Before the 2008 financial crisis, most research in financial mathematics focused on the risk management and the pricing of options without considering effects of counterparties’ default, illiquidity problems, systemic risk and the role of the repurchase agreement (Repo). During the 2008 financial crisis, a frozen Repo market led to a shutdown of short sales in the stock market. Cyclical interdependencies among financial corporations caused that a default of one firm seriously affected other firms and even the whole financial network. In this dissertation, we will consider financial markets which are shaped by financial crisis. This will be done from two distinct perspectives, an investor’s and a regulator’s. From an investor’s perspective, recently models were proposed to compute the total valuation adjustment (XVA) of derivatives without considering a potential crisis in the market. In our research, we include a possible crisis by apply an alternating renewal process to describe a switching between a normal financial status and a financial crisis status. We develop a framework for pricing the XVA of a European claim in this state-dependent framework. We represent the price as a solution to a backward stochastic differential equation and prove the existence and uniqueness of the solution. To study financial networks from a regulator’s perspective, one popular method is the fixed point based approach by L. Eisenberg and T. Noe. However, in practice, there is no accurate record of the interbank liabilities and thus one has to estimate them to use Eisenberg - Noe type models. In our research, we conduct a sensitivity analysis of the Eisenberg - Noe framework, and quantify the effect of the estimation errors to the clearing payments. We show that the effect of the missing specification of interbank connection to clearing payments can be described via directional derivatives that can be represented as solutions of fixed point equations. We also compute the probability of observing clearing payment deviations of a certain magnitude.

Short selling is important for price efficiency as it helps negative information to be incorporated into prices. As short selling requires borrowing stock in advance, the equity lending market plays a central role in price efficiency. For instance, when the costs of borrowing certain equities are high, these stocks are likely to be overpriced. Unfortunately, not much is known about the equity lending market, particularly the Brazilian market. Here, we have investigated a particular feature of the equity lending contract, namely, the lender recall option. Lending contracts either i) allow the lender to recall the stock at an earlier date than initially agreed, or ii) allow no early recall, that is, they are fixed term contracts. We have derived a simple model for recall option pricing and confirmed the model empirically / A venda descoberta desempenha uma importante participação na eficiência da precificação de ativos, pois permite incorporar informações negativas aos seus preços. Como a venda descoberta requer que um ativo seja alugado previamente, o mercado de aluguel de ativos tem um papel central na formação eficiente de preços. Por exemplo, quando os custos de aluguel são altos, ativos estão provavelmente sobrevalorizados. Infelizmente pouco se conhece a fundo sobre o mercado de aluguel de ativos. Neste artigo, investigamos uma característica do aluguel de ações, propriamente dita, a opção de liquidação antecipada pelo doador. Contratos de aluguel, quanto a este aspecto, podem i) permitir que o doador requeira suas ações antes do prazo acordado ou ii) não permitir esta opção, possuindo prazo fixo. Derivamos um modelo simples de precificação desta opção e confirmamos o modelo empiricamente

This thesis examines the application of neural networks in the context of option pricing. Throughout the thesis, different architecture choices and prediction parameters are tested and compared in order to achieve better performance and higher accuracy in option valuation. Two different volatility forecast mechanisms are used to compare neural networks performance with Black Scholes parametric model. Moreover, the performance of a neural network is compared also to more advanced modular neural networks. A new technique of adding rational prediction assumptions to neural network prediction is tested and the thesis shows the importance of adding virtual options fulfilling these assumptions in order to achieve better training of the neural network. This method comes out to increase the prediction power of the network significantly. The thesis also shows the neural network prediction outperforms the traditional parametric methods. The size and number of hidden layers in a neural network is tested with an emphasis to provide a benchmark and a structured way how to choose neural network parameters for future applications in option pricing. JEL Classification C13, C14, G13 Keywords Option pricing, Neural networks, Modular neu- ral networks, S&P500 index options Author's e-mail vach.dominik@gmail.com...

