More recently and, in particular, in the wake of the ﬁnancial crisis, the critique of using a single reference probability measure came from considerations of the so-called Knightian uncertainty, going back to Knight [ 35 ], and describing the model risk as contrasted with ﬁnancial risks captured within a given model. The resulting stream of research aims at extending the probabilistic framework of Dalang et al. [ 16 ] to a framework that allows for a set of possible priors 5 ⊆ 3. The class 5 represents a collection of plausible (probabilistic) models for the market. In continuous-**time** models, this led naturally to the **theory** of quasi-sure stochastic analysis as in Denis and Martini [ 23 ], Peng [ 40 ], and Soner et al. [ 47 , 48 ] and many further contributions; see, for example, Dolinsky and Soner [ 25 ]. In **discrete** **time**, a general approach was developed by Bouchard and Nutz [ 8 ]. Under some technical conditions on the state space and the set 5, they provide a version of the FTAP as well as the superhedging duality. Their framework includes the two extreme cases: the classical case in which 5 {P} is a singleton and, on the other extreme, the case of full ambiguity when 5 coincides with the whole set of probability measures and the description of the model becomes pathwise. Their setup has been used to study a series of related problems; see, for example, Bayraktar and Zhang [ 2 ] and Bayraktar and Zhou [ 3 ].

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of the spreads observed in the markets. Especially after the financial crisis of 2008, bid-ask spreads of many assets were persistently high and at a level that cannot be explained by transaction costs alone. 2 A different approach was taken by Madan and Cherny (2010) which is based on **theory** of conic finance, originating from the work by Cherny and Madan (2009). The basic premise is that the market takes the role of a central counterparty that buys and sells assets from and to investors. The investor buys at the ask price and sells at the bid price. The difference of these prices gives rise to the bid-ask spread observed in financial markets. The central counterparty is viewed as passive in that it does not maximize some utility function, but rather carries out all trades that are acceptable to it. 3

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Chapter 8 presented the Arrow-Debreu asset **pricing** **theory** from the equilibrium perspective. With the help of a number of modeling hypotheses and building on the concept of market equilibrium, we showed that the price of a future contingent dollar can appropriately be viewed as the product of three main components: a pure **time** discount factor, the probability of the relevant state of nature, and an intertemporal marginal rate of substitution reflecting the col- lective (market) assessment of the scarcity of consumption in the future relative to today. This important message is one that we confirmed with the CCAPM of Chapter 9. Here, however, we adopt the alternative **arbitrage** perspective and revisit the same Arrow-Debreu **pricing** **theory**. Doing so is productive precisely because, as we have stressed before, the design of an Arrow-Debreu security is such that once its price is available, whatever its origin and make-up, it pro- vides the answer to the key valuation question: what is a unit of the future state contingent numeraire worth today. As a result, it constitutes the essential piece of information necessary to price arbitrary cash flows. Even if the equilibrium **theory** of Chapter 8 were all wrong, in the sense that the hypotheses made there turn out to be a very poor description of reality and that, as a consequence, the prices of Arrow-Debreu securities are not well described by equation (8.1), it remains true that if such securities are traded, their prices constitute the es- sential building blocks (in the sense of our Chapter 2 bicycle **pricing** analogy) for valuing any arbitrary risky cash-flow.

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As there are no **arbitrage** opportunities in an efficient market, the seller of an option must find a risk neutral price. This thesis examines different characterizations of this option price. In the first characterization, the seller forms a hedging portfolio of shares of the stock and units of the bond at the **time** of the option’s sale so as to reduce his risk of losing money. Before the option matures, the present value stock price fluctuates in **discrete** **time** and, based on those changes, the seller alters the content of the portfolio at the end of each **time** period. The primal linear program captures the seller’s hedging activities. We use linear programming to explore the **pricing** of options for both the Trinomial Asset **Pricing** Model and the General Asset **Pricing** Model, allowing us to consider the **pricing** of any style of option.

