cently, Fouque-Papanicolaou-Sircar observed a fast time scale volatility factor in S&P 500 high frequency data. They use asymptotic analysis to approximate **option** pricing and hedging problems in finance. This motivates our paper to generalize Vecer’s [48] dimension reduction technique on pricing **arithmetic** **average** **Asian** **option** (European style) by relaxing the assumption of constant volatility for the case that volatility is fluctuating and is driven by an auxiliary fast mean-reverting process. Based on the results from asymptotic analysis presented in the Appendix 7.7, the approximated price, or so-called corrected price, is derived. As a consequence, there is no need to estimate the current level of the unobservable stock price volatility. All the param- eters we need to compute the approximated price can be easily calibrated from the observed stock price and the implied volatility surface. Thus, this article describes a robust procedure to correct **Asian** **option** prices by taking the observed implied volatility skew into account. Numerical computation of the corrected **Asian** **option** price is certainly needed.

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underlying price at every node to be a rational number of finite precision. It is partitioned into different parts with varying resolution hence reducing the number of averages considered. It is based on the premise that if the underlying asset prices are multiplied by a constant before pricing the **option**, then **option** value divided by the same constant gives the originally desired **option** value. Convergence is ensured by making sure the underlying asset’s price process simulated mirrors to the continuous time lognormal price process of the underlying. Their work shows that this method converges as the time steps used are increased. In the next section the proposed trinomial model is laid out in more detail.

is **arithmetic** type, which is the most commonly used, though an exact analytical solution for **arithmetic** **average** rate **Asian** options does not existed. This solution is missing primarily because the **arithmetic** **average** of a set of lognormal random variables is not lognormally distributed.

where K is the strike price. Clearly, this is the price of a European **option** with expiry t. ˆ Thus, the price of the **arithmetic** **Asian** **option** (with expiry time T ) is equal to the price of the equivalent European **option** with expiry time t. ˆ This explains why the **Asian** **option** is cheaper than its European counterpart.

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nancially well to do, it consumes time to simulate a price inside reasonable boundaries of accuracy for sure contracts.for reduce computational time, some variance reduction techniques have been suggested. For example, control variates, anti- thetic variates and importance sampling. Con- trol variate method have been famously used for calculative finance as a method of variance re- duction. Kemna and Vorst [13] employed a dis- counted geometric **average** **Asian** **option** payoff less than its price as a control. This technique is effective because the geometric **Asian** **option** has a analytical solution form and the correlation amonge the **arithmetic** and the geometric aver- age is high. This study focus on the analysis of control variates, that is one of the most popular and eicient methods used. This method takes ad- vantage of random variables with positively cor- related with the variable under consideration and known expected value.

Even in the simple Black-Scholes-Merton model, **arithmetic** **Asian** options do not admit an ex- act formula in closed form. This is because the sum of correlated lognormal asset prices is not lognormal anymore. With the focus on the continuous **arithmetic** **average** asset price, Geman and Yor (1993) are the …rst to write the price of the **Asian** **option** as the inverse of its Laplace transform which they derive in analytical form. Thereafter several authors have attempted to compute the inverse transform using standard numerical approaches, including Fourier se- ries expansion, Laguerre series expansion, sequence of Gaver functionals, and deformation of Bromwich contour (for a thorough review of these techniques, see Davies (2002), Chapter 19), and all have encountered signi…cant numerical instabilities for short maturities and low volatil- ities (see Dufresne (2000), Linetsky (2004)). These limitations have been attributed to the slow convergence of the inversion algorithms and computational di¢ culties related to the Kummer con‡uent hypergeometric function appearing in the Laplace transform. Instead, Fusai (2004) and Cai and Kou (2010) obtain analytical expressions for the double Laplace transform of the **option** price, which they invert numerically using a two-sided Euler inversion algorithm. Although the two methods share similarities, Cai and Kou’s inversion technique is faster, for given accuracy, for low asset volatility, e.g., smaller than 0.1, and performs better under jump di¤usion model assumptions.

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Kemna and Vorst [9] show that in the case of the **arithmetic**-**average** **Asian** **option** price, the use of the geometric-**average** **Asian** **option** price can be used as an effective control variate. In a succeeding research, Fu and Madan [5] compare the discrete and the continuous geometric-**average** **Asian** **option** price’s performance as control variate with interesting results. They find that even though the appropriate formula for a dis- crete estimator would be the discrete **average** formula, by using the continuous **average** formula a greater variance reduction can be achieved. In other words, by using a “bi- ased” ( i.e . non-zero expectation) control variate, we not only reduce the variance of our original estimator, but also add an appropriate bias to it, which compensates for the above mentioned discretization error inherent in the basic simulation method. For this

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Part of the conceptual justification for the variation on a traditional covered **call** writing strategy described here pertains to how the **option** premium in question should best be interpreted. For longer-dated options a significant portion of the premium represents disagreement about future earnings, a discussion about which we have argued the individual investor shouldn’t enter. On the other hand, the premiums for very short-term options almost entirely represent speculation about upcoming near-term events which are unlikely to affect the long-term value of the company, short-term price patterns as anticipated through technical analysis, etc. These short-term price fluctuations shouldn’t be of interest to an individual investor (especially if unfavorable fluctuations are likely to be eventually reversed by RTM), and thus it makes good sense to monetize their expected value.

