ON BOUNCING GEOMETRIC BROWNIAN MOTIONS

The Bachelier model for pricing options on futures spreads (OFS) assumes changes in the underlying .futures prices and spread follow unrestricted arithmetic Brownian motion (UABM). The assumption of UABM allows for a convenient analytic solution for the price of an OFS. The same is not possible under the more traditional assumption of geometric Brownian motion (GBM). Given the additional complexity of methods for pricing and hedging OFS using GBM such as Monte Carlo simulation and binomial trees, it is worth investigating how results from the Bachelier model compare to these other methods. The Bachelier model is presented and then extended to price an OFS with three underlying commodities. Hedge parameters for both models are provided. Results indicate that for OFS with sufficiently low volatility, differences between the Bachelier model and methods assuming GBM are quite small.

The aim of this research was to measure the risk of the IHSG stock data using the Value at Risk (VaR). IHSG stock index data typically indicates a jump. However, Geometric Brownian Motion (GBM) model can not catch any of the jumps. To view the jumps, it is necessary that the model was then developed into a Geometric Brownian Motion (GBM) model with Jumps. On the GBM model with Jumps, returns the data are discontinuous. To determine the value of VaR, the value of return to perform the simulation model of GBM with Jumps is required. To represent processes that contain jumps, discontinuous Poisson process using the Peak-Over Threshold is required. To determine the parameters of model, calibration of historical data using the Maximum Likelihood Estimation (MLE) method is performed. VaR value for GBM model with Jumps with a 95% and 99% confidence level are -0,0580 and -0,0818 while VaR value for GBM model with a 95% and 99% confidence level are -0,0101 and -0,0199. VaR for GBM model with Jumps with a confidence level of 95% and 99% show greater than the model VaR for GBM.

The calendar spread options (CSOs) on agricultural commodities, most notably corn, soybeans and wheat, allow market participants to hedge the roll-over risk of futures contracts. Despite the interest from agricultural businesses, there is lack of both theoretical and empirical research on pricing and hedging performances of CSOs. We propose to price and hedge CSOs under geometric Brownian motion (GBM) and stochastic volatility (SV) models. We estimate the model parameters by using implied state-generalized method of moments (IS-GMM) and evaluate the in-sample and out- of-sample pricing and hedging performances. We find that the average pricing errors of the SV model are 0.79% for corn, 0.75% for soybeans and 1.2% for wheat; the pricing and hedging performance of the SV model are mostly superior to the benchmark GBM model, both in and out of sample, with only one exception where the out-of-sample hedging error for the GBM model for market makers is slightly better than the SV model.

Evo-SETI Entropy identifies with Molecular Clock

Financial models consider often stochastic processes satisfying certain differential equations. We show that the solution of a particular geometric Brownian motion observed in discrete time is asymptotically equivalent with a Gaussian white noise model.

This paper considers the introduction of stock options in an (dynamically) incomplete securities market made up of a riskless bond and the stock. The stock price follows a geometric Brownian motion with constant drift. However, there is incomplete information about the unknown stochastic volatility. The option price is determined by a uniform-price auction. Thus an option pricing formula results from the interaction of market participants relying on private information on the unknown stochastic volatility under an explicit market structure. This paper incorporates market microstructure considerations into an extended Black-Scholes model with incomplete information on the underlying volatility. It relies on the growing importance of auctionlike trading rules in financial markets.

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In this paper option pricing theory is used to analyse whether or not to preserve a wilderness area. A numerical approach is demonstrated that can be applied to any generalized stochastic process. The impact of assuming that amenity value follows a logistic process, rather than geometric Brownian motion, is considered. The calculation of critical levels for amenity value necessary to justify preserving a wilderness area such as the Killarney Provincial Park in Ontario or the Headwaters Forest in California is demonstrated. The impact of changing the assumed growth and volatility of amenity value is also examined. JEL Classifcation: D81 Q26Ce mémoire utilise la théorie de la tarification d'une option pour analyser si on doit préserver un espace naturel. On utilise une approche numérique qui peut être appliquée à tout processus stochastique généralisé. On examine l'impact de possibilités que le profil de la valeur de la ressource dans la temps suive un processus logistique plutôt qu'un pattern géométrique de type Brownien. Le texte montre le calcul des niveaux critiques de la valeur de la ressource qui seraient nécessaires pour justifier la préservation d'une espace naturel comme le parc provincial Killarney en Ontario ou les forêts Headwaters en Californie. L'impact de postulats différents quant à la croissance et à la volatilité de la valeur de la ressource est examiné.