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Table 1 exhibits the small MC of DSE when compared to other stock markets in the region as well as in the world. The annual growth rate of MC in DSE was 135.28% in 2007 but somewhat subdued since then as a result of Global Financial Crisis. Consequently, in 2009 the growth rate of MC was 23.9%. In order to test the APT empirically, this study employs pre-specified macroeconomic factor approach developed by Chen et al. (1986) (CRR) that requires the use of two-pass regression methodology originally developed by Fama and Macbeth (1973). The dataset used in this study is a **time**-series data consists of 23 stocks and seven macro-variables for the period 1996-2010. It is noteworthy that CRR results suffer from robustness check as **time**-series of macro-variables contains the possibility of multi- collinearity. To resolve this problem, this study incorporates principal component analysis into the regression model. The results confirm evidence of one significant macroeconomic factor in the Dhaka stock market. Furthermore, this study provides a critic of asset **pricing** models as they fall short of incorporating non-quantifiable factors that affect stock returns. To this effect, this study addresses the issue ‘what constitutes stock returns’ and uses the analysis to explicate the findings. The hypotheses of the present study are as follows:

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menarik sehubungan dengan conditional teori harga arbitase yang di dalamnya mengandung unsur **time**-varying risk atau **time**-varying beta. Fenomena ini didasarkan pada asumsi bahwa beta tidak konstan sehubungan dengan fenomena volatility clustering pada return saham sektoral. **Time**-vary- ing volatility berkaitan dengan adanya informasi baru yang menyebabkan para investor merevisi penilaiannya terhadap intrinsic value dari suatu peluang investasi yang direncanakan (Bodie et al., 2009). Pada fenomena ini terdapat 2 hal yang akan diamati, yaitu berkaitan dengan fenomena struc- tural break dan asymmetric shock. Secara teoritis menyatakan bahwa pada fenomena structural break menunjukkan adanya perbedaan required rate of re- turn dan risk premium faktor sistematis, sehubungan dengan perubahan struktural pada return saham sektoral. Sedangkan pada fenomena asymmetric shock yang menunjukkan adanya asymmetric effect yang merupakan perbedaan respon investor ter- hadap informasi yang bersifat negatif (bad news), dan informasi yang bersifat positif (good news). Secara teoritis asymmetric effect disebabkan oleh faktor leverage effect, sehubungan dengan pertim- bangan adanya financial distress dan faktor volatil- ity feedback effect, dan pertimbangan required rate of return investasi. Kondisi ini menarik untuk dikaji lebih jauh, sehubungan dengan masalah **time**-vary- ing volatility adalah terjadinya krisis keuangan periode 2008, dimana volatilitas return bulanan saham sektoral di Bursa Efek Indonesia mengalami peningkatan secara drastis.

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Since the research’s objective is to analyse Malawi’s equity returns linkage with the chosen economic events, it necessitates **time** series macro-economic data. However to analyse such data it is advised that stationarity tests must performed for each variable involved so as to desist from spurious regressions (Brooks, 2008). So many tools are of service when it comes to testing stationarity amid variables and these include 1979’s ADF test, Phillips (1987) and so many other techniques. These tools are widely utilised in economic arenas, and in this study’s regards, out of other alternatives, ADF test has been applied considering Karamustafa and Kucukkale (2003)’s claim in which they argued that the technique possesses small sample properties which are superior when equated to its options. In this research, at level as well as at their first differences ADF test in logarithm terms was applied to the variables. Actually in ADF we test whether the p-value is equal to zero or not and it can be presented as shown below;

