Approximate Price Research Articles

Pricing of discretely sampled arithmetic Asian options, under the Hull–White interest rate model

This paper studies the pricing of discrete arithmetic Asian options (AAOs) with fixed strikes under the Hull–White interest rate model. For the pricing of AAOs, we first investigate the stochastic dynamics of the price of the underlying asset under the T-forward measure, and then study the distribution of the discrete arithmetic average of the underlying asset price. Specifically, we provide the first three moments of the discrete arithmetic average under the T-forward measure. Then, we derive approximate pricing formulas for AAOs using the three-moment matching method. Furthermore, we calculate the first three conditional moments of the discrete arithmetic average, given the final value of the underlying asset, under the T-forward measure. These conditional moments can be used to improve the accuracy of the approximation of the AAO prices. The numerical results show that our three-moment matching approximations are very accurate. Additionally, the accuracy can be further improved by combining the conditioning approach with the three-moment matching method. Our procedure is also applied to the computation of deltas of AAOs.

A Cheap and Basic Solar-Powered Smart Irrigation System Proposal for Medium and Small-Scale Farming

Nowadays, the need for water consumption for agricultural production is increasing. Economical use of water has become mandatory both to increase agricultural product yield and to eliminate the damage caused by excessive irrigation to the soil. Preferred instead of traditional irrigation, Drip irrigation, sprinkler irrigation, and pivot irrigation systems are now being replaced by “Smart Irrigation Systems” that save more water. In this study, a basic solar energy-supported mobile phone-controlled smart irrigation system, recommended for medium and small-scale agricultural enterprises, is proposed. In the study, the basic elements that make up the system, their approximate prices and circuit connection ways are shown. In the study, the cost, water, energy consumption, and payback periods of smart irrigation systems with traditional drip, sprinkler, and pivot irrigation methods were compared. As a result, although the initial investment cost in smart irrigation systems seems relatively high, it offers significant advantages in terms of resource efficiency and environmental sustainability. It is a fact that modern irrigation systems will make important contributions to national economies in the long term by increasing agricultural production and saving energy and water.

Variance and volatility swaps and options under the exponential fractional Ornstein–Uhlenbeck model

Considering the fair strike values of variance and volatility swaps, we use a stochastic volatility model in which the log volatility is given by a fractional Ornstein–Uhlenbeck process with two versions; a stationary version and a version with a deterministic initial value. Under these versions, the fair strike formulas are obtained in exact form for variance swaps and approximated fair strike formulas are derived for volatility swaps based on the fact that an aggregation of log-normal variables is well-approximated by shifted log-normal or log-normal distribution. In addition, we obtain two approximate pricing formulas for European options on the realized variance and volatility. The accuracy and robustness of the approximated fair strike formulas are examined via Monte-Carlo computations. We conduct calibration experiments to show that the Hurst exponent and the mean-reversion property of the fractional Ornstein–Uhlenbeck process are able to produce various shapes resembling the market term-structures of variance swaps when they are put together.

The Nutritional Implications of High Wheat Prices on Agriculture Households in Pakistan

This study analyzes the potential repercussions of high wheat prices on agriculture household economic and nutritional well-being in Pakistan. This task was accomplished by formulating a theoretical model to capture the economic behavior of agricultural households that produce marketable surpluses of wheat. For empirical verification, eight modules of nationally representative Household Integrated Economic Survey (HIES) were combined with approximate price levels to create a pooled cross-sectional sample of annual price-indices for household expenditure from 2000-01 to 2018-19. The study uses the Linear Approximate of Almost Ideal Demand System (LA-AIDS) model to derive demand elasticities and conduct a comparative static analysis. Finally, the model is extended to assess household nutritional well-being by linking price effects to calorie intake. The empirical findings demonstrated that high wheat prices resulted in high farm revenues for agricultural households. Although wheat consumption has declined, these households have used excess profits to diversify the food basket, which in turn led to an improvement in calorie intake. An indirect spillover effect from changes in wheat prices has also been observed, leading to an increase in demand for domestically produced commodities. Since the overall impact of wheat price increase on agricultural and non-agricultural households is mixed, it is proposed to pursue a careful food security policy to ensure the economic and nutritional well-being of the entire society. This study adopts a two-pronged approach of social to integrating social safety nets that does not hinder agricultural growth to achieve a mutually rewarding outcome.

