Drift and the Risk‐Free Rate

In today’s economy, new technologies rapidly emerge in the durable goods market. Therefore, it is paramount for manufacturing companies to optimize product introduction time for their second-generation product (SGP), which is an upcoming product in the production line, when profitability is at stake. The main factors affecting this timing are technology advancements and changes in customer tastes, which make determining an optimal introduction time for a product a challenging task. The main goal of this research is to find the optimal product introduction time for the SGP that maximizes the net present value (NPV) over a given period while product life cycle (PLC), pricing, and advertising are explicitly being taken into consideration. Demand for each product contains two regimes life cycle, and each regime is defined by a geometric Brownian motion (GBM). Each GBM has a drift rate and volatility. Moreover, there is a correlation between different regimes for different products. Correlated GBMs are discretized using a lattice approach. The Bass model is used to determine demand parameters including drift, volatility, and correlation, while dynamic programming is used to optimize NPV. Flexibilities, such as expansion, contraction, and switching, are identified between two products. Examples are provided to show the applicability of the developed models. Accordingly, the results show as the drift rate, volatility, and initial demand for a first-generation product (FGP) increase, the SGP introduction time has to be delayed. Furthermore, results demonstrate that in decreasing pricing policy, the SGP has to be introduced as early as the FGP. In increasing-decreasing pricing policy, as the increasing pricing rate is increasing, the SGP introduction time is delayed. When the advertising budget percentage increases, the NPV increases up to a certain level and then it will be saturated. Major contributions of this thesis are as follows: first, investigating the product introduction time by integrating marketing and manufacturing aspects; second, developing a model to incorporate the flexibility and production cost of the system for determining the optimal product introduction time. Third, the value of the product introduction time is expressed in terms of dollar value and this would help managers to make decisions easily. The models developed in this research can be used as practical tools for manufacturers to find the optimal product introduction time (PIT) and also the research can be used as a guideline to introduce the second-generation PIT.

ABSTRACTIn this article, we consider an optimization problem of expected utility maximization of continuous-time trading in a financial market. This trading is constrained by a benchmark for a utility-based shortfall risk measure. The market consists of one asset whose price process is modelled by a Geometric Brownian motion where the market parameters change at a random time. The information flow is modelled by initially and progressively enlarged filtrations which represent the knowledge about the price process, the Brownian motion and the random time. We solve the maximization problem and give the optimal terminal wealth depending on these different filtrations for general utility functions by using martingale representation results for the corresponding filtration.

We now return to general continuous-time financial market models in the setting of §6.1, i.e. there are d+1 primary traded assets whose price processes are given by stochastic processes S 0,... , S d , which are assumed to be adapted, right-continuous with left-limits (RCLL) and strictly positive semimartin-gales on a filtered probability space (Ω, F ℙ, F) (as usual F = (F t ) t≤T ). We assume that the market is free of arbitrage, in the sense that there exist equivalent martingale measures, but it contains non-attainable contingent claims, i.e. there are cash flows which cannot be replicated by self-financing trading strategies. In view of Theorem 6.1.5 this means that we do not have a unique equivalent martingale measures. We try to answer the obvious questions in this setting: how should we price the non-attainable contingent claims, i.e. which of the possible equivalent martingale measures should we pick for our valuation formula based on expectation, and, how can we construct hedging strategies for the nonattainable contingent claims to ‘minimise the risk? We try to answer these two questions in the general setting and then consider a prominent example of an incomplete market, a market with stochastic volatility, in more detail.KeywordsTrading StrategyStochastic VolatilityContingent ClaimMartingale MeasureIncomplete MarketThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the classical optimal execution problem, the basic assumption of underlying asset price is Arithmetic Brownian Motion (ABM) or Geometric Brownian Motion (GBM). However, many empirical researches show that the return distribution of assets may have heavy tails than those of normal distribution. The uncertain information impact on financial market may be considered as one of the main reasons for heavy tails of return distribution. To introduce this information impact, our paper proposes a Jump Diffusion model for optimal execution problem. The jumps in our model are described by the compound Poisson process where random jump amplitude depicts the information impact on price process. In particular, the model is simple enough to derive closed-form strategies under risk neutral and Mean-VaR criterion. Simulation analysis of the model is also presented.

How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Δ)≠TAMSD(Δ) and Variance(Δ)≠TAMSD(Δ) at short lag times Δ and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Δ/T≪1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.

