30[NRM88]. F. B. Hanson and D. Ryan, Optimal harvesting with density dependent random effects, Natural Resource Modeling, vol. 2, pp. 439-455, 1988.

Abstract: This paper was presented at the Third Pacific Coast Resource Modeling Conference, University of California, Davis, CA, June 1984. The effect of sudden large, randomly occurring density dependent disasters on the optimal harvest policy and optimal expected return for an exploited population. The population grows logistically with disasters occurring on a time scale very short compared to the natural growth scale. The case of density dependent frequency is also treated. Stochastic dynamic programming is used in the optimization and a refined numerical method is successful in computing the solutions. For a set of realistic field data it is found that random effects typically have a significant effect on both optimal return and optimal effort levels. The effect of density dependence is far more pronounced for optimal return than for optimal effort levels.

64 [MBS98]. F. B. Hanson and D. Ryan, " Optimal Harvesting with Both Population and Price Dynamics," Mathematical BioSciences, vol. 148, issue 2, pp. 129-146, March 1998.

Abstract: We consider the effects of large inflationary price fluctuations on the computed optimal harvest strategy for a randomized Schaefer model. Both prices and population sizes are assumed random with both background (Wiener) and jump (Poisson) components. Population fluctuations are assumed to be density independent, i.e., relative changes are independent of population size. Stochastic dynamic programming is employed to find the optimal harvesting effort and economic return for a realistic set of bioeconomic data for Pacific halibut. It is found that inflationary effects have a pronounced influence on the optimal return, even in a hazardous or disastrous environment. However, optimal harvesting effort levels are much less sensitive to inflationary effects.

76 [ACC01]. F. B. Hanson and J. J. Westman, " Optimal Consumption and Portfolio Policies for Important Jump Events: Modeling and Computational Considerations ," Proceedings of 2001 American Control Conference, pp.4456-4661, 25 June 2001.

Abstract: While the volatility of portfolios are often modeled by continuous Brownian motion processes, discontinuous jump processes are more appropriate for modeling important external events that significantly affect the prices of financial assets. Here the discontinuous jump processes are modeled by state and control dependent compound Poisson processes, such that the random jumps come at the times of a pure Poisson process with jump amplitudes that are randomly distributed. The optimal consumption and investment portfolio policy formulation is in terms of stochastic differential equations with optimal discounted utility objectives. This paper was motivated by a recent paper of Rishel (1999) concerning portfolio optimization when prices are dependent on external events. However, the model has been significantly generalized for realistic computational considerations.

78 [ACC02]. F. B. Hanson and J. J. Westman, " Optimal Consumption and Portfolio Control for Jump-Diffusion Stock Process with Log-Normal Jumps (corrected) ," Proceedings of 2002 American Control Conference, pp. 4256-4261 (1573-1578), 08 May 2002.

Abstract: A jump-diffusion with log-normal jumps is presented. The distribution of the changes in the logarithm of the price of the financial instrument has heavier tails and a higher mode than the usual geometric Brownian motion. Such a distribution is called leptokurtic and is more characteristic of real market distributions than the corresponding log-normal distribution. In addition, this jump-diffusion return distribution can be negatively skewed as is real market returns distribution. The geometric Gaussian mark jump-diffusion process is modeled, basic properties are derived and applied to the problem of portfolio and consumption control optimization.

80 [KU02FM]. Floyd B. Hanson and John J. Westman, " Stochastic Analysis of Jump-Diffusions for Financial Log-Return Processes (corrected)," in Stochastic Theory and Control, Proceedings of a Workshop held in Lawrence, Kansas, October 18-20, 2001, Lecture Notes in Control and Information Sciences, B.Pasik-Duncan (Editor), Springer-Verlag, New York, pp. 169-184, 24 July 2002.

