User s Guide for the Matlab Library Implementing Closed Form MLE for Diffusions

User s Guide for the Matlab Library Implementing Closed Form MLE for Diffusions Yacine Aït-Sahalia Department of Economics and Bendheim Center for Finance Princeton University and NBER This Version: July 19 018 Abstract This document explains the use of the attached Matlab code for estimating the parameters of diffusions using closed-form maximum-likelihood. Copyright c 00-015 Yacine Aït-Sahalia. Redistribution and use in source and binary forms with or without modification are permitted provided that the following conditions are met: i Redistributions of source code must retain the above copyright notice this list of conditions and the following disclaimer; ii Redistributions in binary form must reproduce the above copyright notice this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. DISCLAIMER: THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CON- TRIBUTORS AS IS AND ANY EXPRESS OR IMPLIED WARRANTIES INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PAR- TICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT INDIRECT INCIDENTAL SPECIAL EXEM- PLARY OR CONSEQUENTIAL DAMAGES INCLUDING BUT NOT LIMITED TO PROCURE- MENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE DATA OR PROFITS; OR BUSI- NESS INTERRUPTION HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY WHETHER IN CONTRACT STRICT LIABILITY OR TORT INCLUDING NEGLIGENCE OR OTHERWISE ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 1 References This code implements in Matlab the closed-form maximum-likelihood estimation method for diffusions developed in: Aït-Sahalia Y. 1999 Transition Densities for Interest Rate and Other Nonlinear Diffusions Journal of Finance 54 1361-1395. Aït-Sahalia Y. 00 Maximum-Likelihood Estimation of Discretely-Sampled Diffusions: A Closed- Form Approximation Approach Econometrica 70 3-6. Financial support from the NSF under grants SES-0111140 SES-035077 DMS-053370 and SES-0850533 is gratefully acknowledged. Jia Li provided excellent research assistance in developing this Matlab code. Princeton NJ 08544-101. Phone: 609 58-4015. E-mail: yacine@princeton.edu. 1

Aït-Sahalia Y. 008 Closed-Form Likelihood Expansions for Multivariate Diffusions Annals of Statistics 36 906-937. Extensions to models with latent factors can be found in: Aït-Sahalia Y. and Kimmel R. 007 Maximum Likelihood Estimation of Stochastic Volatility Models Journal of Financial Economics 83 413-45. Aït-Sahalia Y. and Kimmel R. 010 Estimating Affine Multifactor Term Structure Models Using Closed-Form Likelihood Expansions Journal of Financial Economics 98 113 144. Installation This code uses anonymous functions which require Matlab version 7 R14 released in 004 or later..1 Installing the Package The only thing you need to do is to download the file ClosedFormMLE.zip and unzip it somewhere in your computer. You have two options: 1. You can work under the folder ClosedFormMLE\ you may rename this folder this does not matter keeping its subfolders in place and save all your files in that folder.. Or you add this library to your Matlab s search path. We recommend the first option for casual users of Matlab. This package contains the following folders and files: Type Name Usage Folder DERIVESTsuite Numerical differentiation Folder Models A library of closed-form log-likelihood functions Matlab function mymle.m Implementing MLE Matlab function mymle summary.m Post estimation analysis Matlab function example cev.m Demo: CEV model estimation Matlab function logdensityloglik.m A helper function Note: the package DERIVESTsuite was written by John D Errico and downloaded from: http://www.mathworks.com/matlabcentral/fileexchange/13490. It is enclosed here as a convenience for endusers. Please refer to the package for its license. The folder Models contains the formulae for the closed-form approximations of log likelihood functions for various stochastic differential equations SDE that have been implemented in this toolbox. Section 3 below gives the list of models that are curently implemented. Since this is an ongoing project we will keep adding models into the library. Interested users can visit the author s website for updates and request additional models by email. The folder DERIVESTsuite contains routines for numerical differentiation necessary to compute Fisher s information so that we can obtain the asymptotic standard errors of the estimates. Neither one of these folders need to be edited or modified. The two functions mymle.m and mymle summary.m need not be modified either. The simplest way to use this code is to first run example cev.m and then modify the file example cev.m as needed: estimate a different model replace the simulated data with real data etc.

