d2dd5ƒj dƒƒe1j8dd,d6d2dd,ƒZ9erÞe jƒe je9jƒZe je9j dƒddƒZe j!d7ƒned&dd(dd8d+d*d2ƒZ:e:j8dd9d6d2dd,ƒZ;e*e;j dƒƒe*ej.e;j dƒƒƒdZer¢e jƒe je;jƒZe je;j dƒddƒZe j!d:ƒne*e:j<e;ddd…fd;d2ƒƒe:j8dd"d6d2dd4ƒZ=e*d<ƒx1e>d4ƒD]#Z?e*e:j<e=e?d;d2ƒƒqWeƒZ@e@jAdd+d=dd"d6d'd*d2ƒ\ZBZCZDere jƒe jeBjƒZe jeBj dƒddƒZe j!d>ƒe jƒe jeDd?d@dAƒe jeBjEdƒd@dBƒe j!dCƒe jFƒqneddgej4j5ej4j5gƒZGeGjAd!dDddEƒZHe*eHdjIdƒjIdƒƒe*eHdjIƒƒe*eHdj dƒj dƒƒe*eHdj ƒƒe*eHdjJƒe*eHdjIƒƒndS(Gs6getting started with diffusions, continuous time stochastic processes Author: josef-pktd License: BSD References ---------- An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations Author(s): Desmond J. Higham Source: SIAM Review, Vol. 43, No. 3 (Sep., 2001), pp. 525-546 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/3649798 http://www.sitmo.com/ especially the formula collection Notes ----- OU process: use same trick for ARMA with constant (non-zero mean) and drift some of the processes have easy multivariate extensions *Open Issues* include xzero in returned sample or not? currently not *TODOS* * Milstein from Higham paper, for which processes does it apply * Maximum Likelihood estimation * more statistical properties (useful for tests) * helper functions for display and MonteCarlo summaries (also for testing/checking) * more processes for the menagerie (e.g. from empirical papers) * characteristic functions * transformations, non-linear e.g. log * special estimators, e.g. Ait Sahalia, empirical characteristic functions * fft examples * check naming of methods, "simulate", "sample", "simexact", ... ? stochastic volatility models: estimation unclear finance applications ? option pricing, interest rate models iÿÿÿÿ(tprint_functionN(tstatstsignalt DiffusioncBsAeZdZd„Zddddd„Zddddd„ZRS(s:Wiener Process, Brownian Motion with mu=0 and sigma=1 cCsdS(N((tself((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyt__init__=sidicCsp|d|}tj|d|ƒ}tj|ƒtjjd||fƒ}tj|dƒ}||_||fS(s*generate sample of Wiener Process gð?itsize(tnptlinspacetsqrttrandomtnormaltcumsumtdW(RtnobstTtdttnreplttR tW((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyt simulateW@s ( c CsU|jd|d|d|d|ƒ\}}|||ƒ}|jdƒ} || |fS(smget expectation of a function of a Wiener Process by simulation initially test example from RRRRi(Rtmean( RtfuncRRRRRRtUtUmean((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyt expectedsimJs*N(t__name__t __module__t__doc__RtNoneRR(((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR:s  tAffineDiffusioncBsGeZdZd„Zddddd„Zddddddd„ZRS(sò differential equation: :math:: dx_t = f(t,x)dt + \sigma(t,x)dW_t integral: :math:: x_T = x_0 + \int_{0}^{T}f(t,S)dt + \int_0^T \sigma(t,S)dW_t TODO: check definition, affine, what about jump diffusion? cCsdS(N((R((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyResidic Csr|jd|d|d|d|ƒ\}}|jƒ|jƒ|}tj|dƒ}|jdƒ} || |fS(NRRRRii(Rt_driftt_sigRR R( RRRRRRRtdxtxtxmean((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pytsimhs *ic Csk||}|dkr"|j}n|dkr?