statistical data analysis of ﬁnancial time series and...

S.M. Iacus 2011 R in Finance 1 / 113

Statistical data analysis of financial time seriesand option pricing in R

S.M. Iacus (University of Milan)

Chicago, R/Finance 2011, April 29th

Explorative Data Analysis

Explorative DataAnalysis

NYSE data

Estimation of FinancialModels

Likelihood approach

Two stage least squaresestimation

Model selection

Numerical Evidence

Application to real data

The change pointproblem

Overview of the yuimapackage

Option Pricing with R


Cluster Analysis


NYSE data


Likelihood approach


Model selection

Numerical Evidence






Cluster analysis is concerned with discovering group structure among nobservations, for example time series.

Such methods are based on the notion of dissimilarity. A dissimilaritymeasure d is a symmetric: d(A,B) = d(B,A), non-negative, and suchthat d(A,A) = 0.

Dissimilarities can, but not necessarily be, a metric, i.e.d(A,C) d(A,B) + d(B,C).

Cluster Analysis


NYSE data


Likelihood approach


Model selection

Numerical Evidence






A clustering method is an algorithm that works on a n n dissimilaritymatrix by aggregating (or, viceversa, splitting) individuals into groups.

Agglomerative algorithms start from n individuals and, at each step,aggregate two of them or move an individual in a group obtained at someprevious step. Such an algorithm stops when only one group with all the nindividuals is formed.

Divisive methods start from a unique group of all individuals by splittingit, at each step, in subgroups till the case when n singletons are formed.

These methods are mainly intended for exploratory data analysis and eachmethod is influenced by the dissimilarity measure d used.

R offers a variety to clustering algorithm and distances to play with but,up to date, not so much towards clustering of time series.

Real Data from NYSE


Next: time series of daily closing quotes, from 2006-01-03 to 2007-12-31, for thefollowing 20 financial assets:

Microsoft Corporation (MSOFT in the plots) Advanced Micro Devices Inc. (AMD)Dell Inc. (DELL) Intel Corporation (INTEL)Hewlett-Packard Co. (HP) Sony Corp. (SONY)Motorola Inc. (MOTO) Nokia Corp. (NOKIA)Electronic Arts Inc. (EA) LG Display Co., Ltd. (LG)Borland Software Corp. (BORL) Koninklijke Philips Electronics NV (PHILIPS)Symantec Corporation (SYMATEC) JPMorgan Chase & Co (JMP)Merrill Lynch & Co., Inc. (MLINCH) Deutsche Bank AG (DB)Citigroup Inc. (CITI) Bank of America Corporation (BAC)Goldman Sachs Group Inc. (GSACHS) Exxon Mobil Corp. (EXXON)

Quotes come from NYSE/NASDAQ. Source Yahoo.com.

Real Data from NYSE. How to cluster them?


25

35

MSOFT

10

30

AMD

20

26

32

DELL

18

24

INTEL

30

45

HP

405060

SONY

16

22

MOTO

20

35

NOKIA

45

55

EA

2006 2007 2008

15

25

LG

Index

35

BORL

30

40

PHILIPS

15

19

SYMATEC

40

50

JPM

50

80

MLINCH

100

140

DB

30

45

CITI

42

50

BAC

140

220

GSACHS

2006 2007 2008

60

80

EXXON

Index

quotes

How to measure the distance between two time series?


NYSE data


Likelihood approach


Model selection

Numerical Evidence






The are several ways to measure the distance between two time series

the Euclidean distance dEUC

the Short-Time-Series distance dSTS

the Dynamic Time Warping distance dDTW

the Markov Operator distance dMO

the Euclidean distance dEUC


NYSE data


Likelihood approach


Model selection

Numerical Evidence






Let X and Y be two time series, then the usual Euclidean distance

dEUC(X,Y ) =

N

i=1

(Xi Yi)2

is one of the most used in the applied literature. We use it only forcomparison purposes.

the Short-Time-Series distance dSTS


NYSE data


Likelihood approach


Model selection

Numerical Evidence






Let X and Y be two time series, then the usual Euclidean distance

dSTS(X,Y ) =

N

i=1

(

Xi Xi1

Yi Yi1

)2

.

Dynamic Time Warping distance dDTW


NYSE data


Likelihood approach


Model selection

Numerical Evidence






Let X and Y be two time series, then

dDTW (X,Y ) =

N

i=1

(

Xi Xi1

Yi Yi1

)2

.

DTW allows for non-linear alignments between time series notnecessarily of the same length.

Essentially, all shiftings between two time series are attempted and eachtime a cost function is applied (e.g. a weighted Euclidean distancebetween the shifted series). The minimum of the cost function over allpossible shiftings is the dynamic time warping distance dDTW .

The Markov operator


ConsiderdXt = b(Xt)dt+ (Xt)dWt

therefore, the discretized observations Xi form a Markov process and all the mathematicalproperties are embodied in the so-called transition operator

Pf(x) = E{f(Xi)|Xi1 = x}

with f is a generic function, e.g. f(x) = xk.

Notice that P depends on the transition density between Xi and Xi1, so we putexplicitly the dependence on in the notation.

Luckily, there is no need to deal with the transition density, we can estimate P directlyand easily.

The Markov Operator distance dMO


NYSE data


Likelihood approach


Model selection

Numerical Evidence






Let X and Y be two time series, then

dMO(X,Y) =

j,kJ

(P)j,k(X) (P)j,k(Y)

, (1)

where (P)j,k() is a matrix and it is calculated as in (2) separately for Xand Y.

(P)j,k(X) =1

2N

N

i=1

{j(Xi1)k(Xi) + k(Xi1)j(Xi)} (2)

The terms (P)j,k are approximations of the scalar product< Pj , k >b, where b and are the unknown drift and diffusioncoefficient of the model. Thus, (P)j,k contains all the information of theMarkovian structure of X .

Simulations


We simulate 10 paths Xi, i = 1, . . . , 10, according to the combinations of drift bi anddiffusion coefficients i, i = 1, . . . , 4 presented in the following table

1(x) 2(x) 3(x) 4(x)

b1(x) X10, X1 X5b2(x) X2,X3 X4b3(x) X6, X7b4(x) X8

where

b1(x) = 1 2x, b2(x) = 1.5(0.9 x), b3(x) = 1.5(0.5 x), b4(x) = 5(0.05 x)

1(x) = 0.5 + 2x(1 x), 2(x) =

0.55x(1 x)

3(x) =

0.1x(1 x), 4(x) =

0.8x(1 x)

The process X9=1-X1, hence it has drift b1(x) and the same quadratic variation of X1and X10.