. An Example: Estimating the CEV Model on Simulated Data Start Matlab and switch your current directory to ClosedFormMLE\. In Matlab open the file example cev.m and run it by pressing the F5 key in Matlab s editor: Debug >> Run Step 1 of the code simulates a time series according to the CEV model i.e. Model U3 see 3.1.3 with parameters a = 0.08 b = 0.5 c = 0.7 d = 1.5. The sampling interval is set to be 1/5 which should be thought of as weekly data. There are n = 500 observations saved in the variable x. Note: since the sample size is small the estimation is fast but the estimates are inaccurate. You may also want to experiment with n = 5 000 to get statistically better results. Step does the estimation with the function mymle.m. The code is output=mymle@modelu3xdelparam0; where @ModelU3 is the function handle of the log likelihood function x is our data del is our sampling interval = 1/5 and param0 is the initial value of the optimization procedure underlying MLE. The user may need to experiment with different initial values to find the global maximum. In this example we set the initial value as the true value of the parameters that is param0=[0.080.50.71.5]. See Section 3 for the ordering of parameters in each model. If we wanted to estimate the other model say Model U1 see 3.1.1 the syntax would be output=mymle@modelu1xdelparam0foru1; where the initial value param0foru1 should be 3-dimensional since Model U1 has only 3 parameters. The output is a Matlab structure which packages the results of the estimation. You do not have to worry if you are not familiar with this data structure. We provide the output in step 3. As the code runs it first maximize the log likelihood function and then prints the MLE estimates on the screen. It also tells you whether the maximization procedure converges under the default tolerance. After that it moves onto computing the standard error and then prints the estimates standard error constructed from the inverse of Hessian and the misspecification-robust standard error i.e. the sandwich estimate. Step 3 does the post estimation analysis. The code is mymle summaryoutput; where you take the output structure from step directly as your input in this step. The code does two things. First it reprints the table of estimates standard errors robust standard errors and exit status of the maximization procedure as below: 3

389.5 389.5 389 389 388.5 400 00 000 388.5 0.075 0.08 0.085 0 0.5 1 1.5 Parameter 1 Parameter 400 350 300 50 1800 0 0.5 1 1.5 Parameter 3 00 1.4 1.6 1.8 Parameter 4 Figure 1: Example: Log-Likelihood Near the Solution ****************************** COEF SE Robust SE 0.0764 0.006 0.006 0.7961 0.4397 0.4497 0.9755 0.481 0.4875 1.697 0.1917 0.193 ****************************** fminsearch converged to a solution. Secondly it plots the marginal log likelihood for each parameter in a neighborhood of the estimates. See Figure 1. The blue curve is the likelihood function while each red star is our estimate for the corresponding parameter. This gives us a way to visually check whether the maximization is successful of course this says nothing about global optimality!..3 Syntax of the Two Functions for reference only.3.1 mymle Syntax: output = mymlelogdensity x del param0 options; Inputs: logdensity is the function handle of the model for example @ModelU1 @ModelU3 @ModelB4... x: is the data. This is a n k matrix where n is the number of observations and k is the dimensionality of your model. For univariate models k = 1 and for bivariate models k =... 4

del: is the sampling interval of the time series param0 : initial parameter value of the maximization procedure. The dimensionality and ordering of its entries should be consistent with the underlying model listed in Section 3. options : this input is optional. The input is a Matlab option structure for its optimization toolbox. See Matlab s help document of optimset for more details. Output is structure consisting the following fields: exitmsg: exit status of the optimization procedure param: the MLE estimates see Section 3 for the ordering of parameters. variance: covariance estimates variance robust: robust covariance estimates se: standard errors se robust: robust standard errors objfun: the function handle of the negative log likelihood function used internally in the optimization procedure loglik: the optimized value of log likelihood exitflag: exit status reported by fminsearch..3. mymle summary Syntax: mymle summarymle output The input of this function is the output structure from mymle. 3 List of Models Currently Implemented The models below are in no particular order. The 9 canonical affine models and semi-affine models are implemented separately see below. There is of course some duplication since U4 and U7 are affine for instance. 3.1 Univariate Models Univariate models have the following canonical form: dx t = µ X t ; parameters dt + σ X t ; parameters dw t. 3.1.1 Model U1 µ x = ax + bx σ x = σx 3/ Parameters: a b σ. 5