A main source of competitive advantage is derived from the cost efficiency offered by firms’ manufacturing and logistics operations. Consequently, firms typically globalise their operations whereby they may exploit the comparative advantages—defined as production functions—of the nations in which they are present. Production process design thus arises as a significant issue. The research presented in this paper targets two fundamental questions attached to production process design that multinational companies face, namely: (i) should plants that are located in different countries but producing similar products use similar production processes?; and (ii) given that the firm's policy is to use similar production processes, how should the production processes be designed? Among others, the paper shows, by way of a numerical illustration of a binational manufacturing network, that the option of choosing freely upon production process design for the respective facilities in certain cases adds little to firm value. In fact, the value of this option tends to zero as the volatility rate increases when the exchange rate is modelled as a geometric Brownian motion without drift rate, implying that firms should employ similar production processes throughout their manufacturing networks. That is, a market value approach stands up for the so-called copy-exactly approach to production process design in these settings. We furthermore show the effects of economies of scale on the optimal production process design.

Convertible bonds are crucial derivatives in quantitative finance. Their complex terms like automatic redemption and barrier prices, coupled with stocks’ dynamic pricing features, pose precise pricing challenges. Traditional analytical methods struggle to capture the interactions among multiple factors. Thus, this study develops a quantitative pricing framework for complex convertible bonds by integrating the geometric Brownian motion (GBM) model with Monte Carlo simulation. Using Tesla’s closing prices from January 6, 2020, to January 11, 2022, as the sample, this article estimated the GBM parameters (drift rate is 0.4428 and volatility is 1.2866) via log returns. Finally, this paper simulated 10,000 future one-year stock price paths and retained 4,895 valid paths after normal distribution filtering. Based on risk-neutral pricing theory, this paper priced a 13.25% p.a. USD Autocallable Barrier Reverse Convertible issued by Credit Suisse London Branch (face value USD 5,000). The theoretical price was USD 3,593.87, and repeated simulations show the price stabilizes above USD 3,500. Risk analysis reveals that 72.54% of valid paths indicate a stock price decline after one year, and 37.4% of valid paths trigger barrier clauses (consistent with 2022-2023 actual trends), validating the framework’s feasibility.

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.

Under proportional transaction costs, a price process is said to have a consistent price system, if there is a semimartingale with an equivalent martingale measure that evolves within the bid-ask spread. We show that a continuous, multi-asset price process has a consistent price system, under arbitrarily small proportional transaction costs, if it satisfies a natural multi-dimensional generalization of the stickiness condition introduced by Guasoni (2006).

Chapter 35 Overlapping generations

10 - Multifrequency Jump-Diffusions

This research is devoted to studying a geometric Brownian motion with drift switching driven by a 2 × 2 Markov chain. A discrete-time multiplicative approximation scheme was developed, and its convergence in Skorokhod topology to the continuous-time geometric Brownian motion with switching has been proved. Furthermore, in a financial market where the discounted asset price follows a geometric Brownian motion with drift switching, market incompleteness was established, and multiple equivalent martingale measures were constructed.

A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the corresponding price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such "X-factor" we associate a market information process, the values of which we assume are accessible to market participants. Each information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents "noise". The noise term is modelled by an independent Brownian bridge that spans the interval from the present to the time at which the value of the factor is revealed. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the risk-neutral measure, conditional on the information provided by the market filtration. In the case where the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price process. Dividend growth is taken into account by introducing appropriate structure on the market factors. The prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European-style call option is of the Black–Scholes–Merton type. We consider the case where the rate at which information is revealed to the market is constant, and the case where the information rate varies in time. Option pricing formulae are obtained for both cases. The information-based framework generates a natural explanation for the origin of stochastic volatility in financial markets, without the need for specifying on an ad hoc basis the dynamics of the volatility.

An analysis for the value of carbon forestry needs to be provided for the successful establishment of the carbon offset market in Korea. We present an optimal management strategy for a forest owner who participates in the offset market. Given a stochastic process of the timber price following a geometric Brownian motion, the profit maximization problem of the forest owner is solved. The model finds an optimal harvest time in the presence of the carbon and timber revenues with opposing time effects. Sensitivity analysis is performed with respect to the volatility rate of the timber price and the discount rate. The presented model is applied to the study of the Korean larch case to identify the threshold timber price above which it is optimal to harvest trees.

In this paper we introduce a novel trading strategy for trading in continuous-double auctions for risky assets. The principle underlying our strategy is an optimising of the level of leverage using numerical methods. Previous studies have shown that similar strategies can perform well when tested against theoretical models of asset price processes such as geometric Brownian motion, or back-tested against historical empirical price data. However, the former approach fails to account for phenomena such as non-Gaussian return distributions which are observed in real markets, and the latter cannot take into account how other participants in the market would likely respond to a newly introduced trading strategy. In order to account for both of these issues, we test our strategy using an existing agent-based model of financial markets which has previously been shown to replicate many of the statistical features observed in empirical financial time-series data. We analyse our strategy by simulating its behaviour under this model, and find that the key variable which influences its performance is the size of the market.