Derivative pricing, model calibration, and sensitivity analysis are the three main problems in financial modeling. The purpose of this study is to present an algorithm to improve the pricing process, the calibration process, and the sensitivity analysis of the double Heston model, in the sense of accuracy and efficiency. Using the optimized caching technique, our study reduces the pricing computation time by about 15%. Another contribution of this thesis is: a novel application of the Automatic Differentiation (AD) algorithms in order to achieve a more stable, more accurate, and faster sensitivity analysis for the double Heston model (compared to the classical finite difference methods). This thesis also presents a novel hybrid model by combing the heuristic method Differentiation Evolution, and the gradient method Levenberg--Marquardt algorithm. Our new hybrid model significantly accelerates the calibration process.

In this paper we examine energy derivatives pricing. The previous studies considered the same source of uncertainty for the spot and the futures prices. We investigate the problem of futures pricing with two independent sources of risk. In general the structure of the oil and gas futures markets is closely related to some stock indices. Therefore, we develop a model for the futures market and compound derivatives with pricing in accordance with the correspondent index. We derive a framework for energy derivatives pricing, compute the price of the European call option on futures and corresponding hedging strategy. We calculate the price of the European call option adjusted for an index level, study the American put option on futures and corresponding hedging strategies.

In recent years, people pay much attention to derivative pricing subject to credit risk. In this paper, we proposed an autoregressive time series model of log odds ratios to price derivatives. Examples of the proposed model are given via the structural and reduced form approaches. Pricing formulae of the proposed time series models are derived for bonds and options. Furthermore, simulation studies are performed to confirm the accuracy of derived formulae.

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In this manuscript, we develop a nite liquidity framework for two-asset markets. In contrast to the standard multi-asset Black-Scholes framework, trading in our market model has a direct impact on the asset's price. The price impact is incorporated into the dynamics of the first asset through a specific trading strategy, as in large trader liquidity models. We adopt Euler- Maruyama and Milstein scheme in the simulation of asset prices. Exchange and Spread option values are numerically estimated by Monte Carlo with the Margrabe option as a controlled variate. The time complexity of these numerical schemes is included. Finally, we provide some deep learning frameworks to implement these pricing models effectively. / Thesis / Master of Science (MSc)

Collateralization in over-the-counter (OTC) derivatives markets has grown rapidly over the past decade, and even faster in the past few years, due to the impact of the recent financial crisis and the particularly important attention to the counterparty credit risk in derivatives contracts. The addition of collateralization to such contracts significantly reduces the counterparty credit risk and allows to offset liabilities in case of default. We study the problem of valuation of OTC derivatives with payoff in a single currency and with single underlying asset for the cases of zero, partial, and perfect collateralization. We assume the derivative is traded between two default-free counterparties and analyze the impact of collateralization on the fair present value of the derivative. We establish a uniform generalized derivative pricing framework for the three cases of collateralization and show how different approaches to pricing turn out to be consistent. We then generalize the results to include multi-asset and cross-currency arguments, where the underlying and the derivative are in some domestic currency, but the collateral is posted in a foreign currency. We show that the results for the single currency, multi-asset case are consistent with those obtained for the single currency, single asset case.

A design of anautomatic network capacity markets, oftenreferred to as a bandwidth market, is presented. Three topicsare investigated. First, a network model is proposed. Theproposed model is based upon a trisection of the participantroles into network users, network owners, and market middlemen.The network capacity is defined in a way that allows it to betraded, and to have a well defined price. The network devicesare modeled as core nodes, access nodes, and border nodes.Requirements on these are given. It is shown how theirfunctionalities can be implemented in a network. Second, asimulated capacity market is presented, and a statisticalmethod for estimating the price dynamics in the market isproposed. A method for pricing network services based on sharedcapacity is proposed, in which the price of a service isequivalent to that of a financial derivative contract on anumber of simple capacity shares.Third, protocols for theinteraction between the participants are proposed. The marketparticipants need to commit to contracts with an auditableprotocol with a small overhead. The proposed protocol is basedon a public key infrastructure and on known protocols for multiparty contract signing. The proposed model allows networkcapacity to be traded in a manner that utilizes the networkeciently. A new feature of this market model, compared to othernetwork capacity markets, is that the prices are not controlledby the network owners. It is the end-users who, by middlemen,trade capacity among each-other. Therefore, financial, ratherthan control theoretic, methods are used for the pricing ofcapacity. <b>Keywords:</b>Computer network architecture, bandwidthtrading, inter-domain Quality-of-Service, pricing,combinatorial allocation, financial derivative pricing,stochastic modeling