Abstract: A jump-diffusion log-return process with log-normal jump amplitudes is presented. The probability density and other properties of the theoretical model are rigorously derived. This theoretical density is fit to empirical log-returns of Standard & Poor's 500 stock index data. The model repairs some failures found from the log-normal distribution of geometric Brownian motion to model features of realistic financial instruments: (1) No large jumps or extreme outliers, (2) Not negatively skewed such that the negative tail is thicker than the positive tail, and (3) Non-leptokurtic due to the lack of thicker tails and higher mode.

83 [MTMS02FM2]. F. B. Hanson and J. J. Westman, " Computational Methods for Portfolio and Consumption Optimization in Log-Normal Diffusion, Log-Uniform Jump Environments," Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems, 9 pages, 12 August 2002 (invited paper).

Abstract: Computational methods for a jump-diffusion portfolio optimization application using a log-uniform jump distribution are considered. In contrast to the usual geometric Brownian motion problem based upon two parameters, mean appreciation and diffusive volatility, the jump-diffusion model will have at least five, since jump process needs at least a rate, a mean and a variance, depending on the jump-amplitude distribution. As the number number of parameters increases, the computational complexity of the problem of determining the parameter set of the underlying model becomes greater. In a companion stochastic parameter estimation paper, real market data, here a decade of log-returns for Standard and Poor's 500 index closings, is used to fit the jump-diffusion parameters, with constraints based on matching the data mean and variance to keep the unconstrained parameter space to 3 dimensions. A weighted least squares method has been used. The jump-diffusion theoretical distribution and weights has been derived. In this computational paper, the computational features of a new multidimensional, derivative-less global search method used in the companion paper are discussed. The main part of this paper is to discuss the computational solution of an optimal portfolio and consumption finance application with these more realistic parameter results. The constant relative risk aversion (CRRA) canonical model is used to reduce the high dimensionality of the PDE of stochastic dynamic programming problem to something more reasonable. Many computational issues arise due to the jump process part of the model, since several jump integrals arise which are not present in the pure diffusion with drift model. The log-uniformly distributed jumps allow a wider range of portfolio policies than does previous work with normally distributed jumps.

84 [MTMS02FM1]. F. B. Hanson and J. J. Westman, " Jump-Diffusion Stock Return Models in Finance: Stochastic Process Density with Uniform-Jump Amplitude," Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems, 7 pages, 12 August 2002 (invited paper).

Abstract: The stochastic analysis is presented for the parameter estimation problem for fitting a theoretical jump-diffusion model to the log-returns from closing data of the Standard and Poor's 500 (S&P500) stock index during the prior decade 1992-2001. The jump-diffusion model combines a the usual geometric Brownian motion for the diffusion and a space-time Poisson process for the jumps such that the jump amplitudes are uniformly distributed. The uniform jump distribution accounts for the rare large outlying log-returns, both negative and positive in magnitude. The log-normal, log-uniform jump-diffusion density is derived, leading to a jump-diffusion simulator approximation for the case the the log-return time is a small fraction of a year. There are five jump-diffusion parameters that need to be determined, the means and variances for both diffusion and jumps, as well as the jump rate, given the average log-return time. A weighted least squares is used to fit the theoretical jump-diffusion model to the S&P500 data optimizing with respect to three free parameters, with the two other parameters constrained by the mean and variance of the S&P500 data. The weight distribution derives from stochastic methods. The ideal fitted model determines the three free parameters, but the corresponding simulated results resemble the original S&P500 data better. This stochastic analysis paper is a companion to a computational methods and portfolio optimization paper at this conference

85 [CDC02]. F. B. Hanson and J. J. Westman, " Portfolio Optimization with Jump--Diffusions: Estimation of Time-Dependent Parameters and Application," Proceedings of 2002 Conference on Decision and Control, pp. 377-382, 09-13 December 2002, invited paper CDC02-INV0302.

Abstract: This paper treats jump--diffusion processes in continuous time, with emphasis on the jump--amplitude distributions, developing more appropriate 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.

87 [ACC03]. F. B. Hanson and J. J. Westman, " Jump--Diffusion Stock-Return Model with Weighted Fitting of Time-Dependent Parameters," Proceedings of 2003 American Control Conference, pp. 4869-4874, 04 June 2003.

