technical analysis techniques versus mathematical models

8/8/2019 Technical Analysis Techniques Versus Mathematical Models

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Technical Analysis Techniques versusMathematical Models: Boundaries of Their

Validity Domains

Christophette Blanchet-Scalliet1, Awa Diop2, Rajna Gibson3, Denis Talay2,and Etienne Tanre2

1 Laboratoire Dieudonne, universite Nice Sophia-Antipolis Parc Valrose 06108Nice Cedex 2, France [email protected]

2 INRIA, Projet OMEGA, 2004 route des Lucioles, BP93, 06902 Sophia-Antipolis,France {Awa.Diop,Denis.Talay,Etienne.Tanre}@sophia.inria.fr

3

NCCR FINRISK, Swiss Banking Institute, University of Zurich, Plattenstrasse14, Zurich 8032, Switzerland [email protected]

Abstract We aim to compare financial technical analysis techniques to strategieswhich depend on a mathematical model. In this paper, we consider the movingaverage indicator and an investor using a risky asset whose instantaneous rate ofreturn changes at an unknown random time. We construct mathematical strategies.We compare their performances to technical analysis techniques when the model ismisspecified. The comparisons are based on Monte Carlo simulations.

1 Introduction

In the financial industry, there are three main approaches to investment: thefundamental approach, where strategies are based on fundamental economicprinciples, the technical analysis approach, where strategies are based on pastprices behavior, and the mathematical approach where strategies are based onmathematical models and studies. The main advantage of technical analysis isthat it avoids model specification, and thus calibration problems, misspecifi-cation risks, etc. On the other hand, technical analysis techniques have limitedtheoretical justifications, and therefore no one can assert that they are risk-less, or even efficient (see [LMW00]).

Consider a nonstationary financial economy. It is impossible to specify andcalibrate models which can capture all the sources of instability during a longtime interval. Thus it might be useful to compare the performances obtainedby using erroneously calibrated mathematical models and the performances

obtained by technical analysis techniques.


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2 C. Blanchet-Scalliet, A. Diop, R. Gibson, D. Talay, E. Tanre

To our knowledge, this question has not yet been investigated in the lit-erature. The purpose of this paper is to present its mathematical complexityand preliminary results.

Here we consider the case of an asset whose instantaneous expected rate

of return changes at an unknown random time. We compare the performancesof traders who respectively use:

a strategy which is optimal when the model is perfectly specified andcalibrated,

mathematical strategies for misspecified situations, a technical analysis technique.

In all this paper, we limit ourselves to the case in which the traders util-ity function is logarithmic. Of course, it is a severe limitation from a finan-cial point of view. This choice is also questionable from a numerical pointof view because logarithmic utilities tend to smoothen the effects of the dif-ferent strategies. However, we will see that, even in this case and within asimplified model, the analytical formulae are rather cumbersome and that our

analysis requires nonelementary mathematical and numerical tools. See alsothe Remark 1 below.

Our study is divided into two parts: a mathematical part which, when pos-sible, provides analytical formulae for portfolios managed by means of math-ematical and technical analysis strategies; a numerical part which providesquantitative comparisons between all these various strategies.

2 Description of the Setting

The financial market consists of two assets which are traded continuously.The first one is an asset without systematic risk, typically a bond (or a bankaccount), whose price at time t evolves according to

dS0t = S0t rdt,

S00 = 1.(1)

The second asset is a stock subject to systematic risk. We model the evolutionof its price at time t by the linear stochastic differential equation

dSt = St

2 + (1 2)1 (t)

dt + StdBt,S0 = S

0,(2)

where (Bt)0tT is a one-dimensional Brownian motion on a given probabilityspace (, F, P). At the random time , which is neither known, nor directlyobservable, the instantaneous return rate changes from 1 to 2. A simplecomputation shows that

St = S0 exp

Bt + (1

2

2)t + (2 1)

t0

1 (s)ds

=: S0 exp(Rt),


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Technical Analysis Techniques versus Mathematical Models 3

where the process (Rt)t0 is defined as

Rt = Bt +

1

2

2

t + (2 1)

t0

1 (s)ds. (3)

This model was considered by Shiryaev [Shi63] who studied the problem ofdetecting the change time as early and reliably as possible when one onlyobserves the process (St)t0.

Assumptions and Notation

The algebra generated by the observations at time t is denoted by

FSt := (Su, 0 u t) , t [0, T].

