The thesis focuses on methods of option prices calculations using two different pricing models which are Heston and Black-Scholes models. The first part describes theory of these two models and conlcudes with a comparison of the risk-neutral measures of these two models. In the second part, the relations between input parameters and the option price generated by these models are clarified. This part ends up with an analysis of the market data and it answers the question which model predicts better.

Derivatives have a large and significant role on the financial markets today and the popularity of options has increased. This has also increased the demand of finding a suitable option pricing model, since the ground-breaking model developed by Black & Scholes (1973) have poor pricing performance. Practitioners and academics have over the years developed different models with the assumption of non-constant volatility, without reaching any conclusions regarding which model is more suitable to use. This thesis examines four different models, the first model is the Practitioners Black & Scholes model proposed by Christoffersen & Jacobs (2004b). The second model is the Heston´s (1993) continuous time stochastic volatility model, a modification of the model is also included, which is called the Strike Vector Computation suggested by Kilin (2011). The last model is the Heston & Nandi (2000) Generalized Autoregressive Conditional Heteroscedasticity type discrete model. From a practical point of view the models are evaluated, with the goal of finding the model with the best pricing performance and the most practical usage. The model´s robustness is also tested to see how the models perform in out-of-sample during a high respectively low implied volatility market. All the models are effected in the robustness test, the out-sample ability is negatively affected by a high implied volatility market. The results show that both of the stochastic volatility models have superior performances in the in-sample and out-sample analysis. The Generalized Autoregressive Conditional Heteroscedasticity type discrete model shows surprisingly poor results both in the in-sample and out-sample analysis. The results indicate that option data should be used instead of historical return data to estimate the model’s parameters. This thesis also provides an insight on why overnight-index-swap (OIS) rates should be used instead of LIBOR rates as a proxy for the risk-free rate.

Background: Since the emergence of Covid-19 has it reaped and created havoc within every segment of society on a national and global scale. The financial market experienced significant declines and losses but some asset items handled the fluctuations better than others. Moreover, since some asset items are associated with different risk levels will various investors with contrasting risk attitude allocate dissimilar proportion of their disposable capital between these alternatives. Especially during low and high levels of economic uncertainty which is related to the volatile market of Covid-19. Although, little to no research has been conducted aimed at understanding how Covid-19 impacted Swedish students asset allocation prior and post the pandemic with different risk profiles.   Purpose: The purpose of this study is to investigate if students with different risk attitudes (risk-preference, risk-neutral and risk-averse) conduct statistically different asset allocation prior and post the Covid-19 pandemic. Furthermore, investigate shifts in asset holdings prior and post the pandemic. Moreover, in order to fill the identified literature gap and add to the current body of knowledge regarding asset allocation and variability concerning risk attitudes since its exclusion of Swedish student’s risk attitudes and impact of Covid-19 on preferable asset items.                                    Method: This investigative study concerns a quantitative survey of 81 different students attending Jönköping University. The survey was structured in a way to uncover whether students with different risk attitudes conduct asset allocation statistically different prior and post the Covid-19 pandemic. Moreover, incorporate sociodemographic factors of students in order to measure its relation to risk attitudes and uncertainty changes. This will be done through non-parametric tests (distribution free) such as the Chi-square, Kruskal-Wallis and Bonferroni adjusted p-value approach. The data is later discussed and interpreted through various academic sources and in the context of the frame of reference (expected utility theory).                              Conclusion: The impact of Covid-19 resulted into increased asset allocation of less risky and “safe” asset in order to deal with the declining stock market and future economic uncertainty. The study also suggest that students liquidated some of their current/fixed deposits and re-invested their disposable capital into a more conservative money management strategy, which was a continuous identified pattern.  Furthermore, the results indicate that students with different risk attitudes conduct significantly different asset allocation concerning commercial insurance, stocks/funds and various bond types prior to Covid-19. However, post the eruption has the statistical identified differences in bonds asset allocation reduced which refers to that the statistical power and dissimilar allocated proportion amongst asset items has diminished. Further multiple comparison reinsures this conclusion. Thusly, the study implies that the differences between asset allocation and student risk profiles are diminished post Covid-19 and therefore students perceived and allocated more similar capital proportions into various asset items. Hence answer the initial stated research question and empirically state that risk attitude of students impact how they conduct asset allocation prior to and to a lesser extent post Covid-19

