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Portfolio Optimization with Jump–Diffusions: Estimation of Time-Dependent Parameters and Application1 Floyd B. Hanson2 John J. Westman3 Abstract This paper treats jump-diffusion processes in continuous time, with emphasis on the jump-amplitude distributions, developing more ap- propriate models using parameter estimation for the market in one phase and then applying the resulting model to a stochastic optimal portfolio application in a second phase. The new developments are the use of uniform jump-amplitude distributions and time-varying market parameters, introducing more realism into the application model, a Log-Normal-Diffusion, Log-Uniform-Jump model. 1. Introduction The empirical distribution of daily log-returns for actual financial in- struments differ in many ways from the ideal log-normal diffusion process as assumed in the Black-Scholes model [1] and other finan- cial models. The log-returns are the log-differences between two successive trading days, representing the logarithm of the relative size. The most significant difference is that actual log-returns ex- hibit occasional large jumps in value, whereas the diffusion process in Black-Scholes [1] is continuous. Another difference is that the empirical log-returns are usually negatively skewed, since the nega- tive jumps or crashes are likely to be larger or more numerous than the positive jumps for many instruments, whereas the normal distri- bution associated with the diffusion process is symmetric. Thus, the coefficient of skew [2] is negative, � 3 �M 3 =(M 2 ) 1:5 < 0, where M 2 and M 3 are the 2nd and 3rd central moments of the log-return distribution here. A third difference is that the empirical distribution is usually leptokurtic since the coefficient of kurtosis [2] satisfies � 4 �M 4 =(M 2 ) 2 > 3, where the value 3 is the normal distribution kurtosis value and M 4 is the fourth central moment. Qualitatively, this means that the tails are fatter than a normal with the same mean and standard deviation, compensated by a distribution that is also more slender about the mode (local maximum). A fourth difference is that the market exhibits time-dependence in the distributions of log-returns, so that the associated parameters are time-dependent. In 1976, Merton [10, Chap. 9] introduced Poisson jumps with inde- pendent identically distributed random jump-amplitudes with fixed mean and variances into the Black-Scholes model, but the ability to hedge the volatilities was not very satisfactory. Kou [9] uses a jump- diffusion model with a double exponential jump-amplitude distribu- tion with mean � and variance 2�, having leptokurtic and negative skewness properties, although it is difficult to see the empirical jus- 1Work supported in part by the National Science Foundation Grant DMS-99-73231 and DMS-02-07081 at the University of Illinois at Chicago. Preprint of paper to appear in Proceedings of 2002 Conference on Decision and Control, pp. 1-6, Las Vegas, 9-13 December 2002. 2Laboratory for Advanced Computing, University of Illinois at Chicago,851 Morgan St., M/C 249, Chicago, IL 60607-7045, USA, e-mail: hanson@math.uic.edu 3Department of Mathematics and Statistics, Miami University, Oxford, OH 45056, USA and Department of Mathematics, University of California, Los Angeles, e-mail: westmanjj@muohio.edu tification for this distribution. Prior to the Black-Scholes model, Merton [10, Chap. 5-6] analyzed the optimal consumption and investment portfolio with either geo- metric Brownian motion or Poisson noise and examined an example of constant risk-aversion utility having explicit solutions. In [10, Chap. 4], Merton also examined constant risk-aversion problems. Hanson and Westman [5] reformulated an important external events model of Rishel [11] solely in terms of stochastic