Abstract: This paper treats jump-diffusion processes in continuous time, with emphasis on the jump-amplitude distributions, developing more appropriate models using parameter estimation for the market. The proposed method of parameter estimation is weighted least squares of the difference between theoretical and experimental bin frequencies, where the weights or reciprocal variances are chosen as by the theory of jump-diffusion simulation applied to bin frequencies. The empirical data is taken from a decade of Standard & Poor 500 index of stock closings and are viewed as one moderately large simulation. The new developments are the combined use of uniform jump-amplitude distributions, least squares weights and time-varying market parameters, introducing more realism into the model, a Log-Normal-Diffusion, Log-Uniform-Jump financial market model. The optimal parameter estimation is highly nonlinear, computationally intensive, and the optimization is with respect to the three parameters of the log-uniform jump distribution, while the diffusion parameters are constrained by the first two moments of the Standard & Poor's data.

89 [TAC04]. Floyd B. Hanson and John J. Westman, "Optimal Portfolio and Consumption Policies Subject to Rishel's Important Jump Events Model: Computational Methods," Trans. Automatic Control, vol. 49, no. 3, Special Issue on Stochastic Control Methods in Financial Engineering, pp. 326-337, March 2004.

Abstract: This paper illustrates a success in the combined use of stochastic control theory and compution to handle complex jump events. At important events or announcements, there can be large changes in the value of financial portfolios. Events and their corresponding jumps can occur at random or scheduled times. However, the amplitude of the response in either case can be unpredictable or random. While the volatility of portfolios are often modeled by continuous Brownian motion processes, discontinuous jump processes are more appropriate for modeling the response to important external events that significantly affect the prices of financial assets. Discontinuous jump processes are modeled by compound Poisson processes for random events or by quasi-deterministic jump processes for scheduled events. In both cases, the responses are randomly distributed and are modeled in a stochastic differential equation formulation. The objective is the maximal, expected total discounted utility of terminal wealth and instantaneous consumption. This paper was motivated by a paper of Rishel (1999) concerning portfolio optimization when prices are dependent on external events. However, the model has been significantly generalized for more realistic computational considerations with constraints and parameter values. The problem is illustrated for a canonical risk-adverse power utility model, but the usual explicit canonical solution is not strictly valid. Fortunately, iterations about the canonical solution result in computationally feasible approximations.

91 [CM04]. Floyd B. Hanson, John J. Westman and Zongwu Zhu, " Maximum Multinomial Likelihood Estimation of Market Parameters for Stock Jump-Diffusion Models, in Mathematics of Finance: Proc. 2003 AMS-IMS-SIAM Joint Summer Research Conference on Mathematics of Finance, AMS Contemporary Mathematics G. Yin and Q. Zhang (Editors),vol. 351, pp.~155-169, 24 June 2004.

Abstract: The estimated parameters of the log-return density for log-normal-diffusion, log-uniform jump process are found for an observed financial market distribution. When the observed data is collected into bins, it is shown that the appropriate parameter estimation method is the multinomial maximum likelihood estimation. This result is independent of the theoretical distribution, since it is only assumed that the observed distribution is the simulation of independent, identically distributed random variables. For the application to the theoretical jump-diffusion distribution, the estimation procedure is constrained by forcing the first two moments of the theoretical distribution to be the same as that for observed market distribution. The Standard and Poor's 500 stock index for the 1992-2001 decade is used as the observed market data. Numerically, the classical Nelder-Mead and our own direct search method are used to find the maximum likelihood estimation of the parameters. The results and performance of these numerical methods are compared along with the our older weighted least squares estimation method. The results of the two numerical approximations for the multinomial estimation methods were similar, but the weighted least squares results are not as good. In the severe test on the third and fourth moment measures, the multinomial based methods differed significantly from the same measures on the observed data, but did much better than the normal distribution based weighted least squares.