Comparative analysis of data classification methods for prediction of trade-in auto prices

In the presented article, machine learning algorithms are used to predict the price of cars. Forecasting the price of cars is one of the most important issues in modern times, because the number of car users is increasing year by year worldwide. Therefore, it may be interesting for many car owners to know the approximate price of cars in advance. The vehicle’s build date, mileage, size and other parameters are essential data for the machine learning process. Based on these parameters, the data needed to predict the price of any car was classified and a training dataset was created. Based on this data set, some interesting predictions were made. The purpose of the article is to consider the pre-processing of data, to determine and analyze what achievements the direction of artificial intelligence has achieved on the basis of predicting the price of cars. Hybrid forecasting methods using statistical analysis and machine learning methods were used in the research.

Approximate option pricing under a two-factor Heston–Kou stochastic volatility model

Under a two-factor stochastic volatility jump (2FSVJ) model we obtain an exact decomposition formula for a plain vanilla option price and a second-order approximation of this formula, using Itô calculus techniques. The 2FSVJ model is a generalization of several models described in the literature such as Heston (Rev Financ Stud 6(2):327–343, 1993); Bates (Rev Financ Stud 9(1):69–107, 1996); Kou (Manag Sci 48(8):1086–1101, 2002); Christoffersen et al. (Manag Sci 55(12):1914–1932, 2009) models. Thus, the aim of this study is to extend some approximate pricing formulas described in the literature, like formulas in Alòs (Finance Stoch 16(3):403–422, 2012); Merino et al. (Int J Theor Appl Finance 21(08):1850052, 2018); Gulisashvili et al. (J Comput Finance 24(1), 2020), to pricing under the more general 2FSVJ model. Moreover, we provide numerical illustrations of our pricing method and its accuracy and computational advantage under double exponential and log-normal jumps. Numerically, our pricing method performs very well compared to the Fourier integral method. The performance is ideal for out-of-the-money options as well as for short maturities.

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

AbstractIn financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of Monte Carlo (MC) simulations. To obtain a good approximation the MC sample size usually needs to be considerably large resulting in a long computing time to obtain a single approximation. A natural deep learning approach to reduce the computation time when new prices need to be calculated as quickly as possible would be to train an artificial neural network (ANN) to learn the function which maps parameters of the model and of the financial product to the price of the financial product. However, empirically it turns out that this approach leads to approximations with unacceptably high errors, in particular when the error is measured in the ‐norm, and it seems that ANNs are not capable to closely approximate prices of financial products in dependence on the model and product parameters in real life applications. This is not entirely surprising given the high‐dimensional nature of the problem and the fact that it has recently been proved for a large class of algorithms, including the deep learning approach outlined above, that such methods are in general not capable to overcome the curse of dimensionality for such approximation problems in the ‐norm. In this article we introduce a new numerical approximation strategy for parametric approximation problems including the parametric financial pricing problems described above and we illustrate by means of several numerical experiments that the introduced approximation strategy achieves a very high accuracy for a variety of high‐dimensional parametric approximation problems, even in the ‐norm. A central aspect of the approximation strategy proposed in this article is to combine MC algorithms with machine learning techniques to, roughly speaking, learn the random variables (LRV) in MC simulations. In other words, we employ stochastic gradient descent (SGD) optimization methods not to train parameters of standard ANNs but instead to learn random variables appearing in MC approximations. In that sense, the proposed LRV strategy has strong links to Quasi‐Monte Carlo (QMC) methods as well as to the field of algorithm learning. Our numerical simulations strongly indicate that the LRV strategy might indeed be capable to overcome the curse of dimensionality in the ‐norm in several cases where the standard deep learning approach has been proven not to be able to do so. This is not a contradiction to the established lower bounds mentioned above because this new LRV strategy is outside of the class of algorithms for which lower bounds have been established in the scientific literature. The proposed LRV strategy is of general nature and not only restricted to the parametric financial pricing problems described above, but applicable to a large class of approximation problems. In this article we numerically test the LRV strategy in the case of the pricing of European call options in the Black‐Scholes model with one underlying asset, in the case of the pricing of European worst‐of basket put options in the Black‐Scholes model with three underlying assets, in the case of the pricing of European average put options in the Black‐Scholes model with three underlying assets and knock‐in barriers, as well as in the case of stochastic Lorentz equations. For these examples the LRV strategy produces highly convincing numerical results when compared with standard MC simulations, QMC simulations using Sobol sequences, SGD‐trained shallow ANNs, and SGD‐trained deep ANNs.