Trajectories


X1

0.0

0.4

0.8

X2

0.0

0.4

0.8

X3

0.0

0.4

0.8

X4

0.0

0.4

0.8

1 2 3 4 5 6

0.0

0.4

0.8

X5

Time

X6

0.0

0.4

0.8

X7

0.0

0.4

0.8

X8

0.0

0.4

0.8

X9

0.0

0.4

0.8

1 2 3 4 5 6

0.0

0.4

0.8

X10

Time

Simulated diffusions

Dendrograms


Multidimensional scaling


Software


NYSE data


Likelihood approach


Model selection

Numerical Evidence






The package sde for the R statistical environment is freely available athttp://cran.R-Project.org.

It contains the function MOdist which calculates the Markov Operatordistance and returns a dist object.

data(quotes)

d

Multidimensional scaling. Clustering of NYSE data.


Estimation of Financial Models



Examples

Likelihood approach


Model selection

Numerical Evidence






Statistical models


We focus on the general approach to statistical inference for the parameter of models inthe wide class of diffusion processes solutions to stochastic differential equations

dXt = b(,Xt)dt+ (,Xt)dWt

with initial condition X0 and the p-dimensional parameter of interest.

Examples



Examples

Likelihood approach


Model selection

Numerical Evidence






geometric Brownian motion (gBm)

dXt = Xtdt+ XtdWt

Cox-Ingersoll-Ross (CIR)

dXt = (1 + 2Xt)dt+ 3

XtdWt

Chan-Karolyi-Longstaff-Sanders (CKLS)

dXt = (1 + 2Xt)dt+ 3X4t dWt

Examples



Examples

Likelihood approach


Model selection

Numerical Evidence






nonlinear mean reversion (At-Sahalia)

dXt = (1X1t + 0 + 1Xt + 2X

2t )dt+ 1X

t dWt

double Well potential (bimodal behaviour, highly nonlinear)

dXt = (Xt X3t )dt+ dWt

Jacobi diffusion (political polarization)

dXt = (

Xt 1

2

)

dt+

Xt(1Xt)dWt

Examples



Examples

Likelihood approach


Model selection

Numerical Evidence






Ornstein-Uhlenbeck (OU)

dXt = Xtdt+ dWt

radial Ornstein-Uhlenbeck

dXt = (X1t Xt)dt+ dWt

hyperbolic diffusion (dynamics of sand)

dXt =2

2

[

Xt2 + (Xt )2

]

dt+ dWt

...and so on.

Likelihood approach



Likelihood approach

MLE

QMLE


Model selection

Numerical Evidence






Likelihood in discrete time


Consider the modeldXt = b(,Xt)dt+ (,Xt)dWt

with initial condition X0 and the p-dimensional parameter of interest. By Markovproperty of diffusion processes, the likelihood has this form

Ln() =n

i=1

p (, Xi|Xi1) p(X0)

and the log-likelihood is

n() = logLn() =n

i=1

log p (, Xi|Xi1) + log(p(X0)) .

Assumption p(X0) irrelevant, hence p(X0) = 1

As usual, in most cases we dont know the conditional distribution p (, Xi|Xi1)

Maximum Likelihood Estimators (MLE)



Likelihood approach

MLE

QMLE


Model selection

Numerical Evidence






If we study the likelihood Ln() as a function of given the n numbers(X1 = x1, . . . , Xn = xn) and we find that this function has a maximum,we can use this maximum value as an estimate of .

In general we define maximum likelihood estimator of , and weabbreviate this with MLE, the following estimator

n = argmax

Ln()

= argmax

Ln(|X1, X2, . . . , Xn)

provided that the maximum exists.

MLE with R



Likelihood approach

MLE

QMLE


Model selection

Numerical Evidence






It is not always the case, that maximum likelihood estimators can beobtained in explicit form.

For what concerns applications to real data, it is important to know thatMLE are optimal in many respect from the statistical point of view,compared to, e.g. estimators obtained via the method of the moments(GMM) which can produce fairly biased estimates of the models.

R offers a prebuilt generic function called mle in the package stats4which can be used to maximize a likelihood.

The mle function actually minimizes the negative log-likelihood () asa function of the parameter where () = logL().

QUASI - MLE


Consider the multidimensional diffusion process

dXt = b(2, Xt)dt+ (1, Xt)dWt

where Wt is an r-dimensional standard Wiener process independent of the initial valueX0 = x0. Quasi-MLE assumes the following approximation of the true log-likelihood formultidimensional diffusions

n(Xn, ) = 1

2

n

i=1

{

log det(i1(1)) +1

n(Xi nbi1(2))

T1i1

(1)(Xi nbi1(2))

}

(3)

where = (1, 2), Xi = Xti Xti1 , i(1) = (1, Xti), bi(2) = b(2, Xti), = 2, A2 = ATA and A1 the inverse of A. Then the QML estimator of is

n = argmin

n(Xn, )

QUASI - MLE


Consider the matrix

(n) =

( 1nn

Ip 0

0 1nIq

)

where Ip and Iq are respectively the identity matrix of order p and q. Then, is possible toshow that

(n)1/2(n )d N(0, I()1).

where I() is the Fisher information matrix.

The above result means that the estimator of the parameter of the drift converges slowly tothe true value because nn = T slower than n, as n .

QUASI - MLE using yuima package


To estimate a model we make use of the qmle function in the yuima package, whoseinterface is similar to mle. Consider the model

dXt = 2Xtdt+ 1dWtwith 1 = 0.3 and 2 = 0.1

R> library(yuima)

R> ymodel n ysamp yuima set.seed(123)R> yuima

QUASI - MLE. True values: 1 = 0.3, 2 = 0.1


We can now try to estimate the parameters

R> mle1 coef(mle1)

theta1 theta20.30766981 0.05007788R> summary(mle1)Maximum likelihood estimation

Coefficients:Estimate Std. Error

theta1 0.30766981 0.02629925theta2 0.05007788 0.15144393

-2 log L: -280.0784

not very good estimate of the drift so now we consider a longer time series.

