In this article Marena, Roncoroni, and Fusai derive a **closed**-**form** **formula** for the fair value of call and put options written on the arithmetic average of security prices driven by jump diffu- sion processes displaying (possibly pe- riodical) trend, time varying volatility, and mean reversion. The model al- lows one for jointly fitting quoted fu- tures curve and the time structure of spot price volatility. These result ex- tends the no-jump case put forward in [Fusai, G., Marena, M., Roncoroni, A. 2008. Analytical Pricing of Discretely Monitored Asian-Style Options: The- ory and Application to Commodity Mar- kets. Journal of Banking and Finance 32 (10), 2033-2045]. A few tests based on commodity price data assess the im- portance of introducing a jump com- ponent on the resulting option prices.

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Abstract: We introduce a simple, exact and **closed**-**form** **formula** for pricing the arithmetic Asian options. The pricing **formula** is as simple as the classical Black-Scholes **formula**. In doing so, we show that the distribution of the continuous average of log-normal variables is log-normal.

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This paper presents a simple **closed**-**form** **formula** for bond pricing between coupon payments that derives from first principles and is theoretically correct. Our results are more general than the current framework, and we prove that we can retrieve the conventional **formula** for pricing bonds at coupon dates as a special case. We also demonstrate that bond traders’ ‘dirty price’ ef- fectively assumes that interest between coupon payment is simple interest, when mathematical consistency requires that all interest should be coumpounded.

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In general cases, there are two methods to calculate self inductance: the first one is the calculation of inductance of a current- carrying **closed** loop and the second is the calculation of induction of a segment of a current loop using the concept of partial inductance [3]. In this paper, a new method for calculating self inductance is presented. The method results in the Fredholm’s integral equation. We compare our formulae with several well-known ones given in the literature. 2. INTEGRAL EQUATION

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been proposed. In the Bachelier approximation (Wilcox 1990, Shimko 1994, Poitras 1998), one approximates the price difference of the two assets directly as a normal random variable and then uses the Bachelier **formula** for plain-vanilla options to approximate the spread option price. Un- fortunately, the Bachelier approximation is found to be rather crude. Some attempts (Mbanefo (1997)) have been made to improve the accuracy of the Bachelier approximation, usually by in- cluding high-order moments of the price difference or using a Gram-Charlier density function pioneered in finance by Jarrow and Rudd (1982). Kirk (1995) uses the Margrabe **formula** to price spread options by combining the second asset and the fixed spread into a single asset which is then treated as lognormally distributed. His method is equivalent to a linearization of the nonlinear exercise boundary. This method is found to be relatively accurate and thus currently relatively popular among practitioners. Carmona and Durrleman (2003a, 2003b) design a new method to approximate the spread option price by giving the lower and upper price bounds. The Carmona- Durrleman method is generally more accurate than other analytical methods. However, a critical shortcoming is that in this method one needs to solve a nonlinear system of equations which is computationally costly and not completely trivial. Thus, unlike other analytical methods, the Carmona-Durrleman method does not give a **closed**-**form** **formula** for the spread option price.

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Eraker et al. [3] developed a likelihood-based estima- tion strategy and provided estimates of parameters, spot volatility, jump times, and jump sizes using S&P 500 and Nasdaq 100 index returns. Moreover, they examined the volatility structure of the S&P and Nasdaq indices and indicated that models with jumps in volatility are pre- ferred over those without jumps in volatility. But they did not provide a **closed**-**form** **formula** for the price of a European call option.

computing the characters of the representations, Diaconis and Shahshahani showed in [DS81] that the order of mixing for the random transposition walks is n ln n. We adapt this approach to find an upper bound for the mixing time of the n-cycle-to- transpositions shuffle. To obtain a lower bound, we derive the distribution of the number of fixed points for the chain using the method of moments. In the process, we give a nice **closed**-**form** **formula** for the irreducible representation decomposition of tensor powers of the defining representation of S n . Along the way, we also look at

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cient for functions with moderate dimensions, and has been widely used in engineering, finance, atmosphere studies, see [19] the reference therein. Since the function evaluation is expensive for our problems, we need a full control of total function calls in our algorithm: not only in the searching process, but also in the numerical inte- gration. We derive a new **closed** **form** **formula** determin- ing exactly how many point evaluations are needed in the sparse grid method. To the best of our knowledge, only the order of number of points has been shown in the sparse grid literature. Based on our new **formula**, we clearly show that the number of point evaluations needed in the numerical integration increases with the dimension linearly.

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. (6) In the following, we follow the same risk-neutral ap- proach as outlined in the previous subsection to price options when there are restrictions on underlying asset price changes. As seen in the Black-Scholes-Merton framework, when we move from real world into risk- neutral world, volatility remains the same, but expected return is equal to risk-free interest rate. Option prices are then calculated as expected payoff under the risk-neutral measure, further discounted by risk-free interest rate. In the following, we first derive the risk-neutral distribution of asset returns when daily returns follow truncated nor- mal distributions, and then derive a **closed**-**form** **formula** for European call options.

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Blaschke and Müller gave a relation between the Steiner **formula** and the polar moment of inertia around the pole for a moment [6]. A relation to the polar moment of inertia around the origin is demonstrated by Müller [3]. Also the same relation for **closed** functions is inspected by Tölke [7]. Furthermore Kuruoğlu, Düldül and Tutar [8] generalized Müller’s results for homothetic mo- tion.