Nonparametric estimates of option prices via Hermite basis functions

We consider approximate pricing formulas for European options based on approximating the logarithmic return’s density of the underlying by a linear combination of rescaled Hermite polynomials. The resulting models, that can be seen as perturbations of the classical Black-Scholes one, are nonpararametric in the sense that the distribution of logarithmic returns at fixed times to maturity is only assumed to have a square-integrable density. We extensively investigate the empirical performance, defined in terms of out-of-sample relative pricing error, of this class of approximating models, depending on their order (that is, roughly speaking, the degree of the polynomial expansion) as well as on several ways to calibrate them to observed data. Empirical results suggest that such approximate pricing formulas, when compared with simple nonparametric estimates based on interpolation and extrapolation on the implied volatility curve, perform reasonably well only for options with strike price not too far apart from the strike prices of the observed sample.

World Cup Advertising Pricing and Comprehensive Value Analysis for Companies

The contemporary sports industry is rapidly expanding, and sports are assuming greater significance in people's everyday routines. The World Cup is the largest sporting event globally, boasting a massive market that offers a plethora of benefits to companies. This article aims to delve into the advertising pricing system of the World Cup and the driving factors and logic of exhibitors. It studies the division of the World Cup pricing system, the rights enjoyed by each system, and the reasons for corporate sponsorship, as well as the benefits it brings. The significance of this article is to introduce the pricing of World Cup advertising and analyze the various benefits for companies sponsoring the event. Data related to the FIFA World Cup Russia 2018 and the FIFA World Cup Qatar 2022 will be examined to determine the approximate pricing of the FIFA World Cup Russia 2018 and the FIFA World Cup Qatar 2022 advertisements and the overall value of these two World Cups to companies.

Pricing ratchet equity index annuity with mortality risk by complex Fourier series method

In this paper, we consider the pricing of the ratchet equity-indexed annuity with the incorporation of mortality risk by an efficient and accurate method under the general time-changed Lévy processes. The pricing of the contract is a two-dimensional problem: we apply the complex Fourier series method to the log return dimension of the asset and high-order quadrature approximation to the other dimensions. Depending on the subordinating dynamic of the volatility, we use the time-changed Lévy process to model stochastic volatility via the random time change in the Lévy process, which compensates for limitations in the common Lévy process. By combining the recursive algorithm and complex Fourier series method, we obtain the approximate price of the contract. Error analysis proves that the total error can achieve an exponential convergence rate, and numerical examples demonstrate the accuracy and efficiency of the complex Fourier series expansion method.

Valuation of spread options under correlated skew Brownian motions

In this paper, we employ correlated skew Brownian motions to describe the dynamics of the two assets underlying the spread option. In the pricing model, the two underlying assets are exposed to exogenous risks captured by the same Brownian motion, and their endogenous risks are also assumed to be correlated with each other. We obtain an approximate pricing formula of spread options and calibrate model parameters to real data. In addition, we compare the results obtained from the approximate formula with those derived from the exact closed-form formula and from the Monte Carlo method. Finally, we analyze the impact of the skewness parameter on option prices after checking the accuracy of the approximations numerically.

