Total valuation adjustment (XVA) is a new technique which takes multiple material financial factors into consideration when pricing derivatives. This paper explores how funding costs and counterparty credit risk affect pricing the American option based on no-arbitrage analysis. We review previous studies of European option pricing with different funding costs. The conclusions help to compute the no- arbitrage price of the American option in the model with different borrowing and lending rates. Another model with counterparty credit risk is set up, and this pricing approach is referred to as credit valuation adjustment (CVA). A defaultable bond issued by the counterparty is used to hedge the loss from the option's default. We incorporate these two models to assess the XVA of an American option. The collateral, which protects the option investors from default, is considered in our benchmark model. To illustrate our results, numerical experiments are designed to demonstrate the relationship between XVA and parameters, which include the funding rates, bond's rate of return, and number of periods.

This paper has two researches direction, one is in the pricing and application of a probation option, the other is in the pricing and application of an American option. In the research of a probation option, this paper used the concept of the marketing strategy to be the source of financial innovation, and therefore decision a new exotic option. We call this option is a probation option. We introduce the application of this option, and further more to device the value of this option. Beside, this option also can apply to the field of marketing, and to calculate the cost of marketing strategy. In the research of an American, this paper proposes a new method- the implied belief model, to obtain a closed-form solution of the value of the American option. We analyze the value of the American option through the view point of the sellers of the options. By adopting this method, we derive the upper bound for the value of an American option. Then we define the belief value of seller to obtain a closed-form solution of the value of an American option. Finally, we apply the method to S&P 100 American option and deduce the implied belief value.

An option, a type of a financial derivative, is a contract that creates an opportunity for a market player to avoid risks involved in investing, especially in equities. An investor desires to know the accurate value of an option before entering into a contract to buy/sell the underlying asset (stock). There are various techniques that try to simulate real market conditions in order to price or evaluate an option. However, most of them achieved limited success due to high uncertainty in price behavior of the underlying asset. In this study, I propose two new Firefly variant algorithms to compute accurate worth for European and American option contracts and compare them with popular option pricing models (such as Black-Scholes-Merton, binomial lattice, Monte-Carlo, etc.) and real market data. In my study, I have first modelled the option pricing as a multi-objective optimization problem, where I introduced the pay-off and probability of achieving that pay-off as the main optimization objectives. Then, I proposed to use a latest nature-inspired algorithm that uses the bioluminescence of Fireflies to simulate the market conditions, a first attempt in the literature. For my thesis, I have proposed adaptive weighted-sum based Firefly algorithm and non-dominant sorting Firefly algorithm to find Pareto optimal solutions for the option pricing problem. Using my algorithm(s), I have successfully computed complete Pareto front of option prices for a number of option contracts from the real market (Bloomberg data). Also, I have shown that one of the points on the Pareto front represents the option value within 1-2 % error of the real data (Bloomberg). Moreover, with my experiments, I have shown that any investor may utilize the results in the Pareto fronts for deciding to get into an option contract and can evaluate the worth of a contract tuned to their risk ability. This implies that my proposed multi-objective model and Firefly algorithm could be used in real markets for pricing options at different levels of accuracy. To the best of my knowledge, modelling option pricing problem as a multi-objective optimization problem and using newly developed Firefly algorithm for solving it is unique and novel. / October 2016

The standard approach to pricing financial derivatives is to determine the discounted, risk-neutral expected payoff under a model. This model-based approach leaves us prone to model risk, as no model can fully capture the complex behaviour of asset prices in the real world. Alternatively, we could use the prices of some liquidly traded options to deduce no-arbitrage conditions on the contingent claim in question. Since the reference prices are taken from the market, we are not required to postulate a model and thus the conditions found have to hold under any model. In this thesis we are interested in the pricing of American put options using the latter approach. To this end, we will assume that European options on the same underlying and with the same maturity are liquidly traded in the market. We can then use the market information incorporated into these prices to derive a set of no-arbitrage conditions that are valid under any model. Furthermore, we will show that in a market trading only finitely many American and co-terminal European options it is always possible to decide whether the prices are consistent with a model or there has to exist arbitrage in the market.

Motivated by the Guaranteed Minimum Death Benefits (GMDB) in variable annuities, we are interested in valuing equity-linked options whose expiry date is the time of the death of the policyholder. Because the time-until-death distribution can be approximated by linear combinations of exponential distributions or mixtures of Erlang distributions, the analysis can be reduced to the case where the time-until-death distribution is exponential or Erlang. We present two probability methods to price American options with an exponential expiry date. Both methods give the same results. An American option with Erlang expiry date can be seen as an extension of the exponential expiry date case. We calculate its price as the sum of the price of the corresponding European option and the early exercise premium. Because the optimal exercise boundary takes the form of a staircase, the pricing formula is a triple sum. We determine the optimal exercise boundary recursively by imposing the “smooth pasting” condition. The examples of the put option, the exchange option, and the maximum option are provided to illustrate how the methods work. Another issue related to variable annuities is the surrender behavior of the policyholders. To model this behavior, we suggest using barrier options. We generalize the reflection principle and use it to derive explicit formulas for outside barrier options, double barrier options with constant barriers, and double barrier options with time varying exponential barriers. Finally, we provide a method to approximate the distribution of the time-until-death random variable by combinations of exponential distributions or mixtures of Erlang distributions. Compared to directly fitting the distributions, my method has two advantages: 1) It is more robust to the initial guess. 2) It is more likely to obtain the global minimizer.

