# Pricing average price advertising options when underlying spot market prices are discontinuous

## TL;DR

This paper develops a new framework for pricing advertising options based on average prices over time and models market prices with jump-diffusion processes, addressing limitations of previous models that assumed continuous prices and point-in-time payoffs.

## Contribution

It introduces a path-dependent pricing model using power means and jump-diffusion stochastic processes, along with a Monte Carlo algorithm and explicit formulas for geometric mean payoffs.

## Key findings

- The new model captures jumps and spikes in market prices.
- Monte Carlo simulation effectively prices the options under the new framework.
- Explicit formulas are derived for geometric mean-based payoffs.

## Abstract

Advertising options have been recently studied as a special type of guaranteed contracts in online advertising, which are an alternative sales mechanism to real-time auctions. An advertising option is a contract which gives its buyer a right but not obligation to enter into transactions to purchase page views or link clicks at one or multiple pre-specified prices in a specific future period. Different from typical guaranteed contracts, the option buyer pays a lower upfront fee but can have greater flexibility and more control of advertising. Many studies on advertising options so far have been restricted to the situations where the option payoff is determined by the underlying spot market price at a specific time point and the price evolution over time is assumed to be continuous. The former leads to a biased calculation of option payoff and the latter is invalid empirically for many online advertising slots. This paper addresses these two limitations by proposing a new advertising option pricing framework. First, the option payoff is calculated based on an average price over a specific future period. Therefore, the option becomes path-dependent. The average price is measured by the power mean, which contains several existing option payoff functions as its special cases. Second, jump-diffusion stochastic models are used to describe the movement of the underlying spot market price, which incorporate several important statistical properties including jumps and spikes, non-normality, and absence of autocorrelations. A general option pricing algorithm is obtained based on Monte Carlo simulation. In addition, an explicit pricing formula is derived for the case when the option payoff is based on the geometric mean. This pricing formula is also a generalized version of several other option pricing models discussed in related studies.