3.1. Model U µ x = a + bx σ x = dx Parameters: a b d. 3.1.3 Model U3 µ x = b a x σ x = cx ρ Parameters: a b c ρ. Note: do not let the parameter ρ go below 1. Special cases for ρ = 1 and 3/. 3.1.4 Model U4 µ x = κ α x σ x = σx 1/ Parameters: κ α σ. 3.1.5 Model U5 µ x = θ 0 + θ 1 x + θ x + θ 3 x 3 σ x = γx ρ Parameters: γ ρ θ 0 θ 1 θ θ 3. Note: do not let the parameter ρ go below 1. Special cases for ρ = 1 and 3/. 3.1.6 Model U6 µ x = a + bx + cx + dx 3 σ x = f Parameters: a b c d f. 3.1.7 Model U7 µ x = κ α x σ x = σ Parameters: κ α σ. 6

3.1.8 Model U8 µ x = a 1 x + a 0 + a 1 x + a x σ x = σx ρ Parameters: a 1 a 0 a 1 a σ ρ. Note: do not let the parameter ρ go below 1. Special cases for ρ = 1 and 3/. 3.1.9 Model U9 µ x = a 1 x + a 0 + a 1 x + a x σ x = b 0 + b 1 x + b x b3 1/ Parameters: a 1 a 0 a 1 a b 0 b 1 b b 3. 3.1.10 Model U10 Note: do not let the parameter ρ go below 1. µ x = a 1 x + a 0 + a 1 x + a x σ x = b 0 + b 1 x + b x b3 Parameters: a 1 a 0 a 1 a b 0 b 1 b b 3. 3.1.11 Model U11 µ x = a + bx σ x = f + dx Parameters: a b f d. 3.1.1 Model U1 µ x = β x αx3 σ x = γx 1/ Parameters: β α γ. 3.1.13 Model U13 µ x = a 0 + a 1 x + a 1x + a x + a 3 x 3 σ x = σx ρ Parameters: a 0 a 1 a 1 a a 3 σ ρ. Note: do not let the parameter ρ go below 1. Special cases for ρ = 1 and 3/. 7

3.1.14 Model U14 µ x = h + ax + bx σ x = vx 3/ Parameters: h a b v. 3. Bivariate Models Consider a -dimensional process X t = X 1t X t determined by or written componentwise dx1t dx t = µ1 X 1t X t dt + µ X 1t X t dx t = µ X t dt + Σ X t dw t Σ 11 X 1t X t Σ 1 X 1t X t Σ 1 X 1t X t Σ X 1t X t dw1t. The Brownian motions are independent. Any correlation between the components is introduced by the off-diagonal terms in the Σ matrix. dw t 3..1 Model B1 a + bx µ x 1 x = c + dx Σ x 1 x = Parameters: r h a b c d. r x 0 h 1 r x 3.. Model B a0 + a 1 x 1 + a x µ x 1 x = b 0 + b 1 x 1 + b x Σ x 1 x = c 0 + c 1 x 1 + c x 0 0 d 0 + d 1 x 1 + d x Parameters: a 0 a 1 a b 0 b 1 b c 0 c 1 c d 0 d 1 d. 3..3 Model B3 µ x / µ x 1 x = α + βx x 0 Σ x 1 x = σρx γ σ 1 ρ x γ Parameters: µ α β σ ρ γ. 8

