Pricing in a Competitive Insurance Market Driven by Fractional Noise

Motivated by the empirical evidence of the long-range dependency found within the Greek motor insurance market, we formulate a particular stochastic pricing model in a continuous framework. We assume the structure of a competitive insurance market where the business volume of each company is directly related to the existing relativity between the company’s premium and the market’s average premium. Using a simple demand function and modeling the movements of the market via a fractional Brownian motion, we derive the optimal premium control strategy. Finally, we support the importance of the specific approach by a short application. It is shown that the optimal premium strategy is considerably different under the absence or existence of the long-range dependency. KEYwORdS Fractional Brownian motion, stochastic linear optimal control, competitive markets, insurance pricing Variance_Zimbidis.indd 55 9/22/11 8:11 AM Variance Advancing the Science of Risk 56 CASUALTY ACTUARIAL SOCIETY VOLUME 5/ISSUE 1 tion that determines the volume of business under the different levels of premium. Such a model had been initially proposed by Taylor (1986) in a discrete framework and revisited in a continuous framework by Emms, Haberman, and Savoulli (2007). The demand functions appearing in those works all contain the concept of the “average premium of the market,” which is modeled either on a deterministic basis or stochastically via the standard Brownian motion. In this paper, we adopt the pricing framework model of a competitive market using the fractional Brownian motion as a modeling tool for the movement of the market’s average premium. This is supported by the empirical evidence of the data gathered from the Greek motor insurance market within the last decade. This paper is organized as follows. In Section 2, we briefly discuss the concept of the competitive insurance markets. Section 3 provides a short background on fractional Brownian motion. In Section 4, we formulate the general model, and in Section 5 we solve a special case. Section 6 contains some insights into the structure of the Greek motor insurance market, identifies the phenomenon of long-range dependency, and provides a short numerical application. In Section 7, we add some final comments. 2. Competitive insurance markets The structure of a competitive insurance market can be extremely complicated. Here, we consider only the effect of the premium relativities, and how these quantities determine the market share of each company. We assume that the volume of business is determined via the function V t f p t p t t ( ) ( ( ), ( ), ( )), =  (1) where V(t) is the volume of business, p(t) is the premium rate of the company, p t ( ) is the market’s aver