Katsushige Sawaki

A continuous-time dynamic pricing model knowing the competitor’s pricing strategy

In this paper we consider a dynamic pricing model for a firm knowing that a competitor adopts a static pricing strategy. We establish a continuous time model to analyze the effect of dynamic pricing on the improvement in expected revenue in the duopoly. We assume that customers arrive to purchase tickets in accordance with a geometric Brownian motion. We derive an explicit closed-form expression for an optimal pricing policy to maximize the expected revenue. It is shown that when the competitor adopts a static pricing policy, dynamic pricing is not always effective in terms of maximizing expected revenue compared to a fixed pricing strategy. Moreover, we show that the size of the reduction in the expected revenue depends on the competitor’s pricing strategy. Numerical results are presented to illustrate the dynamic pricing policy.

Recent Advances in Stochastic Operations Research

Operations research uses quantitative models to analyze and predict the behavior of systems and to provide information for decision makers. Two key concepts in such research are optimization and uncertainty. Typical models in stochastic operations research include queueing models, inventory models, financial engineering models, reliability models, and simulation models. This book contains a collection of peer-reviewed papers from the International Workshop on Recent Advances in Stochastic Operations Research (2007 RASOR Nanzan) held on March 5–6, 2007, at Nanzan University, Nagoya, Japan. It enables advanced readers to understand the recent topics and results in stochastic operations research.

Optimal policies in continuous time inventory control models with limited supply

Abstract In this paper, we deal with the problem of a fixed number of units of a certain perishable commodity over a continuous time horizon. Airline seats, hotel rooms, advertising space in newspapers, and some seasonal products that must be sold before the end of the season are examples of such commodities that cannot be carried over and are not storable for consumers. This paper considers such a problem of continuous time perishable inventory control by applying semi-Markov decision processes over a finite time horizon. It is shown that there is an optimal policy that is simple and stationary. Furthermore, some analytical properties of this optimal policy and its value function are explored under certain specific assumptions.

The pricing and optimal strategies of callable warrants

In this paper, we introduce a valuation model of callable warrants under a setting of the optimal stopping problem between the holder (investor) and the issuer (firm). A warrant is the right to purchase new shares at a predetermined price. When the new stocks are issued, the value of the stock is diluted. We consider the model taking the dilution into account. After identifying optimal policies for the issuer and the investor, we explore the analytical properties of the optimal exercise and call boundaries for the holder and the issuer, respectively. Furthermore, the value of such a callable warrant and the optimal critical prices are examined numerically using the binomial method.

THE VALUATION OF CALLABLE-PUTTABLE REVERSE CONVERTIBLE BONDS

Many companies issue some complex structured bonds. A reverse convertible bond is one of such structured bonds. In this paper we consider a valuation model of callable-puttable reverse convertible bonds which have the complex payoff in a setting of the optimal stopping problem between the issuer and the investor. Reverse convertible bonds issued by a company can be exchanged for the shares of another company. We analyze the pricing of reverse convertible bonds with call and put clauses and explore analytical properties of the value of the reverse convertible bond and optimal call and put boundaries by the issuer and the investor, respectively. Furthermore, we investigate how the call and put clauses affect the value and the optimal strategies for both of them.

THE VALUATION OF RUSSIAN OPTIONS FOR DOUBLE EXPONENTIAL JUMP DIFFUSION PROCESSES

In this paper, we derive closed form solution for Russian option with jumps. First, we discuss the pricing of Russian options when the stock pays dividends continuously. Secondly, we derive the value function of Russian options by solving the ordinary differential equation with some conditions (the value function is continuous and differentiable at the optimal boundary for the buyer). And we investigate properties of optimal boundaries of the buyer. Finally, some numerical results are presented to demonstrate analytical properties of the value function.

Callable Russian Options and Their Optimal Boundaries

We deal with the pricing of callable Russian options. A callable Russian option is a contract in which both of the seller and the buyer have the rights to cancel and to exercise at any time, respectively. The pricing of such an option can be formulated as an optimal stopping problem between the seller and the buyer, and is analyzed as Dynkin game. We derive the value function of callable Russian options and their optimal boundaries.

OPTIMAL ORDERING POLICIES WITH STOCHASTIC DEMAND AND PRICE PROCESSES

In this paper, we consider an inventory model in which a firm uses the spot market for procurement in order to accomplish the minimization of total discounted costs. The model can be formulated as impulse control problem where the demand and spot price follow diffusion stochastic processes. We explore sufficient conditions under which an optimal policy exists. Furthermore, we derive an optimal policy as an (s, S) policy where s and S are uniquely determined as a solution of simultaneous equation. Finally, we show some analytical properties of the optimal policy. Some numerical examples are also presented.

Transformation of partially observable Markov decision processes into piecewise linear ones

Abstract This paper considers how partially observable Markov decision processes may be transformed into piecewise linear ones, which have many advantages in that they are easily represented in a computer. Also we specify how to find the products of simple partitions on which cost functions are piecewise linear.

The Valuation of Callable Financial Commodities with Two Stopping Boundaries

AbstractIn this paper we consider a model of valuing callable financial commodities which enable both an issuer and an investor to exercise their rights, respectively. We show that such a model can be formulated as a coupled stochastic game for the optimal stopping problem with two reflecting barriers. It is also shown that there exists a pair of optimal stopping rules and the value of the stochastic game. Most previous work concerning American options, Israeli options, convertible bonds and callable derivatives has required the specific payoff function when either of the issuer or the investor has exercise their option. However, we deal with a rather general payoff function of the underlying asset price and the time. We also explore some analytical properties of optimal stopping rules of the issuer and the investor.