Note that the Brownian motion (Bt)0tT is not adapted to the filtration(FSt )t0.

The Brownian motion (Bt)t0 and the random variable are independent. The change time follows an exponential law 4 of parameter :

P ( > t) = et, t 0. (4)

The value of the portfolio at time t is denoted by Wt. We denote by Ft the conditional a posteriori probability (constructed by

means of the observation of the process S) that the change time hasoccurred within the interval [0, t]:

Ft := P

t/FSt

. (5)

We denote by (Lt)t0 the following exponential likelihood-ratio process :

Lt = exp 1

2(2 1)Rt

122

(2 1)2 + 2(2 1)(1

2

2)

t

.

(6)

Finally, the parameters 1, 2, > 0 and r 0 are such that

1 2

2< r < 2

2

2.

4Any other law is allowed to derive our main results.


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3 A Technical Analysis Detection Strategy

3.1 Introduction

Technical analysis is an approach which is based on the prediction of the fu-ture evolution of a financial instrument price using only its price history. Thus,technical analysts compute indicators which result from the past history oftransaction prices and volumes. These indicators are used as signals to antic-ipate future changes in prices (see, e.g., the book by Steve Achelis [Ach00]).

Here, we limit ourselves to the moving average indicator because it issimple and often used to detect changes in return rates. To obtain its value,one averages the closing prices of the stock during the most recent timeperiods.

3.2 Moving Average Based on the Prices

Our trader takes decisions at discrete times. We thus consider a regular par-

tition of the interval [0, T] with step t = TN:

0 = t0 < t1 < .. . < tN = T, tn = nt.

We denote by t {0, 1} the proportion of the agents wealth invested in therisky asset at time t, and by Mt the moving average of the prices. Therefore,

Mt =1

tt

Su du. (7)

We suppose that, at time 0, the agent knows the history of the risky assetprices before time 0 and has enough data to compute M0 .

At each tn, n [1 N], the agent invests all his/her wealth into the riskyasset if the price Stn is larger than the moving average M

tn . Otherwise, he/sheinvests all the wealth into the riskless asset. Consequently,

tn = 1 (StnMtn). (8)

Denote by x the initial wealth of the trader. The wealth at time tn+1 is

Wtn+1 = Wtn

Stn+1

Stntn +

S0tn+1S0tn

(1 tn)

,

and therefore, since S0tn+1/S0tn = exp(rt),

WT = x

N1

n=0

tn exp(Rtn+1 Rtn) exp(rt) + exp(rt) . (9)


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3.3 The Particular Case of the Logarithmic Utility Function

One of our key results is

Proposition 1. The expectation of the logarithmic utility function of the

agents wealth is

E log(WT) = log(x) + rT +

2

2

2 r

T p

(1)

+t

2

2

2 r

1 eT1 et

(p

(2) p(1) )e +p(3)

t(2 1)(et t) 1 e

T

1 etp(3) ,

where we have set

p(1) =

0

y

z23/2

2ye(2//2)

22

(1+z2)22y i2/2

z

2y

dzdy, (10)

p(2) =

0

R4

1

y2

z1y1

+z2

z23/222y2

e(2//2)2(v)

2 (1+z22)22y2

i2(v)/2

z2

2y2

z13/21

2y1e(1//2)

2v2

(1+z21)22y1 i2v/2

z1

2y1

evdy1dz1dy2dz2dv, (11)

p(3) =

0

y

z13/2

2ye(1//2)

22

(1+z2)22y i2/2

z

2y

dzdy, (12)

iy(z) =ze

2/4y

y

0

ez cosh(u)u2/4y sinh(u)sin(u/2y)du. (13)

Proof. The tedious calculation involves an explicit formula, due to Yor [Yor01],

for the density of (t0

exp(2Bs)ds,Bt). See [BSDG+05] for details.

3.4 Empirical Determination of a Good Windowing

One can optimize the choice of by using Proposition 1 and deterministicnumerical optimization procedures, or by means of Monte Carlo simulations.In this subsection we present results obtained from Monte Carlo simulations,which show that bad choices of may weaken the performance of the technicalanalyst strategy. For each value of we have simulated 500,000 trajectoriesof the asset price and computed the expectation E log(WT) by a Monte Carlomethod. The parameters used to obtain Figure 1(a) and Figure 1(b) are all

equal but the volatility. It is clear from the figures that the optimal choice of varies. When the volatility is 5 percent, the optimal choice of is around 0.3


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whereas, when the volatility is 15 percent, the optimal choice of is around0.8.