QUASI - MLE. True values: 1 = 0.3, 2 = 0.1


R> n ysamp yuima set.seed(123)R> yuima mle1 coef(mle1)

theta1 theta20.3015202 0.1029822R> summary(mle1)Maximum likelihood estimation

Coefficients:Estimate Std. Error

theta1 0.3015202 0.006879348theta2 0.1029822 0.114539931

-2 log L: -4192.279

Two stage least squares estimation



Likelihood approach


Model selection

Numerical Evidence






At-Sahalias interest rates model



Likelihood approach


Model selection

Numerical Evidence






At-Sahalia in 1996 proposed a model much more sophisticated than theCKLS to include other polynomial terms.

dXt = (1X1t + 0 + 1Xt + 2X

2t )dt+ 1X

t dBt

He proposed to use the two stage least squares approach in the followingmanner.




Likelihood approach


Model selection

Numerical Evidence







dXt = (1X1t + 0 + 1Xt + 2X

2t )dt+ 1X

t dBt

He proposed to use the two stage least squares approach in the followingmanner. First estimate the drift coefficients with a simple linear regressionlike

E(Xt+1 Xt|Xt) = 1X1t + 0 + (1 1)Xt + 2X2t




Likelihood approach


Model selection

Numerical Evidence







dXt = (1X1t + 0 + 1Xt + 2X

2t )dt+ 1X

t dBt


E(Xt+1 Xt|Xt) = 1X1t + 0 + (1 1)Xt + 2X2t

then, regress the residuals 2t+1 from the first regression to obtain theestimates of the coefficients in the diffusion term with

E(2t+1|Xt) = 0 + 1Xt + 2X3t




Likelihood approach


Model selection

Numerical Evidence







dXt = (1X1t + 0 + 1Xt + 2X

2t )dt+ 1X

t dBt


E(Xt+1 Xt|Xt) = 1X1t + 0 + (1 1)Xt + 2X2t

then, regress the residuals 2t+1 from the first regression to obtain theestimates of the coefficients in the diffusion term with

E(2t+1|Xt) = 0 + 1Xt + 2X3t

and finally, use the fitted values from the last regression to set the weightsin the second stage regression for the drift.

Example of 2SLS estimation


Stage one: regression

E(Xt+1 Xt|Xt) = 1X1t + 0 + (1 1)Xt + 2X2t

R> library(Ecdat)R> data(Irates)R> X Y stage1 coef(stage1)(Intercept) I(1/Y) Y I(Y^2)0.253478884 -0.135520883 -0.094028341 0.007892979

Example of 2SLS estimation


Stage regression now we extract the squared residuals and run a regression on those

E(2t+1|Xt) = 0 + 1Xt + 2X3t

R> eps2 mod w stage2 coef(stage2)(Intercept) I(1/Y) Y I(Y^2)0.189414678 -0.104711262 -0.073227254 0.006442481R> coef(mod)

b0 b1 b2 b30.00001000 0.09347354 0.00001000 0.00001000

Model selection



Likelihood approach


Model selection

Idea

AIC example

Lasso

Numerical Evidence






Idea


The aim is to try to identify the underlying continuous model on the basis of discreteobservations using AIC (Akaike Information Criterion) statistics defined as (Akaike1973,1974)

AIC = 2n(

(ML)n

)

+ 2dim(),

where (ML)n is the true maximum likelihood estimator and n() is the log-likelihood

Model selection via AIC


Akaikes index idea AIC = 2n(

(ML)n

)

+ 2dim() is to penalize this value

2n(

(ML)n

)

with the dimension of the parameter space

2 dim()

Thus, as the number of parameter increases, the fit may be better, i.e. 2n(

(ML)n

)

decreases, at the cost of overspecification and dim() compensate for this effect.

When comparing several models for a given data set, the models such that the AIC islower is preferred.

Model selection via AIC


In order to calculateAIC = 2n

(

(ML)n

)

+ 2dim(),

we need to evaluate the exact value of the log-likelihood n() at point (ML)n .

Problem: for discretely observed diffusion processes the true likelihood function is notknown in most cases

Uchida and Yoshida (2005) develop the AIC statistics defined as

AIC = 2n(

(QML)n

)

+ 2dim(),

where (QML)n is the quasi maximum likelihood estimator and n the local Gaussianapproximation of the true log-likelihood.

A trivial example



Likelihood approach


Model selection

Idea

AIC example

Lasso

Numerical Evidence






We compare three models

dXt = 1(Xt 2)dt+ 1

XtdWt (true model),

dXt = 1(Xt 2)dt+

1 + 2XtdWt (competing model 1),

dXt = 1(Xt 2)dt+ (1 + 2Xt)3dWt (competing model 2),

We call the above models Mod1, Mod2 and Mod3.

We generate data from Mod1 with parameters

dXt = (Xt 10)dt+ 2

XtdWt ,

and initial value X0 = 8. We use n = 1000 and = 0.1.

We test the performance of the AIC statistics for the three competingmodels

Simulation results. 1000 Monte Carlo replications



Likelihood approach


Model selection

Idea

AIC example

Lasso

Numerical Evidence






dXt = (Xt 10)dt+ 2

XtdWt (true model),

dXt = 1(Xt 2)dt+ 1

XtdWt (Model 1)

dXt = 1(Xt 2)dt+

1 + 2XtdWt (Model 2)

dXt = 1(Xt 2)dt+ (1 + 2Xt)3dWt (Model 3)

Model selection via AICModel 1 Model 2 Model 3

(true)99.2 % 0.6 % 0.2 %

QMLE estimates under the different models1 2 1 2 3

Model 1 1.10 8.05 0.90Model 2 1.12 8.07 2.02 0.54Model 3 1.12 9.03 7.06 7.26 0.61

LASSO estimation


The Least Absolute Shrinkage and Selection Operator (LASSO) is a useful and wellstudied approach to the problem of model selection and its major advantage is thesimultaneous execution of both parameter estimation and variable selection (seeTibshirani, 1996; Knight and Fu, 2000, Efron et al., 2004).To simplify the idea: take a full specified regression model

Y = 0 + 1X1 + 2X2 + + kXk

perform least squares estimation under L1 constraints, i.e.

= argmin

{

(Y X)T (Y X) +k

i=1

|i|}

model selection occurs when some of the i are estimated as zeros.

The SDE model



Likelihood approach


Model selection

Idea

AIC example

Lasso

Numerical Evidence






Let Xt be a diffusion process solution to

dXt = b(,Xt)dt+ (,Xt)dWt

= (1, ..., p) p Rp, p 1

= (1, ..., q) q Rq, q 1

b : p Rd Rd, : q Rd Rd Rm and Wt, t [0, T ], is astandard Brownian motion in Rm.

We assume that the functions b and are known up to and .

We denote by = (, ) p q = the parametric vector and with0 = (0, 0) its unknown true value.