Model of stock price movements that follow stochastic process can be formulated in Stochastic Diferential Equation (SDE). The exact solution of SDE model is called Geometric Brownian Motion (GBM) model. Determination the optimal portfolio of three asset that follows Multidimensional GBM model is to be carried out in this research.Multidimensional GBM model represents stock price in the future is affected by three parameter, there are expectation of stock return, risk stock, and correlation between stock return. Therefore, theory of portfolio Markowitz is used on formation of optimal portfolio. Portfolio Markowitz formulates three of same parameter that is calculated on Multidimensional GBM model. The result of this research are optimal portfolio reaches with the proportion of fund are 39,38% for stock BBCA, 59,82% for stock ICBP, and 0,80% for stock INTP. This proportion of fund represents value of parameters that is calculated on modelling stock price.

We study the extremal properties of a stochastic process x t defined by the Langevin equation , in which ξ t is a Gaussian white noise with zero mean and D t is a stochastic ‘diffusivity’, defined as a functional of independent Brownian motion B t . We focus on three choices for the random diffusivity D t : cut-off Brownian motion, D t ∼ Θ(B t ), where Θ(x) is the Heaviside step function; geometric Brownian motion, D t ∼ exp(−B t ); and a superdiffusive process based on squared Brownian motion, . For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process x t on the time interval t ∈ (0, T). We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (D t = D 0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.

Geometric Brownian Motion and structural breaks in oil prices: A quantitative analysis

As an important part of structured products, Barrier Reverse Convertibles also occupies a large market share and attracts many investors, which makes an accurate valuation of it very significant. This paper does an application of Monte Carlo Simulation in Pricing of Barrier Reverse Convertibles. The author intends to verify the feasibility of Geometric Brownian Motion (GBM) and Monte Carlo Simulation on Barrier Reverse Convertibles linked to high technology companies. In this paper, taking 15.00% USD Barrier Reverse Convertible linked to Intel Corporation as an example, GBM and Monte Carlo Simulation are used to estimate final stock prices of intel corporation. By payoff function and present value function, stock prices are transformed into product prices. This article uses three years Intel corporation stock adjusted price, which is from 20 May 2022 to 20 May 2025 as historical data and 2025 2 years U.S. Treasury Bond coupon rate 0.0388 as risk free rate. The estimated price on fixing day is $ 983.21, from which a distinct overpricing in real market can be seen. At most of time points and stock price levels, Deltas and Gammas are relatively high. Given the above results, this article proves that it is feasible to use GBM and Monte Carlo Simulation in valuation of Barrier Reverse Convertibles, but obvious errors still exist. This article provides the future development direction of valuation methods of Barrier Reverse Convertibles.

Modeling and forecasting stock prices of public corporates are important studies in financial analysis, due to their stock price characteristics. Stocks investments give a wide variety of risks. Taking a portfolio of several stocks is one way to minimize risk. Stochastic process of single stock price movements model can be formulated in Geometric Brownian Motion (GBM) model. But for a portfolio that consist more than one corporate stock, we need an expansion of GBM Model. In this paper, we use multidimensional Geometric Brownian Motion model. This paper aims to model and forecast two stock prices in a portfolio. These are PT. Matahari Department Store Tbk and PT. Telekomunikasi Indonesia Tbk on period January 4, 2016 until April 21, 2017. The goodness of stock price forecast value is based on Mean Absolute Percentage Error (MAPE). As the results, we conclude that forecast two stock prices in a portfolio using multidimensional GBM give less MAPE than using GBM for single stock price respectively. We conclude that multidimensional GBM is more appropriate for modeling stock prices, because the price of each stock affects each other.

This paper proposes a way to forecast the future closing price of small size companies in Bursa Malaysia by using geometric Brownian motion (GBM). Forecasting is restricted to short term investment because most of the investors aim to gain profit in short period of time. The reasons of choosing small size companies are because the asset prices are lower, hence the asset are affordable for all level of investors. In this paper, we suggest that GBM which involves randomness, volatility, and drift can help investor in making their investment decision wisely. This research shows the model is highly accurate model in forecasting stock prices and it is proven by the lower value of mean absolute percentage error (MAPE). Although it is highly accurate, we try to find the suitable volatility measurements that give the forecast value closer to the actual movement of stock prices. The result shows that by using highs-lows-close volatility, the forecast stock prices are closest to the actual prices. This volatility measurement and GBM model are suggested to the investor to forecast future prices for a maximum of two week investment.