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unique measures of risk from the daily **time**-series observations in the first half-period (1962 to 1972) and used them to explain the daily cross-section returns for the second half-period (1972 to 1981). DFG&G were concerned about the question o f how the number o f factors that are significant (on the first stage) and /or priced (on the second stage) varies with the sizes o f securities groups or the length of the **time** series. They showed that tests results appeared to be extremely sensitive to the number o f securities used in two stages of the tests o f the APT model. The tests also indicated that unique risk was fully as important as common risk. In another study, Dhrymes, Friend, Gultekin, and Gultekin (1985a) presented a comprehensive set of tests of the implications of the APT. They found that the risk premia was not significant in most groups (at least 36 out of 42), indicating a lack o f a linear relationship between the expected rates of return and the measures of risk parameters implied by the APT model. Furthermore, unique variance measures o f risk, while generally making only small contributions to the explanation of asset returns, turned out to be as significant as frequently as the covariance measures of risk - which was inconsistent with the APT model. These intercept tests were more mixed, but provided only limited support to the model. One o f the important implications of the model is that the intercept terms are, on average, the same in all groups which would be true if the intercepts were either the risk-free or zero-beta rates of return. Such an implication was not rejected by their study; on the other hand, the same evidence suggested that on average the intercept term was insignificantly different from zero for most groups. Moreover, these intercepts were significantly different from the risk free rate interpreted as the appropriate Treasury Bill rate.

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Martikainen et al. (1991) tested APT for the Finish Stock Market using monthly data. They used two different approaches: an exploratory factor analysis and a pre-specified macroeconomic factor approach. They tested how many factors there were that affected finish stocks in the two **time** periods 1977-81 and 1982-86. In the first step of the test they used principal components analysis and varimax rotation to get the factor loadings. Then, OLS regressions were made where factor loadings were independent variables and the average return on stock was the dependent variable. The purpose was to find how many factors that were priced in the market. In the second step of the test they used 11 pre-specified macroeconomic factors to test the APT model. They used different stock market indices, price indices, interest rates and other national economic variables such as the GNP and money supply. They could find only one priced factor for the first subperiod. In the second subperiod all of the factors become priced. This was an encouraging result that supported the **theory** that the equilibrium stock returns were generated by an economic factor model.

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Childs et al. (2001-2002-2002-2004) have provided interesting concepts, well adapted to the real estate specificities, and allowing to handle the uncertainty on H(0) (they employ the term “noisy asset”, coming from real options **theory** in this situation). The tools they have developed can be used in the management of research and development projects, for the exploitation of corporate assets as mines, and for real estates. A common point between these fields is the ignorance of the exact prices unless an irreversible action is undertaken (beginning of the production or the exploitation, or sale completion). The principles of the modelisation are as follows: The real price process x(t) is unknown for investors because of a noise process y(t). The only thing that can be observed is a noisy estimation z(t) of the true value x(t) , the relation can be for instance z(t) = x(t) + y(t) or z(t) = x(t) y(t). At any **time** t, the available information is represented by a filtration {I(t)}, and investors estimate x(t) given this filtration, by m(t) = E [x(t) | I(t)] . This value m(t) is an appraisal value conditional on all the relevant information (for instance sale prices for similar real estate), and is named the **time**-filtered asset value. Subsequently using Lipster and Shiryayev differential equations, the dynamic of m(t) can be expressed with the parameters present in the dynamics of x(t) and y(t). Once this is done Childs et al price an option on a noisy asset x(t) with a version of Black- Scholes formula but using m(t) as underlying. Unfortunately, the same concerns discussed above for the PDE could be raised for this **pricing** methodology. m(t) is not a tradable asset contrary to the stock in the Black-Scholes world. Moreover, at maturity the payoff is not m(T), but x(T), making the construction of a riskless portfolio difficult, which puts the validity of the **pricing** formula under question. Similarly, for t < T, m(t) is an optimal appraisal price, it is not the real price needed to build an actual **arbitrage** portfolio. In Childs et al, a **pricing** formula is not really the aim, the analysis rather stresses the importance of

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has been stressed, among others, by Rudebusch, Sack and Swanson (2006) and Diebold, Rudebusch and Aruoba (2006). As far as our dataset is con- cerned, the statistical tests we perform on a battery of models strongly reject the overidentifying restriction of independence, not only under the historical probability measure, but also under the risk-neutral (**pricing**) one. We also …nd that relaxing overidentifying restrictions produces material di¤erences in estimated risk premia, impulse response functions and variance decom- positions. We use our canonical representation to carry out a speci…cation analysis in the spirit of that conducted by Dai and Singleton (2000). Be- sides testing the validity of the aforementioned overidentifying restrictions, we also conduct statistical tests to …nd the optimal number of unobservable factors and lags of the observable macro variables. Our …ndings suggest that the best model is a fully parametrized one with three unobservable and only one lag of the observable variables.