|d|}n|jd|d|d|d|ƒ\}}|j} tj|d|ƒ}||} ||} tj|| fƒ} |} || dd…df*N(RRRRRR$R0(((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRTs  tExactDiffusioncBs/eZdZd„Zddd„Zd„ZRS(stDiffusion that has an exact integral representation this is currently mainly for geometric, log processes cCsdS(N((R((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR§sgð?ic Csˆtj||||ƒ}tj|j |ƒ}tjjd||fƒ}|j|ƒ|j|ƒ|}tj dgd| g|ƒS(s`ddt : discrete delta t should be the same as an AR(1) not tested yet Rgð?( RRtexptlambdR R t _exactconstt _exactstdRtlfilter( RR%RtddtRRtexpddttnormrvstinc((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyt exactprocessªs  cCsStj|j |ƒ}|||j|ƒ}|j|ƒ}tjd|d|ƒS(Ntloctscale(RR2R3R4R5Rtnorm(RR%Rtexpnttmeanttstdt((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyt exactdist»s(RRRRR;RB(((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR1 s tArithmeticBrowniancBsDeZdZd„Zd„Zd„Zdddd„Zd„ZRS( s2 :math:: dx_t &= \mu dt + \sigma dW_t cCs||_||_||_dS(N(R%tmutsigma(RR%RDRE((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRÇs  cOs|jS(N(RD(Rtargstkwds((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRÌscOs|jS(N(RE(RRFRG((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR Îsgð?icCs|dkr|j}ntj||||ƒ}tjjd||fƒ}|j|jtj|ƒ|}|tj |dƒS(s7ddt : discrete delta t not tested yet RiN( RR%RRR R Rt_sigmaR R (RRR%R7RRR9R:((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR;Ðs   !cCsPtj|j |ƒ}|j|}|jtj|ƒ}tjd|d|ƒS(NR<R=(RR2R3RRHR RR>(RR%RR?R@RA((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRBÝs N( RRRRRR RR;RB(((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRCÁs     tGeometricBrowniancBs)eZdZd„Zd„Zd„ZRS(sýGeometric Brownian Motion :math:: dx_t &= \mu x_t dt + \sigma x_t dW_t $x_t $ stochastic process of Geometric Brownian motion, $\mu $ is the drift, $\sigma $ is the Volatility, $W$ is the Wiener process (Brownian motion). cCs||_||_||_dS(N(R%RDRE(RR%RDRE((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRðs  cOs|d}|j|S(NR"(RD(RRFRGR"((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRõs cOs|d}|j|S(NR"(RE(RRFRGR"((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR øs (RRRRRR (((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRIäs   t OUprocesscBsSeZdZd„Zd„Zd„Zd„Zddd„Zd„Zd „Z RS( s¢Ornstein-Uhlenbeck :math:: dx_t&=\lambda(\mu - x_t)dt+\sigma dW_t mean reverting process TODO: move exact higher up in class hierarchy cCs(||_||_||_||_dS(N(R%R3RDRE(RR%RDR3RE((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR s   cOs|d}|j|j|S(NR"(R3RD(RRFRGR"((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRs cOs|d}|j|S(NR"(RE(RRFRGR"((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR s cCsYtj|j |ƒ}|||jd||jtjd||d|jƒ|S(Nig@(RR2R3RDRER (RR%RR9R?((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pytexactsgð?ic CsËtj||||ƒ}tj|j |ƒ}tj|j |ƒ}tjjd||fƒ}ddlm} |jd||j tj d||d|jƒ|} | j dgd| g| ƒS(s¢ddt : discrete delta t should be the same as an AR(1) not tested yet # after writing this I saw the same use of lfilter in sitmo Riÿÿÿÿ(Rig@gð?