The parameters used to obtain Figure 1(b) and Figure 1(c) are all identicalbut the maturity. The optimal choice of is around 0.3 when the maturity is

2 years, and is around 0.4 when the maturity is 3 years.The empirical variance of log(WT) is around 0.04. Thus, the Monte Carlo

error on E log(WT) is of order 5.104 with probability 0.99. The number of

trajectories used for these simulations seems to be too large; however, con-sidered as a function of , the quantity E log(WT) varies very slowly, so thatwe really need a large number of simulations to obtain the smooth curves(Figure 1).

4.72

4.73

4.74

4.75

4.76

4.77

4.78

4.79

4.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

E(log

(W_

T))

order of moving average = delta

(a)

4.77

4.78

4.79

4.8

4.81

4.82

4.83

4.84

4.85

4.86

4.87

4.88

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

E(log

(W_

T))


(b)

5.01

5.02

5.03

5.04

5.05

5.06

5.07

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

E

(log(W_

T))


(c)

Fig. 1. E(log(WT)) as a function of.

Parameter 1 2 r T Figure 1(a)

0.2 0.2 2 0.15 0.0 2.0

Figure 1(b) 0.2 0.2 2 0.05 0.0 2.0Figure 1(c) 0.2 0.2 2 0.05 0.0 3.0


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Figure 2 below illustrates the impact of the parameters 1, 2 and onthe optimal choice of .

0 0.5 1 1.5 24.74

4.76

4.78

4.8

4.82

4.84

E[log(W

T)

]

1=0.2

2=0.2

0 0.5 1 1.5 24.76

4.78

4.8

4.82

4.84

4.86

1=0.1

2=0.2

0 0.5 1 1.5 24.65

4.66

4.67

4.68

4.69

4.7

E[log(W

T)]

delta (years)

1=0.2

2=0.1

0 0.5 1 1.5 24.66

4.67

4.68

4.69

4.7

delta (years)

1=0.1

2=0.1

Fig. 2. Volatility and Optimal Moving Average Window Size: Plot of ExpectedValue of the Log of Terminal Wealth vs. Window size with T = 2, = 2, - = 0.1,

- - = 0.15, and -. = 0.2

Remark 1. One can observe that the empirical optimal choices of are closeto the classical values used by the technical analysts, that is, around 200 daysor 50 days. One can also observe from Monte Carlo simulations that theseoptimal values also hold when the traders utility function belongs to theHARA family: see [BSDG+05].

4 The Optimal Portfolio Allocation Strategy

4.1 A General Formula

In this section our aim is to make explicit the optimal wealth and strategy ofa trader who perfectly knows all the parameters 1, 2, and . Of course,


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this situation is unrealistic. However it is worth computing the best financialperformances that one can expect within our model. To be able to comparethis optimal strategy to a technical analyst strategy, we impose constraints onthe portfolio. Indeed, a technical analyst is only allowed to invest all his/her

wealth in the stock or the bond. Therefore the proportions of the traderswealth invested in the stock are constrained to lie within the interval [0, 1].

To compute the constrained optimal wealth we use the martingale ap-proach to stochastic control problems as developed by Karatzas, Shreve, Cvi-tanic, etc. More precisely, we follow and carefully adapt the martingale ap-proach to the celebrated Merton problem [Mer71]. We emphasize that oursituation differs from the Merton problem by two aspects:

The drift coefficient of the dynamics of the risky asset is not constant overtime (since it changes at the random time ).

Here we must face some subtle measurability issues since the traders strat-egy needs to be adapted with respect to the filtration generated by (St): asalready noticed, the drift change at the random time makes this filtration

different from the filtration generated by the Brownian motion (Bt).Let t be the proportion of the traders wealth invested in the stock at time

t; the remaining proportion 1t is invested in the bond. For a given nonran-dom initial capital x > 0, let Wx, denote the wealth process correspondingto the portfolio (). Let A(x) denote the set of admissible portfolios, that is,

A(x) := { FSt progressively measurable process s.t.Wx,0 = x, W

x,t > 0 for all t > 0, [0, 1]}.

The investors objective is to maximize his/her expected utility U of wealthat the terminal time T. The value function thus is

V(x) := supA(x)E

U(W

T).