Let Hn(Xn, ) = n() from the local Gaussian approximation.

Quasi-likelihood function


The classical adaptive LASSO objective function for the present model is then

min,

Hn(, ) +

p

j=1

n,j |j |+q

k=1

n,k|k|

n,j and n,k are appropriate sequences representing an adaptive amount of shrinkage foreach element of and .

Idea of Quadratic Approximation


By Taylor expansion of the original LASSO objective function, for around n (theQMLE estimator)

Hn(Xn, ) = Hn(Xn, n) + ( n)Hn(Xn, n) +1

2( n)Hn(Xn, n)( n)

+op(1)

= Hn(Xn, n) +1

2( n)Hn(Xn, n)( n) + op(1)

with Hn and Hn the gradient and Hessian of Hn with respect to .

The Adaptive LASSO estimator



Likelihood approach


Model selection

Idea

AIC example

Lasso

Numerical Evidence






Finally, the adaptive LASSO estimator is defined as the solution to thequadratic problem under L1 constraints

n = (n, n) = argmin

F().

with

F() = ( n)Hn(Xn, n)( n) +p

j=1

n,j |j |+q

k=1

n,k|k|

The lasso method is implemented in the yuima package.

How to choose the adaptive sequences



Likelihood approach


Model selection

Idea

AIC example

Lasso

Numerical Evidence






Clearly, the theoretical and practical implications of our method rely tothe specification of the tuning parameter n,j and n,k.

The tuning parameters should be chosen as is Zou (2006) in the followingway

n,j = 0|n,j|1 , n,k = 0|n,j|2 (4)

where n,j and n,k are the unpenalized QML estimator of j and krespectively, 1, 2 > 0 and usually taken unitary.

Numerical Evidence



Likelihood approach


Model selection

Numerical Evidence






A multidimensional example



Likelihood approach


Model selection

Numerical Evidence






We consider this two dimensional geometric Brownian motion processsolution to the stochastic differential equation

(

dXtdYt

)

=

(

1 11Xt + 12Yt2 + 21Xt 22Yt

)

dt+

(

11Xt 12Yt21Xt 22Yt

)(

dWtdBt

)

with initial condition (X0 = 1, Y0 = 1) and Wt, t [0, T ], andBt, t [0, T ], are two independent Brownian motions.

This model is a classical model for pricing of basket options inmathematical finance.

We assume that = (11 = 0.9, 12 = 0, 21 = 0, 22 = 0.7) and = (11 = 0.3, 12 = 0, 21 = 0, 22 = 0.2)

, = (, ).

Results


11 12 21 22 11 12 21 22True 0.9 0.0 0.0 0.7 0.3 0.0 0.0 0.2Qmle: n = 100 0.96 0.05 0.25 0.81 0.30 0.04 0.01 0.20

(0.08) (0.06) (0.27) (0.15) (0.03) (0.05) (0.02) (0.02)

Lasso: 0 = 0 = 1, n = 100 0.86 0.00 0.05 0.71 0.30 0.02 0.01 0.20(0.12) (0.00) (0.13) (0.09) (0.03) (0.05) (0.02) (0.02)

% of times i = 0 0.0 99.9 80.2 0.0 0.3 67.2 66.7 0.1

Lasso: 0 = 0 = 5, n = 100 0.82 0.00 0.00 0.66 0.29 0.01 0.00 0.20(0.12) (0.00) (0.00) (0.09) (0.03) (0.03) (0.01) (0.02)

% of times i = 0 0.0 100.0 99.9 0.0 0.4 86.9 89.7 0.2

Qmle: n = 1000 0.95 0.03 0.21 0.79 0.30 0.04 0.01 0.20(0.07) (0.04) (0.25) (0.13) (0.03) (0.06) (0.02) (0.02)

Lasso: 0 = 0 = 1, n = 1000 0.88 0.00 0.08 0.73 0.30 0.02 0.01 0.20(0.08) (0.00) (0.16) (0.09) (0.03) (0.05) (0.01) (0.02)

% of times i = 0 0.0 99.7 72.1 0.0 0.1 67.5 66.6 0.1

Lasso: 0 = 0 = 5, n = 1000 0.86 0.00 0.00 0.68 0.29 0.01 0.00 0.20(0.09) (0.00) (0.01) (0.06) (0.03) (0.04) (0.01) (0.02)

% of times i = 0 0.0 100.0 99.4 0.0 0.2 87.8 89.9 0.2




Likelihood approach


Model selection

Numerical Evidence






Interest rates LASSO estimation examples



Likelihood approach


Model selection

Numerical Evidence






rates

1950 1960 1970 1980 1990

05

10

15



LASSO estimation of the U.S. Interest Rates monthly data from 06/1964 to 12/1989.These data have been analyzed by many author including Nowman (1997), At-Sahalia(1996), Yu and Phillips (2001) and it is a nice application of LASSO.

Reference Model Merton (1973) dXt = dt+ dWt 0 0Vasicek (1977) dXt = (+ Xt)dt+ dWt 0Cox, Ingersoll and Ross (1985) dXt = (+ Xt)dt+

XtdWt 1/2

Dothan (1978) dXt = XtdWt 0 0 1Geometric Brownian Motion dXt = Xtdt+ XtdWt 0 1Brennan and Schwartz (1980) dXt = (+ Xt)dt+ XtdWt 1

Cox, Ingersoll and Ross (1980) dXt = X3/2t dWt 0 0 3/2

Constant Elasticity Variance dXt = Xtdt+ Xt dWt 0

CKLS (1992) dXt = (+ Xt)dt+ Xt dWt




Likelihood approach


Model selection

Numerical Evidence






Model Estimation Method Vasicek MLE 4.1889 -0.6072 0.8096

CKLS Nowman 2.4272 -0.3277 0.1741 1.3610

CKLS Exact Gaussian 2.0069 -0.3330 0.1741 1.3610(Yu & Phillips) (0.5216) (0.0677)

CKLS QMLE 2.0822 -0.2756 0.1322 1.4392(0.9635) (0.1895) (0.0253) (0.1018)

CKLS QMLE + LASSO 1.5435 -0.1687 0.1306 1.4452with mild penalization (0.6813) (0.1340) (0.0179) (0.0720)

CKLS QMLE + LASSO 0.5412 0.0001 0.1178 1.4944with strong penalization (0.2076) (0.0054) (0.0179) (0.0720)

LASSO selected: Cox, Ingersoll and Ross (1980) model

dXt =1

2dt+ 0.12 X3/2t dWt

Lasso in practice


An example of Lasso estimation using yuima package. We make use of real data withCKLS model

dXt = (+ Xt)dt+ Xt dWt

R> library(Ecdat)R> data(Irates)R> rates plot(rates)R> require(yuima)R> X mod yuima

Example of Lasso estimation


R> lambda10 start low upp lasso10 round(lasso10$mle, 3) # QMLEsigma gamma alpha beta0.133 1.443 2.076 -0.263

R> round(lasso10$lasso, 3) # LASSOsigma gamma alpha beta0.117 1.503 0.591 0.000

dXt = (+ Xt)dt+ Xt dWt

dXt = 0.6dt+ 0.12X3

2

t dWt

The change point problem



Likelihood approach


Model selection

Numerical Evidence



Least Squares

QMLE




Change point problem



Likelihood approach


Model selection

Numerical Evidence



Least Squares

QMLE




Discover from the data a structural change in the generating model.