differential equa- tions and applied it to the computation of the optimal portfolio and consumption policies problem for a portfolio of stocks and a bond. The stock prices depend on both scheduled and unscheduled jump external events. The computations were illustrated with a simple log- bi-discrete jump-amplitude model, either negative or positive jumps, such that both stochastic and quasi-deterministic jump magnitudes were estimated. In [6], they constructed a jump-diffusion model with marked Poisson jumps that had a log-normally distributed jump- amplitude and rigorously derived the density function for the log- normal-diffusion and log-normal-jump stock price log-return model. In [7], this financial model is applied to the optimal portfolio and consumption problem for a portfolio of stocks and bonds including computational results. In this paper, the log-normal-diffusion, log-uniform-jump problem is treated. In Section 2, the jump-diffusion density is rigorously de- rived using a modification of our prior theorem [6]. In Section 3, the time dependent parameters for this log-return process are estimated using this theoretical density and the S&P500 Index daily closing data for the prior decade. In Section 4, the optimal portfolio and consumption policy application is presented and then solved. Con- cluding remarks are given in Section 5. 2. Log-Return Density for Log-Normal-Diffusion, Log-Uniform Jump Let S(t) be the price of a single financial instrument, such as a stock or mutual fund, that is governed by a Markov, geometric jump- diffusion stochastic differential equation (SDE) with time-dependent coefficients, dS(t) = S(t) [� d (t)dt+ � d (t)dZ(t) + J(t)dP (t)] ; (1) with S(0) = S 0 ; S(t) > 0, where � d (t) is the appreciation return rate at time t, � d (t) is the diffusive volatility,Z(t) is is a continuous, one-dimensional Gaussian process, P (t) is a discontinuous, one- dimensional standard Poisson process with jump rate �(t), and asso- ciated jump-amplitude J(t) with log-return mean � j (t) and variance � 2 j (t). The stochastic processes Z(t) and P (t) are assumed to be Markov and pairwise independent. The jump-amplitude J(t), given that a Poisson jump in time occurs, is also independently distributed. The stock price SDE (1) is similar in our prior work [6, 7], except that time-dependent coefficients introduce more realism here. The continuous, differential diffusion process dZ(t) is standard, so has zero mean and dt variance. The symbolic notation for the discontinuous space-time jump process, J(t)dP (t), is better defined in terms of the Poisson random measure, P(dt; dq), by the stochastic integral, J(t)dP (t) = R Q b J(t; q)P(dt; dq), where Q = q is the Poisson spatial mark variable for the jump ampli- tude process, and bJ(t; q) is the kernel of the Poisson operator J(t), such that �1 < bJ(t; q) <1 so that a single jump does not make the underlying non-positive. The infinitesimal moments of the jump process are E[J(t)dP (t)] = �(t)dtR Q b J(t; q)� Q (q; t)dq and Var[J(t)dP (t)] = �(t)dt R Q b J 2 (t; q)� Q (q; t)dq, neglecting O 2 (dt) here, where � Q (q; t) is the Poisson amplitude mark den- sity. The differential Poisson process is a counting process with the probability of the jump count given by the usual Poisson distri- bution, p k (�(t)dt) = exp(��(t)dt)(�(t)dt) k =k!, k = 0; 1; 2; : : :, with parameter �(t)dt > 0. Since the stock price process is geometric, the common multiplica- tive factor of S(t) can be transformed away yielding the SDE of the stock price log-return using the stochastic chain rule for Markov processes in continuous time, d[ln(S(t))] = � ld (t)dt+ � d (t)dZ(t) + ln(1 + J(t))dP (t); (2) where � ld (t) � � d (t)� � 2 d (t)=2 is the log-diffusion drift and ln(1 + b J(t; q)) the stock log-return jump-amplitude is the logarithm of the relative