A closed-form approximation for pricing spread options on futures under a mean-reverting spot price model with multiscale stochastic volatility

Abstract Commodity spot prices tend to revert to some long-term mean level and most commodity derivatives are based on futures prices, not on spot prices. So, we consider spread options on futures instead of spot or spot index, where the log spot price follows a mean-reverting process. The volatility of the mean-reverting process is driven by two different (fast and slow) scale factors. We use asymptotic analysis to obtain a closed-form approximation of the futures prices and a closed-form formula for the approximate prices of spread options on the futures. The overall improvement of our analytic formula over the classical Kirk–Bjerksund–Sternsland (KBS) formula is discussed via numerical experiments.

Closed-form approximate solutions for stop-loss and Russian options with multiscale stochastic volatility

<abstract><p>In general, derivation of closed-form analytic formulas for the prices of path-dependent exotic options is a challenging task when the underlying asset price model is chosen to be a stochastic volatility model. Pricing stop-loss and Russian options is studied under a multiscale stochastic volatility model in this paper. Both options are commonly perpetual American-style derivatives with a lookback provision. We derive closed-form formulas explicitly for the approximate prices of these two exotic options by using multiscale asymptotic analysis and partial differential equation method. The formulas can be efficiently computed starting with the Black-Scholes option prices. The accuracy of the analytic approximation is verified via Monte-Carlo simulations and the impacts of the multiscale stochastic volatility on the corresponding Black-Scholes option prices are revealed. Also, the performance of the model is compared with that of other models.</p></abstract>

Bonds and Options Valuation Using a Conditioning Factor Approach

This paper looks at methods to calculate prices (or approximate prices) of bonds (zero coupon as well as coupon bearing) where the interest rates follow a log-normal distribution using two different waysthe first method makes use of a conditioning variable similar to the approach of Basu (1999) and Rogers and Shi (1995), while the second method (only applicable in case of the zero - coupon bond case) is by making use of a direct expansion technique. The conditioning factor based method is then used to approximate the price of the bond (in fact the lower bound to the price of the bond) for the case of coupon carrying bonds - both non-defaultable as well as defaultable ones. Finally, the conditioning factor method is used to value European options on assets with stochastic volatility. The approach used is not one based on numerical solutions to partial differential equations but rather on an approximation to the price of the option that can be arrived at using the conditioning factor approach. As is shown in the tables at the end of the paper, the results obtained by using the conditioning factor based approach is quite accurate - in fact in some cases it is exactly equal to the true prices themselves.

Efficiency of Black and Scholes Model for Pricing Options at Indian Stock Market

Black and Scholes model was developed for valuing the options in derivatives market in the year 1973. This model became the prime tool for pricing options but because of its deficiencies it forced researchers to find out new model for pricing options (Rubinstein, 1985; Hull and White, 1987; Wiggins, 1987; Dumas, Fleming and Whaley, 1998). Option pricing is proven to be very challenging task during high volatility situations whereas Black and Scholes option pricing model is not proven a successful predictor for determining approximate price of the options. Even empirical evidences support that Black and Scholes model produces some biasness in estimating the prices. The main objective of this paper is to find out the efficiency of the Black and Scholes option pricing model for predicting the option prices at Indian Stock Exchanges for which data set from Indian Stock Exchanges will be chosen, market price of option contracts are considered as sample for the year 2008-2015. The actual market price will be compared with the price derived with the help of Black and Scholes model formula.

Valuation of European-style vulnerable options under the non-affine stochastic volatility and double exponential jump

The pricing of European-style vulnerable option when the price process of the underlying asset follows non-affine stochastic volatility and double exponential jump is investigated. An approximate expression for the joint characteristic function of the log-price of underlying asset and the log-value of counterparty asset is derived. An analytical approximate price of European-style vulnerable option is also obtained by means of Fourier-cosine method. Numerical experiments are given to confirm the accuracy and efficiency of the proposed result for pricing the European-style vulnerable option compared with Monte Carlo simulation. Finally, sensitivity analysis is presented to further explain the theoretical results.