Likelihood approach


Model selection

Numerical Evidence



Least Squares

QMLE




Discover from the data a structural change in the generating model.Next famous example shows historical change points for the Dow-Jonesindex.




Likelihood approach


Model selection

Numerical Evidence



Least Squares

QMLE




Discover from the data a structural change in the generating model.Next famous example shows historical change points for the Dow-Jonesindex.

750

850

950

1050

1972 1973 1974

Dow-Jones closings-0.06

0.00

0.04

1972 1973 1974

Dow-Jones returns

break of gold-US$ linkage (left) Watergate scandal (right)

Least squares approach



Likelihood approach


Model selection

Numerical Evidence



Least Squares

QMLE




De Gregorio and I. (2008) considered the change point problem for theergodic model

dXt = b(Xt)dt+(Xt)dWt, 0 t T,X0 = x0,

observed at discrete time instants. The change point problem isformulated as follows

Xt =

{

X0 + t0 b(Xs)ds+

1

t0 (Xs)dWs, 0 t

X + t b(Xs)ds+

2

t (Xs)dWs,

< t T

where (0, T ) is the change point and 1, 2 two parameters to beestimated.




Likelihood approach


Model selection

Numerical Evidence



Least Squares

QMLE




Let Zi =Xi+1 Xi b(Xi)n

n(Xi)be the standardized residuals. Then the

LS estimator of is n = k0/n where k0 is solution to

k0 = argmaxk

k

n Sk

Sn

, Sk =k

i=1

Z2i .




Likelihood approach


Model selection

Numerical Evidence



Least Squares

QMLE




Let Zi =Xi+1 Xi b(Xi)n

n(Xi)be the standardized residuals. Then the

LS estimator of is n = k0/n where k0 is solution to

k0 = argmaxk

k

n Sk

Sn

, Sk =k

i=1

Z2i .

If we assume to observe the following SDE with unknown drift b()

dXt = b(Xt)dt+dWt,

we can still estimate b() non parametrically and obtain a good changepoint estimator.

This approach has been used in Smaldone (2009) to re-analyze the recentglobal financial crisis using data from different markets.

June 30July 4

July 7July 11

July 14July18

August 18August 22

August 25August 29

Sept. 1Sept. 5

Sept. 8Sept. 12

Sept. 15Sept. 19

Sept. 22Sept. 26

Sept. 29Oct. 3

Oct. 6Oct. 10

Oct. 20Oct 24

LehmanBrothers

DJ Stoxx Global 1800 Banks

S&P MIBNikkei 225

FTSEDJ Stoxx 600

LehmanBrothers

DJ Stoxx 600 Banks

GoldmanSachs

Deutsche BankHSBC Nasdaq

Dj Stoxx Asia Pacific 600 BanksJP Morgan Chase

NyseDow JonesS&P 500FTSEDAXS&P MIBCACIBEXSMINikkei 225DJ Stoxx 600 Banks

DJ Stoxx Global 1800MCSI WorldMorgan StanleyBank of AmericaBarclaysRBSUnicreditIntesa SanpaoloDeutsche Bank (GER)Commerzbank

NyseDow JonesS&P 500FTSEDAXS&P MIBCACIBEXSMINikkei 225DJ Stoxx 600 Banks

DJ Stoxx Global 1800MCSI WorldMorgan StanleyBank of AmericaBarclaysRBSUnicreditIntesa SanpaoloDeutsche Bank (GER)Commerzbank

DJ Stoxx America 600 BanksDJ Stoxx 600 BanksDeutsche BankHBSCBarclaysDeutsche Bank (GER)CAC

DJ Stoxx America 600 BanksDJ Stoxx 600 BanksDeutsche BankHBSCBarclaysDeutsche Bank (GER)CAC

Time

cpoint function in the sde package


0 50 100 150 200

0.

04

0.02

0.00

0.02

0.04

log returns

Index

X

16

18

20

22

S [20040723/20050505]

16

18

20

22

Last 20.45Bollinger Bands (20,2) [Upper/Lower]: 21.731/20.100

Lug 232004

Ott 012004

Dic 012004

Feb 012005

Apr 012005

> S require(sde)> cpoint(S)$k0[1] 123$tau0[1] "2005-01-11"$theta12005-01-120.1020001$theta2[1] 0.3770111attr(,"index")[1] "2005-01-12"

Volatility Change-Point Estimation



Likelihood approach


Model selection

Numerical Evidence



Least Squares

QMLE




The theory works for SDEs of the form

dYt = btdt+ (Xt, )dWt, t [0, T ],

where Wt a r-dimensional Wiener process and bt and Xt aremultidimensional processes and is the diffusion coefficient (volatility)matrix.

When Y = X the problem is a diffusion model.

The process bt may have jumps but should not explode and it is treated asa nuisance in this model and it is completely unspecified.

Change-point analysis



Likelihood approach


Model selection

Numerical Evidence



Least Squares

QMLE




The change-point problem for the volatility is formalized as follows

Yt =

{

Y0 + t0 bsds+

t0 (Xs,

1)dWs for t [0, )

Y + t bsds+

t (Xs,

2)dWs for t [, T ].

The change point instant is unknown and is to be estimated, along with1 and

2, from the observations sampled from the path of (X,Y ).