3..4 Model B4 a 0 + a 1 x µ x 1 x = b a x + λgx β a + f x a 1 r a + f x a r a + f x a Σ x 1 x = Parameters: a 0 a 1 a b λ g β f r. 0 gx β 3..5 Model B5 bx1 µ x 1 x = c dx hx 1 x 0 Σ x 1 x = gr x g 1 r x Parameters: b c d h g r. 3..6 Model B6 This is the Heston model with x 1 being the log of the asset price and x its stochastic volatility. m x / µ x 1 x = a bx x 0 Σ x 1 x = sr x s 1 r x Parameters: m a b s r. or Note that this model is observationally equivalent to the ones with 1 r x r x Σ x 1 x = 0 s x Σ x 1 x = r x 1 r x s x 0. 3..7 Model B7 0 µ x 1 x = a 1 a x x1 γ Σ x 1 x = ηx 1 x γ x 0 Parameters: a 1 a γ η. 9

3..8 Model B8 a + bx1 µ x 1 x = cx dx γ 1 Σ x 1 x = ex 0 0 f Parameters: a b c d γ f. 3..9 Model B9 θ5 1 µ x 1 x = ex θ θ 1 x e x 0 Σ x 1 x = θ 3 θ 4 1 θ 4 θ 3 Parameters: θ 1 θ θ 3 θ 4 θ 5. 3..10 Model B10 b1 a 1 x 1 µ x 1 x = b a x g 1 0 Σ x 1 x = 0 g x Parameters: a 1 a b 1 b g 1 g. 3..11 Model B11 k1 + k x µ x 1 x = κ θ x 1 ρ x ρ x Σ x 1 x = 0 σx Parameters: k 1 k ρ κ θ σ. 3..1 Model B1 ax1 µ x 1 x = bx cx 1 e x 0 Σ x 1 x = dr d 1 r Parameters: a b c d r. 10

3..13 Model B13 b11 a 1 x 1 + b 1 a x µ x 1 x = b 1 a 1 x 1 + b a x σ 11 σ 1 Σ x 1 x = σ 1 σ Parameters: a 1 a b 11 b 1 b 1 b σ 11 σ 1 σ 1 σ. 3..14 Model B14 k1 x x 1 µ x 1 x = k θ x σ 1 x1 0 Σ x 1 x = 0 σ x Parameters: k 1 k θ σ 1 σ. 3..15 Model B15 a + bx1 µ x 1 x = fx 1 + dx x1 0 Σ x 1 x = h 1 + gx1 Parameters: g h a b f d. 3..16 Model B16 a + bx1 + gx µ x 1 x = d + ηx 1 + fx x1 0 Σ x 1 x = 0 x Parameters: a b g d η f. 3..17 Model B17 a00 a 1 + a x / + n 0 1 g 1 + nu 1 g 1 a 1 + a x b+d µ x 1 x = a 01 + a 11 x + nu 1 g 11 a 1 + a x b+d 1 g Σ x 1 x = 1 a1 + a x g 1 a1 + a x 0 g 11 a 1 + a x b Parameters: a 00 a 01 a 11 A 1 A n 0 nu 1 g 1 g 11 b d. 11

3..18 Model B18 3..19 Model B19 b1 x 1 µ x 1 x = a + b x g 11 e x1 0 Σ x 1 x = g r g 1 r Parameters: b 1 a b g 11 g r. b1 x 1 µ x 1 x = a + b x e x 0 Σ x 1 x = g r g 1 r Parameters: b 1 a b g r. 3..0 Model B0 a1 + b 1 x 1 µ x 1 x = a + b x x 0 Σ x 1 x = gr x g 1 r x P arameters : a 1 b 1 a b g r. 3..1 Model B1 a 1 b 1 x 1 µ x 1 x = a 1 b 1 x 1 + a b x x1 0 Σ x 1 x = g 1 x1 g x1 Parameters: a 1 b 1 a 1 a b g 1 g. 3.. Model B k1 + k x µ x 1 x = k 3 a x 1 r x r x Σ x 1 x = 0 s x b Parameters: k 1 k k 3 a r s b. 3.3 Trivariate Models Again the Brownian motions are independent. Any correlation between the components is introduced by the off-diagonal terms in the Σ matrix. Some models are specified by the local variance matrix ΣΣ directly ΣΣ is what matters not Σ. 1