<p>A design of anautomatic network capacity markets, oftenreferred to as a bandwidth market, is presented. Three topicsare investigated. First, a network model is proposed. Theproposed model is based upon a trisection of the participantroles into network users, network owners, and market middlemen.The network capacity is defined in a way that allows it to betraded, and to have a well defined price. The network devicesare modeled as core nodes, access nodes, and border nodes.Requirements on these are given. It is shown how theirfunctionalities can be implemented in a network. Second, asimulated capacity market is presented, and a statisticalmethod for estimating the price dynamics in the market isproposed. A method for pricing network services based on sharedcapacity is proposed, in which the price of a service isequivalent to that of a financial derivative contract on anumber of simple capacity shares.Third, protocols for theinteraction between the participants are proposed. The marketparticipants need to commit to contracts with an auditableprotocol with a small overhead. The proposed protocol is basedon a public key infrastructure and on known protocols for multiparty contract signing. The proposed model allows networkcapacity to be traded in a manner that utilizes the networkeciently. A new feature of this market model, compared to othernetwork capacity markets, is that the prices are not controlledby the network owners. It is the end-users who, by middlemen,trade capacity among each-other. Therefore, financial, ratherthan control theoretic, methods are used for the pricing ofcapacity.</p><p><b>Keywords:</b>Computer network architecture, bandwidthtrading, inter-domain Quality-of-Service, pricing,combinatorial allocation, financial derivative pricing,stochastic modeling</p>

We develop expansion methods as a new computational approach towards high-dimensional partial differential equations (PDEs), particularly of such type as arising in the valuation of financial derivatives. The proposed methods are extended from [41] and use principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. They enable calculation of highly accurate approximate solutions with computational complexity polynomial in the number of dimensions for PDEs with a low number of dominant principal components. For the case of PDEs with constant coefficients, we show existence of expansion solutions and prove theoretical error bounds. We give a precise characterisation of when our methods can be applied and construct specific examples of a first and second order version. We provide numerical results showing that the empirically observed convergence speeds are in agreement with the theoretical predictions. For the case of PDEs with varying coefficients, we give a heuristic motivation using the Parametrix approach and empirically test the methods' accuracy for a range of variable parameter stock models. We demonstrate the applicability of our expansion methods to real-world securities pricing problems by considering path-dependent and early-exercise options in the LIBOR market model. Using the example of Bermudan swaptions and Ratchet floors, which are considered difficult benchmark problems, we give a careful analysis of the numerical accuracy and computational complexity. We are able to demonstrate that for problems with medium to high dimensionality, around 60-100, and moderate time horizons, the presented PDE methods deliver results comparable in accuracy to benchmark state-of-the-art Monte Carlo methods in similar or (significantly) faster run time.

This thesis is concerned with the mathematical analysis of electricity and carbon emission markets. We introduce a novel, versatile and tractable stochastic framework for the joint price formation of electricity spot prices and allowance certificates. In the proposed framework electricity and allowance prices are explained as functions of specific fundamental factors, such as the demand for electricity and the prices of the fuels used for its production. As a result, the proposed model very clearly captures the complex dependency of the modelled prices on the aforementioned fundamental factors. The allowance price is obtained as the solution to a coupled forward-backward stochastic differential equation. We provide a rigorous proof of the existence and uniqueness of a solution to this equation and analyse its behaviour using asymptotic techniques. The essence of the model for the electricity price is a carefully chosen and explicitly constructed function representing the supply curve in the electricity market. The model we propose accommodates most regulatory features that are commonly found in implementations of emissions trading systems and we analyse in detail the impact these features have on the prices of allowance certificates. Thereby we reveal a weakness in existing regulatory frameworks, which, in rare cases, can lead to allowance prices that do not conform with the conditions imposed by the regulator. We illustrate the applicability of our model to the pricing of derivative contracts, in particular clean spread options and numerically illustrate its ability to "see" relationships between the fundamental variables and the option contract, which are usually unobserved by other commonly used models in the literature. The results we obtain constitute flexible tools that help to efficiently evaluate the financial impact current or future implementations of emissions trading systems have on participants in these markets.