本研究結果的重點如下：一、各樣本金融機構經由模型計算出的費率差異 頗大，顯示財務結構的不同，確實應課收不同保費。而現行的單一費率對 於好壞金融機構收取的費率一視同仁無法反應金融機構其經營風險的差異 ，也就無法避免低風險金融機構補貼高風險金融機構的弊病。因此應實施 差別費率較為合理。二、根據研究結果顯示，若以金融機構性質為分類， 其費率估計負擔由低至高依序為：一般民營銀行、國家行局、省市屬行庫 、民營中小企業銀行、信託投資公司、外商銀行、信用合作社、漁會信用 部、農會信用部。而此法的一大優點在於可對個別機構評估其相對風險性 的高低，即可依各自經營情況、組織結構分別評估其風險，以達精算上的 合理性。三、由各家樣本金融機構應負擔之估計存保費率可知，只要金融 機構資產結構稍有不同，便會使存保公司的負擔不同。換言之，因金融機 構體質的互異，其為存保公司所帶來的承保風險也有別，為避免資源配置 不當的缺失，實施以個別金融機構風險為基礎的差別費率是較公平、合理 的方式。四、承保比率亦是影響費率估計的一項因素，由實證可知承保比 率對費率估計有顯著的影響。即承保比率高的金融機構，其估計費率與全 部承保時的差異較小；承保比率低者，其估計費率與全部承保時的差異較 大。

Derivative pricing, and in particular the pricing of options, is an important area of current research in financial mathematics. Experts debate on the best method of pricing and the most appropriate model of a price process to use. In this thesis, a ``Switching Black-Scholes'' model of a price process is proposed. This model is based on the standard geometric Brownian motion (or Black-Scholes) model of a price process. However, the drift and volatility parameters are permitted to vary between a finite number of possible values at known times, according to the state of a hidden Markov chain. This type of model has been found to replicate the Black-Scholes implied volatility smiles observed in the market, and produce option prices which are closer to market values than those obtained from the traditional Black-Scholes formula. As the Markov chain incorporates a second source of uncertainty into the Black-Scholes model, the Switching Black-Scholes market is incomplete, and no unique option pricing methodology exists. In this thesis, we apply the methods of mean-variance hedging, Esscher transforms and minimum entropy in order to price options on assets which evolve according to the Switching Black-Scholes model. C programs to compute these prices are given, and some particular numerical examples are examined. Finally, filtering techniques and reference probability methods are applied to find estimates of the model parameters and state of the hidden Markov chain. / Thesis (Ph.D.)--Applied Mathematics, 2003.

<p>In this thesis the double exponential jump-diffusion model is considered and the Laplace transform is used as a method for pricing both plain vanilla and path-dependent options. The evolution of the underlying stock prices are assumed to follow a double exponential jump-diffusion model. To invert the Laplace transform, the Euler algorithm is used. The thesis includes the programme code for European options and the application to the real data. The results show how the Kou model performs on the NASDAQ OMX Stockholm Market in the case of the SEB stock.</p>

Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

In this thesis the double exponential jump-diffusion model is considered and the Laplace transform is used as a method for pricing both plain vanilla and path-dependent options. The evolution of the underlying stock prices are assumed to follow a double exponential jump-diffusion model. To invert the Laplace transform, the Euler algorithm is used. The thesis includes the programme code for European options and the application to the real data. The results show how the Kou model performs on the NASDAQ OMX Stockholm Market in the case of the SEB stock.

本研究係利用財務理論定價模型之實質選擇權擴張情境模式，以三項式選擇權評價方法，評估現行上市的不動產投資信託商品（REITs）的合理價值。並選定富邦一號與二號、國泰一號與二號、新光一號等5檔REITs商品進行評價分析。 　　經研究結果，其中有4檔REITs評價價值與其2010年財報淨值非常接近；此外，並有其中3檔REITs的評價價值，相對於財報的每股淨值，更為接近實際股票市場交易的最高價格，證明本研究的三項式選擇權評價模型可適用於REITs商品的評價方法。 / This paper employs the Trinomial Real Option Pricing Model for the valuation of Real Estate Investment Trusts (REITs). The following five REITs in Taiwan (or T-REITs) are selected for empirical analysis: Fubon No.1 and No.2 REITs, Cathay No.1 and No.2 REITs, and Shin Kong No.1 REITs. Results show that the values of four T-REITs values from the valuation model are very close to their book value in the end of 2010, and three T-REITs values are also similar to their highest prices in the exchange market. Conclusions of this study imply that the Trinomial Real Option Pricing Model may serve as a good approach for the valuation of REITs prices.

An in depth study of the pricing of Game contingent claims under a general diffusion market model, in which interest rate is non constant, is presented. With the idea of providing a few numerical examples of the valuation of such claims, we present a detailed description of a Bootstrapping procedure to obtain interest rate information from Swaps rates. We also present a Stripping procedure that can be used to obtain initial spot (caplet) volatility from Market quotes on Caps/FLoors. These methods are of general application and could be used in the calibration of diffusion models of interest rate. Then we show several examples of calibration of the Hull--White model of interest rates. Our calibration examples are later used in the numerical approximation of the value of a particular form of Game option.