Fast and Accurate Pricing and Hedging of Long-Dated CMS Spread Options

The main objective of the present thesis is to provide efficient methods and tools for calibrating Libor Market Models (LMM) to market-prices of caps, swaptions and CMS spread options, and in this way to extract the volatility and correlation information implicitly contained in these products.

This paper introduces parallel computation for spread options using two-dimensional Fourier transform. Spread options are multi-asset options whose payoffs depend on the difference of two underlying financial securities. Pricing these securities, however, cannot be done using closed-form methods; as such, we propose an algorithm which employs the fast Fourier Transform (FFT) method to numerically solve spread option prices in a reasonable amount of short time while preserving the pricing accuracy. Our results indicate a significant increase in computational performance when the algorithm is performed on multiple CPU cores and GPU. Moreover, the literature on spread option pricing using FFT methods documents that the pricing accuracy increases with FFT grid size while the computational speed has opposite effect. By using the multi-core/GPU implementation, the trade-off between pricing accuracy and speed is taken into account effectively.

Pricing of interest rate derivatives, such as CMS spread or mid-curve options, depends on the modelling of the underlying single rates. For flexibility and realism, these rates are often described in the framework of stochastic volatility models. In this paper we allow rates to be modelled within a class of general stochastic volatility models, which includes the SABR, ZABR, free SABR and Heston models. We provide a versatile technique called Effective Markovian Projection, which allows a tractable model to be found that mimics the distribution of the more complex models used to price multi-rate derivatives. Three different numerical approaches are outlined and applied to relevant examples from practice. Finally, a new method that involves moment-matching of Johnson distributions is applied to facilitate closed-form pricing formulas.

Replication (pricing) is widely used in the interest rate derivatives pricing, in particular for constant maturity swap (CMS) products. This chapter focuses on this subject to examine some of the practical issues, as well as practical solutions. It explains how the replication techniques can be used effectively to deal with CMS convexity and CMS derivatives pricing. The chapter also examines replication consistent copula pricing of CMS spread option products widely used in yield curve shape products. In the context of CMS spread options, CMS rates, though derived from the same yield curve and the same swaption volatilities, can be viewed as correlated individual components in a basket. The marginal distributions of CMS rates are important in volatility smile/skew calibration and analysing joint statistical characteristics. The chapter demonstrates that a good model exhibits self-consistency to the vanilla markets, in terms of underlyings calibrations as well as their marginal distributions (volatility smile/skew).

Valuing spread options with counterparty risk and jump risk

Pricing of American lookback spread options

<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.

This paper explores the application of Machine Learning techniques for pricing high-dimensional options within the framework of the Uncertain Volatility Model (UVM). The UVM is a robust framework that accounts for the inherent unpredictability of market volatility by setting upper and lower bounds on volatility and the correlation among underlying assets. By leveraging historical data and extreme values of estimated volatilities and correlations, the model establishes a confidence interval for future volatility and correlations, thus providing a more realistic approach to option pricing. By integrating advanced Machine Learning algorithms, we aim to enhance the accuracy and efficiency of option pricing under the UVM, especially when the option price depends on a large number of variables, such as in basket or path-dependent options. In this paper, we consider two approaches based on Machine Learning. The first one, termed GTU, evolves backward in time, dynamically selecting at each time step the most expensive volatility and correlation for each market state. Specifically, it identifies the particular values of volatility and correlation that maximize the expected option value at the next time step, and therefore, an optimization problem must be solved. This is achieved through the use of Gaussian Process regression, the computation of expectations via a single step of a multidimensional tree and the Sequential Quadratic Programming optimization algorithm. The second approach, referred to as NNU, leverages neural networks and frames pricing in the UVM as a control problem. Specifically, we train a neural network to determine the most adverse volatility and correlation for each simulated market state, generated via random simulations. The option price is then obtained through Monte Carlo simulations, which are performed using the values for the uncertain parameters provided by the neural network. The numerical results demonstrate that the proposed approaches can significantly improve the precision of option pricing and risk management strategies compared with methods already in the literature, particularly in high-dimensional contexts.