As in Karatzas-Shreve [KS98], we introduce an auxiliary unconstrainedmarket defined as follows. We first decompose the process R in its own filtra-tion as

dRt =

(1

2

2) + (2 1)Ft

dt + dBt,

where B is the innovation process, i.e., the FSt - Brownian motion defined as

Bt =1

Rt (1

2

2)t (2 1)

t0

Fsds

, t 0,

where F is the conditional a posteriori probability (5).Let D the subset of the {FSt } progressively measurable processes :

[0, T] R such that


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E

T0

(t)dt < , where (t) := inf(0, (t)).

The bond price process S0() and the stock price S() satisfy

S0t () = 1 + t0 S0u()(r + (u))du,St() = S0 +

t0

Su()

(1 + (2 1)Fu + (u) + (u))du + dBu

.

For each auxiliary unconstrained market driven by a process , the valuefunction is

V(, x) := supA(,x)

ExU(WT()),

where

dWt () = W

t ()

(r + (t))dt + t

(t)dt + (2 1)Ftdt + (1 r)dt + dBt

.

Proposition 2. If there exists such thatV(

, x) = inf

DV(, x) (14)

then there exists an optimal portfolio for which the optimal wealth is

Wt = W

t (). (15)An optimal portfolio allocation strategy is

t := 1

1 r + (2 1)Ft + (t)

+

t

Ht Wt ertt0 (s)ds

, (16)

where Ft defined in (5) satisfies

Ft =etLt

t0

esL1s ds

1 + etLt

t

0esL1s ds

,

and Ht is the exponential process defined byHt = exp

t

0

1 r + (s)

+

(2 1)Fs

dBs

12

t0

1 r + (s)

+

(2 1)Fs

2ds

,

and is aFSt adapted process which satisfies

E

HTerTT0 (t)dt(U)1(HTerTT0 (t)dt) / FSt

= x +

t0

sdBs.

Here, v is the Lagrange multiplier which makes the expectation of the left handside equal to x for all x.

Proof. See Karatzas-Shreve [KS98, p. 275] to prove (15). We obtain (16) bysolving the classical unconstrained problem for .


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4.2 The Particular Case of the Logarithmic Utility Function

Proposition 3. If U() = log() and the initial endowment is x, then theoptimal wealth process and strategy are

W,xt =xer(Tt)+

Tt(t)dt

Ht ,t =

1 r + (2 1)Ft + (t)

2

,

(17)

where

(t) =

(1 r + (2 1)Ft) if 1 r + (2 1)Ft2

< 0,

0 if1 r + (2 1)Ft

2 [0, 1],

2 (1 r + (2 1)Ft) otherwise,(18)

and, as above, (t) = inf (0, (t)) .Remark 2. The optimal strategies for the constrained problem are the projec-tions on [0, 1] of the optimal strategies for the unconstrained problem.

Remark 3. In the case of the logarithmic utility function, when t is small andthus before the change time with high probability, one has Ft close to 0; as,by hypothesis, one also has 1r2 0, the optimal strategy is close to 0 ; afterthe change time , one has Ft close to 1, and the optimal strategy is close tomin(1, 2r2 ). In both cases, we approximately recover the optimal strategiesof the constrained Merton problem with drift parameters equal to 1 or 2respectively.

Using (18) one can obtain an explicit formula for the value function cor-responding to the optimal strategy:

E log(WT) = log(x) + rT

+

T0

0

1 r + (2 1) a

1 + a

2

2

1

a> 2 1 + r

2 2 + r

1

2

1 r + (2 1) a

1 + a

21

1 r2 r


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Technical Analysis Techniques versus Mathematical Models 11t0

exp

2 1

Bs (2 1)

2

22s + s

ds,

where B is a Brownian Motion. This density admits a rather complex explicit

expression which involves the function iy(z) defined as in (13): see Yor [Yor01]and Borodin and Salminen [BS02].

5 A Model and Detect Strategy

The optimal strategy of the preceding section assumes that the trader haschosen a mathematical model and controls the investment policy to optimizethe expected utility function of his/her wealth, and that the investment policyis time continuous, whereas the technical analyst does not control the policyand invests at discrete times according to a rupture detection rule. We nowconsider the case of a trader who chooses a mathematical model and wantsto reinvest the portfolio only once, namely at the time where the change time

is optimally detected owing to the price history. In this section we describethe wealth of such a trader, supposing that the reinvestment rule is the sameas the technical analysts one: at the detected change time from 1 to 2, allthe portfolio is reinvested in the risky asset. We also continue to suppose fora while that the trader perfectly knows all the parameters of the model.