Example of Volatility Change-Point Estimation


Consider the 2-dimensional stochastic differential equation

(

dX1tdX2t

)

=

(

1X1t3X2t

)

dt+

[

1.1 X1t 0 X1t0 X2t 2.2 X2t

](dW 1tdW 2t

)

X10 = 1.0, X20 = 1.0,

with change point instant at time = 4

x1

0.5

1.5

0 2 4 6 8 10

13

57

x2

the CPoint function in the yuima package



Likelihood approach


Model selection

Numerical Evidence



Least Squares

QMLE




the object yuima contains the model and the data from the previous slideso we can call CPoint on this yuima object

> t.est > t.est$tau[1] 3.99

Overview of the yuima package



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



The yuima object



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



The main object is the yuima object which allows to describe the modelin a mathematically sound way.

Then the data and the sampling structure can be included as well or, justthe sampling scheme from which data can be generated according to themodel.

The package exposes very few generic functions like simulate, qmle,plot, etc. and some other specific functions for special tasks.

Before looking at the details, let us see an overview of the main object.

YuimaSampling

randomdeterministic

tick timesspace disc.

...

regularirregular

multigridasynch.

Dataunivariate

multivariatets, xts, ...

Model

diffusionLvy

fractionalBM

MarkovSwitching HMM

yuima

yuima-class

yuima-class

model

data

sampling

characteristic

functional

data

yuima.data

yuima.data

original.data

zoo.data

model

yuima.model

yuima.model

drift

diffusion

hurst

measure

measure.type

state.variable

parameter

state.variable

jump.variable

time.variable

noise.number

equation.number

dimension

solve.variable

xinit

J.flag

sampling

yuima.sampling

yuima.sampling

Initial

Terminal

n

delta

grid

random

regular

sdelta

sgrid

oindex

interpolation

characteristic

yuima.characteristic


equation.number

time.scale

functional

yuima.functional

yuima.functional

F

f

xinit

e

Yuima

SimulationExact

Euler-Maruyama

Spacediscr.

NonparametricsCovariation p-variation

ParametricInference

High freq.Low freq.

QuasiMLEDiff,

Jumps,fBM

AdaptiveBayes

MCMCChange

point

Modelselection

Akaikes

LASSO-type

HypothesesTesting

Optionpricing

Asymptoticexpansion

MonteCarlo

The model specification



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



We consider here the three main classes of SDEs which can be easilyspecified. All multidimensional and eventually parametric models.




Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it




Diffusions dXt = a(t,Xt)dt+ b(t,Xt)dWt




Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it





Fractional Gaussian Noise, with H the Hurst parameter

dXt = a(t,Xt)dt+ b(t,Xt)dWHt




Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it





Fractional Gaussian Noise, with H the Hurst parameter

dXt = a(t,Xt)dt+ b(t,Xt)dWHt

Diffusions with jumps, Lvy

dXt = a(Xt)dt+ b(Xt)dWt +

|z|>1

c(Xt, z)(dt, dz)

+

0

dXt = 3Xtdt+ 11+X2t dWt



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



> mod1




Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



> mod1 str(mod1)Formal class yuima.model [package "yuima"] with 16 slots

..@ drift : expression((-3 * x))

..@ diffusion :List of 1

.. ..$ : expression(1/(1 + x^2))

..@ hurst : num 0.5

..@ jump.coeff : expression()

..@ measure : list()

..@ measure.type : chr(0)

..@ parameter :Formal class model.parameter [package "yuima"] with 6 slots

.. .. ..@ all : chr(0)

.. .. ..@ common : chr(0)

.. .. ..@ diffusion: chr(0)

.. .. ..@ drift : chr(0)

.. .. ..@ jump : chr(0)

.. .. ..@ measure : chr(0)

..@ state.variable : chr "x"

..@ jump.variable : chr(0)

..@ time.variable : chr "t"

..@ noise.number : num 1

..@ equation.number: int 1

..@ dimension : int [1:6] 0 0 0 0 0 0

..@ solve.variable : chr "x"

..@ xinit : num 0

..@ J.flag : logi FALSE




Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



And we can easily simulate and plot the model like

> set.seed(123)> X plot(X)

0.0 0.2 0.4 0.6 0.8 1.0

0.

8

0.6

0.

4

0.2

0.0

0.2

t

x




Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



The simulate function fills the slots data and sampling

> str(X)




Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it




> str(X)yuima

yuima-class

yuima-class

model

data

sampling

characteristic

functional

data

yuima.data

yuima.data

original.data

zoo.data

model

yuima.model

yuima.model

drift

diffusion

hurst

measure

measure.type

state.variable

parameter

state.variable

jump.variable

time.variable

noise.number

equation.number

dimension

solve.variable

xinit

J.flag

sampling

yuima.sampling

yuima.sampling

Initial

Terminal

n

delta

grid

random

regular

sdelta

sgrid

oindex

interpolation

characteristic



equation.number

time.scale

functional

yuima.functional

yuima.functional

F

f

xinit

e




Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it




> str(X)

Formal class yuima [package "yuima"] with 5 slots..@ data :Formal class yuima.data [package "yuima"] with 2 slots.. .. ..@ original.data: ts [1:101, 1] 0 -0.217 -0.186 -0.308 -0.27 ..... .. .. ..- attr(*, "dimnames")=List of 2.. .. .. .. ..$ : NULL.. .. .. .. ..$ : chr "Series 1".. .. .. ..- attr(*, "tsp")= num [1:3] 0 1 100.. .. ..@ zoo.data :List of 1.. .. .. ..$ Series 1:zooreg series from 0 to 1

..@ model :Formal class yuima.model [package "yuima"] with 16 slots

(...) output dropped

..@ sampling :Formal class yuima.sampling [package "yuima"] with 11 slots.. .. ..@ Initial : num 0.. .. ..@ Terminal : num 1.. .. ..@ n : num 100.. .. ..@ delta : num 0.1.. .. ..@ grid : num(0).. .. ..@ random : logi FALSE.. .. ..@ regular : logi TRUE.. .. ..@ sdelta : num(0).. .. ..@ sgrid : num(0).. .. ..@ oindex : num(0).. .. ..@ interpolation: chr "none"

Parametric model: dXt = Xtdt+ 11+Xt dWt



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it




..@ drift : expression((-theta * x))


.. ..$ : expression(1/(1 + x^gamma))

..@ hurst : num 0.5





.. .. ..@ all : chr [1:2] "theta" "gamma"

.. .. ..@ common : chr(0)

.. .. ..@ diffusion: chr "gamma"

.. .. ..@ drift : chr "theta"

.. .. ..@ jump : chr(0)

.. .. ..@ measure : chr(0)






..@ dimension : int [1:6] 2 0 1 1 0 0

..@ solve.variable : chr "x"

