Approximate Price Research Articles

EFFICIENT LAPLACE INVERSION, WIENER-HOPF FACTORIZATION AND PRICING LOOKBACKS

We construct fast and accurate methods for (a) approximate Laplace inversion, (b) approximate calculation of the Wiener-Hopf factors for wide classes of Lévy processes with exponentially decaying Lévy densities, and (c) approximate pricing of lookback options. In all cases, we use appropriate conformal change-of-variable techniques, which allow us to apply the simplified trapezoid rule with a small number of terms (the changes of variables in the outer and inner integrals and in the formulas for the Wiener-Hopf factors must be compatible in a certain sense). The efficiency of the method stems from the properties of functions analytic in a strip (these properties were explicitly used in finance by Feng and Linetsky 2008). The same technique is applicable to the calculation of the pdfs of supremum and infimum processes, and to the calculation of the prices and sensitivities of options with lookback and barrier features.

A simple derivation of Kirk’s approximation for spread options

Ever since Kirk proposed an approximate price formula for a European call spread option in 1995, Kirk’s approximation has become the most widely used among the practitioners, especially in the energy markets. It is well known that Kirk’s approximation extends from Margrabe’s exchange option formula but no explicit derivation is available or has ever been published. In this paper we apply the idea of WKB method to provide a simple derivation of Kirk’s approximation and discuss its validity.

A Simple Derivation of Kirk's Approximation for Spread Options

Ever since Kirk proposed an approximate price formula for a European call spread option in 1995, Kirk's approximation has become the most widely used among the practitioners, especially in the energy markets. It is well known that Kirk's approximation extends from Margrabe's exchange option formula but no explicit derivation is available or has ever been published. In this paper we apply the idea of the WKB method to provide a simple derivation of Kirk's approximation and discuss its validity.

Taylor approximation pricing for K -user multiple-input multiple-output (MIMO) interference channels

This study considers the immense problem of sum rate maximisation in K-user multiple-input multiple-output interference channels (IFC). In order to address this problem, the authors propose an iterative Taylor approximation algorithm to find the optimal transmit covariance matrices, where each user maximises the sum rate function through Taylor expansion. Owing to the distributed nature of the ITA algorithm, Taylor terms can be treated as prices, where each user collects current prices from the others to perform update. This process further leads us modelling it as a more general concave n-person game to analyse the existence and uniqueness of the Nash equilibrium (NE). Then an iterative covariance-price updating (ICP) algorithm is proposed for flexible updating and establishing a convergence process to obtain the optimal solution under NE uniqueness conditions. Simulation results show that both algorithms come with similar performance and both significantly outperform the conventional iterative waterfilling algorithm.

PRICING DERIVATIVES ON MULTISCALE DIFFUSIONS: AN EIGENFUNCTION EXPANSION APPROACH

Using tools from spectral analysis, singular and regular perturbation theory, we develop a systematic method for analytically computing the approximate price of a large class of derivative-assets. The payoff of the derivative-assets may be path-dependent. In addition, the process underlying the derivatives may exhibit killing (i.e., jump to default) as well as combined local/nonlocal stochastic volatility. The nonlocal component of volatility may be multiscale, in the sense that it may be driven by one fast-varying and one slow-varying factor. The flexibility of our modeling framework is contrasted by the simplicity of our method. We reduce the derivative pricing problem to that of solving a single eigenvalue equation. Once the eigenvalue equation is solved, the approximate price of a derivative can be calculated formulaically. To illustrate our method, we calculate the approximate price of three derivative-assets: a vanilla option on a defaultable stock, a path-dependent option on a nondefaultable stock, and a bond in a short-rate model.

Investigation on Fish Marketing Pattern in Mymensingh

An investigation was carried out on fish marketing system of Mymensingh town to find out different marketing channels, fish trading, marketing cost and margins of fish at different levels. The investigation was conducted from October, 2010 to September, 2011 in sixteen markets of Mymensingh town. In Mymensingh town a large number of people were involved in the fish marketing channel as bepari, aratdar, paiker and retailer. Paiker and bepari bought fish from producer and finally sold fish to retailer through aratdar. In the study area it was estimated that total marketing cost of fish for aratdar was Tk. 1.32, paiker, Tk. 6.05, and retailer, Tk. 1.83 per kg of fish. Cost item of fish for different intermediaries was Tk. 10.49 total. The highest marketing costs (Tk. 6.05) were for paiker because they had paid higher aratdari commission and higher transportation cost. Average Net profit was Tk. 3.19 for paikers at peak period and Tk. 3.36 at lean period, for aratdars Tk. 1.60 at peak period and Tk. 1.75 lean period and for retailers Tk. 5.06 at peak period and Tk. 6.24 at lean period per kg of fish. Ilish (Tenualosa ilisha) had high demand to consumers and available in the market with approximate price of Tk. 250 to Tk. 300 per piece DOI: http://dx.doi.org/10.3329/jesnr.v5i1.11554 J. Environ. Sci. & Natural Resources, 5(1): 61-68, 2012