92 [CDC04]. Floyd B. Hanson and Zongwu Zhu, " Comparison of Market Parameters for Jump-Diffusion Distributions Using Multinomial Maximum Likelihood Estimation, Proceedings of 43nd IEEE Conference on Decision and Control, pp. 3919-3924, invited paper, December 2004.

Abstract: Previously, we have shown that the proper method for estimating parameters from discrete, binned stock log returns is the multinomial maximum likelihood estimation, and its performance is superior to the method of least squares. Also, useful formulas have been derived for the density for jump-diffusion Previously, we have shown that the proper method for estimating parameters from discrete, binned stock log returns is the multinomial maximum likelihood estimation, and its performance is superior to the method of least squares. Also, useful formulas have been derived for the density for jump-diffusion distributions. Numerically, the parameter estimation can be a large scale nonlinear optimization, but we have successfully implemented variants of multi-dimension direct search methods. In this paper, three jump-diffusion models using different jump-amplitude distributions are compared. These jump-amplitude distributions are the normal, uniform and double-exponential distribution. The parameters of all three models are fit to the Standard and Poor's 500 log-return market data, given the same first moment and second central moments. Our main results are first that uniform jump distribution has superior qualitative performance since it produces genuine fat tails that are typical of market data, whereas the other two have exponentially thin tails. Secondly, the uniform distribution is quantitatively better overall as measured by the closeness of both the skewness and kurtosis coefficients to the data, although the double-exponential is best on skewness while worst on kurtosis. However, the log-normal model has a big advantage in computational costs of parameter estimation compared with the others, while the double-exponential is most costly due to having one more model parameter to fit.

95 [CDC05zh]. Zongwu Zhu and Floyd B. Hanson, " A Monte-Carlo Option-Pricing Algorithm for Log-Uniform Jump-Diffusion Model , Proceedings of Joint 44nd IEEE Conference on Decision and Control and European Control Conference, pp. 1-6, 12 December 2005.

Abstract: A reduced European call option pricing formula by risk-neutral valuation is given. It is shown that the European call and put options for jump-diffusion models are worth more than that for the Black-Scholes (diffusion) model with the common parameters. Due to the complexity of the jump-diffusion models, obtaining a closed option pricing formula like that of Black-Scholes is not viable. Instead, a Monte Carlo algorithm is used to compute European option prices. Monte Carlo variance reduction techniques such as both antithetic and control variates are used. The numerical results show that this is a practical, efficient and easily implementable algorithm.

96 [ACC06]. Guoqing Yan and Floyd B. Hanson, " Option Pricing for a Stochastic-Volatility Jump-Diffusion Model with Log-Uniform Jump-Amplitudes," Proceedings of American Control Conference, pp. 2989-2994, 14 June 2006.

Abstract: An alternative option pricing model is proposed, in which the stock prices follow a diffusion model with square root stochastic volatility and a jump model with log-uniformly distributed jump amplitudes in the stock price process. The stochastic-volatility follows a square-root and mean-reverting diffusion process. Fourier transforms are applied to solve the problem for risk-neutral European option pricing under this compound stochastic-volatility jump-diffusion (SVJD) process. Characteristic formulas and their inverses simplified by integration along better equivalent contours are given. The numerical implementation of pricing formulas is accomplished by both fast Fourier transforms (FFTs) and more highly accurate discrete Fourier transforms (DFTs) for verifying results and for different output.

97 [Sethi06] Zongwu Zhu and Floyd B. Hanson, " Optimal Portfolio Application with Double-Uniform Jump Model," Stochastic Processes, Optimization, and Control Theory:Applications in Financial Engineering, Queueing Networks and Manufacturing Systems/A Volume in Honor of Suresh Sethi, International Series in Operations Research & Management Sciences, Vol. 94, H. Yan, G. Yin, Q. Zhang (Eds.), Springer Verlag, New York, pp. 331-358, Invited chapter, June 2006.