..@ xinit : num 0


Parametric model: dXt = Xtdt+ 11+Xt dWt



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



And this can be simulated specifying the parameters

> simulate(mod2,true.param=list(theta=1,gamma=3))

2-dimensional diffusions with 3 noises



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



dX1t = 3X1t dt+ dW 1t +X2t dW 3tdX2t = (X1t + 2X2t )dt+X1t dW 1t + 3dW 2t

has to be organized into matrix form

(

dX1tdX2t

)

=

(

3X1tX1t 2X2t

)

dt+

(

1 0 X2tX1t 3 0

)

dW 1tdW 2tdW 3t

> sol a b mod3

2-dimensional diffusions with 3 noises



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



dX1t = 3X1t dt+ dW 1t +X2t dW 3tdX2t = (X1t + 2X2t )dt+X1t dW 1t + 3dW 2t

> str(mod3)Formal class yuima.model [package "yuima"] with 16 slots

..@ drift : expression((-3 * x1), (-x1 - 2 * x2))


.. ..$ : expression(1, 0, x2)

.. ..$ : expression(x1, 3, 0)

..@ hurst : num 0.5





.. .. ..@ all : chr(0)

.. .. ..@ common : chr(0)


.. .. ..@ drift : chr(0)

.. .. ..@ jump : chr(0)

.. .. ..@ measure : chr(0)




..@ noise.number : int 3


..@ dimension : int [1:6] 0 0 0 0 0 0

..@ solve.variable : chr [1:2] "x1" "x2"

..@ xinit : num [1:2] 0 0


Plot methods inherited by zoo



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



> set.seed(123)> X plot(X,plot.type="single",col=c("red","blue"))

0.0 0.2 0.4 0.6 0.8 1.0

3

2

1

01

t

x1

Multidimensional SDE



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



Also models likes this can be specified

dX1t = X2t

X1t

2/3dW 1t ,

dX2t = g(t)dX3t ,

dX3t = X3t (dt+ (dW

1t +

1 2dW 2t )),

where g(t) = 0.4 + (0.1 + 0.2t)e2t

The above is an example of parametric SDE with more equations thannoises.

Fractional Gaussian Noise dYt = 3Ytdt+ dWHt



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



> mod4




Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it




..@ drift : expression((3 * y))


.. ..$ : expression(1)

..@ hurst : num 0.3





.. .. ..@ all : chr(0)

.. .. ..@ common : chr(0)


.. .. ..@ drift : chr(0)

.. .. ..@ jump : chr(0)

.. .. ..@ measure : chr(0)






..@ dimension : int [1:6] 0 0 0 0 0 0

..@ solve.variable : chr "y"

..@ xinit : num 0





Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



> set.seed(123)> X plot(X, main="Im fractional!")

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

t

y

Im fractional!

Jump processes



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



Jump processes can be specified in different ways in mathematics (andhence in yuima package).

Let Zt be a Compound Poisson Process (i.e. jumps follow somedistribution, e.g. Gaussian)

Then is is possible to consider the following SDE which involves jumps

dXt = a(Xt)dt+ b(Xt)dWt + dZt

Next is an example of Poisson process with intensity = 10 andGaussian jumps.

In this case we specify measure.type as CP (Compound Poisson)

Jump process: dXt = Xtdt+ dWt + Zt



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



> mod5 set.seed(123)> X plot(X, main="Im jumping!")

0.0 0.2 0.4 0.6 0.8 1.0

8

6

4

2

0

t

x

Im jumping!

Jump processes



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



Another way is to specify the Lvy measure. Without going into too muchdetails, here is an example of a simple OU process with IG Lvy measuredXt = Xtdt+ dZt

> mod6 set.seed(123)> plot( simulate(mod6, Terminal=10, n=10000), main="Im also jumping!")

0 2 4 6 8 10

05

1015

20

t

x

Im also jumping!

The setModel method



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it



Models are specified viasetModel(drift, diffusion, hurst = 0.5, jump.coeff, measure,

measure.type, state.variable = x, jump.variable = z, time.variable

= t, solve.variable, xinit) in

dXt = a(Xt)dt+ b(Xt)dWt + c(Xt)dZt

The package implements many multivariate RNG to simulate Lvy pathsincluding rIG, rNIG, rbgamma, rngamma, rstable.

Other user-defined or packages-defined RNG can be used freely.

The setModel method



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it









The setModel method



Likelihood approach


Model selection

Numerical Evidence




yuima object

functionalities

how to use it







The package implements many multivariate RNG to simulate Lvy pathsincluding rIG, rNIG, rbgamma, rngamma, rstable.Other user-defined or packages-defined RNG can be used freely.




Likelihood approach


Model selection

Numerical Evidence





Advertizing


R options for option pricing



Likelihood approach


Model selection

Numerical Evidence





Advertizing


There are several good packages on CRAN which implement the basicoption pricing formulas. Just to mention a couple (with apologizes to theauthors pf the other packages)

Rmetrics suite which includes: fOptions, fAsianOptions,fExoticOptions

RQuanLib for European, American and Asian option pricing

But if you need to go outside the standard Black & Scholes geometricBrownian Motion model, the only way reamins Monte Carlo analysis.And for jump processes, maybe, yuima package is one of the currentframeworks you an think to use.

In addition, yuima also offers asymptotic expansion formulas for generaldiffusion processes which can be used in Asian option pricing.

Asymptotic expansion



Likelihood approach


Model selection

Numerical Evidence





Advertizing


The yuima package can handle asymptotic expansion of functionals ofd-dimensional diffusion process

dXt = a(Xt , )dt+ b(X

t , )dWt, (0, 1]

with Wt and r-dimensional Wiener process, i.e. Wt = (W 1t , . . . ,Wrt ).

The functional is expressed in the following abstract form

F (Xt ) =

r

=0

T

0f(X

t , d)dW

t + F (X

t , ), W

0t = t

Estimation of functionals. Example.



Likelihood approach


Model selection

Numerical Evidence





Advertizing


Example: B&S asian call option

dXt = Xt dt+ X

t dWt

and the B&S price is related to E

{

max

(

1

T

T

0Xt dtK, 0

)}

. Thus

the functional of interest is

F (Xt ) =1

T

T

0Xt dt, r = 1




Likelihood approach


Model selection

Numerical Evidence





Advertizing


Example: B&S asian call option

dXt = Xt dt+ X

t dWt

and the B&S price is related to E

{

max

(

1

T

T

0Xt dtK, 0

)}

. Thus

the functional of interest is

F (Xt ) =1

T

T

0Xt dt, r = 1

withf0(x, ) =

x

T, f1(x, ) = 0, F (x, ) = 0

in

F (Xt ) =r

=0

T

0f(X

t , d)dW

t + F (X

t , )




Likelihood approach


Model selection

Numerical Evidence





Advertizing


So, the call option price requires the composition of a smooth functional

F (Xt ) =1

T

T

0Xt dt, r = 1

with the irregular function

max(xK, 0)

Monte Carlo methods require a HUGE number of simulations to get thedesired accuracy of the calculation of the price, while asymptoticexpansion of F provides unexpectedly accurate approximations.

