Mathematics aild Computers in Simulation 33 (1992) 575-580 North-Holland

575

of Some Continuous

ome iblonte Ca

Y. K. Tse

National UniveGty of Singapore

iMany financial models are based on assumptions concerning the stochastic movements

of some security prices. % evaluate the price of a derivative security, the parameters

dking the stochastic process of the underlying asset have to be estimated. Lo (1988)

studied the theory of maximum likelihood estimation of Ito processes, He established

a characterization theorem of the exact likelihood function of data that are sampled

at discrete time points which may or may not be equally apart. Under standard

asymptot,: theoYes, the maximum likelihood estimators (MLE) of these processes

ca be straightforward!y established.

While it is possible LO charP.cterize the exact likelihood function of an Ito process,

the existence and the derivation of it are by no means guaranteed. Hence in many

models of application, researchers have to rely 0~1 a discretized approximation of

the likelib~od function. As pointed out by Lo, the discretized MLE is in general

inconsistcn:. The magnitude of the inconsistency depends on the sampling interval

and typically decreases as the sampling interval is shortened. It is thus important

to know the consequences of the approximation, especially its effect on the size of

the inconGstency. In a recent paper Tse (191) derived closed form solutions for the

.LILE oi uome diffu$on processes commonly used to describe interest rate mcvements,

providing asvmptotic results of the discretized MILE as w4 as the exact MLE. In

lhis pa!Jrr we present so~ne XIonte Carlo results for comparing the perfor .nauw o!’ the

discretized nud csact 4lLE of the Ornstein-Uhlenbeck process. The comparison is

extended to include the geometric Brownian motion for :Ttocl. price mc,vements. jve

find that the discretized MLE gives results comparable to the exact MLE, provided

the dala are based on a judicious choice of sampling interval supported by appropriate

salllp]e sizcb. it is alsO fount’ thaw the rates of converi;:l;<e of the estimalcd parameters.

witllin the sitme model, to tlwir asymptotic distribution can be drastically diKerent

IT c!stiInaLiug volatility is me main concern, our findings favour choosing data wit11 a

short sampling intcrva!.

0378-1754/92/$05.00 0 1992 - Elscvicr Scicncc Publishers B.V. All righls rcscrvcd

516 X.K. Tse I MLE of continuous time financial models

nterest

Two types of diffusion processes are considered in this paper: stock price model and

interest rate model. We denote S, and & as stock pri ce and interest rate, respectively.

For stock price movement, we consider the geometric Brownian (GB) motion given

by the following equation

dSt = pSt dt + US, dll, (11

where dIVt is a Wiener process. For convenience, we assume that the observations are

equally spaced and let h be the sampling interval. It is we!l-known that (see, e.g.,

Hull (1989, p.83)) ln(&/S,_,) are IID N((p - (u”/2))h,a2h). Thus, the following

,MLE can be obtained by solving the the exact log-likelihood function

where

it = ln($) - l4SJSo)

t-1 n *

The discretized version of t:le GB process can be written as

S, - St-1 - ph -!- Et,

St-1 -

where Et w IID N(0, &‘h). Thus, the MLE of the discretized process are

(2)

(3)

(4)

For interest rate movement, we consider the Ornstein-Uhlenbeck (OU) process

given by the following equation

dR, = ct(p - R,) dt -!- is dl;V,. (6)

This process has the property of reverting to the mean, where p represents the steady-

state mean level and r~ is the speed af adjustment coefficient. From results given by

Vasicclk (1977), conditional on R6_1, & is normally distributed with mean c-Ohfi!t_i +

p( 1 - esrzh) and variance o?( 1 - cmJoh )/(2o). The exact MLE, denoted by c%,F and

Y.K. Tsc I MLE qf continuous time financial models 577

ir’, are given in TV (1991). The discrctized QU process can be represented by the

equation

Rt - RI-1 = a(p - Rt_l)h + ct. (7)

The formula for the MLE of the discretized process, denoted by &,$ and 2, can be

found in Tse (1991), where it has been established that /; = i. Besides, the following

points have also been noted: (1) while b is consistent, both (j: and 6’ are incc nsistent,

(2) the asymptotic biases of & and 6* are negative and increase with cr to the first

order in h. For 8*, the asymptotic bias also increases with cr’, (3) the asymptotic

variances of the exact MLE do not depend on ~1: and (4) the asymptotic variance of

& does not depend on u*.

