Subordination, Self-Similarity, and Option Pricing

In this paper, we try to investigate how the current account to gross domestic product (CAGDP) ratio in Turkey reacts to variations in the real effective exchange rate (REER) over the period of 1991:Q4-2007:Q3. For this purpose, we consider two different nonparametric regression models. In both approaches the TIME variable is specified as nonparametrically, but the REER variable is different. Firstly, we discuss a semi-parametric regression model where the parametric part is REER variable. Secondly, we consider an additive model where one of the nonparametric parts is a smooth function of REER variable and, therefore, the model falls within the class of additive models. The obtained findings showed that the response of the CAGDP ratio is negative nonlinear function of REER. Further, our estimations demonstrate that the coefficient giving the response of the CAGDP ratio to a change in REER declines over time when we assume a semi-parametric model.

<title>Abstract</title> This study explores the put-call duality under the Truncated Normal Distribution (TND) model in option pricing which assumes that the log-prices of the underlying assets are bounded above and below. The model is compared with famous Black \&amp; Scholes Model which has been consistently been considered as the benchmark model for option pricing. Using the model, we derived a model-consistent put-call duality identity and applied it in the construction of an objective function that improves the accuracy and consistency of option pricing. This is in conjunction with the various modifications that have been put forward to "improve" the option price. The purpose of this study was to extend Zhu \&amp; He's work by studying the the put-call duality relation of the proposed model and compare it with the results obtained using the classical Black \&amp; Scholes (BS) Model. Similarly, this study investigated the implied volatility of the proposed model and compared it with the BS model. From the duality constraint calibration, the proposed model performed significantly better than the traditional Black-Scholes model. A numerical analysis was done using real market data for S\&amp;P 500 stock price data retrieved on June 27, 2025. Model performance check such as AIC, BIC and pricing errors indicated that outstanding performance of the TND model. From the methodology and numerical analysis, there is evidence that the duality relation holds for the Truncated Normal Distribution model as is the case with the Black \&amp; Scholes model. The implied volatility indicated that the TND model was sensitive for in-the-money and out-of-the money options. Therefore, the enforcement of the duality constraint in the pricing of European option improved the market consistency ensures that no arbitrage opportunities exist in the market that cab provide enormous risk-free earnings in the market.

Empirical Performance of GARCH Models with Heavy-Tailed Innovations

Using daily closing price data spreading over 3 April 1990, to 5 May 2020, this study explores the skewness and excess kurtosis behaviour across energy, metals, and agricultural commodity futures. Subsequently, it compares the fitting of an empirical distribution under normal distribution assumption with those under the Generalized Hyperbolic distribution. The generalized hyperbolic distribution includes Hyperbolic distribution, Variance-Gamma distribution, and Normal Inverse Gaussian distribution. The results show that the Normal Inverse Gaussian distribution for natural gas, gold, platinum, copper, sugar, and feeder cattle futures captures skewness as well as excess kurtosis of the daily logarithmic returns. The findings are robust to the sub-sample analysis.

When powerful numerical tools like the finite element method encounter their limits for the evaluation of physical systems it is very common to use surrogate models as an approximation. There are many possible choices concerning the model approach, among which Gaussian process models are the most popular ones due to their clear statistical basis. A very desirable attribute of such surrogates is a high flexibility for making them applicable to a great class of underlying problems while obtaining interpretable results. To achieve this Gaussian processes are used as basis functions of an additive model in this work. Another important property of a surrogate is stability, which can be especially challenging when it comes to the estimation of the correlation parameters. To solve this we use a Bayesian approach where a reference prior is assigned to each component of the additive model assuring robust correlation matrices. Due to the additive structure of the model a simplified parameter estimation process is proposed that reduces the usually high-dimensional optimization problem to a few sub-routines of low dimension. Finally, we demonstrate this concept by modeling the magnetic field of a magnetic linear position detection system.

The normal inverse Gaussian (NIG) distribution has previously been shown to be a versatile tool to model heavy-tailed processes. A cumulant-matching estimator of the NIG parameters was introduced in Oigard and Hansen (2001). Here, we analyze the performance of this estimator. Next, we study whether NIG models are appropriate models for multi-user interference. Finally, we empirically test the goodness of fit of squared-NIG models to tele-traffic data, which have been fitted with alpha-stable models.

One of the harder new derivatives problems in recent years is estimating the probability distribution of credit losses on a portfolio of risky bonds or credit default swaps. This is the essential valuation problem for securitized debt instruments, including CDOs, CLOs, and CBOs, as well as exotic derivative contracts like "nth to default" swaps. The standard Gaussian copula model is not entirely satisfactory because it constrains correlation in the tails of the distribution in such a way that it is typically very hard to calibrate it to market tranche prices. The Student t-distribution has more plausible tail behavior, but because it is not stable under convolution, it becomes computationally very unwieldy as the number of variables increases. This article proposes the Normal Inverse Gaussian (NIG) distribution as a more tractable alternative. Comparison tests on several standard CDS index portfolios show that the NIG distribution has better tail characteristics than the Normal and it is much more efficient for large scale computations than the multivariate Student t. <b>TOPICS:</b>Credit default swaps, CLOs, CDOs, and other structured credit, statistical methods

This thesis contains investigation of the effect of the use of nonsimultaneous data in the pricing of futures-style options. Both the option pricing model, and the put-call parity relation specified by Asay (1982) provide the media for empirical testing of the research question. All share price index futures trades, and futures-style option trades, which occurred on the floor of the Sydney Futures Exchange, between 4 January 1993 and 31 August 1995, are examined. The study conducts analysis of option pricing errors resulting from the use of both time-matched data, and daily closing prices. Comparison of errors provides evidence to support the contention that the use of nonsimultaneous data introduces significant bias to option pricing. Additionally, evidence is presented to indicate that the Asay option pricing model exhibits a tendency to overprice options which are out-of-the-money and underprice options which are in-the-money. The model also exhibits significant pricing error for options which are near to maturity. Finally, evidence is presented regarding violation of put-call parity. It is shown that, when allowance is made for transaction costs, opportunity for arbitrage profit is scarce.