post-jump-amplitude. This log-return SDE (2) will be the model that will used for comparison to the S&P500 log-returns. Since bJ(t; q) > �1, it is convenient to select the mark process to be the jump-amplitude random variable, Q = ln � 1 + b J(t;Q) � , on the mark space Q = (�1;+1). Though this is a convenient mark selection, it implies the time-independence of the jump-amplitude, so bJ(t;Q) = bJ 0 (Q) or J(t) = J 0 . Since market jumps are rare and the tails are relatively flat, a reasonable approximation is uni- form jump-amplitude distribution with density � Q (q; t) on the finite, time-dependent mark interval [Q a (t); Q b (t)], � Q (q; t) � H(Q b (t)� q)�H(Q a (t)� q) Q b (t)�Q a (t) ; (3) where H(x) is the Heaviside, unit step function. The density � Q (q; t) yields a mean E Q [Q] = � j (t) = (Q b (t) +Q a (t))=2 and variance Var Q [Q] = � 2 j (t) = (Q b (t)�Q a (t)) 2 =12, which define the basic log-return jump amplitude moments. It is assumed that Q a (t) < 0 < Q b (t), to make sure that both negative and positive jumps are represented, which was a problem for the log-normal jump-amplitude distribution in [7]. The uniform distribution is treated as time-dependent in this paper, so Q a (t), Q b (t), � j (t) and � 2 j (t) all depend on t. The difficulty in separating out the small jumps about the mode or maximum of real market distributions is explained by the fact that a diffusion approximation for small marks can be used for the jump process that will be indistinguishable from the continuous Gaussian process anyway. The basic moments of the stock log-return differential are M (jd) 1 � E[d[ln(S(t))]] = (� ld (t) + �(t)� j (t))dt; (4) M (jd) 2 � Var[d[ln(S(t))]] = � � 2 d (t) + �(t) � � 2 j (t)(1 + �(t)dt)� 2 j (t) �� dt; (5) where the O2(dt) term has been retained in the variance, rather than being neglected as usual, since the discrete return time, dt = �t, the daily fraction of one trading year (about 250 days), will be small, but not negligible. The log-normal-diffusion, log-uniform-jump density can be found by basic probabilistic methods following a slight modification for time-dependent coefficients of constant coefficient theorem found our paper [6], Theorem: The probability density for the log-normal-diffusion, log-uniform-jump amplitude log-return differential d[ln(S(t))] specified in the SDE (2) with time-dependent coefficients is given by � d ln(S(t)) (x) = p 0 (�(t)dt)� (n) � x;� ld (t)dt; � 2 d (t)dt � + P 1 k=1 p k (�(t)dt) k(Q b (t)�Q a (t)) � h � (n) � kQ b (t)� x+ � ld (t)dt; 0; � 2 d (t)dt � � � (n) � kQ a (t)� x+ � ld (t)dt; 0; � 2 d (t)dt � i ; (6) �1 < z < +1, where p k (�(t)dt) is the Poisson distribution and the normal distribution with mean � ld dtr and variance �2 d dt is � (n) (x;� ld dt; � 2 d dt) = Z x �1 � (n) (y;� ld dt; � 2 d dt)dy associated with d ln(S(t)), the diffusion part of the log-return pro- cess, � � ld dt+� d dZ(t) (x) = � (n) (x;� ld (t)dt; � 2 d (t)dt) : The proof, which is only briefly sketched here, follows from the den- sity of a triad of independent random variables, � + � � � given the densities of the three component processes �, �, and �. Here, (1) � = � ld (t)dt+ � d (t)dZ(t) is the log-normal plus log-drift diffu- sion process, (2) � = Q = ln(1 + bJ 0 (Q)) is the log-uniform jump- amplitude, and (3) � = dP (t) is the differential Poisson process. The density of a sum of independent random variables, as in the sum operation of � + (� � �), is very well-known and is given by a convolution of densities � �+�� (z) = R +1 �1 � � (z � y)� �� (y)dy (see Feller [3]). However, the distribution of the product of two random variables � � � is not so well-known [6] and has the density, � �� (x) = p 0 (�(t)dt)�(x) + P 1 k=1 p k (�(t)dt)[H(Q b (t)�x=k)�H(Q a (t)�x=k)] k(Q b (t)�Q a (t)) ; (7) for the log-uniform-jump process. The probabilistic mass