AGGREGATION, HETEROGENEOUS AUTOREGRESSION AND VOLATILITY OF DAILY INTERNATIONAL TOURIST ARRIVALS AND EXCHANGE RATES*

Tourism is a major source of service receipts for many countries, including Taiwan. The two leading tourism countries for Taiwan are Japan and USA, which are sources of short and long haul tourism, respectively. As a strong domestic currency can have adverse effects on international tourist arrivals through the price effect, daily data from 1 January 1990 to 31 December 2008 are used to model the world price, exchange rates, and tourist arrivals from the world, USA and Japan to Taiwan, and their associated volatility. Inclusion of the exchange rate and its volatility captures approximate daily and weekly price and price volatility effects on world, US and Japanese tourist arrivals to Taiwan. The Heterogeneous Autoregressive (HAR) model is used to approximate the slowly decaying correlations associated with the long memory properties in daily and weekly exchange rates and international tourist arrivals, to test whether alternative short and long run estimates of conditional volatility are sensitive to the long memory in the conditional mean, to examine asymmetry and leverage in volatility, and to examine the effects of temporal and spatial aggregation. The approximate price and price volatility effects tend to be different, with the exchange rate typically having the expected negative impact on tourist arrivals to Taiwan, whereas exchange rate volatility can have positive or negative effects on tourist arrivals to Taiwan. For policy purposes, the empirical results suggest that an arbitrary choice of data frequency or spatial aggregation will not lead to robust findings as they are generally not independent of the level of aggregation used.

RELATING TOP-DOWN WITH BOTTOM-UP APPROACHES IN THE EVALUATION OF ABS WITH LARGE COLLATERAL POOLS

This paper uses a conditional law of large numbers and a conditional central limit theorem to provide simplified asymptotic valuation formulas for credit derivatives on baskets, including synthetic and cash-flow CDOs. In particular, approximate pricing procedures are provided for synthetic and cash-flow CDOs. In the process, this paper also clarifies the relation between the "top-down" and "bottom-up" approaches for pricing credit derivatives.

Pricing perpetual American puts under multi-scale stochastic volatility

In this paper, we consider the problem of pricing perpetual American put options with volatility driven by two other processes. By using a perturbation approach, we obtain approximate but explicit closed-form pricing formulae for the option and optimal exercise prices, respectively, under a general multi-scale SV (stochastic volatility) model. A key feature of the expansion methodology employed here is to balance the two SV processes, while dealing with the free boundary conditions properly. It turns out that in the current formulae, the fast volatility factor does not play an explicit role, while the slow factor is quite crucial, a phenomenon that is shown to be quite reasonable through our discussions.

Integrated Optimization of Procurement, Processing, and Trade of Commodities

We consider the integrated optimization problem of procurement, processing, and trade of commodities in a multiperiod setting. Motivated by the operations of a prominent commodity processing firm, we model a firm that procures an input commodity and has processing capacity to convert the input into a processed commodity. The processed commodity is sold using forward contracts, while the input itself can be traded at the end of the horizon. We solve this problem optimally and derive closed-form expressions for the marginal value of input and output inventory. We find that the optimal procurement and processing decisions are governed by price-dependent inventory thresholds. We use commodity markets data for the soybean complex to conduct numerical studies and find that approximating the joint price processes of multiple output commodities using a single, composite output product and using the approximate price process to determine procurement and processing decisions is near optimal. Compared to a myopic spread-option-based heuristic, the optimization-based dynamic programming policy provides significant benefits under conditions of tight processing capacities and high price volatilities. Finally, we propose an approximation procedure to compute heuristic policies and an upper bound to compare the heuristic against, when commodity prices follow multifactor processes.

Mean-Variance Hedging under Additional Market Information

In this paper we analyse the mean-variance hedging approach in an incomplete market under the assumption of additional market information, which is represented by a given, finite set of observed prices of non-attainable contingent claims. Due to no-arbitrage arguments, our set of investment opportunities increases and the set of possible equivalent martingale measures shrinks. Therefore, we obtain a modified mean-variance hedging problem, which takes into account the observed additional market information. Solving this by means of the techniques developed by Gourieroux, Laurent and Pham (1998), we obtain an explicit description of the optimal hedging strategy and an admissible, constrained variance-optimal signed martingale measure, that generates both the approximation price and the observed option prices.

Pricing of First Touch Digitals under Normal Inverse Gaussian Processes

We calculate prices of first touch digitals under normal inverse Gaussian (NIG) processes, and compare them to prices in the Gaussian model with the same instantaneous variance. Numerical results are produced to show that for typical parameters values, the relative error of the Gaussian approximation to NIG-price can be several dozen percent if the spot price is at the distance 0.05-0.2 from the barrier (normalized to one). A fast approximate pricing formula under NIG is derived.

