Use of Stochastic Differential Equation Models in Financial Time Series Analysis : Monitoring and Control of Currencies in Exchange Market.
T. Ozaki, J.C. Jimenez, M. Iino, Z. Shi and S. Sugawara
Proceedings of Japan-US joint seminar on statistical time series analysis, Kyoto, Japan, June 2001 (2001) 17-24.
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Use of Stochastic Differential Equation Models in Financial Time Series Analysis : Monitoring and Control of Currencies in Exchange Market. T. Ozaki*, J.C. Jimenez**, M. Iino†, Z. Shi†, S. Sugawara†
* The Institute of Statistical Mathematics, 4-6-7 Minami Azabu, Minato-ku, Tokyo 106-8569, Japan. E-mail:
[email protected] ** The Institute of Cybernetics, Mathematics & Physics, Cuba. † Mizuho Trust & Banking Co.Ltd., Japan. Abstract: This paper considers technical analysis used in financial analysis as a kind of control method based on monitoring of the market and presents a mathematical way of doing a similar job. For this purpose a stochastic differential equation model called the Micro-Market Structure model for the dynamics of currency exchange market is considered as an illustrative example. A use of filtered estimates of the unobserved variable of the model is suggested for monitoring the market to see whether the currency concerned is over-valued or under-valued in the currency exchange market. An example of the use of the filtered estimates of the variables for the currency allocation is illustrated. Keywords: technical analysis, asset allocations, exchange rate dynamics, micro-market structure model, stochastic differential equations, multiplicative noise, innovation approach, maximum likelihood method, nonlinear Kalamn filter, monitoring and control
1. Introduction In financial practice “technical analysis” has been recognized to be useful for prediction and control of financial assets for many years (the oldest technique seemed to be attributed to Charles Dow in the late 19-th century). On the other hand it has been widely known that in most cases of financial time series analysis, prediction errors are more or less equivalent to the price difference P(t)-P(t-1), suggesting that financial time series exhibits random walk behavior. This forms a source of controversy and disagreement between financial academic groups and financial industry engineers. It has been argued that the difference between fundamental analysis and technical analysis is not unlike the difference between astronomy and astrology. Among some circles, technical analysis is known as “voodoo finance.” Lo et al(2000) say “One of the GREATEST GULFS between academic finance and industry practice is the separation that exists between technical analysts and their academic critics.” Obviously the topic of prediction and control of financial commodities in the markets is one of the most important subjects in practical financial studies. Although technical analysis has been proved to be useless from the view point of Wiener-Kolmogorov prediction theory(Neftci(1991)), it does not mean it is useless for monitoring purposes. One of the most widely used technical
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rules is the “moving-average-oscillator”(Brock et al(1992), Skouras, S.(2001)), which appears to be a kind of control method based on monitoring a system. Here technical analysis suggests that buy and sell signals are generated by two moving averages of the index level --- a long-period average and a short period average. Sell and buy decisions are made based on these two types of moving averages. Here the main concern is whether the price is likely to go up or down in future. The predictor, the actual size of the rising (or falling) price is not the main concern. If such an empirical monitoring and control method is useful in practice, this may imply that even though the improvement of control performance by reducing the prediction error is not very hopeful, monitoring the past and present price may give rise to a useful method of controlling assets in the market. This idea of “monitoring and control” is reminiscent of the PID control method which is one of the most widely used methods in control industries. In control engineering there are two types of control methods; one is forward-looking control and another is backward-looking control. A typical example of forward-looking control is the predictive control method, based on predictions of the process (Kalman(1960)). A typical example of backward-looking control is the PID control, based on monitoring errors. Both methods work well and have been used in practice for many years(Astrom(1995)). The difference is that predictive control leads to sharper and more efficient results than PID control. This advantage is attained at the cost of the effort involved in identifying a good prediction model for the process(Peng et al(2000)). PID control is simpler because what is required is monitoring the deviation of the current ( and past) states from the target, and a simple control rule can be given based on measured deviations. It does not use prediction of the process. Both PID control and predictive control are defined unambiguously using mathematical language. To fill the gap between the two groups in financial studies we need to clarify the ideas of technical analysis, to correct mis-interpretations and to give a mathematical form to what is being done and attained by technical analysis. In the present paper we show that monitoring the market using a mathematical model provides us with a control method which performs reasonably well without resorting to predicting the process. Here we consider the Yen-Dollar currency exchange market as an illustrative example. We employ a three dimensional stochastic differential equation model, called the micro-market structure model for the US Dollar-Japanese Yen daily exchange rate time series. The model dynamics are explained in section 2. The estimation method and estimation results of the unobserved state variables of the model are given in section 3. The model identification method is briefly explained in section 4. The application of the model to the monitoring of the Yen-Dollar(US) exchange market and allocation of the currency is discussed in section 5.
