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Geometric Brownian Motion (geometric + brownian_motion)
Selected AbstractsGeometric Brownian motion as a model for river flowsHYDROLOGICAL PROCESSES, Issue 7 2002Mario Lefebvre Abstract Let X(t) be the flow of a certain river at time t. A geometric Brownian motion process is used as a model for X(t) and is found to give very good forecasts of future flows. The forecasted values generated by this one-dimensional model are compared with those provided by a deterministic model that requires the evaluation of 18 entries. Based on two important criteria, the stochastic model is superior, on average, to the deterministic model for forecasts up to 4 days ahead. Copyright © 2002 John Wiley & Sons, Ltd. [source] The rate of learning-by-doing: estimates from a search-matching modelJOURNAL OF APPLIED ECONOMETRICS, Issue 6 2010Julien Prat We construct and estimate by maximum likelihood a job search model where wages are set by Nash bargaining and idiosyncratic productivity follows a geometric Brownian motion. The proposed framework enables us to endogenize job destruction and to estimate the rate of learning-by-doing. Although the range of the observations is not independent of the parameters, we establish that the estimators satisfy asymptotic normality. The structural model is estimated using Current Population Survey data on accepted wages and employment durations. We show that it accurately captures the joint distribution of wages and job spells. We find that the rate of learning-by-doing has an important positive effect on aggregate output and a small impact on employment. Copyright © 2009 John Wiley & Sons, Ltd. [source] A MULTINOMIAL APPROXIMATION FOR AMERICAN OPTION PRICES IN LÉVY PROCESS MODELSMATHEMATICAL FINANCE, Issue 4 2006Ross A. Maller This paper gives a tree-based method for pricing American options in models where the stock price follows a general exponential Lévy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American-type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Lévy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path-dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Lévy process has infinite activity. [source] A Dynamic Investment Model with Control on the Portfolio's Worst Case OutcomeMATHEMATICAL FINANCE, Issue 4 2003Yonggan Zhao This paper considers a portfolio problem with control on downside losses. Incorporating the worst-case portfolio outcome in the objective function, the optimal policy is equivalent to the hedging portfolio of a European option on a dynamic mutual fund that can be replicated by market primary assets. Applying the Black-Scholes formula, a closed-form solution is obtained when the utility function is HARA and asset prices follow a multivariate geometric Brownian motion. The analysis provides a useful method of converting an investment problem to an option pricing model. [source] Testing range estimators of historical volatilityTHE JOURNAL OF FUTURES MARKETS, Issue 3 2006Jinghong Shu This study investigates the relative performance of various historical volatility estimators that incorporate daily trading range: M. Parkinson (1980), M. Garman and M. Klass (1980), L. C. G. Rogers and S. E. Satchell (1991), and D. Yang and Q. Zhang (2000). It is found that the range estimators all perform very well when an asset price follows a continuous geometric Brownian motion. However, significant differences among various range estimators are detected if the asset return distribution involves an opening jump or a large drift. By adding microstructure noise to the Monte Carlo simulation, the finding of S. Alizadeh, M. W. Brandt, and F. X. Diebold (2002),that range estimators are fairly robust toward microstructure effects,is confirmed. An empirical test with S&P 500 index return data shows that the variances estimated with range estimators are quite close to the daily integrated variance. The empirical results support the use of range estimators for actual market data. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:297,313, 2006 [source] The valuation of European options when asset returns are autocorrelatedTHE JOURNAL OF FUTURES MARKETS, Issue 1 2006Szu-Lang Liao This article derives the closed-form formula for a European option on an asset with returns following a continuous-time type of first-order moving average process, which is called an MA(1)-type option. The pricing formula of these options is similar to that of Black and Scholes, except for the total volatility input. Specifically, the total volatility input of MA(1)-type options is the conditional standard deviation of continuous-compounded returns over the option's remaining life, whereas the total volatility input of Black and Scholes is indeed the diffusion coefficient of a geometric Brownian motion times the square root of an option's time to maturity. Based on the result of numerical analyses, the impact of autocorrelation induced by the MA(1)-type process is significant to option values even when the autocorrelation between asset returns is weak. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:85,102, 2006 [source] Option pricing with a non-zero lower bound on stock priceTHE JOURNAL OF FUTURES MARKETS, Issue 8 2005Ming Dong Black, F. and Scholes, M. (1973) assume a geometric Brownian motion for stock prices and therefore a normal distribution for stock returns. In this article a simple alternative model to Black and Scholes (1973) is presented by assuming a non-zero lower bound on stock prices. The proposed stock price dynamics simultaneously accommodate skewness and excess kurtosis in stock returns. The feasibility of the proposed model is assessed by simulation and maximum likelihood estimation of the return probability density. The proposed model is easily applicable to existing option pricing models and may provide improved precision in option pricing. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:775,794, 2005 [source] Optimal No-Arbitrage Bounds on S&P 500 Index Options and the Volatility SmileTHE JOURNAL OF FUTURES MARKETS, Issue 12 2001Patrick J. Dennis This article shows that the volatility smile is not necessarily inconsistent with the Black,Scholes analysis. Specifically, when transaction costs are present, the absence of arbitrage opportunities does not dictate that there exists a unique price for an option. Rather, there exists a range of prices within which the option's price may fall and still be consistent with the Black,Scholes arbitrage pricing argument. This article uses a linear program (LP) cast in a binomial framework to determine the smallest possible range of prices for Standard & Poor's 500 Index options that are consistent with no arbitrage in the presence of transaction costs. The LP method employs dynamic trading in the underlying and risk-free assets as well as fixed positions in other options that trade on the same underlying security. One-way transaction-cost levels on the index, inclusive of the bid,ask spread, would have to be below six basis points for deviations from Black,Scholes pricing to present an arbitrage opportunity. Monte Carlo simulations are employed to assess the hedging error induced with a 12-period binomial model to approximate a continuous-time geometric Brownian motion. Once the risk caused by the hedging error is accounted for, transaction costs have to be well below three basis points for the arbitrage opportunity to be profitable two times out of five. This analysis indicates that market prices that deviate from those given by a constant-volatility option model, such as the Black,Scholes model, can be consistent with the absence of arbitrage in the presence of transaction costs. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:1151,1179, 2001 [source] A scenario-based stochastic programming model for the control or dummy wafers downgrading problemAPPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 3 2009Shu-Hsing Chung Abstract The subject of this paper is to study a realistic planning environment in wafer fabrication for the control or dummy (C/D) wafers problem with uncertain demand. The demand of each product is assumed with a geometric Brownian motion and approximated by a finite discrete set of scenarios. A two-stage stochastic programming model is developed based on scenarios and solved by a deterministic equivalent large linear programming model. The model explicitly considers the objective to minimize the total cost of C/D wafers. A real-world example is given to illustrate the practicality of a stochastic approach. The results are better in comparison with deterministic linear programming by using expectation instead of stochastic demands. The model improved the performance of control and dummy wafers management and the flexibility of determining the downgrading policy. Copyright © 2008 John Wiley & Sons, Ltd. [source] An Importance Sampling Method to Evaluate Value-at-Risk for Assets with Jump Risk,ASIA-PACIFIC JOURNAL OF FINANCIAL STUDIES, Issue 5 2009Ren-Her Wang Abstract Risk management is an important issue when there is a catastrophic event that affects asset price in the market such as a sub-prime financial crisis or other financial crisis. By adding a jump term in the geometric Brownian motion, the jump diffusion model can be used to describe abnormal changes in asset prices when there is a serious event in the market. In this paper, we propose an importance sampling algorithm to compute the Value-at-Risk for linear and nonlinear assets under a multi-variate jump diffusion model. To be more precise, an efficient computational procedure is developed for estimating the portfolio loss probability for linear and nonlinear assets with jump risks. And the titling measure can be separated for the diffusion and the jump part under the assumption of independence. The simulation results show that the efficiency of importance sampling improves over the naive Monte Carlo simulation from 7 times to 285 times under various situations. We also show the robustness of the importance sampling algorithm by comparing it with the EVT-Copula method proposed by Oh and Moon (2006). [source] PORTFOLIO OPTIMIZATION WITH DOWNSIDE CONSTRAINTSMATHEMATICAL FINANCE, Issue 2 2006Peter Lakner We consider the portfolio optimization problem for an investor whose consumption rate process and terminal wealth are subject to downside constraints. In the standard financial market model that consists of d risky assets and one riskless asset, we assume that the riskless asset earns a constant instantaneous rate of interest, r > 0, and that the risky assets are geometric Brownian motions. The optimal portfolio policy for a wide scale of utility functions is derived explicitly. The gradient operator and the Clark,Ocone formula in Malliavin calculus are used in the derivation of this policy. We show how Malliavin calculus approach can help us get around certain difficulties that arise in using the classical "delta hedging" approach. [source] Pricing American options by canonical least-squares Monte CarloTHE JOURNAL OF FUTURES MARKETS, Issue 2 2010Qiang Liu Options pricing and hedging under canonical valuation have recently been demonstrated to be quite effective, but unfortunately are only applicable to European options. This study proposes an approach called canonical least-squares Monte Carlo (CLM) to price American options. CLM proceeds in three stages. First, given a set of historical gross returns (or price ratios) of the underlying asset for a chosen time interval, a discrete risk-neutral distribution is obtained via the canonical approach. Second, from this canonical distribution independent random samples of gross returns are taken to simulate future price paths for the underlying. Third, to those paths the least-squares Monte Carlo algorithm is then applied to obtain early exercise strategies for American options. Numerical results from simulation-generated gross returns under geometric Brownian motions show that the proposed method yields reasonably accurate prices for American puts. The CLM method turns out to be quite similar to the nonparametric approach of Alcock and Carmichael and simulations done with CLM provide additional support for their recent findings. CLM can therefore be viewed as an alternative for pricing American options, and perhaps could even be utilized in cases when the nature of the underlying process is not known. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:175,187, 2010 [source] Pension funding problem with regime-switching geometric Brownian motion assets and liabilitiesAPPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 2 2010Ping Chen Abstract This paper extends the pension funding model in (N. Am. Actuarial J. 2003; 7:37,51) to a regime-switching case. The market mode is modeled by a continuous-time stationary Markov chain. The asset value process and liability value process are modeled by Markov-modulated geometric Brownian motions. We consider a pension funding plan in which the asset value is to be within a band that is proportional to the liability value. The pension plan sponsor is asked to provide sufficient funds to guarantee the asset value stays above the lower barrier of the band. The amount by which the asset value exceeds the upper barrier will be paid back to the sponsor. By applying differential equation approach, this paper calculates the expected present value of the payments to be made by the sponsor as well as that of the refunds to the sponsor. In addition, we study the effects of different barriers and regime switching on the results using some numerical examples. The optimal dividend problem is studied in our examples as an application of our theory. Copyright © 2009 John Wiley & Sons, Ltd. [source] Regularity of the free boundary of an American option on several assets,COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2009Peter Laurence We establish the C, regularity of the free boundary for an American option on several assets in the case where the payoff is convex and the assets follow correlated geometric Brownian motions. Our work builds on results concerning the qualitative properties and initial regularity of the free boundary by Broadie and Detemple; Jaillet, Lamberton, and Lapeyre; and Villeneuve. © 2008 Wiley Periodicals, Inc. [source] | |||||||||||||