3.3.1 Model T1 µ x 1 x x 3 = Σ x 1 x x 3 = k 11 m 1 x 1 + k 1 m x k 1 m 1 x 1 + k m x k 33 m 3 x 3 a 1 + b 1 x 1 1/ 0 0 0 a + b x 1/ 0 0 0 a 3 + b 3 x 3 Parameters: m 1 m m 3 k 11 k 1 k 1 k k 33 a 1 b 1 a b a 3 b 3. 3.3. Model T µ x 1 x x 3 = Σ x 1 x x 3 Σ x 1 x x 3 = a 1 x + x 3 x 1 a + b x a 3 + b 3 x 3 s 1x 1 s 1 s r 1 x 1 s 1 s 3 r 13 x 1 s s s 3 r 3 s 3 Parameters: a 1 a b a 3 b 3 s 1 s s 3 r 1 r 13 r 3. 3.3.3 Model T3 µ x 1 x x 3 = Σ x 1 x x 3 Σ x 1 x x 3 = k 11 g 1 x 1 + k 1 g x + k 13 g 3 x 3 k g x k 33 g 3 x 3 s 11 x 1 s 1 x s 13 x 3 0 s x 0 0 0 s 33 x 3 Parameters: g 1 g g 3 k 11 k 1 k 13 k k 33 s 11 s 1 s 13 s s 33. 3.3.4 Model T4 b 1 x 1 µ x 1 x x 3 = b x b 3 x 3 Σ x 1 x x 3 = d 11 x 1 0 0 d 1 x x 1 d x 0 d 31 x 3 x 1 d 3 x 3 x d 33 x 3 Parameters: b 1 b b 3 d 11 d d 33 d 1 d 31 d 3 13

3.3.5 Model T5 µ x 1 x x 3 = x 3 x / v 1 v x r 1 r x 3 x 0 0 ρ Σ x 1 x x 3 = 1 v 3 x 1 ρ 1 v 3 x 0 ρ r 3 ρ 13 r 3 ρ 13ρ 1 3 r 1 ρ 3 1 ρ 13 ρ 3 ρ 13ρ 1 1 1 ρ 1 Parameters: v 1 v v 3 r 1 r r 3 ρ 1 ρ 13 ρ 3. 3.3.6 Model T6 µ x 1 x x 3 = Σ x 1 x x 3 = a 1 b 1 x 1 a 1 b 1 x 1 + a b x a 31 b 1 x 1 + a 3 b x + a 3 b 3 x 3 x1 0 0 g 1 x1 g x1 0 g 31 x1 g 3 x1 g 33 x1 Parameters: a 1 b 1 a 1 a b a 31 a 3 a 3 b 3 g 1 g g 31 g 3 g 33. 3.4 Canonical Affine Models Dimensions 1 to 3 3.4.1 Model AY01 µ x 1 = κ 11 x 1 σ x 1 = 1 Pamameters: κ 11. 3.4. Model AY11 µ x 1 = κ 11 θ 1 x 1 σ x 1 = x 1/ 1 Parameters: θ 1 κ 11. 3.4.3 Model AY0 κ 11 x 1 µ x 1 x = κ 1 x 1 κ x 1 0 Σ x 1 x = 0 1 Parameters: κ 11 κ 1 κ. 14

3.4.4 Model AY1 κ 11 θ 1 x 1 µ x 1 x = κ 1 θ 1 x 1 κ x x 1/ 1 0 Σ x 1 x = 0 1 + β 1 x 1 1/ Parameters: θ 1 κ 11 κ 1 κ β 1. 3.4.5 Model AY κ11 θ 1 x 1 + κ 1 θ x µ x 1 x = κ 1 θ 1 x 1 + κ θ x x 1/ 1 0 Σ x 1 x = 0 x 1/ Parameters: θ 1 θ κ 11 κ 1 κ 1 κ. 3.4.6 Model AY03 κ 11 x 1 µ x 1 x x 3 = κ 1 x 1 κ x κ 31 x 1 κ 3 x κ 33 x 3 1 0 0 Σ x 1 x x 3 = 0 1 0 0 0 1 Parameters: κ 11 κ 1 κ κ 31 κ 3 κ 33. 3.4.7 Model AY13 µ x 1 x x 3 = Σ x 1 x x 3 = κ 11 θ 1 x 1 κ 1 θ 1 x 1 κ x κ 3 x 3 κ 31 θ 1 x 1 κ 3 x κ 33 x 3 x 1/ 1 0 0 0 1 + β 1 x 1 1/ 0 0 0 1 + β 31 x 1 1/ Parameters: θ 1 κ 11 κ 1 κ κ 3 κ 31 κ 3 κ 33 β 1 β 31. 15