2. Monitoring the Market Price through Micro-Market Structure Model We are interested in a model for market monitoring which may lead to a useful control method for currency
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allocation in the market. A combination of monitoring and control through an efficient dynamic model may yield a useful method in financial engineering similar to the PID control method in control engineering. A useful model for characterizing the market tendency of the price is the micro-market structure model (Bouchaud & Cont(1998), Iino & Ozaki(2000)). The model is given by the following stochastic differential equations: d φ = (α1 + β1φ )dt + γ 1 dw1 (t) d λ =(α 2 + β2 λ )dt + γ 2 dw2 (t)
(1)
dP = φ exp( λ)dt + γ 3 exp(λ / 2)dw3 (t)
Here P(t) is the price of the currency in the market. dw1(t), dw2(t), dw3(t) are increments of Brownian motion. The state variable φ(t) characterizes the state of the market, and whether the market is over-valued or under-valued: in other words, whether the currency (US Dollars) is over-valued or under-valued against the Japanese Yen. If φ(t) >0, it means the market is over-valued, and if φ(t) 0) and if the estimate of φt|t is less than a threshold -θ sell Dollars and buy Yen. A critical problem here is how to obtain the estimate of φt|t using the past price data, which is a filtering problem.
3. Estimation of Unobserved States The filtering problem of the micro-market structure model can be solved numerically by employing a nonlinear filtering technique developed for the continuous-discrete filtering problem (Ozaki(1992,1993,1994), Jimenez & Ozaki(2000a, 2000b)). The method has been introduced and used in some areas such as control engineering, brain sciences, mechanical engineering and geophysics for many years. For example in Valdes et al(1999), filtering estimates of an 11 dimensional states of nonlinear stochastic dynamical system( called the Zetterberg model) for the neural mass dynamics are calculated from scalar time series using Ozaki’s local linearization technique. By using the same technique, the estimation of the unobserved state φ(t) and λ(t) of the micro-market structure model (1) is given through the following continuous-discrete state space Markov representation,
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dz = f ( z)dt + g(z)dW (t)
(2)
Pt = h(z t ) + j( z t )ε t
Here z(t)={φ(t), λ(t), P(t)}’, dW(t)=(dw1(t),dw2(t),dw3(t))’ and α1 + β1φ f (z) = α 2 + β2 λ φ exp(λ)
γ 1 0 g (z) = 0 γ 2 0 0
0 γ 3 exp(λ / 2) 0
h(zt)=Pt and j(zt)=0.