In this thesis we use nonstandard methods for the valuation of derivatives on electricity. We model the dynamics of electricity spot price as mean reverting processes on the hyperfinite binomial tree and by switching to the risk-neutral world we derive analytical formulas for the price of forward contracts. Both of our models are fitted to the German electricity market and forward price predictions are compared with forward products traded on the exchange. We conclude that both the Ornstein-Uhlenbeck and the Schwartz one factor model fit long-term forward contracts well while our prediction results for short-term forward prod- ucts are not conclusive due to low liquidity and alternative approaches might be suitable. 1

Esta tese promove uma integração entre Finanças e Teoria de Informação para criação de um ambiente alternativo para a determinação do preço racional da opção européia simples ordinária sobre ação e ativo de renda fixa (bond). Uma das características deste novo ambiente de determinação de preço racional é poder continuar utilizando o cálculo newtoniano em vez do estocástico. Cria uma notação matemática precisa e completa para a Teoria da Informação e a integra com a teoria de Finanças em condições de incerteza. Integra as abordagens entrópicas de determinação do preço racional da opção européia simples ordinária de Gulko (1998 e 1998a) e de Yang (1997). Define precisamente o mundo com preço da incerteza neutralizado (risk-neutral world), o mundo martingale, o mundo informacionalmente eficiente e o mundo entrópico e suas implicações para a Ciência do Investimento e, mais especificamente, para a determinação do preço racional de ativos básicos e derivativos. Demonstra detalhadamente a fórmula do preço racional da opção européia simples ordinária de Black-Scholes-Merton, melhorando a notação matemática, simplificando (eliminando a abordagem martingale) e complementando a demonstração feita por Baxter & Rennie (1998). Interrompe uma sucessão de trabalhos que estabelecem uma forma equivocada de calcular o preço da opção européia simples ordinária. Esse erro teve sua origem, muito provavelmente, numa edição de Brealey & Myers, que equivocadamente utilizou um resultado de Cox & Rubinstein (1985); esse resultado facilitava o cálculo do preço racional da opção européia simples ordinária por meio de uma tabela que evita o uso direto da fórmula de Black-Scholes-Merton. Brealey & Myers (desde a quarta edição de 1991), Luehrman (nos seus dois artigos da HBR de 1998 e um caso de 1995 pela HBS) e Edleson (caso publicado em 1994 pela HBS) ensinam que o valor percentual encontrado nessa tabela deve ser multiplicado pelo preço do valor mobiliário, quando deveria ser multiplicado pelo valor presente do preço de exercício. Os resultados mais importantes desta tese para Finanças são: (i) desenvolvimento de um método alternativo, robusto e parcimonioso, baseado no princípio da máxima entropia da Teoria da Informação e do Sistema de Distribuições de Pearson para obtenção de uma única medida de probabilidade neutralizadora do preço da incerteza (risk-neutral probability), (ii) obtenção de fórmula prática para a determinação do preço racional da opção européia simples ordinária para ação, (iii) validação da fórmula de Black-Scholes-Merton para ação, (iv) obtenção de uma fórmula adequada para a determinação do preço racional da opção européia simples ordinária sobre um título de renda fixa (bond), (v) estimação da volatilidade implícita entrópica do preço do valor mobiliário e (vi) definição e estimação do valor em risco (value at risk) entrópico. Há ainda dois resultados importantes para a Teoria da Informação e Economia: (i) distinção mais precisa entre incerteza e risco e (ii) desenvolvimento da medida de ganho informacional da previsão aprimorando o resultado de Theil (1967) e Benish (1999) pela utilização do conceito de divergência de Kullback-Leibler. / This thesis integrates Finance and Information Theory in order to create an alternative environment to the calculation of the rational price of the simple ordinary European option over stocks and bonds. One of the features of this new environment is to allow us to continue using the Newtonian calculus instead of the stochastic one. It creates a precise and complete mathematical notation for the Information Theory and integrates it with the Finance Theory under uncertainty conditions. It integrates Gulko’s (1998 and 1998a) and Yang’s (1997) entropic approaches to the calculation of the rational price of the simple ordinary European option. It precisely defines the uncertainty-price-neutral world (risk-neutral world), the martingale world, the informationally efficient world and the entropic world and their implications to the Investment Science and, more specifically, to the calculation of the rational price of ordinary assets and derivatives. It demonstrates with details the Black-Scholes-Merton formula of the rational price of the simple ordinary European option, improves the mathematical notation, simplifies it (by eliminating the martingale approach) and completes the demonstration done by Baxter & Rennie (1998). It breaks a succession of works that established a mistaken way to calculate the price of the simple ordinary European option. This mistake had its origin, much probably, in an edition of Brealey & Myers, who erroneously used a result from Cox & Rubinstein (1985). This result facilitates the calculation of the rational price of the simple ordinary European option by using a table that avoids the direct usage of the Black-Scholes-Merton formula. Brealey & Myers (since the 1991 fourth edition), Luehrman (in his two 1998 articles in HBR and in a 1995 case in HBS) and Edleson (1994 case published in HBS) teach that the percentage value found in this table must be multiplied by the price of the asset, when in reality it should have been multiplied by the present value of the strike price. The most important results of this thesis for Finance are: (i) development of a robust and economic alternative method, based on the maximum-entropy principle of the Information Theory and on Pearson’s Distribution System, to the calculation of a unique uncertainty-price-neutral probability measure (risk-neutral probability), (ii) achievement of a practical formula to the calculation of the rational price of the simple ordinary European option on stocks, (iii) validation of the Black-Scholes-Merton formula on stocks, (iv) achievement of an adequate formula to the calculation of the rational price of the simple ordinary European option on bonds, (v) estimation of the implied entropic volatility of the price of an asset and (vi) definition and estimation of the entropic value-at-risk. There are still two important results to the Information Theory and to Economics: (i) a more precise distinction between uncertainty and risk and (ii) development of the forecast informational gain, an enhancement of the result of Theil (1967) and Benish (1999) by using the Kullback-Leibler divergence concept.