The yuima package provides functions to construct the functional F ,and automatic asymptotic expansion based on Malliavin calculus startingfrom a yuima object.

setFunctional method



Likelihood approach


Model selection

Numerical Evidence





Advertizing


> diff.matrix model T xinit f F e yuima yuima

setFunctional method



Likelihood approach


Model selection

Numerical Evidence





Advertizing


> diff.matrix model T xinit f F e yuima yuima str(yuima)Formal class yuima [package "yuima"] with 5 slots

..@ data :Formal class yuima.data [package "yuima"] with 2 slots

..@ model :Formal class yuima.model [package "yuima"] with 16 slots

..@ sampling :Formal class yuima.sampling [package "yuima"] with 11 slots

..@ functional :Formal class yuima.functional [package "yuima"] with 4 slots

.. .. ..@ F : num 0

.. .. ..@ f :List of 2

.. .. .. ..$ : expression(x/T)

.. .. .. ..$ : expression(0)

.. .. ..@ xinit: num 1

.. .. ..@ e : num 0.3




Likelihood approach


Model selection

Numerical Evidence





Advertizing


Then, it is as easy as> F0 F0[1] 1.716424> max(F0-K,0) # asian call option price[1] 0.7164237




Likelihood approach


Model selection

Numerical Evidence





Advertizing


Then, it is as easy as> F0 F0[1] 1.716424> max(F0-K,0) # asian call option price[1] 0.7164237

and back to asymptotic expansion, the following script may work> rho get_ge

Add more terms to the expansion


The expansion of previous functional gives> asymp asymp$d0 + e * asymp$d1 # asymp. exp. of asian call price

[1] 0.7148786

> library(fExoticOptions) # From RMetrics suite> TurnbullWakemanAsianApproxOption("c", S = 1, SA = 1, X = 1,+ Time = 1, time = 1, tau = 0.0, r = 0, b = 1, sigma = e)Option Price:

[1] 0.7184944

> LevyAsianApproxOption("c", S = 1, SA = 1, X = 1,+ Time = 1, time = 1, r = 0, b = 1, sigma = e)Option Price:

[1] 0.7184944

> X mean(colMeans((X-K)*(X-K>0))) # MC asian call price based on M=1000 repl.

[1] 0.707046

Multivariate Asymptotic Expansion



Likelihood approach


Model selection

Numerical Evidence





Advertizing


Asymptotic expansion is now also available for multidimensionaldiffusion processes like the Heston model

dX1,t = aX1,t dt+ X

1,t

X2,t dW1t

dX2,t = c(dX2,t )dt+

X2,t

(

dW 1t +

1 2dW 2t)

i.e. functionals of the form F (X1,, X2,).

Advertizing


Iacus (2008) Simulation and Inference for Stochastic Differential Equations with RExamples, Springer, New York.

Iacus (2011) Option Pricing and Estimation of Financial Models with R, Wiley &Sons, Chichester.

Explorative Data AnalysisCluster AnalysisCluster AnalysisReal Data from NYSEReal Data from NYSE. How to cluster them?How to measure the distance between two time series?the Euclidean distance dEUCthe Short-Time-Series distance dSTSDynamic Time Warping distance dDTWThe Markov operatorThe Markov Operator distance dMOSimulationsTrajectoriesDendrogramsMultidimensional scalingSoftwareMultidimensional scaling. Clustering of NYSE data.

Estimation of Financial ModelsStatistical modelsExamplesExamplesExamples

Likelihood approachLikelihood in discrete timeMaximum Likelihood Estimators (MLE)MLE with RQUASI - MLEQUASI - MLEQUASI - MLE using yuima packageQUASI - MLE. True values: 1=0.3, 2=0.1QUASI - MLE. True values: 1=0.3, 2=0.1

Two stage least squares estimationAt-Sahalia's interest rates modelExample of 2SLS estimationExample of 2SLS estimation

Model selectionIdeaModel selection via AICModel selection via AICA trivial exampleSimulation results. 1000 Monte Carlo replicationsLASSO estimationThe SDE modelQuasi-likelihood functionIdea of Quadratic ApproximationThe Adaptive LASSO estimatorHow to choose the adaptive sequences

Numerical EvidenceA multidimensional exampleResults

Application to real dataInterest rates LASSO estimation examplesInterest rates LASSO estimation examplesInterest rates LASSO estimation examplesLasso in practiceExample of Lasso estimation

The change point problemChange point problemLeast squares approachLeast squares approachFinancial crisis 2008cpoint function in the sde packageVolatility Change-Point EstimationChange-point analysisExample of Volatility Change-Point Estimationthe CPoint function in the yuima package

Overview of the yuima packageThe yuima objectNice pictNice pictNice pictNice pictThe yuima object: overviewNice pictNice pictNice pictNice pictNice pictNice pictThe model specificationdXt = -3 Xt dt + 11+Xt2dWtdXt = -3 Xt dt + 11+Xt2dWtdXt = -3 Xt dt + 11+Xt2dWtdXt = -3 Xt dt + 11+Xt2dWtdXt = -3 Xt dt + 11+Xt2dWtParametric model: dXt = -Xt dt + 11+XtdWtParametric model: dXt = -Xt dt + 11+XtdWt2-dimensional diffusions with 3 noises2-dimensional diffusions with 3 noisesPlot methods inherited by zooMultidimensional SDEFractional Gaussian Noise dYt = 3 Yt dt + dWtHFractional Gaussian Noise dYt = 3 Yt dt + dWtHJump processesJump process: dXt = -Xt dt + dWt + ZtJump processesThe setModel methodThe setModel methodThe setModel methodThe setModel methodThe setModel method

Option Pricing with RR options for option pricingAsymptotic expansionEstimation of functionals. Example.Estimation of functionals. Example.setFunctional methodsetFunctional methodEstimation of functionals. Example.Add more terms to the expansionMultivariate Asymptotic ExpansionAdvertizing

statistical data analysis of ﬁnancial time series and...

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