Simulation of nonlinear interest rates in quantum finance: Libor Market Model

The simulation of the Libor Market Model (LMM) is extensively studied in the framework of quantum finance. The imperfectly correlated Libor rates are simulated based on a Gaussian quantum field and a recursion equation of nontrivial stochastic drift. The Libor options are studied using both the simulation method and the analytical formula. The caplet price of simulation is compared with Black’s caplet formula which can be exactly derived from the LMM. The invariance of caplet price for different forward bond numeraire is verified by using the simulation. The simulation results for coupon bond options and swaptions are compared with the approximate price, which are limited for the reason that the approximate price is derived using the small volatility expansion. The simulation method is shown to have great potential in the application of pricing interest rate instruments.

A new approximate CVA of interest rate swap in the SABR/LIBOR market model: an asymptotic expansion approach

Abstract The author presents a new approximate pricing formula for the credit valuation adjustment of interest rate swap in the SABR/LIBOR market model using an asymptotic expansion method. He compare it with Monte Carlo simulation. It is shown that the new method makes computations extremely fast.

Pricing Derivatives on Multiscale Diffusions: An Eigenfunction Expansion Approach

Using tools from spectral analysis, singular and regular perturbation theory, we develop a systematic method for analytically computing the approximate price of a large class of derivative‐assets. The payoff of the derivative‐assets may be path‐dependent. In addition, the process underlying the derivatives may exhibit killing (i.e., jump to default) as well as combined local/nonlocal stochastic volatility. The nonlocal component of volatility may be multiscale, in the sense that it may be driven by one fast‐varying and one slow‐varying factor. The flexibility of our modeling framework is contrasted by the simplicity of our method. We reduce the derivative pricing problem to that of solving a single eigenvalue equation. Once the eigenvalue equation is solved, the approximate price of a derivative can be calculated formulaically. To illustrate our method, we calculate the approximate price of three derivative‐assets: a vanilla option on a defaultable stock, a path‐dependent option on a nondefaultable stock, and a bond in a short‐rate model.

Trying Harder and Doing Worse: How Grocery Shoppers Track In-Store Spending

Although almost one in three U.S. households shops on a budget, it remains unclear whether and how shoppers track their in-store spending to stay within those budgets. A field study and two laboratory studies offer four key generalizations about budget shoppers in grocery stores: (1) They predominantly use mental computation strategies to track their in-store spending, (2) they adapt their mental computation strategy to the dominant range of price endings of items in their shopping baskets, (3) those who try to calculate the exact total price of their basket are less accurate than those who estimate the approximate price, and (4) motivated shoppers are less accurate than less motivated shoppers (because they tend to calculate rather than estimate the total basket price). A second field study demonstrates that shoppers who underestimate the total price of their basket are more likely to overspend, leading to negative store satisfaction.

Trying Harder and Doing Worse: How Grocery Shoppers Track In-Store Spending

Although almost one in three U.S. households shops on a budget, it remains unclear whether and how shoppers track their in-store spending to stay within those budgets. A field study and two laboratory studies offer four key generalizations about budget shoppers in grocery stores: (1) They predominantly use mental computation strategies to track their in-store spending, (2) they adapt their mental computation strategy to the dominant range of price endings of items in their shopping baskets, (3) those who try to calculate the exact total price of their basket are less accurate than those who estimate the approximate price, and (4) motivated shoppers are less accurate than less motivated shoppers (because they tend to calculate rather than estimate the total basket price). A second field study demonstrates that shoppers who underestimate the total price of their basket are more likely to overspend, leading to negative store satisfaction.

A simple approach to valuing Asian rainbow options

Asian rainbow options, also called options on the minimum or the maximum of two or several average prices, are found widely applicable in both risk management and the design of incentive contracts for executive compensation. However, their valuation is very complicated. This paper proposes a new and simple approach to valuing such options on dividend-paying assets. Starting from an observation that the price of an Asian option in some cases can become that of a vanilla European option and then using the method of Johnson (1987), we can easily obtain a closed-form valuing formula for the option with two geometric averaging. Moreover, this method can be directly extended to obtain a closed-form solution for the case of several geometric averaging. We also derive a closed-form approximation pricing formula for an Asian rainbow option with two arithmetic averaging based on the obtained price formula of its counterpart geometric averaging Asian rainbow option and the approach of Zhang (1998). Numerical comparison shows the efficiency and accuracy of our approximation price and the different effects of the interest rate, volatility, dividend and observation frequency on the price of Asian rainbow options.

Parametrix Approximation of Diffusion Transition Densities

A new analytical approximation tool, derived from the classical PDE theory, is introduced in order to build approximate transition densities of diffusions. The tool is useful for approximate pricing and hedging of financial derivatives and for maximum likelihood and method of moments estimates of diffusion parameters. The approximation is uniform with respect to time and space variables. Moreover, easily computable error bounds are available in any dimension.

Multi-layer model of correlated energy prices

In this article we develop an extension of the affine jump–diffusion modeling framework and use it to build an intuitive and tractable model of an energy price complex. The development is motivated by the need to model prices of electricity while capturing their dependence on the price of other energy commodities. Such a model is essential for valuing a range of typical derivatives traded in the electricity markets: cross-commodity spread options, cross-location spread options, fuel-switching powerplants, etc. We give an approximate pricing method for these derivatives together with precise error bound estimates.