Of course the state space representation is not unique in the nonlinear case. Another representation for the same process with a two dimensional state vector is given by using the state z(t)={φ(t), λ(t)}’, dW(t)={dw1(t),dw2(t)}’ εt=γ3{w3(t)-w(t-1)} α1 + β1φ γ 1 0 f (z) = g(z) = α 2 + β2 λ 0 γ 2
h(z t ) = φ t exp( λt ) j( zt ) = γ 3 exp( λt )
Iino & Ozaki(2000) employed the latter representation, while we employ the former representation in the present paper. Filtering of both representations may be regarded as a special form of filtering problem with general nonlinear observations, nonlinear dynamics, multiplicative system noise and multiplicative observation noise. A detailed description of the solution and algorithm for the calculation of the general continuous-discrete nonlinear filtering is given in Jimenez and Ozaki(2000a, 2000b). Here we are interested in seeing how these methods are useful in the present micro-market structure model estimations. 510
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Fig.2; From 5th January,1994 to 26th December, 1997
Fig.1 shows the log-transformed daily Dollar-Yen exchange rate data (100×log(xt)) from 17th January,1990 to 5th January, 1994 and Fig.2 shows the data from 5th January,1994 to 26th December, 1997. We applied the filtering method using a model with appropriate parameter values (obtained from the model identified by the Maximum Likelihood method in section 4) to the data of Fig.1. The data of Fig.2 is used in later sections for testing the estimated model. Fig.3 and 4 show the filtered estimates of the two unobserved states φ(t) and λ(t) for the data of Fig.1. The upward slope and downward slope in the data of Fig.1 are corresponding to the positive and negative periods of φt|t of Fig.3 respectively, implying the Dollar is over-valued when the Dollar is going up
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and vice versa. The λt|t of Fig.4 seems to show the volatility of the market price. 0.06
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Fig.4: Filtered estimates λ t|t for the data of Fig.1
4. Model Identifications Model identification of the state space model is fairly simple. Since both system noise and the observation noise are assumed to be Gaussian we can expect that the innovations
νt =Pt −E[Pt |Pt−1,Pt−2,Pt−3,.... are approximately Gaussian distributed even though the state dynamics are nonlinear. We can write down the (-2)log-likelihood using the Gaussian likelihood function as the following, (-2)log - likelihood = (-2)logp(P1 ,P2 ,...,PN | θ ) = (−2)log P(ν 1 ,ν 2 ,...,ν N | θ ) N
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where σ ν t = E[νt ] . We can obtain the maximum likelihood estimates of the system parameters and the state initial values φ0|0 and λ 0|0 using a nonlinear optimization technique. The innovations νt, the innovation variances σ2t and other estimates such as the filtered estimates φt|t and λ t|t are obtained using recursive nonlinear filtering techniques (see Jimenez & Ozaki(2000a, 2000b) for the detail). The maximum likelihood )
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estimates of the parameters are α 1 = 0 , β1 = −0.0794, γ 1 = −0.0316, )
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α 2 = −0.0018, β 2 = −0.0053, γ 2 = 0.1732,
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γ 3 = 0.6128 φ 0|0 = 0 λ 0|0 = 0.2468. Log-likelihood = -961.13 , AIC=1942.2
An interesting point may be to see whether such a simple model as micro-market structure model is useful in practice for monitoring purposes. To see this we fix the estimated parameters from the training data of Fig.1 and applied the same model for the filtering of the testing data of Fig.2. The results are shown in Fig.5 and Fig.6. The filtered estimates of φ(t) and λ(t) look reasonable: the period of positive φt|t of Fig.5 corresponds to the period of upward slope of Fig.2 and negative φt|t corresponds to the downward slope of the price in Fig.2. 6
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Fig.6: Filtered estimates λ t| t for the data of Fig.2
5. Currency Allocations based on the Monitoring We are interested in seeing how the filtered estimate φt1t provides us with a useful information for allocating the future currency at time point t+1. This could be checked by combining the estimated φt|t's with some allocation rules. For example we could divide the region of the domain of φt|t's into three regions; one the Dollar region above a threshold θ1, the second the Yen region below a threshold θ2, and the third region is a neutral region between θ1 and θ2. We could set a set of rules to allocate currencies at t+1 according to the following principles: 1) If φt|t > θ1, keep 100 per cent Dollars. 2) If φt|t < θ2, keep 100 per cent Yen. 3) If If θ2
