Paper: Dynamic pricing in retail with diffusion process demand | IMA Journal of Management Mathematics

Abstract
When randomness in demand affects the sales of a product, retailers use dynamic pricing strategies to maximize their profits. In this article, we formulate the pricing problem as a continuous-time stochastic optimal control problem and find the optimal policy by solving the associated Hamilton–Jacobi–Bellman (HJB) equation. We propose a new approach to modelling the randomness in the dynamics of sales based on diffusion processes. The model assumes a continuum approximation to the stock levels of the retailer which should scale much better to large-inventory problems than the existing Poisson process models in the revenue management literature. The diffusion process approach also enables modelling of the demand volatility, whereas Poisson process models do not. We present closed-form solutions to the HJB equation when there is no randomness in the system. It turns out that the deterministic pricing policy is near-optimal for systems with demand uncertainty. Numerical errors in calculating the optimal pricing policy may, in fact, result in a lower profit on average than with the heuristic pricing policy.

1. Introduction

Consider a monopolist retailer who wants to design a dynamic pricing policy for a product over a given period, in order to maximize their total revenue and minimize the cost associated with handling unsold items at a given terminal time. Two important components in the decision process are the ability to take into account the uncertainty associated with future cost and demand and to optimally adjust for new knowledge as it arrives. We will illustrate how to address both components in this article, focusing on large-inventory limits and multiplicative demand uncertainty. Assume the retailer sells large volumes of its product at high frequency compared to the total pricing period. In such a setting, it is appropriate to model the sales process in a continuum limit, both for product volume and for time, similar to Kalish (1983).

In the revenue management literature, most efforts to model demand uncertainty in continuous time have focused on Poisson processes. See, for example the overviews by Bitran & Caldentey (2003) or Aviv & Vulcano (2012). We stress that in other communities, such as financial markets, both diffusion processes and jump diffusions are common for modelling demand and spot prices (Benth et al., 2014). The survey by Carmona & Coulon (2014) gives an indication of how flexible more advanced models used for commodity markets can be. With this article, we wish to inspire the revenue management community to take advantage of this research when modelling uncertainty in their own domain. For the remainder of this section, however, we mainly focus on modelling in the revenue management literature.

A pure Poisson process assumption is not compatible with taking a time-continuum limit for sales volume with demand uncertainty and may be better suited for demand modelling in industries with lower product sales volumes, such as the airline and hotel industries. Maglaras & Meissner (2006), Schlosser (2015b) and Schlosser (2015a) propose pricing heuristics similar to this article, by considering a deterministic model based on the asymptotic continuum limit of Poisson processes. To the author’s knowledge, however, few attempts have been made to unify demand uncertainty with the continuum limit. For example, Raman & Chatterjee (1995) and Wu & Wu (2016) model demand uncertainty as increments of a Brownian motion. As we will show in this article, their approaches lead to demand processes that admit negative sales, with a probability approaching over infinitesimally small time periods. We believe this is an important factor for why so little research has been done in this area. This article proposes a different approach to modelling demand uncertainty in order to remedy this. In our approach, the parameters of the system are described as diffusion processes that are solutions to stochastic differential equations (SDEs). This enables modelling of the demand volatility, which Poisson processes do not. Our proposed approach can be combined with parameter estimation methods already used by retailers and also extends naturally to multiple products. Modelling the demand over time as a diffusion process has previously been done by Chambers (1992) at macroeconomic scale with UK national data. His focus was on the data assimilation aspect and was not applied to a setting of optimal control.

The retailer’s dynamic pricing policies are stochastic processes that control the SDE which describes the depletion of stock. In this article, we seek an optimal pricing policy that maximizes the expected value of the profit over a given pricing period. One way to find an optimal pricing policy is to solve an associated non-linear partial differential equation (PDE), known as the Hamilton-Jacobi-Bellman (HJB) equation (Pham, 2009). We provide closed-form solutions for the HJB equation in the deterministic case, for both linear and exponential demand functions. The solution identifies two pricing regimes, one where the retailer maximizes their profits without depleting the inventory and another where the retailer aims to maximize the price and still deplete its inventory. Xu & Hopp (2006) have considered a continuous-time pricing problem where the uncertainty is modelled by a geometric Brownian motion (GBM). Their expected demand function is unbounded, with the result that the optimal pricing strategy ensures that all stock is sold by the terminal time. The demand functions we consider are bounded, and by introducing a penalty on unsold stock we capture the pricing regime change that does not arise in Xu & Hopp (2006).

In financial markets traders face a similar problem to that presented in this article, known as the optimal execution, or liquidation, problem. There, a trader tries to sell, or purchase, a particular amount of an asset by a predetermined time. See, for example Cartea et al. (2015) for an overview of this problem. Instead of controlling the price, the focus of the retailer pricing problem, the trader directly controls how much of the product to sell at a particular time. If the expected demand model is invertible, the retailer’s pricing problem can be reformulated to control the expected amount of stock to sell at each time. This reformulation is sometimes chosen in the revenue management community as well, see, for example Bitran & Caldentey (2003). In this article we focus on the formulation that writes the expected demand model in terms of the price.

By investigating the terms in the HJB equation, we identify the cases where the deterministic-case solution is appropriate. Potentially significant changes to the pricing policy for the stochastic system are at the interface between the two pricing regimes; far away from this interface the deterministic pricing policy is near-optimal. For example, the expected price path is decreasing when one takes into account uncertainty, while it is not for the deterministic heuristic. For a risk-neutral decision maker, however, the differences in profit are insignificant for most cases that may be relevant in industry.

The article is structured as follows: in Section 2, we describe the modelling of the system and compare the new parameter uncertainty approach to the existing Brownian increments approach. Then, a formulation of the pricing problem and the associated HJB equation is given in Section 3. We also propose a method to estimate the multiplicative factor in our model, in order to implement the pricing policy in practice. The optimal pricing policy in the deterministic limit is covered in Section 4, and the comparison to the stochastic system is shown in Section 5. Extensions to the problem, such as other models for uncertainty, and risk aversion, are discussed in Section 6. Finally, we conclude and suggest avenues for further research in Section 7.