本研究依據Hu(1998)的理論模型來探討證券市場的存在對所得分配的影響，並且以跨國橫斷面迴歸方式進行實證分析，利用Deininger and Squire(1996)所整理的吉尼係數資料，探討五十二國1986-1990年間的平均股市發展、資本市場完全程度與所得分配之間的關係；此外，我們也對Kuzents的倒U型假設進行驗證，以分析不同經濟發展程度與所得分配之間的關係；最後，本研究進一步探討先進工業化與開發中子樣本群中，股票市場對所得分配的影響是否不同。 模型推導的結論為股市的存在並不會使所得分配惡化，產生惡化效果的原因是因為資本市場的不完全性，使窮人受限於借貸限制與賣空限制，無法取得公平的投資機會；而投資人對股市看法的分歧也是造成所得分配惡化的原因。實證分析的結果與理論模型之結論一致，也就是股市的發展並不會惡化所得分配，甚至有益於所得分配，而資本市場限制愈大的國家其所得分配愈為惡化；Kuzents的倒U型假設也獲得了支持。股市的發展程度與資本市場的完全性對開發中國家的所得分配有顯著的影響，然而這種效果在先進工業化國家中並不明顯；而教育對先進國家之所得分配有顯著影響力，對開發中國家則無。本研究的結果隱含著政府除了消除貧窮與實行所得重分配政策外，更應致力於健全資本市場，將市場的不完全性降到最低，讓市場上的資訊充分流通，使每個投資人都能得到公平的投資機會，如此便可改善所得分配。