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Portfolio **theory** and option **pricing** **theory**, which are broadly two of the most active areas of research in mathematical finance, typically involve the study of decision-making under un- certainty. For instance, the following questions arise: How much of one’s wealth should be utilised for consumption or allocated between investments in stocks and bonds? How does one hedge away the risk and determine the valuation of an option? The ongoing research in these areas aims to develop more realistic models that reflect the dynamics of financial markets and accurately describe the strategies to be taken by decision-makers. The present study concentrates on analysing the impact of transaction costs in portfolio **theory** and op- tion **pricing** **theory**. Pioneering research in portfolio **theory** and option **pricing** **theory**, which ignored the presence of transaction costs, derived exact closed-from solutions but may lead to unrealistic investment or hedging strategies. On the other hand, the inclusion of trans- action costs in subsequent financial market models often resulted in equations that did not allow exact solutions. In order to solve these equations, one had to employ mathematical analysis and techniques to obtain analytical, numerical or approximate solutions.

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This paper regards film as an ordinary capital asset, introduces APT into the prediction model setting of movie asset investment returns for the first **time**, and strengthens the theoretical basis for analyzing the risk and return of cinematographic production relationship with multivariable linear regression model, which provides a new perspective for the research in this field. Based on the empirical research of domestic and foreign movie box office prediction, this paper localizes the factors of the box office prediction model and establishes a multi-factor box office prediction model to carry out empirical research and quantitative analysis. And the empirical results and the feedback results are basically in accordance with the realistic laws. The research of box office forecasting model has great practical significance: For movie investors, the establishment of box office forecasting model, which relies on key elements such as movie project type, director, starring, and schedule before the movie is released,can allow investors to have a more intuitive judgment, greatly reducing the investment risk of investors; for the film’s founding group, they can apply the box office prediction model based on the relevant factors of the movie planning to calculate the expected revenue of the project, and hence use this as a basis to finance. The box office prediction model has a key guiding significance for the film and television financial industry: All along, it has been difficult to realize the asset securitization of the film industry. The main reason is that the value of the film project and the cash flow generated in the future are difficult to measure. The box office prediction model can effectively solve this problem. So as to design film and television financial products reasonably.

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The first route is to retain a traditional dynamic equilibrium asset **pricing** model, but not fully endogenize the constraints. This is done in Gromb & Vayanos (2002), who study how leverage constraints affect arbitrageurs’ ability to provide liquidity. The main result in terms of welfare is that equilibrium can fail to be constrained efficient and a reduction in arbitrageur positions can make all agents better off. The intuition is as follows. Following an adverse shock, arbitrageurs incur capital losses and are forced to liquidate positions because their leverage constraints tighten. As a result, they find themselves more constrained and less able to provide liquidity—at a **time** where liquidity is low and its provision profitable. Ex-ante, arbitrageurs account for this possibility and engage in risk management by keeping some capital in cash to exploit episodes of low liquidity. However, they fail to account for the impact of their liquidations on other arbitrageurs during such episodes. Indeed, liquidation by one arbitrageur hurts other arbitrageurs because it lowers the price at which they can liquidate. This can hurt not only arbitrageurs but also outside investors because of the reduced liquidity that they receive.

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The study is going to employ the Vector Auto Regression (VAR) model to model the relationship between stock prices and certain macroeconomic variables. A VAR is an economic model that is used to capture the innovation and interdependency of multiple times series variables. The VAR model, developed by Sims (1980) represents dynamic models of a group of **time** series. In a VAR model each variable will have its own equation explaining the changes in that variable in question, in response to its own current and past values and the current and past values of all the variables in the model. Unit root tests will be conducted to test the data for stationarity. If the variables are integrated of the same order then a cointegration test will be performed.