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We remark that (2.8) tells us that the d − dimensional Fourier transform (d odd) of the original Wendland function (designed for IR d with smoothness param- eter k) coincides with the d − 1 dimensional Fourier transform of the missing Wendland function, designed for IR d − 1 with smoothness parameter k + 1 2 . We note that this observation is not a surprising one since it is known that the derivatives of the Wendland family, when written in f − **form**, also remain in the Wendland family, see [3] for further details and [16] for practical implications.

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Most methodologies in the basket option pricing literature are either restricted to a simple lognor- mal setting or to a model-dependent framework. In this paper we introduced novel methods which allow for a fast and reliable approximation of basket options, via lower and upper bounds. Such bounds rely on a rather weak assumption, i.e. the characteristic function of the vector of log-prices is known. This assumption is very general as most models which are commonly found in the literature allow for an explicit expression of this quantity either by means of the L´ evy-Khintchine **formula** for L´ evy processes or systems of (generalized) Riccati equations for affine processes. However, our approach is not restricted to those classes of models, provided the multivariate characteristic func- tion is known. We study the case of strictly positive basket weights as well as the negative one, i.e. the so-called basket spread option. We test our methodologies on different models: a Gaussian model, a jump diffusion model, a mean reverting jump diffusion model, a multi-factor stochastic volatility model with common factors among the underlyings and finally a stochastic correlation model. In particular we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. Both approximations are particularly appealing for higher dimensional problems, versus most existing methods in the literature that cannot be applied when the basket dimension is large, due to the significant computational cost. Moreover, by using one of our bounds as a control variate, we can largely improve the accuracy of the Monte Carlo estimate.

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Age of children to administer tablets should be considered. Management of the use of drug dosage **form** for children requires knowledge of pharmacology with respect to the period of child development. 6 The current study indicated that physicians prescribed compounding because of the parents requested. Patients may be more practical and easier in administering the compounding **form**. However, the problem of bitter taste was expressed by physicians as patients complaint. Development of drug dosage formulations for oral use in children remains a challenge for scientists. 6,7 The bitter taste problem should be noted. There is a lack of suitable and safe drug

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The first is the electrostatic interpretation of the zeros of polynomials satisfying second-order diﬀerential equations 15–17, the second is a simple curve fitting of numerical data, and[r]

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RESULTS AND DISCUSSION In this section, we compute the Omega and the Sadhana polynomial of the nanosheet C4 C6 C8 2n,m, where 2n is the number of hexagons arrangement row wise and m is t[r]

The convenience of analytical tractability, paired with maximum stability, make the GEV distribution an attractive choice to model incomplete information structures. Analytical tractability entails the additional benefit in that it permits **closed** **form** expressions for comparative static results, e.g. characterizing the eﬀect of an increase in the number of bidders on the expected revenue in an IPV auction. Such results, next to the well-known analytical expressions for own and cross elasticities in logit demand models, are of great value, e.g. in applied competition analysis. In the microeconometric analysis of incomplete information games, the data typically only capture the value of the winning agents’ optimal strategies, e.g the winning bid or the price reached in the final of a sequence of bargaining episodes. The values of rival agents’ strategies along and oﬀ the equilibrium path, such as losing bids and inferior bargaining matches, typically are not observed. To the extent that agents’ optimal strategies are constrained by, and hence depend on, such values, structural econometric analysis needs to properly account for them. This can be done relatively eﬃciently if they can be replaced - or imputed - by expectations, conditional on observables; see, for example, Beckert, Smith and Takahashi (2015). An analytical E (or expectation) step does not only avoid additional computations necessary in simulation and numerical approximations, but it also improves the precision of resulting estimators relative to their simulation-assisted counterparts 8 .

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Having reached a formal solution, there is no general recipe for transforming this into a more amenable **form** and it is hard even to know if such a **form** exists. Symbolic algebra software may be helpful in simple cases but for this example did not lead to simplification of the result. However, a literature search led to a relatively obscure result known as the Mehler **formula** (Srivastava & Manocha, 1984), that contains bilinear sums of Hermite polynomials:

It is well known that p-nilpotent groups form a subgroup closed saturated formation and that the intersection of two subgroup closed saturated formations is a subgroup closed saturated f[r]

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which are in the physiologically expected range. However, in their study, iterative procedures such as steepest descent algorithm were used to find minimum of the RLS cost function and a **closed** **form** solution was not provided. Bias and variance of the estimator were also estimated using some Monte-Carlo simulations.

794 predicted European option prices and market traded options prices, the smile curve, can be accounted for by stochastic volatility models. Modelling volatility as a stochastic process is motivated a priori by empirical studies of the stock price returns in which estimated volatility is observed to exhibit random characteristics. Additionally, the effects of transaction costs show up, under many models, as uncertainty in the volatility; fat tailed returns distributions can be simulated by stochastic volatility. The assumption of constant volatility is not reasonable, since one requires different values for the volatility parameter for different strikes and different expiries to match market prices. The volatility parameter that is required in the Black Scholes **formula** to reproduce market prices is called the implied volatility. This is a critical internal inconsistency, since the implied volatility of the underlying should not be dependent on the specifications of the contract. Thus to obtain market prices of options maturing at a certain date, volatility needs to be a function of the strike. This function is the so called volatility skew or smile. Furthermore for a fixed strike one also needs to different volatility parameters to match the market prices of options maturing on different dates written on the same underlying, hence volatility is a function of both the strike and the expiry date of the derivative security. This bivariate function is called the volatility surface. There are two prominent ways of working around this problem, namely, local volatility models and stochastic volatility models. For local volatility models the assumption of constant volatility made in Black and Scholes is relaxed. The underlying risk-neutral stochastic process becomes:

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