Abstract: This paper treats jump-diffusion processes in continuous time, with emphasis on the jump-amplitude distributions, developing more appropriate 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 double-uniform jump-amplitude distributions and time-varying market parameters, introducing more realism into the application model -- a log-normal diffusion, log-double-uniform jump-amplitude model. Although unlimited borrowing and short-selling play an important role in pure diffusion models, it is shown that borrowing and shorting is limited for jump-diffusions, but finite jump-amplitude models can allow very large limits in contrast to infinite range models which severely restrict the instant stock fraction to [0,1]. Among all the time-dependent parameters modeled, it appears that the interest and discount rate have the strongest effects.

99 [ACC07]. Floyd B. Hanson and Guoqing Yan, " American Put Option Pricing for Stochastic-Volatility, Jump-Diffusion Models," Proceedings of 2007 American Control Conference, pp. 384-389, 11 September 2007; in invited WeA12.1: Stochastic Theory and Control in Finance I session.

Abstract: The numerical treatment for the American put option pricing is discussed for a stochastic-volatility, jump-diffusion (SVJD) model with log-uniform jump amplitudes. Heston's (1993) mean reverting, square-root stochastic volatility model is used along with our uniform jump-amplitude model. However, computation is needed for nonlinear and multidimensional terms with dependence on the combined stock and volatility state space. A systematic finite difference formulation of the American put integro-partial differential complementary problem (PIDCP) is implemented using a successive over-relaxtion method projected on the maximum payoff function. Interpolation is used to construct the smooth transition to the payoff of the corresponding free boundary problem. Also, a fast, but less accurate, heuristic quadratic approximation, originally due to MacMillan (1986), is corrected and extended from pure diffusion models. The fast and simple quadratic approximation is compared with a more accurate PIDCP formulation. The simple quadratic approximation is briefly compared with market data.

100 [SIAMbook]. Floyd B. Hanson, Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis and Computation, SIAM Books: Advances in Design and Control Series, published 03 October 2007.

WORK SUBMITTED OR NEARLY SUBMITTED

104 [IISC07]. Floyd B. Hanson, "Stochastic Processes and Control for Jump-Diffusions," under revision, 44 pages, 22 October 2007.

Abstract: An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions using Wiener processes. Then jump perturbations are added using simple Poisson processes constructing the theory of simple jump-diffusions. Next, marked-jump-diffusions are treated using compound Poisson processes to include random marked jump-amplitudes in parallel with the equivalent Poisson random measure formulation. Otherwise, the approach is quite applied, using basic principles with no abstractions beyond Poisson random measure. This treatment is suitable for those in classical applied mathematics, physical sciences, quantitative finance and engineering, but having trouble getting started with the abstract measure-theoretic literature. The approach here builds upon the treatment of continuous functions in the regular calculus and associated ordinary differential equations by adding non-smooth and jump discontinuities to the model. Finally, the stochastic optimal control of marked-jump-diffusions is developed, emphasizing the underlying assumptions. The survey concludes with applications in biology and finance, some of which are canonical, dimension reducible problems and others are genuine nonlinear problems.

105 [BFS08]. Floyd B. Hanson, Optimal Portfolio Problem for Stochastic-Volatility, Jump-Diffusion Models with Jump-Bankruptcy Condition: Practical Theory and Computation, Fifth World Congress of Bachelier Finance Society, 2008, 27 pages, revised 11 July 2008. ( See this page for abstract and SSRN paper download.)

Abstract: This paper treats the risk-averse optimal portfolio problem with consumption in continuous time with a stochastic-volatility, jump-diffusion (SVJD) model of the underlying risky asset and the volatility. The new developments are the use of the SVJD model with double-uniform jump-amplitude distributions and time-varying market parameters for the optimal portfolio problem. Although unlimited borrowing and short-selling play an important role in pure diffusion models, it is shown that borrowing and short-selling are constrained for jump-diffusions. Finite range jump-amplitude models can allow constraints to be very large in contrast to infinite range models which severely restrict the optimal instantaneous stock-fraction to [0,1]. The reasonable constraints in the optimal stock-fraction due to jumps in the wealth argument for stochastic dynamic programming jump integrals remove a singularity in the stock-fraction due to vanishing volatility. Main modifications for the usual constant relative risk aversion (CRRA) power utility model are for handling the partial integro-differential equation (PIDE) resulting from the additional variance independent variable, instead of the ordinary integro-differential equation (OIDE) found for the pure jump-diffusion model of the wealth process. In addition to natural constraints due to jumps when enforcing the positivity of wealth condition, other constraints are considered for all practical purposes under finite market conditions. Computational, result are presented for optimal portfolio values, stock fraction and consumption policies. Also, a computationally practical solution of Heston's (1993) square-root-diffusion model for the underlying asset variance is derived. This shows that the non\-negativity of the variance is preserved through the proper singular limit of a simple perfect-square form. An exact, non\-singular solution is found for a special combination of the Heston stochastic volatility parameters.