現實社會中，由於人並非如傳統學派所聲稱完全理性制定決策，自1980年以來即產生諸多傳統學派無法說明的現象，因此行為財務學派興起。本篇導入行為財務學的模型，探討當經理人具過度自信與過度樂觀特質時，經理人的特性會如何影響公司價值。研究假設當經理人為風險中立者，可達到公司價值極大化first-best value。若經理人為風險趨避，產生的效用成本將使其無法達成此目標，但此時經理人若具過度自信，則可抵消風險趨避帶來的公司價值減損，而達成股東所希望達到的公司價值極大。除此之外，根據Heaton(2003)模型所聲稱，過度樂觀的經理人無法達成公司價值極大，而本篇修改其模型，得出當公司經理人具過度樂觀特性，是有可能符合股東利益，而達到公司價值極大化的目標。 / In real world, people don’t make decisions depend on rationality. Therefore, there exists many facts that traditional researchers can’t explain since 1980’s and that’s why behavioral finance school arises. In this paper, we use behavioral finance models to discuss when managers are overconfident or optimistic, how their personality will affect the value of company. We find that when a manager is risk-neutral, he can maximize the firm value that we called “first-best value.” However, when a manager is risk-averse, the utility cost will be the huge obstacle to attain the goal. However, if the manager is overconfident, this characteristic will counterbalance the drawback that risk-averse will decrease company value. In addition, according to Heaton’s (2003) model, an optimistic manager can’t maximize firm value. This paper modifies Heaton’s model and finds that when managers are optimistic, it is likely that a manager can meet shareholder’s needs and maximize the firm value.

Quanto EIA是一種具有選擇權特性且能連結至外幣投資的保險年金商品.以往針對權益連動年金所做的文獻中,均未考慮Quanto的特性.本文利用風險中立評價法求算出六種具有Quanto特性的鎖高型權益連動年金商品的評價公式,並進一步利用數值分析來探討各個契約及市場參數對契約價值的影響. / Quanto Ratchet EIAs link to foreign investments and provide options-like properties. The literature covers the pricing of the EIAs that are not quantos. This paper intends to fill the hole. To derive the pricing formulas, we added an exchange rate model as well as a foreign risk-free rate model to the pricing framework of Black and Scholes. Our formulas cover quanto ratchet EIA products for both compound and simple versions that may have a return cap and employ two types of geometric return averaging. We further provide numerical analyses on how contract features and market parameters affect the contract value.

<p>The present mini-thesis seeks to explore and investigate the mathematical theory and concepts that underpins the valuation of derivative securities, particularly European plainvanilla options. The main argument that we emphasise is that novel models of option pricing, as is suggested by Hull and White (1987) [1] and others, must account for the discrepancy observed on the implied volatility &ldquo / smile&rdquo / curve. To achieve this we also propose that market volatility be modeled as random or stochastic as opposed to certain standard option pricing models such as Black-Scholes, in which volatility is assumed to be constant.</p>

Pricing options quickly and accurately is a well known problem in finance. Quantum computing is being researched with the hope that quantum computers will be able to price options more efficiently than classical computers. This research extends the quantum binomial option pricing model proposed by Zeqian Chen to European put options and to Barrier options and develops a quantum algorithm to price them. This research produced three key results. First, when Maxwell-Boltzmann statistics are assumed, the quantum binomial model option prices are equivalent to the classical binomial model. Second, options can be priced efficiently on a quantum computer after the circuit has been built. The time complexity is O((N − τ)log(N − τ)) and it is in the BQP quantum computational complexity class. Finally, challenges extending the quantum binomial model to American, Asian and Bermudan options exist as the quantum binomial model does not take early exercise into account. / May 2009