Option is an asymmetric contract between two parties with future payoff derived from the price of underlying asset. Methods of pricing di erent types of options under more or less general assumptions have been extensively studied since the Nobel price winning works of Black and Scholes [1] and Merton [12] were published in 1973. A new way of pricing options with the use of Mellin transforms have been recently introduced by Panini and Srivastav [15] in 2004. This thesis offers a brief introduction to option pricing with Mellin transforms and a revision of some of the recent research in this field.

We propose a dynamic programming (DP) approach for pricing American options over a finite time horizon. We model uncertainty in stock price that follows geometric Brownian motion (GBM) and let interest rate and volatility be fixed. A procedure based on dynamic programming combined with piecewise linear interpolation approximation is developed to price the value of options. And we introduce the free boundary problem into our model. Numerical experiments illustrate the relation between value of option and volatility.

Stochastic volatility models on option pricing have received much study following the discovery of the non-at implied surface following the crash of the stock markets in 1987. The most widely used stochastic volatility model is introduced by Heston (1993) because of its ability to generate volatility satisfying the market observations, being non-negative and mean-reverting, and also providing a closed-form solution for the European options. However, little research has been done on Heston model used to price early-exercise options. This presumably is largely due to the absence of a closed-form solution and the increase in computational requirement that complicates the required calibration exercise. This thesis examines the performance of the Heston model versus the Black-Scholes model for the American Style equity option of Microsoft and the index option of S&P 100 index. We employ a finite difference method combined with a Projected Successive Over-relaxation method for pricing an American put option under the Black-Scholes model, while an Alternating Direction Implicit method is utilized to decompose a multi-dimensional partial differential equation into several one dimensional steps under the Heston model. For the calibration of the Heston model, we apply a two step procedure where in the first step we apply an indirect inference method to historical stock prices to estimate diffusion parameters under a probability measure and then use a least squares method to estimate the instantaneous volatility and the market risk premium which are used to switch from working under the probability measure to working under the risk-neutral measure. We find that option price is positively related with the value of the mean reverting speed and the long-term variance. It is not sensitive to the market price of risk and it is negatively related with the risk free rate and the volatility of volatility. By comparing the European put option and the American put option under the Heston model, we observe that their implied volatility generally follow similar patterns. However, there are still some interesting observations that can be made from the comparison of the two put options. First, for the out-of-the-money category, the American and European options have rather comparable implied volatilities with the American options' implied volatility being slightly bigger than the European options. While for the in-the-money category, the implied volatility of the European options is notably higher than the American options and its value exceeds the implied volatility of the American options. We also assess the performance of the Heston model by comparing its result with the result from the Black-Scholes model. We observe that overall the Heston model performs better than the Black-Scholes model. In particular, the Heston model has tendency of underpricing the in-the-money option and overpricing the out-of-the-money option. Whereas, the Black-Scholes model is inclined to underprice both the in-the-money option and the out-of-the-money option.b

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 of Science / Department of Industrial & Manufacturing Systems Engineering / Chih-Hang Wu / In recent years the stock markets have shown tremendous volatility with significant spikes and drops in the stock prices. Within the past decade, there have been numerous jumps in the market; one key example was on September 17, 2001 when the Dow industrial average dropped 684 points following the 9-11 attacks on the United States. These evident jumps in the markets show the inaccuracy of the Black-Scholes model for pricing options. Merton provided the first research to appease this problem in 1976 when he extended the Black-Scholes model to include jumps in the market. In recent years, Kou has shown that the distribution of the jump sizes used in Merton’s model does not efficiently model the actual movements of the markets. Consequently, Kou modified Merton’s model changing the jump size distribution from a normal distribution to the double exponential distribution. Kou’s research utilizes mathematical equations to estimate the value of an American put option where the underlying stocks follow a jump-diffusion process. The research contained within this thesis extends on Kou’s research using Monte Carlo simulation (MCS) coupled with least-squares regression to price this type of American option. Utilizing MCS provides a continuous exercise and pricing region which is a distinct difference, and advantage, between MCS and other analytical techniques. The aim of this research is to investigate whether or not MCS is an efficient means to pricing American put options where the underlying stock undergoes a jump-diffusion process. This thesis also extends the simulation to utilize copulas in the pricing of baskets, which contains several of the aforementioned type of American options. The use of copulas creates a joint distribution from two independent distributions and provides an efficient means of modeling multiple options and the correlation between them. The research contained within this thesis shows that MCS provides a means of accurately pricing American put options where the underlying stock follows a jump-diffusion. It also shows that it can be extended to use copulas to price baskets of options with jump-diffusion. Numerical examples are presented for both portions to exemplify the excellent results obtained by using MCS for pricing options in both single dimension problems as well as multidimensional problems.