If the market is incomplete, we have several choices of equivalent martingale measures to price contingent claims .Recall that we didn't impose conditions on the preferences of the economic agents other than that they prefer more to less. So it is quite natural to specify the preferences further in order to select one of the equivalent martingale measures.The specification of the investor's attitude towards risk is done in terms of a utility function. Credit risk is increasingly being recognised as a significant area of risk and arbitrage, yet there exists relatively little research on it. In this paper we show that operational risk represents a fundamental risk to option hedging (1) and investigate it by proposing a new theoretical model. We derive an exposure indicator for the operational risk of option hedging and the resulting operational risk distribution. We obtain analytical results for various risk measures including the Value at Risk equation; this includes deriving a new analytical result for the option function of the half-normal distributions (which will be of interest to Statisticians in general). We determine an analytical solution to the price of options under operational risk (2). We conduct numerical experiments on empirical option data to validate our model and estimate the operational Value at Risk for option hedging. There exists currently a good deal of literature on hedging methods for the pricing options. It is well known that in this case a straightforward Monte Carlo simulation algorithm will be time-consuming and yield unstable results for the prices and especially the sensitivities. The knock-in/out features in the barrier option payoffs lead to slower convergence of the Monte Carlo algorithm. To address this problem the following (semi-)analytical approaches have been developed for specific models. In any case, theoretically, long-run uncertainty about demographic changes and shocks restricts the land supply and raises the land price. The consequences for housingpolicies may be significant, especially in countries (likeFrance) where policymakers attempt to develop the housingstock and to make housing more affordable for the less well-off. Now, the spatial demographic risk is local by nature: a shortfall in one place entails a surplus somewhere else; therefore, it is an insurable risk. Consequently, a future insurance market may be a tool with which to improve the way the housing market works. Empirically, we conduct the analysis for a French department, Nord. The data base used in this study is made up of individual transactions for developable land for residential purposes(19,495observations) or secondary or tertiary activities (1,667 observations) between 1989 and 2002.The results show that, during the upward trend in the land market, the price of developable land is significantly higher when the change in population in the vicinity is more volatile, and so more difficult to predict. The plot price is 7.8% (housing market) or 15.3% (office and factory market) higher when population change in the 10 nearest communes increases by one standard deviation. 2. A General Option-Pricing Formula we compare our results with those given by Merton (3).We first explain Merton's results. Merton restricted his discussion to discrete probability distributions. Let an asset have terminal value

Spread options are a fundamental class of derivative contracts written on multiple assets and are widely traded in a range of financial markets. There is a long history of approximation methods for computing such products, but as yet there is no preferred approach that is accurate, efficient, and flexible enough to apply in general asset models. The present paper introduces a new formula for general spread option pricing based on Fourier analysis of the payoff function. Our detailed investigation, including a flexible and general error analysis, proves the effectiveness of a fast Fourier transform implementation of this formula for the computation of spread option prices. It is found to be easy to implement, stable, efficient, and applicable in a wide variety of asset pricing models.

Local Risk Minimization on Supply Curve Model with Market Impacts

In today's volatile financial market the demand for an accurate option price forecaster has been a focal point for researchers. The purpose of this study is to forecast option prices using neural networks. Initially simple neural network was implemented using twenty year period data from S&P 500 index call option prices. The prediction result was better than that of traditional Black-Scholes model. A hybrid neural network was developed that utilized aspects of Black- Scholes model into the neural network and tested against the traditional approach and simple neural network. The hybrid neural network outperformed performance of the tradition forecasting model and improved prediction results of simple neural networks.

On reset option pricing in binomial market with both fixed and proportional transaction costs

Exponential Lévy models generalize the classical Black and Scholes setup by allowing the stock prices to jump while preserving the independence and stationarity of returns. This article introduces most common examples of such models and discusses the pricing and hedging of options in this framework.

The empirical characteristics of the underlying asset prices should be taken into account for the pricing and hedging of options. In this paper, we show how to price basket options when assets follow the “shifted asymmetric jump-diffusion” process. The methodology is based on the Hermite polynomial expansion that can match exactly the first m moments of the model implied-probability distribution. The resultant pricing and hedging formulae are in closed-form and similar to the Black and Scholes ones.