In the full report [BSDG+05], we consider two detection methods proposedby Karatzas [Kar03] and by Shiryaev [Shi02]. Here, we limit ourselves toconsider Karatzas optimal stopping rule K which minimizes the expectedmiss

R() := E| | (20)over all stopping rules , where is a positive random variable. In view ofthe results in [Kar03] one has here

Proposition 4. The stopping rule K that minimizes the expected miss E|| over all the stopping rules withE() < is

K = inf

t 0

etLt t0

esL1s ds p

1p

,

where Lt is defined as in (6), and p is the unique solution in (12 , 1) of the

equation 1/20

(1 2s)e/s(1 s)2+ s

2ds =

p1/2

(2s 1)e/s(1 s)2+ s

2 ds

with = 2

2

/(2 1)2

.


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We thus are in a position to compute the wealth of the trader who usesthe strategy consisting in investing all of his/her money in the bond until K

and in the stock after K . The value of the portfolio at maturity T is

WT =xS0

K

SKST 1 (KT) + xS0T 1 (K>T).

In the particular case of the logarithmic utility function, one can exhibit anexact formula for E(log(WT)). Unfortunately this new formula has a com-plexity similar to (19): see [BSDG+05]. However we can numerically comparethe performances of the two change time detection strategies (one based ontechnical analysis, the other one based on a mathematical model), and theoptimal portfolio allocation strategy. Figure 3 illustrates, based on the typicalresults that we have obtained so far, that the methods using mathematicalmodels have better performances than the technical analyst method.

t

E[log(W)

]

4.6

4.65

4.7

4.75

4.8

4.85

0 0.4 0.6 0.8 1 1.2 1.6 1.8 2

Time0.2 1.4

Optimal allocation

Model and detectTechnical Analyst

Fig. 3. Comparison

6 The Performances of the Strategies Based on

Misspecified Models

6.1 Introduction

In practice, it is extremely difficult to know parameters exactly. If one mayhope to calibrate 1 and relatively well owing to historical data, the valueof 2 cannot be determined a priori (i.e. before the occurrence of the driftchange), and the law of cannot be calibrated accurately because of the lackof data concerning .

Consider a trader who believes that the stock price is

dSt = St

2 + (1 2) 1 (t)

dt + StdBt, (21)


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where the law of is exponential with parameter . We suppose that thetrue stock price is given by (2). Our aim is to study the misspecified optimalallocation strategy and the misspecified model and detect strategy.

Notation. As above we set Rt = log(St), where St is the actual price. We

define Lt and Ft as follows:

Lt = exp

1

2(2 1)Rt

1

22

(2 1)2 + 2(2 1)(1

2

2)

t

,

Ft =etLt

t0

esL1

s ds

1 + etLtt0

esL1

s ds.

6.2 On the Misspecified Optimal Allocation Strategy

Observing the stock price St, the trader computes a pseudo optimal allocationby using the erroneous parameters 1, 2, and . Thus the value of his/hermisspecified optimal allocation strategy is

t = proj[0,1] (1 r + (2 1)Ft)

2,

and the corresponding wealth is

W

t = ert exp

t0

ud(eruSu)

.

Numerical Example

In this section, we compare numerically the performance of two traders whorespectively use a misspecified model and the true model. We fix the valueof 1 = 0.2, = 0.15, r = 0.0 and = 2.0, and we assume that theyare perfectly known by the trader. A contrario 2 is misspecified. Its truevalue is 2 = 0.2. Figure 4 shows the functions t E(log(Wt)) for threevalues of 2. It suggests that it is better to overestimate 2 (2 > 2) thanto underestimate it (2 < 2).

6.3 On Misspecified Model and Detect Strategies

The erroneous stopping rule is

K

= inf

t 0, etLt

t0

esL1

s ds p

1p

where p is the unique solution in (1

2, 1) of the equation

1/20

(1 2s)e/s(1 s)2+ s

2ds =p1/2

(2s 1)e/s(1 s)2+ s

2 ds


14/17


E[log(W)

]t

= 0.22

= 0.32

= 0.12

Time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 24.6

4.65

4.7

4.75

4.8

4.85

Fig. 4. Error on 2 for the optimal trader

with = 22/(2 1)2.The value of the corresponding portfolio is

WT = xS0

K

STS

K

1

(KT)

+ xS0T 1 (K>T).