3 onte esults

Optimality properties of the MLE, such as consistency and asymptotic efficiency, are

applicable only when the sample size is sufficiently large. As the rate of convergenw

to the asymptotic distributicn varies according to the underlying model, it is difficult

to provide rule of thumb that is suitable for all models. It is thus important to

consider small sample distributions of the MLE in order to obtain come information

regarding the sample size needed to justify the application of asymptotic results. This

information may also affect the choice of sampling interval. In this section we examine

these issues using a Monte Carlo experiment.

For !,he GB process: we consider p = 0.12,0.18; cr2 = 0.0625: h = 1,4,13 (in

weeks) znd n = 100,200,~OO. Random sampies of observations were generated based

on the exact diffusion process. The discretized and the exact MLE were calculated for

each sample, and this was repeated 1000 times. To conserve space, not all results of

the experiment are reported. Selected findings are summarized in Table 1, which gives

the means and the standard deviations of the MLE frc~nl the Monte Carlo sample. It

can be observed that b and fi are quite similar, except that, as expected, fi is larger

on average and shows signs of upward bias for h equals 1 and 13. The precision of

estimates of p is rather low, and it decreases with h. We note that 100 observations

of four-weekly data achieve the same accuracy as 400 observations of weekly data.

as far as the estimation of p is concerned. In contrast, the standard deviations of

estimates of 2’ are quite small and they do not vary with h. It can be seen that

6’ is upward biased when h is large . When h = 13, the relative bias is about 10

percent. To examine the convergence to normality, we calculated the nominal 95

percent conlidence interval for each ;Ilontc Carlo sample. based on the asynlptotic

norlnal distribution al1c.i cstimatcs of variance. The results (not reported hcrc) shol\

that the asymptotic distribution is in good approximation for 71 = 100.

578 Y.K. Tse I MLE of contintcous time financial models

Table 1

Monte Carlo Estimates of GB Process

/L = 0.18, u2 = 0.0625

Estimates

h(weeks) n *

1 100 0.1718 0.0618

O.lSOO O.OOY9 400 0.1758 0.0623

0.0917 0.0045

4 100 0.1816 0.0624 0.1832 0.0643

0.0897 0.0085 0.0910 0.0089

400 0.1807 0.06;20 0.1320 0.0639 0.0454 0.00.!3 0.0460 0.0046

13 100 O.lXd

0.0501 -100 0.1803

0.0216

0.061s

0.00% 0.0624

0.0045

0.1724 0.0622

0 !8?5 0.0090

6.1762 0.0627 0.0920 0.0045

O.lS32 0.0651

0.0524 0.0103 0.1845 O.OGS9

0.0257 0.0053

Note: The llmtc Carlo sample size is 1000. For each case, the first number refers

to the sample mean and the second number refers to the sample standard deviation.

Y.K. Tse / MLE iIf continuous time financial models 579

Table 2

Monte Carlo Estimates of OU Process

h(weeks) n Estimates

Exact Discretized 6 p=j 2-y x100) B fY( x 100)

Panel A: CI = 0.8,~ = 0.07,~~’ = 0.1225 x 10s2

1 100 3.5791 0.0527 0.12:7 3.4029 0.1165 2.5193 0.4228 0.0175 2.2830 0.0162

400 1.3990 0.0696 0.1237 1.3755 0.1205 0.7213 0.0154 0.0086 0.6962 0.0064

-l 100 1.111s o.oioo 0.1255 1.31s9 0.7504 O.OlSl O.Old4 0.6506

100 0.5261 0.0698 0.1227 0.8931 0.2694 0.0075 CL0087 0.2436

13 100 0.9903 0.0701 0.1256 0.3504 O.OOS6 0.0204

400 O.t34S6 0.0699 0.1233 0.1502 0.0012 0.0098

0.8656 0.2621 0 7f.F 0: 12;;