( RRR2R3R R tscipyRRDRER R6( RR%RR7RRR?R8R9RR:((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR;s.cCstj|j |ƒ}|||jd|}|jtjd||d|jƒ}ddlm}|jd|d|ƒS(Nig@iÿÿÿÿ(RR<R=( RR2R3RDRER RLRR>(RR%RR?R@RAR((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRB2s )cCs×t|ƒd}tjtj|ƒ|d fƒ}tjj||dddƒ\}}}}|\} } ||d} tj| ƒ |} tj| dtj| ƒd| d|ƒ} | d| }|| | fS(sNassumes data is 1d, univariate time series formula from sitmo iiÿÿÿÿtrcondg@i(tlenRt column_stacktonestlinalgtlstsqtlogR (RtdataRRtexogtparesttrestranktsingtconsttslopeterrvarR3RERD((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pytfitls;s"+ 1( RRRRRR RKR;RBR](((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRJýs      t SchwartzOnecBsJeZdZd„Zd„Zd„Zddd„Zd„Zd„ZRS( sÏthe Schwartz type 1 stochastic process :math:: dx_t = \kappa (\mu - \ln x_t) x_t dt + \sigma x_tdW \ The Schwartz type 1 process is a log of the Ornstein-Uhlenbeck stochastic process. cCs1||_||_||_||_||_dS(N(R%RDtkappaR3RE(RR%RDR_RE((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRVs     cCs%d||j|jdd|jS(Niig@(RDRER_(RR?((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR4]scCs'|jtjd||d|jƒS(Nig@(RERR R_(RR?((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR5`sgð?icCsFtj|ƒ}t|j|ƒj||d|d|ƒ}tj|ƒS(s/uses exact solution for log of process R7R(RRStsupert __class__R;R2(RR%RR7Rtlnxzerotlnx((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR;cs*cCs\tj|j |ƒ}tj|ƒ||j|ƒ}|j|ƒ}tjd|d|ƒS(NR<R=(RR2R3RSR4R5Rtlognorm(RR%RR?R@RA((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyRBjs cCsFt|ƒd}tjtj|ƒtj|d ƒfƒ}tjj|tj|dƒddƒ\}}}}|\} } ||d} tj| ƒ |} tj| | dtjd| |ƒƒ} | dtj| |ƒ| dd| }tj |ƒdkr|d}ntj | ƒd kr9| d} n|| | fS( sNassumes data is 1d, univariate time series formula from sitmo iiÿÿÿÿRMg@iþÿÿÿii(i(i( RNRRORPRSRQRRR R2tshape(RRTRRRURVRWRXRYRZR[R\R_RERD((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR]qs+4 ,,  ( RRRRR4R5R;RBR](((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR^Ks     tBrownianBridgecBs#eZd„Zdddd„ZRS(cCsdS(N((R((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyR†sigð?cCs‡|d}|d|}tj||||ƒ}tj|||ƒ}||d||g} d|||} ||||} |tj|d||ƒ} |tj||||||ƒ} tj||fƒ}||dd…df<| tjjd||fƒ}xetd|ƒD]T}|dd…|df| || ||dd…|f|dd…|fætRiôsBrownian Motion - expi g @R%gð?RDg{®Gáz„?REgà?idsGeometric Brownians$Geometric Brownian - log-transformedgš™™™™™©?sArithmetic BrownianR3gš™™™™™¹?Rii R7sOrnstein-UhlenbeckR_i2s Schwartz OneRs true: mu=1, kappa=0.5, sigma=0.1icsBrownian Bridgetrtlabelt theoreticalt simulatedsBrownian Bridge - Variancei Ni(ii (KRt __future__RtnumpyRRLRRtmatplotlib.pyplottpyplottplttobjectRRR1RCRIRJR^RfRrRtdoplotRtexamplestwRtwstfiguretplotRttmpRttitleRRtusRQR>R2tinftaverrRytgbR0tgbsRStabtabstoutousRKR R toueRR;touestsotsosR]tsos2RgRptbbRqtbbsRRjtstdtlegendtcptcpsR(R(((s@lib/python2.7/site-packages/statsmodels/sandbox/tsa/diffusion.pyt2sÆ L!#N:7    "  5    ' ! $&+ ! ) ! 0    '