Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Lévy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Lévy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points

On the calibration of the 3/2 model

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The Amnesiac Lookback Option: Selectively Monitored Lookback Options and Cryptocurrencies

This study proposes a strategy to make the lookback option cheaper and more practical, and suggests the use of its properties to reduce risk exposure in cryptocurrency markets through blockchain enforced smart contracts and correct for informational inefficiencies surrounding prices and volatility. This paper generalizes partial, discretely-monitored lookback options that dilute premiums by selecting a subset of specified periods to determine payoff, which we call amnesiac lookback options. Prior literature on discretely-monitored lookback options considers the number of periods and assumes equidistant lookback periods in pricing partial lookback options. This study by contrast considers random sampling of lookback periods and compares resulting payoff of the call, put and spread options under floating and fixed strikes. Amnesiac lookbacks are priced with Monte Carlo simulations of Gaussian random walks under equidistant and random periods. Results are compared to analytic and binomial pricing models for the same derivatives. Simulations show diminishing marginal increases to the fair price as the number of selected periods is increased. The returns correspond to a Hill curve whose parameters are set by interest rate and volatility. We demonstrate over-pricing under equidistant monitoring assumptions with error increasing as the lookback periods decrease. An example of a direct implication for event trading is when shock is forecasted but its timing uncertain, equidistant sampling produces a lower error on the true maximum than random choice. We conclude that the instrument provides an ideal space for investors to balance their risk, and as a prime candidate to hedge extreme volatility. We discuss the application of the amnesiac lookback option and path-dependent options to cryptocurrencies and blockchain commodities in the context of smart contracts

Analysis of high-frequency financial data over different timescales: a Hilbert-Huang transform approach

This thesis provides a better understanding of the complex dynamics of high-frequency financial data. We develop a methodology that successfully and simultaneously character¬izes both the short and the long-term fluctuations latent in a time series. We extensively investigate the applications of the empirical mode decomposition (EMD) and the Hilbert transform to the analysis of intraday financial data. The applied methodology reveals the time-dependent amplitude and frequency attributes of non-stationary and non-linear time series. We uncover a scaling law that links the amplitude of the oscillating components to their respective period. We relate such scaling law to distinctive properties of financial markets. This research is relevant because financial data contain patterns specific to the observa¬tion frequency and are thus, of interest to different type of market agents (market traders, intraday traders, hedging strategist, portfolio managers and institutional investors), each characterized by a different reaction time to new information and by the frequency of its intervention in the market. Understanding how the investment horizons of these agents in¬teract may reveal significant details about the physical processes that generate or influence financial time series. We use the EMD to estimate volatility, generalising the idea of the popular realised volatility estimator by decomposing financial time series into several timescales compo¬nents which are related to different investment horizons. We also investigate the dynamic correlation at different timescales and at different time-lags, revealing a complex structure of financial signals. Following the multiscale analysis approach, we propose a novel empirical method to es¬timate a time-dependent scaling parameter in analogy to the scaling exponent for self-similar processes. Using numerical simulations, we investigate the robustness of our estimator to heavy-tailed distributions. We apply the scaling estimator to intraday stock market prices and uncover scaling properties which differ from what would be expected from a random walk. We also introduce a novel entropy-like measure which estimates the regularity of a time series. This measure of complexity can be used to identify periods of high and low volatility x which could help investors to choose the appropriate time for investment. Finally, we pro¬pose a multistep-ahead forecasting framework based on EMD combined with support vector regression. The originality of our models is the inclusion of a coarse-to-fine reconstruction step to analyse the forecasting capabilities of a combination of oscillating functions. We compare our models with popular benchmark models which do not use the EMD as a pre¬processing tool, obtaining better results with our proposed framework. Part of the research developed on this thesis is published in Physica A: Statistical Me¬chanics and its Applications [137] and in the European Physical Journal, Special Topics [136]. It was also presented at international conferences, including the 20th annual work¬shop on the Economic Science with Heterogeneous Interacting Agents (WEHIA) 2015 and the 21st Computing in Economics and Finance (CEF) conference 2015