Conventional wisdom decrees that in order for insurers to provide cover, they require capital. One of the many methods of calculating capital requirements of short-term insurers is the insolvency put option framework. This technique was originally introduced by Merton (1977). The general argument is that bankruptcy occurs when shareholders exercise a valuable put option. Indeed, the corporation was introduced to protect shareholders from, mainly contractual, liabilities of persons who trade with the corporation. The corporation thus introduced the idea of limited liability of shareholders or as is often called the corporate veil. However, if a company defaults on its debt then equity holders have decided to allow an embedded call option to expire unexercised. As a result shareholders will behave as if they in fact hold a call option, which creates a different incentive than that suggested by the insolvency put idea. This study examines the role of capital and the influence of the insolvency put option within a short-term insurer. Specifically, it is argued that capital is not the cornerstone of a short-term insurer. Moreover, using Brownian motion and Itō calculus as well as continuous time financial models a more complete mathematical description of an insurance company is articulated by explicitly taking the embedded equity call option into account.

<p>In this work we consider a problem of the</p><p>computation of the components of the hedging portfolio structure. In</p><p>literature often one can find valuations and estimations of the</p><p>fair price of American options. But the formulas for hedging portfolio</p><p>are interesting as well and are known for very particular cases</p><p>only. In our work we study different cases of American Put and Russian</p><p>options on finite and infinite horizon.</p>

<p>In this thesis a new method for the option pricing will be introduced with the help of the Mellin transforms. Firstly, the Mellin transform techniques for options on a single underlying stock is presented.After that basket options will be considered. Finally, an improvement of existing numerical results applied to Mellin transforms for 1-basket and 2-basket American Put Option will be discussed concisely. Our approach does not require either variable transformations or solving diffusion equations.</p> / thesis

In this thesis a new method for the option pricing will be introduced with the help of the Mellin transforms. Firstly, the Mellin transform techniques for options on a single underlying stock is presented.After that basket options will be considered. Finally, an improvement of existing numerical results applied to Mellin transforms for 1-basket and 2-basket American Put Option will be discussed concisely. Our approach does not require either variable transformations or solving diffusion equations. / thesis

In this work we consider a problem of the computation of the components of the hedging portfolio structure. In literature often one can find valuations and estimations of the fair price of American options. But the formulas for hedging portfolio are interesting as well and are known for very particular cases only. In our work we study different cases of American Put and Russian options on finite and infinite horizon.

Lookback option is a well-known path-dependent option where its payoff depends on the historical extremum prices. The thesis focuses on the binomial pricing of the American floating strike lookback put options with payoff at time $t$ (if exercise) characterized by \[ \max_{k=0, \ldots, t} S_k - S_t, \] where $S_t$ denotes the price of the underlying stock at time $t$. Build upon the idea of \hyperlink{RBCV}{Reiner Babbs Cheuk and Vorst} (RBCV, 1992) who proposed a transformed binomial lattice model for efficient pricing of this class of option, this thesis extends and enhances their binomial recursive algorithm by exploiting the additional combinatorial properties of the lattice structure. The proposed algorithm is not only computational efficient but it also significantly reduces the memory constraint. As a result, the proposed algorithm is more than 1000 times faster than the original RBCV algorithm and it can compute a binomial lattice with one million time steps in less than two seconds. This algorithm enables us to extrapolate the limiting (American) option value up to 4 or 5 decimal accuracy in real time.

Lookback option is a well-known path-dependent option where its payoff depends on the historical extremum prices. The thesis focuses on the binomial pricing of the American floating strike lookback put options with payoff at time $t$ (if exercise) characterized by \[ \max_{k=0, \ldots, t} S_k - S_t, \] where $S_t$ denotes the price of the underlying stock at time $t$. Build upon the idea of \hyperlink{RBCV}{Reiner Babbs Cheuk and Vorst} (RBCV, 1992) who proposed a transformed binomial lattice model for efficient pricing of this class of option, this thesis extends and enhances their binomial recursive algorithm by exploiting the additional combinatorial properties of the lattice structure. The proposed algorithm is not only computational efficient but it also significantly reduces the memory constraint. As a result, the proposed algorithm is more than 1000 times faster than the original RBCV algorithm and it can compute a binomial lattice with one million time steps in less than two seconds. This algorithm enables us to extrapolate the limiting (American) option value up to 4 or 5 decimal accuracy in real time.

美式賣權已經存在很長的時間，由於沒有公式解，目前只能利用數值分析方法(numerical analysis approach)和解析近似法(analytic approximations) 來評價它。這類的評價方法在文獻中相當多，但對這些方法的完整的比較卻相當貧乏。本文整理了27種評價方法和186種在文獻中常被引用的美式賣權契約，這些契約包含了各種不同狀態(有股利、沒有股利、價內、價平、價外、短到期日、長到期日)，後續的研究者可以用這些美式賣權契約來驗證他們的方法。本文實作其中14種方法並應用於上述的186種美式賣權契約上。這14種方法包含了樹狀法、有限差分法、蒙地卡羅法與解析近似法。從這些數值的結果中，本文根據精確度與計算效率整理出各種方法的優缺點與適用的時機。 　由本文之數值分析，我們得到下列幾點結論：1.Binomial Black and Scholes with Richardson extrapolation of Broadie and Detemple (1996)與Extrapolated Flexible Binomial Model of Tian (1999)這二種方法在這14種方法中，在速度與精確度的考量下是最好的方法；2.在精確度要求在root mean squared relative error大約1%的情形下，解析近似法是最快的方法；3.Least-Squares Simulation method of Longstaff and Schwartz (2001)在評價美式賣權方面並不是一個有效的方法。 / American put option has existed for a long time. They cannot be valued by closed-form formula and require the use of numerical analysis methods and analytic approximations. There exists a great deal of methods for pricing American put option in related literatures. But a complete comparison of these methods is lacking. From literatures, we survey 27 methods and 186 commonly cited option contracts, including options on stock with dividend, non-dividend, in-the-money, at-money and out-of-money, short maturity and long maturity. In addition, we implement 14 methods, including lattice approaches, finite difference methods, Monte Carlo simulations and analytic approximations, and apply these methods to value the 186 option contracts above. From the numerical results, we summarize the advantages and disadvantages of each method in terms of speed and accuracy: 1.The binomial Black and Scholes with Richardson extrapolation of Broadie and Detemple (1996) and the extrapolated Flexible Binomial Model of Tian (1999) are both efficient improvements over the binomial method. 2.With root mean squared relative error about 1%, the analytic approximations are faster than the numerical analysis methods. 3.The Least-Squares Simulation method of Longstaff and Schwartz (2001) is not an effective method for pricing American put options.

Master-Arbeit Univ. St. Gallen, 2006.

Master-Arbeit Univ. St. Gallen, 2006.