APPROXIMATE OPTION PRICING FORMULA FOR BARNDORFF-NIELSEN AND SHEPHARD MODEL

For the Barndorf-Nielsen and Shephard model, we present approximate expressions of call option prices based on the decomposition formula developed by [T. Arai (2021) Alos type decomposition formula for Barndor-Nielsen and Shephard model, Journal of Stochastic Analysis 2 (2), 3]. Besides, some numerical experiments are also implemented to make sure how effective our approximations are.

Approximate pricing of American exchange options with jumps

AbstractThis paper presents a numerical method to price American exchange options based on jump‐diffusion processes. We first derive a closed‐form expression for the value of European exchange options, then decompose the value function of an American exchange option into a European counterpart, and an early exercise premium that is derived analytically. The early exercise boundary for an American exchange option approximately satisfies an algebraic equation that can be quickly numerically solved. Consequently, a formula is obtained for efficiently pricing American exchange options. The numerical results reveal that our pricing formula is robust and accurate.

Erratum

The publisher, American Accounting Association, regrets an error in “Can Management Accounting Help Aid Associations Make Tough Choices in Haiti” by Krumweide, Paik, and Walden in the February 2018 issue (Volume 33, Number 1, pp. 1–16: https://meridian.allenpress.com/iaetn/article/33/1/1/212579/Can-Management-Accounting-Help-Aid-Associations) of Issues in Accounting Education, Teaching Notes. There was an error in the price for and computation of the cost of peanuts as presented in the “Quantitative Analysis of Costs versus Benefits” section. The correct computations are shown below.The online version of the article has been corrected. The error only exists in the printed version.Kip R. KrumwiedeInstitute of Management AccountantsGyung H. (Daniel) PaikW. Darrell WaldenUniversity of RichmondThere is not a lot of cost information in the case, but there is enough to estimate the extra cost of local peanuts. We know from farmer Samuel Hilaire that the market price for peanuts last year was $0.74 per kg and he barely broke even. This year, the market price is only $0.61 per kg (about 18 percent less than the $0.74 price). If we assume that $0.74 is also the farmer's break-even price this year, then it seems reasonable to assume that $0.74 is the approximate price per kg that HHC pays to local farmers. We can check this from the financial statements in Exhibit 2 of the case:As shown, the average cost of peanuts per kg for HHC has ranged from $0.742 to $0.749 during the past two years due to fluctuating prices. However, this is per kg of total MM produced. We must first compute the price paid per kg of peanuts, and then estimate the total incremental cost of buying local peanuts, as follows:a Since the MM weight of 268,905 kg consists of about 76 percent peanuts, the estimated amount of peanuts purchased in FY2014 is 268,905 kg × 76% = 204,368 kg of peanuts purchased. Thus, the estimated price paid to farmers is $201,379 ÷ 204,368 kg = $0.985.

On the robustness of the approximate price of anarchy in generalized congestion games

One of the main results shown through Roughgarden's notions of smooth games and Robust Price of Anarchy is that, for any sum-bounded utilitarian social function, the worst-case Price of Anarchy of Coarse Correlated Equilibria coincides with that of Pure Nash Equilibria in the class of weighted congestion games with non-negative and non-decreasing latency functions and that such a value can always be derived through the, so called, smoothness argument. We significantly extend this result by proving that, for a variety of (even non-sum-bounded) utilitarian and egalitarian social functions, and for a broad generalization of the class of weighted congestion games with non-negative (and possibly decreasing) latency functions, the worst-case Price of Anarchy of ϵ-approximate Coarse Correlated Equilibria still coincides with that of ϵ-approximate Pure Nash Equilibria, for any ϵ≥0. As a byproduct of our proof, it also follows that such a value can always be determined by making use of the primal-dual method we introduced in a previous work.