3.4.8 Model AY3 µ x 1 x x 3 = Σ x 1 x x 3 = κ 11 θ 1 x 1 + κ 1 θ x κ 1 θ 1 x 1 + κ θ x κ 31 θ 1 x 1 + κ 3 θ x κ 33 x 3 x 1/ 1 0 0 0 x 1/ 0 0 0 1 + β 31 x 1 + β 3 x 1/ Parameters: θ 1 θ κ 11 κ 1 κ 1 κ κ 31 κ 3 κ 33 β 31 β 3. 3.4.9 Model AY33 µ x 1 x x 3 = Σ x 1 x x 3 = κ 11 θ 1 x 1 + κ 1 θ x + κ 13 θ 3 x 3 κ 1 θ 1 x 1 + κ θ x + κ 3 θ 3 x 3 κ 31 θ 1 x 1 + κ 3 θ x + κ 33 θ 3 x 3 x 1/ 1 0 0 0 x 1/ 0 0 0 x 1/ 3 Parameters: θ 1 θ θ 3 κ 11 κ 1 κ 13 κ 1 κ κ 3 κ 31 κ 3 κ 33. 3.5 Semi-Affine Models Dimensions 1 to 3 3.5.1 Model SA11 µ x = a + bx + d x σ x = x Parameters: a b d 3.5. Model SA1 a + b 11 x 1 + d 1 x1 µ x 1 x = b 1 x 1 + b x + d 1 + fx1 x1 0 Σ x 1 x = 0 1 + fx1 Parameters: a b 11 d 1 b 1 b d f 3.5.3 Model SA a 1 + b 11 x 1 + b 1 x + d 1 x1 µ x 1 x = a + b 1 x 1 + b x + d x x1 0 Σ x 1 x = 0 x Parameters: a 1 b 11 b 1 d 1 a b 1 b d 16

3.5.4 Model SA13 a + b 11 x 1 + d 1 x1 µ x 1 x x 3 = b 1 x 1 + b x + b 3 x 3 + d 1 + e1 x 1 b 31 x 1 + b 3 x + b 33 x 3 + d 3 1 + e x 1 x1 0 0 Σ x 1 x x 3 = 0 1 + e1 x 1 0 0 0 1 + e x Parameters: a b 11 d 1 b 1 b b 3 d e 1 b 31 b 3 b 33 d 3 e. 3.5.5 Model SA3 µ x 1 x x 3 = a 1 + b 11 x 1 + b 1 x + d 1 x1 a + b 1 x 1 + b x + d x b 31 x 1 + b 3 x + b 33 x 3 + d 3 1 + e1 x 1 + e x x1 0 0 Σ x 1 x x 3 = 0 x 0 0 0 1 + e1 x 1 + e x Parameters: a 1 b 11 b 1 d 1 a b 1 b d b 31 b 3 b 33 d 3 e 1 e. 3.5.6 Model SA33 µ x 1 x x 3 = a 1 + b 11 x 1 + b 1 x + b 13 x 3 + d 1 x1 a + b 1 x 1 + b x + b 3 x 3 + d x a 3 + b 31 x 1 + b 3 x + b 33 x 3 + d 3 x3 x1 0 0 Σ x 1 x x 3 = 0 x 0 0 0 x3 Parameters: a 1 b 11 b 1 b 13 d 1 a b 1 b b 3 d a 3 b 31 b 3 b 33 d 3. 17