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

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.Comment: IEEE Transactions on Knowledge and Data Engineering, 201

Stochastic model genetic programming: Deriving pricing equations for rainfall weather derivatives

Rainfall derivatives are in their infancy since starting trading on the Chicago Mercantile Exchange (CME) in 2011. Being a relatively new class of financial instruments there is no generally recognised pricing framework used within the literature. In this paper, we propose a novel Genetic Programming (GP) algorithm for pricing contracts. Our novel algorithm, which is called Stochastic Model GP (SMGP), is able to generate and evolve stochastic equations of rainfall, which allows us to probabilistically transform rainfall predictions from the risky world to the risk-neutral world. In order to achieve this, SMGP's representation allows its individuals to comprise of two weighted parts, namely a seasonal component and an autoregressive component. To create the stochastic nature of an equation for each SMGP individual, we estimate the weights by using a probabilistic approach. We evaluate the models produced by SMGP in terms of rainfall predictive accuracy and in terms of pricing performance on 42 cities from Europe and the USA. We compare SMGP to 8 methods: its predecessor DGP, 5 well-known machine learning methods (M5 Rules, M5 Model trees, k-Nearest Neighbors, Support Vector Regression, Radial Basis Function), and two statistical methods, namely AutoRegressive Integrated Moving Average (ARIMA) and Monte Carlo Rainfall Prediction (MCRP). Results show that the proposed algorithm is able to statistically outperform all other algorithms

Data-Driven Models: An Alternative Discrete Hedging Strategy

Options hedging is a critical problem in financial risk management. The prevailing approach in financial derivative pricing and hedging has been to first assume a parametric model describing the underlying price dynamics. An option model function is then calibrated to current market option prices and various sensitivities are computed and used to hedge the option risk. It has been recognized that computing hedging position from the sensitivity of the calibrated model option value function is inadequate in minimizing the variance of the option hedging risk, as it fails to capture the model parameter dependence on the underlying price. We propose several data-driven approaches to directly learn a hedging function from the historical market option and underlying data by minimizing certain measures of the local hedging risk and total hedging risk. This thesis will focus on answering the following questions: 1) Can we efficiently build direct data-driven models for discrete hedging problems that outperform existing state-of-art parametric hedging models based on the market prices? 2) Can we incorporate feature selection and feature extraction into the data-driven models to further improve the performance of the discrete hedging? 3) Can we build efficient models for both the one-step local risk hedging problem and multi-step total risk hedging problem based on the state-of-art learning framework such as deep learning framework and kernel learning framework? Using the S&P 500 index daily options data for more than a decade ending in August 2015, we first propose a direct data-driven approach based on kernel learning framework and we demonstrate that the proposed method outperforms the parametric minimum variance hedging method, as well as minimum variance hedging corrective techniques based on stochastic volatility or local volatility models. Furthermore, we show that the proposed approach achieves significant gain over the implied Black-Sholes delta hedging for weekly and monthly hedging. Following the direct data-driven kernel learning approach, we propose a robust encoder-decoder Gated Recurrent Unit (GRU) model, for optimal discrete option hedging. The proposed model utilizes the Black-Scholes model as a pre-trained model and incorporates sequential information and feature selection. Using the S&P 500 index European option market data from January 2, 2004, to August 31, 2015, we demonstrate that the weekly and monthly hedging performance of the proposed model significantly surpasses that of the data-driven minimum variance (MV) method, the regularized kernel data-driven model, and the SABR-Bartlett method. In addition, the daily hedging performance of the proposed model also surpasses that of MV methods based on parametric models, the kernel method, and the SABR-Bartlett method. Lastly, we design multi-step data-driven models to hedge the option discretely until the expiry. We utilize the SABR model and Local Volatility Function (LVF) to augment existing market data and thus alleviate the problem of scarcity in market option prices. The augmented market data is used to train a sufficient total risk hedging model

Rational Expectations in the Macro Model

macroeconomics, macro model

Hysteresis and economics - taking the economic past into account

The goal of this article is to discuss the rationale underlying the application of hysteresis to economic models. In particular, we explain why many aspects of real economic systems are hysteretic is plausible. The aim is to be explicit about the difficulties encountered when trying to incorporate hysteretic effects into models that can be validated and then used as possible tools for macroeconomic control. The growing appreciation of the ways that memory effects influence the functioning of economic systems is a significant advance in economic thought and, by removing distortions that result from oversimplifying specifications of input-output relations in economics, has the potential to narrow the gap between economic modeling and economic reality