I address the dichotomy between American put option pricing theory and the numerical algorithms designed to estimate American put option prices. The literature has focused only on pricing error. However, early exercise is the essence of American option pricing (exercising) and with it comes the possibility of early exercise error and its opportunity costs. I introduce an economically viable metric that identifies all the errors of American put option pricing algorithms. The accuracy of such algorithms can thereby be fully assessed. A rational option pricing result generalises the usual integral equation and motivates pure pricing error. This provides new intuition for the optimality condition for early exercise. Early exercise error is motivated by comparing discounted expected profits generated by the estimated model vs. the optimal early exercise model. The error measure applies to any put pricing algorithm and any benchmark. We illustrate our total error measure with a new algorithm.

As one of the world’s largest securities markets, the Hong Kong stock market plays a significant role in facilitating the development of Chinese economy. In this paper, we investigate a suite of widely-used models, the GARCH models in risk management of the Hong Kong stock market returns. To account for conditional volatilities, we consider a new type of fat-tailed distribution, the normal reciprocal inverse Gaussian distribution (NRIG), and compare its empirical performance with two other popular types of fat-tailed distribution, the Student’s t distribution and the normal inverse Gaussian distribution (NIG). We show that the NRIG distribution performs slightly better than the other two types of distribution. Also, our results indicate that it is important to introduce both GJR-terms and the NRIG distribution to improve the models’ performance. Our results illustrate that the asymmetric GARCH NRIG model has practical advantages in quantitative risk management, and serves as a very useful tool for industry participants.

In the analysis of data from magnetoencephalography (MEG) and electroencephalography (EEG), it is common practice to arithmetically average event-related magnetic fields (ERFs) or event-related electric potentials (ERPs) across single trials and subsequently across subjects to obtain the so-called grand mean. Comparisons of grand means, e.g. between conditions, are then often performed by subtraction. These operations, and their statistical evaluation with parametric tests such as ANOVA, tacitly rely on the assumption that the data follow the additive model, have a normal distribution, and have a homogeneous variance. This may be true for single trials, but these conditions are rarely met when ERFs/ERPs are compared between subjects, meaning that the additive model is seldom the correct model for computing grand mean waveforms. Here, we summarize some of our recent work and present new evidence, from auditory-evoked MEG and EEG results, that the non-normal distributions and the heteroscedasticity observed instead result because ERFs/ERPs follow a mixed model with additive and multiplicative components. For peak amplitudes, such as the auditory M100 and N100, the multiplicative component dominates. These findings emphasize that the common practice of simply subtracting arithmetic means of auditory-evoked ERFs or ERPs is problematic without prior adequate transformation of the data. Application of the area sinus hyperbolicus (asinh) transform to data following the mixed model transforms them into the requested additive model with its normal distribution and homogeneous variance. We therefore advise checking the data for compliance with the additive model and using the asinh transform if required.

Option Pricing with Economic Regime Shifts

Optimization of complex real-time control systems often requires efficient response to any system changes over time. By combining pattern search optimization with a fast estimated Gaussian Process model, we are able to perform global optimization more efficiently for response surfaces with multiple local minimums or even dramatic changes over the design space. Our approach extends pattern search for global optimization problems by incorporating the global and local information provided by an additive global and local Gaussian Process model. We further develop a global search method to identify multiple promising local regions for parallel implementation of local pattern search. We demonstrate our methods on a standard test problem.

Hospital emergency arrivals are often modeled as Poisson processes because of the similarity of the problem to a telephone exchange model, i.e. applications that involve counting the number of times a random event occurs in a given time, where the interval between individual counts follows the exponential distribution. In this paper, we propose a statistical modeling of hospital emergency ward arrivals, using self-similar processes. This modeling takes into account the fact that it is known empirically that the emergency time series consists of periods marked by "bursts" of high level demand peaks, followed by periods of lower demand. This is explained neither by a Poisson during fixed-lengths intervals nor by a lognormal distribution. We show that a self-similar distribution can model this phenomenon. We also show that the commonly used Poisson models seriously underestimate the "burstiness" of emergency arrivals over a wide range of time scales, and that the emergency time series cannot be modeled by a lognormal distribution. The self-similar distribution was tested by the Hurst parameter, which we have calculated using five different methods, all of which agree on the value of the parameter.

Heterogeneity in beliefs and time preferences among investors make stock volatility stochastic, even though the volatility of the underlying dividend is constant. Prices of the European options written on this stock admit closed-form solutions, hence their hedging deltas. The Black–Scholes implied volatility surface, which depends on wealth distribution, investors' beliefs, and time preferences, exhibits observed patterns that are widely documented in various options markets. Along with benchmark models, the model is calibrated weekly to the S&amp;P 500 index options from January 1996 to April 2006. It shows comparable performance to the stochastic volatility and jump model and outperforms the traders' rules and two no-arbitrage models (stochastic volatility, and stochastic volatility and stochastic interest rate) in terms of out-of-sample pricing errors. This paper was accepted by Wei Xiong, finance.