at x = 0, represented by the Dirac �(x) and corresponds to the zero jump event case. Finally, applying the convolution formula for density of the sum � + (��) leads to the density for the random variable triad � + �� given in (6) of the theorem. Using the log-normal jump-diffusion log-return density in (6), the third and fourth central moments with finite return time dt = �t are computed, for later use for skew and kurtosis coefficients, respec- tively, yielding the jump-diffusion higher moments [6], M (jd) 3 � E � � d[ln(S(t))]�M (jd) 1 � 3 � = 6� j (t)(�(t)dt) 2 � 2 j (t) + (3� j (t)� 2 j (t) + � 3 j (t))�(t)dt ; (8) M (jd) 4 � E � � d[ln(S(t))]�M (jd) 1 � 4 � = 3(� 2 j (t)) 2 (�(t)dt) 4 + (6� 2 j (t)� 2 j (t) +18(� 2 j (t)) 2 )(�(t)dt) 3 + (3� 4 j (t) + 30� 2 j (t)� 2 j (t) +21(� 2 j (t)) 2 + 6� 2 j (t)dt� 2 j (t))(�(t)dt) 2 +(� 4 j (t) + 6� 2 j (t)dt� 2 j (t) + 6� 2 j (t)� 2 j (t) +6� 2 j (t)� 2 j (t)dt+ 3(� 2 j (t)) 2 )�dt+ 3(� 2 j (t)) 2 dt 2 : (9) 3. Jump-Diffusion Parameter Estimation Given the log-normal-diffusion, log-uniform-jump density (6), it is necessary to fit this theoretical model to realistic empirical data to estimate the parameters of the log-return model (2) for d[ln(S(t))]. For realistic empirical data, the daily closings of the S&P500 Index during the decade from 1992 to 2001 are used from data available on-line [13]. The data consists of n (sp) = 2522 daily closings. The S&P500 data can be viewed as an example of one large mutual fund rather than a single stock. The data has been transformed into the discrete analog of the continuous log-return, i.e., into changes in the natural loga- rithm of the index closings, �[ln(SP i )] � ln(SP i+1 )� ln(SP i ) for i = 1; : : : ; n(sp) � 1 daily closing pairs. For the decade, the mean is M(sp) 1 ' 4:015 � 10 �4 and the variance is M (sp) 2 ' 9:874 � 10 �5 , the coefficient of skewness is � (sp) 3 �M (sp) 3 =(M (sp) 2 ) 1:5 ' �0:2913 < 0, demonstrating the typical negative skewness property, and the coefficient of kurtosis is �(sp) 4 �M (sp) 4 =(M (sp) 2 ) 2 ' 7:804 > 3, demonstrating the typical leptokurtic behavior of many real markets. The S&P500 log-returns, �[ln(SP i )] for i = 1 : n(sp) decade data points, are partitioned into 10 yearly data sets, �[ln(SP(spy) j y ;k )] for k = 1 : n(sp) y;j y yearly data points for j y = 1 : 10 years, where P 10 j y =1 n (sp) y;j y = n (sp) . For each of these yearly sets, the pa- rameter estimation objective is to find the least sum of squares of the deviation between the empirical S&P500 log-return his- tograms for the year and the analogous theoretical log-normal- diffusion, log-uniform-jump distribution histogram based upon the same bin structure. Since jumps are rare, 100 centered bins within the log-return domain [x a ; x b ] were used. Since the most ex- treme log-returns are the same as the most extreme jumps, the log-return domain is selected to coincide with the time-dependent uniform distribution domain, i.e., [x a (t); x b (t)] = [Q a (t); Q b (t)], both dimensionless, where Q a (t) = min k (�[ln(SP (spy) j y ;k )]) and Q b (t) = max k (�[ln(SP (spy) j y ;k )]) with t = T j y = Year j y + 0:5, say, assigning the yearly value to the mid-year with steps of dt = �T j y , for each j y = 1 : 10. For a given t = T j y year, fixed [Q a (t);Q b (t)] implies fixed uniform distribution parameters � j (t) = (Q b (t) +Q a (t))=2 and �2 j (t) = (Q b (t)�Q a (t)) 2 =12. However, the Poisson jump rate �(t) is still a free parameter for the jump component of the log-return process. Further to keep the number of free parameters as small as practical, we require that the mean and variances of the yearly log-returns be the same for both empirical and theoretical distributions, i.e., M (spy) 1;j y � Mean n (sp) y;j y k=1 h � h ln � SP (spy) j y ;k �ii = M (jdy) 1;j y using (4) and M (spy) 