0.1128 0.0166 0.1113 o.ooi9

O.OYS9 0.0145 (3.1004 0.0072

Panel B: cy = 0.4. p = 0.03. u2 = 0.1225 x 10s2

1 100 3.2402 0.0240 0.1245 3.0650 0.1171 IL.5350 0 . 1’31 _- 0.0177 2.2314 0.0165

600 1.0320 0.0255 0.1232 1.0!80 0.120s 0.6463 0.0506 O.OOK 0.62S3 0.0084

4 100 1.0653 0.0259 0.1345 I.0017 0.1241 0.7257 0.04s: 0.0166 0.6415 0.0167

400 0.5539 0 . O”96 _ 0.1231 0.5405 3.1180 O.Zl’-’ 0.015s O.OOSY 0.2103 0.008::

13 100 0.5S63 0.0290 0 l”33 . ” 0.5378 0.1064

O.“o;S7 0.016s 0.0184 i).Z’l 0.0151 2400 0.4‘I11 G.0300 P. 12% 0.:1:66 0.1102

0.106s O.OOG iI.OOY.5 0.0949 O.OOSl

Note: Tlic Mlontc! Carlo samplr size is . ‘000. I;br each case, the lirst number refers

to the sample mc’an and the second number refers to the sample standard deviakw.

580 Y.K. Tse / W.E of continuous time jkancicl models

To investigate the use of daily data, we condccted further experiments. The

results show that with 100 observations of daily data, the standard deviations of the

estimates of o2 are approximately the same as those obtained from 100 observations

of weekly data. Thus, for the purpose of obtaining estimates of d as input to the

Bla&-Scholes formula, daily data are recommended. Since only a few months of past

data are required, nonstationarity of the variance parameter is unlikely to pose asy

serious difficulty.

For the OU process, we set up the parameters as follows: Q* = Q.001225, and

(a,~) = (O.S.O.O7), (0.4,0.03) and (1.2,O.ll). b ‘imilar to the GB process, we consider

h = 1,4,13 (in weeks) and n = 100,200,400. Selected findings are summarized in

Table 2. We observe that 6’ and fi(fi) converge to their asymptotic meai 5 from

helow, while &‘, 2u aud & converge to their asyrl.ptotic means from above, The

rates of convergence are very different foi estimates of &.&rent parameters. For

example. the rate of convergence of & and iu appear to be very slow and standard

deviations of these estimates are quite large. To investigate further the behaviour

of & and ti we performed an extra experiment with (a,~) = (OS, 0.07) for n =

600. d00.1000. V;hen n = 1fklO and h = 1, the mean of 6 is 1.0395 and its standard

deviation is 0.3573. Thus, & is still far from its asymptotic value of 0.8. The resuhs,

hewever, do show tnat there is a tendency of convergence to :he true parameter value.

The results also verify that & is asymptotically downward biased, as is predicted

from the anaiysis. In contrast, estimares of p and q2 converge very quickly to their

asymptotic means. The experiment also shows that Es2 is asymptotically downward

biased. As the asymptotic variance of estimates of the parameters depend on a, which

cannot be estimated precisely, statistical inference concerning the parameters using

the asymptotic approximation should be interpreted with care.

eferences

[l] Hull. J.. 1959. Options, Murea. und Other Deriuatiae Swurities, Eng!cwood

Cliffs: Prentice-Hall.

[‘) Lo. A.\$!., 19%. *‘Maximum lil~elihood estimation of generalized Ito processes

wilh discreteI!. sampled data.” Eoo7~or:ze~ric Z%tor~ 4. 231 - 247.

[:J] Tse. Y.K.. 1991, “On estimating some co;Kicuous time models for inturcst late

movements.” mimeo.

111 Va~ic& 0.X.. 1977, “:1n equilibrium charzclerization of the term structurr,‘,”

Jvtimul of’ F’inunciul Economics 5. 177 - 188.