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In this paper we proposed a new family of recursive **average** (RAV) operators (i.e. the recursive weighted power mean), expanding in particular on the **arithmetic** RAV. The motivation for the new operator family is the desire to provide an intuitive path to aggregating data (evidence) in conjunction with the powerful structure of a fuzzy measure. We showed how the proposed RAV avoids potential challenges, such as the non- symmetry of outputs for symmetrically flipped inputs that can occur when employing a FI in conjunction with a FM, and thus how the **arithmetic** RAV can provide a useful alternative to FIs. We provided both synthetic and real world data fusion

This article is my response to both the White attorney who wanted to know my unique value or calling in the Deep South and to my Black classmate who wanted to know if an **Asian** lawyer had a place or role here. Thus, this paper explores the **call** for and role of **Asian** lawyers in the Deep South. As will be discussed later in this article, categorization of those of **Asian** heritage is not simple. Some are **Asian** Americans, born or naturalized in the United States. Other Asians are present in the United States, but are not US citizens. Moreover, as is noted below, those of **Asian** heritage are quite diverse. Here, I choose to use the term Asians to refer to all of **Asian** descent, regardless of particular country identity and regardless of American citizenship status.

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worth noting (Jarvis, 2005) that, in language education, we see the technology impacting on the subject matter itself with computer-mediated-communication varieties of English emerging. How significant is this? It’s probably too early to say;; perhaps we need to w8nc (wait and see)! There is no novelty value to **CALL** for these web generation learners who access the internet and other programs all the time in their daily lives. In short, “unconscious acquisition” arising out of frequent access to authentic English through globally networked environments using any number of C-bMs, frequently in combination, suggests a need to go beyond **CALL**.

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It is also clear that the utility-indifference price tends to the super-replication price, which is the supremum of the expectations of H over all equivalent martingale measures, as the risk-aversion tends to infinity, price converges to the super-hedging price, that is, worst price under the viability assumption. On the other hand, as the absolute risk-aversion decreases to zero, the utility indifference price tends to a risk-neutral valuation under the minimal entropy martingale measure P ∗ which is the lower bound of the interval of arbitrage-free price i.e. infimum of expectation of H over all equivalent martingale measures. Above theorem for **Asian** put **option** and arguments portrays that MEMM P ∗ is the appropriate choice for pricing generalized **Asian** options.

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The rest of the paper is organized as follows. In Sec- tion 2, we briefly discuss the model descriptions for the **option** pricing. The relationship between stochastic dif- ferential equations and partial differential equations for the jump-diffusion process with jump stochastic volatil- ity is presented in Section 3. Finally, a closed-form for- mula for a European **call** **option** in terms of characteristic functions is presented.

**Option** value is composed of connotation value and time value. Connotation value refers to the immediate profit obtained from the exercise of the **option**. The time value of the **option** is the part of the **option** value minus the connotation value. The traditional definition of arbitrage refers to the activity of profiting by buying and selling the same subject matter at the same favorable price in different markets at the same time. Options portfolio trading strategies include buy **call** **option** to sell the subject matter, buy put **option** to sell the subject matter.

The Video Module installs in any available **option** slot in the ATLAS 800 Series chassis. You can view the status of the module itself, as well as the circuits to which it interfaces. The terminal menus are accessible using a VT100 terminal connection (through the CONTROL port) or a Telnet session (through the Ethernet port). Use the terminal menu to configure the Video Module and to download application soft- ware.

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A basket **option** has two or more underlying assets and the holder of the **option** has the right to purchase a specified amount of these at the expiry date. When determinig the value of such an **option** using the Black-Scholes model, the dimensionality of the problem grows linearly with the number of underlying assets.

The objective of this paper is to explore the possibilities of incorporating twinkling within genetic algorithms. Thiswork is divided as follows. Section 2 explains the twinkling paradigm, anda brief overview of the simple genetic algorithm is presented in Section 3. The weighted **average** normally-distributed **arithmetic** crossover (NADX) is introduced in Section 4. Section 5 describes the process of incorporating twinkling in the SGA crossover operation. The steps of the algorithms proposed in this paper are presented in Section 6. A detailed assessment of the influence of the various components of the pro- posed algorithms on the search progression is discussed in Section 7. Evaluation of the proposed algorithms using various benchmark test functions and engineering design problems is included in Section 8. Finally, conclusions and recommendations for future work are provided in Section 9.

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We propose a modified path simulation model to value an **Asian** American **option** under VG process. Several scenarios have been investigated, especially by comparing **Asian** American **option** prices with Plain Vanilla and **Asian** European **option** prices. It was found that the proposed method gives **option** prices that inline with the characterization of each **option** type. Some scenarios have also been developed to confirm the **option** prices ob- tained using the modified path simulation method for different values of parameters. In general, we can con- clude that the proposed method performs quite well and can be used as an alternative for determining the **option** prices, especially for **Asian** American options.

Eﬀ ects of the commissions on the strategy performance have also been tested. The commis- sions and trading costs varies depending on the broker providing the market access. The **average** commission for trading plain vanilla options is around 50 cents per contract (according to Bloomberg statistics). Those for **Asian** options are lower, because even the prices are lower. Gamma- delta-theta hedging strategy is strongly sensitive to the eﬀ ects of spreads and commissions. Even commission such as 1 cent eliminated possibility of any proﬁ t. Theirs eﬀ ect is even stronger during the last month before maturity, when time value of both **call** and put options is declining faster.

Pricing of the fixed-strike **Asian** has been the subject of much research over the last ten years and academic interest in these options has experienced a revival recently, see Carr and Schr¨ oder [6], Donati-Martin, Ghomrasni and Yor [9]. Early work used the fact that the distribution of the geometric **average** of a set of lognormal distributions is also lognormal, see Conze and Visvanathan [8] and Turnbull and Wakeman [24]. A second popular line of research is to price the fixed-strike **Asian** by direct numerical methods, including Monte Carlo simulation (Kemna and Vorst [16], Lapeyre and Teman [18]) or numerically solving the PDE (Rogers and Shi [22], Alziary et al. [1] ).