2;j y � Var n (sp) y;j y k=1 h � h ln � SP (spy) j y ;k �ii = M (jdy) 2;j y using (5), for each j y = 1 : 10 years. This, in turn, implies con- straints on the log-diffusion parameters, � ld;j y = � M (spy) 1;j y � (� dt� j ) j y � =�T j y ; (10) � 2 d;j y = � M (spy) 2;j y � (�dt((1 + �dt)� 2 j + � 2 j ) j y ) � =�T j y ; (11) with �2 d;j y > 0 for each j y = 1 : 10 years. Of the six parameters f� ld;j y ; � 2 d;j y ; � j;j y ; � 2 j;j y ; � j y ;�T j y g, needed for each year j y to specify the jump-diffusion log-return distribution, only the jump rate � j y needs to be estimated by least squares. The time step dt = �T j y is the reciprocal of the number of trading days per year, close to 250 days, but varies a little for j y = 1 : 10 and has values lying in the range, [0.003936, 0.004050], used here for parameter estimation. Thus, we have a one dimensional global minimization problem for a highly complex discretized jump-diffusion density function (2). The analytical complexity indicates that a general global optimization method that does not require derivatives would be useful. For this purpose, such a method, Golden Super Finder (GSF) [8], was developed for [7] and implemented in MATLABTM , since simple techniques are desirable in financial engineering. The GSF method is an extensive modification to the Golden Section Search method [4], extended to multi-dimensions and allowing search beyond the initial hyper-cube domain by including the endpoints in the local optimization test with the two golden section interior points per dimension, moving rather than shrinking the hypercube when the local optimum is at an edge or corner. The method, as a general method, is slow, but systematically moves the search until the uni-modal optimum is found at a interior point and then approaches the optimum if within the original search bounds. Additional constraints can be added to the objective function, such as (10,11). If the diffusion coefficient vanishes, � 2 d ! 0 + , then (11) implies a maximum jump count constraint, max[� � dt] = 0:5( p ((� 2 j + � 2 j ) 2 + 4� 2 j �M 2 )� (� 2 j + � 2 j ))=� 2 j . An additional compatibility constraint, � j (t) > 0, does not need enforcement as long as Q a (t) < Q b (t) and is not violated here. The jump-diffusion estimated parameter results in this log-normal- diffusion, log-uniform-jump amplitude case are summarized in Ta- ble 1. The jump rate estimates and their variability are summarized in Table 2. A hybrid value-position stopping criterion with a toler- ance, tol = 5:e–3 was used, and all yearly iterations converged in at most 13 iterations each, out of a maximum limit of 20, except for the year 1999� which exhibited little evidence of the long and flat tails of other years, with a limiting behavior indicating a zero jump rate value, � 8 ' 2:52e � 4 ' 0:0 for Year 8 = 1999 � when j y = 8. Table 1: Summary of yearly coefficients for Log-Normal- Diffusion, Log-Uniform-Jump estimated parameters by least squares (variance of deviation between S&P500 and jump-diffusion histograms) with respect to the variable �dt given dt = �T j y and constraints mentioned in the text. Year j y � d;j y � d;j y � j;j y � j;j y � j y 1992 4.1e-2 7.3e-2 -1.6e-3 9.9e-3 36. 1993 6.7e-2 7.0e-2 -2.6e-3 1.3e-2 15. 1994 -1.5e-2 7.6e-2 -9.1e-4 1.3e-2 22. 1995 3.0e-1 5.8e-2 1.5e-3 9.9e-3 25. 1996 1.7e-1 9.7e-2 -6.0e-3 1.5e-2 17. 1997 2.8e-1 1.5e-1 -1.1e-2 3.5e-2 7.1 1998 2.2e-1 1.5e-1 -1.0e-2 3.5e-2 14. 1999� 1.9e-1 1.8e-1 3.1e-3 1.8e-2 2.5e-4� 2000 -1.2e-1 1.9e-1 -6.8e-3 3.1e-2 14. 2001 -1.1e-1 1.8e-1 -7.9e-4 2.9e-2 15. In Figure 1 a sample comparison can be made of the empirical S&P500 histogram on the left for the relatively noisy year of 2000 with the corresponding theoretical jump-diffusion histogram on the right using the fitted, optimized parameters and the same number of Table