106 [SIAMCT11]. Floyd B. Hanson, Stochastic-Volatility, Jump-Diffusion Optimal Portfolio Problem with Jumps in Returns and Volatility , SIAM Conference on Control and Its Applications (CT11), 26 pages, to be presented in Invited Session MS22 Stochastic Methods in Finance, Baltimore, MD. ( See this page for abstract and SSRN paper download.) (And/Or Slides?)

Abstract: This paper treats the risk-averse optimal portfolio problem with consumption in continuous time for a stochastic-jump-volatility, jump-diffusion (SJVJD) model of the underlying risky asset and the volatility. The new developments are the use of the SJVJD model with double-uniform jump-amplitude distributions and time-varying market parameters for the optimal portfolio problem. Although unlimited borrowing and short-selling play an important role in pure diffusion models, it is shown that borrowing and short-selling are constrained for jump-diffusions. Finite range jump-amplitude models can allow constraints to be very large in contrast to infinite range models which severely restrict the optimal instantaneous stock-fraction to [0,1]. The reasonable constraints in the optimal stock-fraction due to jumps in the wealth argument for stochastic dynamic programming jump integrals remove a singularity in the stock-fraction due to vanishing volatility. Main modifications for the usual constant relative risk aversion (CRRA) power utility model are for handling the partial integro-differential equation (PIDE) resulting from the additional variance independent variable, instead of the ordinary integro-differential equation (OIDE) found for the pure jump-diffusion model of the wealth process. In addition to natural constraints due to jumps when enforcing the positivity of wealth condition, other constraints are considered for all practical purposes under finite market conditions. Computational results are presented for optimal portfolio values, stock fraction and consumption policies.

107[MF05zh]. Zongwu Zhu and Floyd B. Hanson, " Risk-Neutral Option Pricing for Log-Uniform Jump-Amplitude Jump-Diffusion Model ," under revision, pp. 1-44, 20 August 2005.

Abstract: Reduced European call and put option formulas by risk-neutral valuation are given. It is shown that the European call and put options for log-uniform jump-amplitude jump-diffusion models are worth more than that for the Black-Scholes (diffusion) model with the common parameters. Due to the complexity of the jump-diffusion models, obtaining a closed option pricing formula like that of Black-Scholes is not tractable. Instead, a Monte Carlo algorithm is used to compute European option prices. Monte Carlo variance reduction techniques such as both antithetic and optimal control variates are used to accelerate the calculations by allowing smaller sample sizes. The numerical results show that this is a practical, efficient and easily implementable algorithm.

This material is based upon work supported by the National Science Foundation under Grants Numbers 880699, 9102343, 9301107, 9626692, 9973231, 0207081 in the Computational Mathematics Program.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or other agency.

Supercomputing research support was provided by San Diego Supercomputing Center NPACI Account UIL203, National Center for Supercomputing Applications Grant Numbers DCR860001N, DMS890009N, DMS900016N, DMS920003N, DMS960002N; Pittsburgh Supercomputing Center Grant Numbers DMS940001P and DMS9400011P; Argonne National Laboratory's Advanced Computing Research Facility; Los Alamos National Laboratory's Advanced Computing Laboratory ACL-Accounts Z803835 and gwqfbh-d.

23 September 2003