Mathematical modelling in production, operations and finance

Abstract

Mathematical modelling translates real-world problems into mathematical expressions to help people in industries develop scientific insights and make rational decisions. In this thesis, efforts are devoted to mathematical modelling in production, operations and finance.

Production and operations management includes supply chain management and inventory control problems. The issue of advertising outsourcing and production planning for a manufacturer facing asymmetric information and uncertain market demand is investigated in this thesis. A contract taking into account both advertising effort level and payment is introduced. Furthermore, a quantitative model with the goal of maximizing the manufacturer's net profit is proposed. Optimal strategy for outsourcing and production is obtained. Also, the inventory control problem for deteriorating items is studied. Six mathematical models are proposed for an inventory system with deterioration rate depending on the maximum lifetime of items. Two replenishment policies: (i) quantity-based policy and (ii) time-based policy, and three inspection scenarios: (i) no inspection, (ii) one inspection and (iii) continuous monitoring have been applied to these models. Examples and sensitivity analysis are given for each model. The results indicate that the quantity-based policy should be adopted under some circumstances. Although continuous monitoring can increase the long-run average profit, it should be dropped if it costs too much.

Mathematical models have been used in financial industry for a long time, such as derivative pricing, risk management and portfolio selection. Option pricing under different mathematical models is studied in this thesis. A quantitative model is introduced to describe the endogenous risk incurred by financial institution's fire sales, in which the option writer's asset value is subject to the price pressure from distressed selling. The change of num\'{e}raire techniques are applied to derive an analytical pricing formula for European call options. The impacts of distressed selling on the option price are discussed via numerical experiment. The results indicate that the price of a European call option subject to market impact of distressed selling is higher than those without market impact. Furthermore, the valuation of vulnerable option under a Markov-modulated jump-diffusion model is studied. A semi-analytical pricing formula for vulnerable European option is obtained in the presence of distressed selling and regime switching effect. The impacts of distressed selling and regime switching on the option price are also discussed via numerical experiments. In addition, a mathematical model is proposed to price vulnerable European options when the dynamic of the underlying asset value follows a jump-diffusion process with fast mean-reverting stochastic volatility. An analytical approximation formula for the option price is derived through the multi-scale asymptotic method and the two-dimensional Laplace transform. Numerical results are given to demonstrate the price movement by using the Euler inversion.