6.4 A Comparison Between Misspecified Strategies and the

Technical Analysis Technique

Our main question is: Is it better to invest according to a mathematical strat-egy based on a misspecified model, or according to a strategy which does notdepend on any mathematical model? Because of the analytical complexity ofall the explicit formulae that we have obtained for the various expected utili-ties of wealth at maturity, we have not yet succeeded to find a mathematicalanswer to this question, even in asymptotic cases (when 212 is large, e.g.).

As this part of our work is still in progress, we present here a few numericalresults obtained from Monte Carlo simulations. Consider the following studycase.

Parameters of the model 1 2 rTrue values 0.2 0.2 2 0.15 0.0

Parameters used by the trader 1 2 rMisspecified values (case I) 0.3 0.1 1.0 0.25 0.0Misspecified values (case II) 0.3 0.1 3.0 0.25 0.0

Figure 5 shows that the technical analyst overperforms misspecified opti-mal allocation strategies when the parameter is underestimated.

We have looked for other cases where the technical analyst is able to

overperform the misspecified optimal allocation strategies. Consider the casewhere the true values of the parameters are in Table 1. Table 2 summarizesour results. It must be read as follows. For the misspecified values 2 = 0.1,


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t

MSP Case II

MSP Case I

E[log(

W)

]

4.6

4.65

4.7

4.75

4.8

4.85

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time

TechnicalAnalyst

Fig. 5. A technical analyst may overperform misspecified optimal allocation strate-gies

= 0.25, = 1, if the trader chooses 1 in the interval (0.5,0.05) thenthe misspecified optimal strategy is worse than the technical analysts one.In fact, other numerical studies show that a single misspecified parameter isnot sufficient to allow the technical analyst to overperform the Model andDetect traders. Astonishingly, other simulations show that the technical an-

Table 1. True values of the parameters

Parameter True Value

1 -0.22 0.2 0.15 2

Table 2. Misspecified values and range of the parameters

1 (-0.5,-0.05) 1 -0.3 1 -0.3 1 -0.32 0.1 2 (0,0.13) 2 0.1 2 0.1 0.25 0.25 (0.2,) 0.25

1 1 1 (0,1.5)

alyst may overperform the misspecified optimal allocation strategy but notthe misspecified model and detect strategy. One can also observe that, when2/1 decreases, the performances of well specified and misspecified modeland detect strategies decrease. See [BSDG+05].


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7 Conclusions and Remarks

We have compared strategies designed from possibly misspecified mathemat-ical models and strategies designed from technical analysis techniques. We

have made explicit the traders expected logarithmic utility of wealth in allthe cases under study. Unfortunately, the explicit formulae are not propitiousto mathematical comparisons. Therefore we have used Monte Carlo numeri-cal experiments, and observed from these experiments that technical analysistechniques may overperform mathematical techniques in the case of severemisspecifications. Our study also brings some information on the range ofmisspecifications for which this observation holds true.

Jointly with M. Martinez (INRIA) and S. Rubenthaler (University of NiceSophia Antipolis) we are now considering the infinite time case where theinstantaneous expected rate of return of the stock changes at the jump timesof a Poisson process and the values after each change time are unknown. Wealso plan to consider technical analysis techniques different from the movingaverage considered here.

Acknowledgment

This research is part of the Swiss national science foundation research programNCCR Finrisk which has funded A. Diop during her postdoc studies at INRIAand University of Zurich.

References

[Ach00] S. Achelis. Technical Analysis from A to Z. McGraw Hill, 2000.[BS02] A. N. Borodin and P. Salminen. Handbook of Brownian MotionFacts

and Formulae. Probability and its Applications. Birkhauser Verlag,Basel, second edition, 2002.

[BSDG+05] C. Blanchet-Scalliet, A. Diop, R. Gibson, D. Talay,E. Tanre, and K. Kaminski. Technical Analysis Comparedto Mathematical Models Based Methods Under Misspecifi-cation. NCCR-FINRISK Working Paper Series 253, 2005.http://www.nccr-finrisk.unizh.ch/wp/index.php

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technical analysis techniques versus mathematical models

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