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Even though these two models share all the other parameters except the collateral rate and the collateralization rate, which are unique in the model with collateral, the no-**arbitrage** prices in these two models are very different from each other. The upper bound and the lower bound for the no-**arbitrage** price interval in the model with collateral is relatively larger. The reason is that the collateralized option gives the buyer the right to use the collateral to hedge the loss from default. On the other hand, the uncollateralized option is fully exposed to the risk from the counterparty credit risk.

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endogenously. **Discrete** **time** dynamic models are derived in Jagannathan and Viswanathan (1988), Bossaerts and Green (1989), Connor and Korajczyk (1989), and Hollifield (1993). Even if the factor loadings (betas) for the cash flow process are **time** invariant, the betas of asset returns (relative to factor-mimicking portfolios) will be functions of the current information set. In the static APT we can replicate the priced payoff from a security with the riskless asset and factor-mimicking portfolios. Jagannathan and Viswanathan (1988) show that in a multiperiod economy there is, in general, a different riskless asset for every maturity (i.e., a discount bond with that maturity). Thus, even though the risky components of assets' payoffs are driven by, for example, a one-factor model, asset returns may follow an infinite factor structure (corresponding to a portfolio mimicking the single factor, plus the returns on discount bonds for every maturity).

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In this paper, I have examined the relation between expected returns and measures of systematic risk stemming from macroeconomic factors studied by Chen, Roll and Ross (1986, hereafter CRR) for a different **time** period (1978-2007) and different formation of portfolios (based on ME and BE/ME). Like CRR, I’ve used a version of Fama and MacBeth’s (1973) two-pass cross- sectional regression (CSR) methodology. Apparently, changing the **time** period and formation of portfolio lead to noticeably different conclusions. Using the same macrofactors as CRR only factor related to the change in expected inflation (DEI) is significantly priced in the overall period. The sample mean of the Industrial production factor (MP), a highly significant factor in CRR, is insignificant, although positive, for this period. Adding a sixth factor that captures the investor’s confidence in the market is quite insensitive to other marcofactors. However, both the five factor by CRR and proposed six factor model show evidence of joint significance, which is a new property entered in this paper.

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Credit Risk. In the classical framework, it is also assumed that both parties to a financial derivative contract are non-defaultable. That is, the agreed-upon payments will be made fully at the agreed-upon **time**. However, it is possible for either party to default on the contract, in which case the non-defaulting party will receive a partial settlement, or zero. As such, it is necessary to consider the possibility of future default when **pricing** the contract at its outset. We will refer to the risk of default by the counterparty as credit risk, and default by the investor as debit risk.

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The introduction of jumps into financial models present various advantages, either in terms of **time** series analysis or in terms of option **pricing**. The deviation of histori- cal returns from the Black Scholes (1973) hypothesis of Gaussianity is now well known and documented, see for example Eberlein and Prause (2002), Bouchaud and Potters (2003) or Embrecht et al. (2005). It led to the development of various **time** series models usually combining a GARCH component with a non-Gaussian distribution or a jump process. The use of Lévy processes is by no means specific to **time** series analysis: var- ious continuous **time** finance models are built on the combination of a process of the Heston (1993) type for the conditional variance with a jump component, either in the dynamics of the variance or of the log-returns. Examples of this sort are available in Merton (1973), Bates (1996) or Duffie et al. (2000). When the obvious interest of jumps in **time** series analysis lies within the fit of the tails of returns’ distribution, the incor- poration of jumps into a continuous **time** finance option **pricing** model is essential to the **pricing** of short term options: the convexity of the implied volatility smile is sharper for such options and stochastic volatility models are unable to replicate this stylized fact without a jump component. On this point, we refer to Duffie et al. (2000).

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