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Diploma thesis deals with models of asset pricing. We investigated in detail three classical models: binomial, Black-Scholes and Merton model. These models are widely used to date, although they were first published a few decades ago. It is because they are relatively simple and easy-to-use. The models were originally derived for option pricing however they can be used for the wide range of financial instruments. The theoretical part of the thesis includes an introduction to options and models derivation. The practical part consists of the sensitivity analyst and empirical test of the models. S&P 500 index options data were used for this purpose. The result is that Merton model seems to be the most accurate.

We derive at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a selection of exponential Lévy models, restricting our attention to asset-price models whose log returns structure is a Lévy process. We consider two main problems. First, we consider very general Lévy models that are in the domain of attraction of a stable random variable. Under some relatively minor assumptions, we give first-order at-the-money call-price and implied volatility asymptotics. In the case where our Lévy process has Brownian component, we discover new orders of convergence by showing that the rate of convergence can be of the form t¹/ᵃℓ(t) where ℓ is a slowly varying function and $\alpha \in (1,2)$. We also give an example of a Lévy model which exhibits this new type of behavior where ℓ is not asymptotically constant. In the case of a Lévy process with Brownian component, we find that the order of convergence of the call price is √t. Second, we investigate the CGMY process whose call-price asymptotics are known to third order. Previously, measure transformation and technical estimation methods were the only tools available for proving the order of convergence. We give a new method that relies on the Lipton-Lewis formula, guaranteeing that we can estimate the call-price asymptotics using only the characteristic function of the Lévy process. While this method does not provide a less technical approach, it is novel and is promising for obtaining second-order call-price asymptotics for at-the-money options for a more general class of Lévy processes.

Portfolio Management with Multiple Benchmarks Bc. Robert Navrátil Abstract: In this thesis, we study a maximal volatility portfolio that treats all assets in a symmetric way and related option contract. To preserve symmetry we need numeraire that treats all assets symmetrically. We choose market index with equal weights. In case of two assets we focus on a variation of a passport option on the portfolio. The optimal strategy for the investor is the mentioned maximal volatility portfolio. We extend the known optimal strategy for the option to a richer class of convex payoff functions. We also show a modification of the optimal strategy for maximizing the probability of ending above or at a desired level. We later extend the symmetric market model to case of three assets, which can be even further extended to an arbitrary number of assets. The three asset model requires more parameters than are observable from the data, however we show indistinguishably of the model on the choice of parameters under very natural conditions. Both numerical simulations and an application on real data is provided. 1

本文在不完美財務市場(Imperfect Markets)的假設下，探討採取不同的選擇權的避險策略與對選擇權評價模式的影響，並分析最適避險策略。在這裡所提到的不完美市場指的是無法連續時點的交易、交易時產生交易成本及異質訊息交易者。結果在不完美因素的考量下，其避險策略將不再是完美避險(Delta Hedge)，應適當考慮避險策略。不同於Leland(1985)的分析方式，在此透過不同的避險策略分析去探討比較不完美市場產生的差異，分別以最小變異數避險分析及平均數--變異數避險分析，探討不完美市場對選擇權評價的影響。 不完美市場下選擇權的評價將受其他參數的影響，例如：股票預期報酬及個人風險偏好的影響，本文則嘗試在模型中解釋這兩項因子的角色。 Figlewski(1989a)透過模擬分析探討不完美財務市場對選擇權的影響，並提出不完美市場選擇權的評價影響應考慮對股票預期報酬及個人風險偏好的影響，然而並沒有提出有關的模擬分析及模型的探討。 當採取不同的避險策略考量會有很大的差異，若市場不完美僅是在無法連續性避險則產生的影響相對較小，在最小變異數分析下僅修正相關參數即可，若考量平均數變異數分析將產生買賣價差。但若不完美的因素尚包括交易成本，將明顯影響結果，此時將不再可任意時點避險交易，因為任何交易皆存在交易成本，無限次的交易將使得交易成本趨近無限大。 最後得到調整避險比例時點，結果發現與Whalley 和Wilmott(1997)所推導避險帶為一致的，但在本文模型中將更有彈性的運用，並在數理分析簡化及及計算時間上較為省時的處理。 目次 頁次 第一章 緒論 第一節 前言………………………………………………………………1 第二節 本文內容與架構……………………….……..………………….4 第二章 相關文獻 第一節 完美財務市場……………………………………….……..…….5 第二節 不完美財務市場………………………………………....………7 第三節 不完美避險…………………………………………..….………10 第三章 基本模型分析 第一節 前言…………………..…….………………………………….…11 第二節 市場不完美……………………………………………….…..….11 第三節 最小變異數避險策略……..…………………….……….………14 第四節 平均數變異數避險策略…..………………………………..……20 第五節 極限的情況………………..……………………………….…….26 第四章 交易成本 第一節 前言………………………..……………………………….……29 第二節 最小變異數分析法………….………………………………..…30 第三節 平均數變異數分析法………….……………………..…………33 第四節 極限的情況………………………………………………………36 第五章 不對稱訊息分析 第一節 前言……..………………………………………………..………38 第二節 異質訊息的形成…………………………………………………38 第三節 異質訊息的評價…………………………………………………42 第六章 最適避險區間 第一節 前言………………………………………………………………49 第二節 最佳避險區間……………………………………………………50 第三節 本章小結…………………………………………………………58 第七章 結論與建議……………………………………………………………59 數學附錄 附錄A…………………………………………………………………………60 附錄B…………………………………………………………………………61 附錄C…………………………………………………………………………66 附錄D…………………………………………………………………………68 附錄E……………………………………………………………………….…70 附錄F………………………………………………………………………….71 附錄G…………………………………………………………………………73 附錄H…………………………………………………………………………74 附錄I………………………………………………………………….…….…75 參考文獻 一、國內文獻…………………………………………………………………79 二、國外文獻..…………………………………………………………..……79

<p>The purpose of this study is to identify an impact on an option pricing within NASDAQ OMX Stockholm Market, if the underlying</p><p>asset prices include jumps. The current financial crisis, when jumps are much more evident than ever, makes this issue very actual and important in the global sense for the portfolio hedging and other risk management applications for example for the banking sector. Therefore, an investigation is based on OMXS30 Index and SEB A Bank. To detect jumps the Barndorff-Nielsen and Shephard non-parametric bipower variation test is used. First it is examined on simulations, to be finally implemented on the real data. An affirmation of a jumps occurrence requires to apply an appropriate model for the option pricing. For this purpose the Kou model, a double exponential jump-diffusion one, is proposed, as it incorporates essential stylized facts not available for another models. Th parameters in the model are estimated by a new approach - a combined cumulant matching with lambda taken from the Barrndorff-Nielsen and Shephard test. To evaluate how the Kou model manages on the option pricing, it is compared to the Black-Scholes model and to the real prices of European call options from the Stockholm Stock Exchange. The results show that the Kou model outperforms the latter.</p>

The objective of this study is to implement a Java applet for calculating Bermudan/American-Asian call option prices and to obtain their respective optimal exercise strategies. Additionally, the study presents a computational time analysis and the effect of the variables on the option price.

The purpose of this study is to identify an impact on an option pricing within NASDAQ OMX Stockholm Market, if the underlying asset prices include jumps. The current financial crisis, when jumps are much more evident than ever, makes this issue very actual and important in the global sense for the portfolio hedging and other risk management applications for example for the banking sector. Therefore, an investigation is based on OMXS30 Index and SEB A Bank. To detect jumps the Barndorff-Nielsen and Shephard non-parametric bipower variation test is used. First it is examined on simulations, to be finally implemented on the real data. An affirmation of a jumps occurrence requires to apply an appropriate model for the option pricing. For this purpose the Kou model, a double exponential jump-diffusion one, is proposed, as it incorporates essential stylized facts not available for another models. Th parameters in the model are estimated by a new approach - a combined cumulant matching with lambda taken from the Barrndorff-Nielsen and Shephard test. To evaluate how the Kou model manages on the option pricing, it is compared to the Black-Scholes model and to the real prices of European call options from the Stockholm Stock Exchange. The results show that the Kou model outperforms the latter.

Thesis (MComm (Mathematics))--University of Stellenbosch, 2007. / The word numéraire refers to the unit of measurement used to value a portfolio of assets. The change of numéraire technique involves converting from one measurement to another. The foreign exchange markets are natural settings for interpreting this technique (but are by no means the only examples). This dissertation includes elementary facts about the change of numeraire technique. It also discusses the mathematical soundness of the technique in the abstract setting of Delbaen and Schachermayer’s Mathematics of Arbitrage. The technique is then applied to financial pricing problems. The right choice of numéraire could be an elegant approach to solving a pricing problem or could simplify computation and modelling.