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Basket Approximation
  • Generalizing the Black-Scholes formula to multivariate contingent claims This paper provides approximate formulas that generalize the Black-Scholes formula in all dimensions. Pricing and hedging of multivariate contingent claims are achieved by computing lower and upper bounds. These bounds are given in closed form in the same spirit as the classical one-dimensional Black-Scholes formula. Lower bounds perform remarkably well. Like in the onedimensional case, Greeks are also available in closed form. We discuss an extension to basket options with barrier. , V.Durrleman, R.Carmona (2006)
  • Heterogeneous Basket Options Pricing Using Analytical Approximations This paper proposes the use of analytical approximations to price an heterogeneous basket option combining commodity prices, foreign currencies and zero-coupon bonds. We examine the performance of three moment matching approximations: inverse gamma, Edgeworth expansion around the lognormal and Johnson family distributions. Since there is no closed-form formula for basket options, we carry out Monte Carlo simulations to generate the benchmark values. We perform a simulation experiment on a whole set of options based on a random choice of parameters. Our results show that the lognormal and Johnson distributions give the most accurate results , G.Dionne, G.Gauthier, N.Ouertani, N.Tahani (2006)
  • Markovian Projection Method for Volatility Calibration We present the Markovian projection method, a method to obtain closed-form approximations to European option prices on various underlyings that, in principle, is applicable to any (diffusive) model. Successful applications of the method have already appeared in the literature, in particular for interest rate models (short rate and forward Libor models with stochastic volatility), and interest rate/FX hybrid models with FX skew. The purpose of this note is thus not to present other instances where the Markovian projection method is applicable (even though more examples are indeed given) but to distill the essence of the method into a conceptually simple plan of attack, a plan that anyone who wants to obtain European option approximations can follow. , V.Piterbarg (2006)
  • Pricing of Arithmetic Basket Options by Conditionning Determining the price of a basket option is not a trivial task, because there is no explicit analytical expression available for the distribution of the weighted sum of prices of the the assets in the basket. However, by using a conditioning variable, this price can be decomposed in two parts, one of which can be computed exactly. For the remaining part we first derive a lower and an upper bound based on comonotonicity, and another upper bound equal to that lower bound plus an error term. Secondly, we derive an approximation by applying some moment matching method. Keywords: basket option; comonotonicity; analytical bounds; moment matching; Asian basket option; Black & Scholes model , G.Deelstra,J.Liinev, M.Vanmaele (2004)
  • Reconstruction of Volatility: Pricing Index Options by the Steepest Descent Approximation We propose a formula for calculating the implied volatility of index options based on the volatility skews of the options on the underlying stocks and on a given correlation matrix for the basket. The derivation uses the steepest-descent approximation for the multivariate probability distribution function of forward prices. A simple financial justification is provided. We apply the formula to compute the implied volatilities of liquidly-traded options on exchange-traded funds (ETF) across different strikes. Our theoretical results were found to be in good agreement with contemporaneous quotes on the Chicago Board of Options Exchange (CBOE) and the American Stock Exchange (AMEX). , M.Avellaneda, D.Boyer-Olson (2002)
  • Pricing Asian and Basket Options Via Taylor Expansion Asian options belong to the so-called path-dependent derivatives. They are among the most di?cult to price and hedge both analytically and numerically. Basket op- tions are even harder to price and hedge because of the large number of state variables. Several approaches have been proposed in the literature, including Monte Carlo simula- tions, tree-based methods, partial di?erential equations, and analytical approximations among others. The last category is the most appealing because most of the other meth- ods are very complex and slow. Our method belongs to the analytical approximation class. It is based on the observation that though the weighted average of lognormal vari- ables is no longer lognormal, it can be approximated by a lognormal random variable if the ?rst two moments match the true ?rst two moments. To have a better approx- imation, we consider the Taylor expansion of the ratio of the characteristic function of the average to that of the approximating lognormal random variable around zero volatility. We include terms up to ?6 in the expansion. The resulting option formulas are in closed form. We treat discrete Asian option as a special case of basket options. Formulas for continuous Asian options are obtained from their discrete counterpart. Numerical tests indicate that the formulas are very accurate. Comparisons with all other leading analytical approximations show that our method has performed the best overall in terms of accuracy for both short and long maturity options. Furthermore, unlike some other methods, our approximation treats basket (portfolio) and Asian op- tions in a uni?ed way. Lastly, in the appendix we point out a serious mathematical error of a popular method of pricing Asian options in the literature. , N.Ju (2002)
  • Approximated Moment-Matching Dynamics for Basket-Options Simulation The aim of this paper is to present two moment matching procedures for basketoptions pricing and to test its distributional approximations via distances on the space of probability densities, the Kullback-Leibler information (KLI) and the Hellinger distance (HD). We are interested in measuring the KLI and the HD between the real simulated basket terminal distribution and the distributions used for the approximation, both in the lognormal and shifted-lognormal families. We isolate influences of instantaneous volatilities and instantaneous correlations, in order to assess which configurations of these variables have a major impact on the KLI and HD and therefore on the quality of the approximation. , D. Brigo, F. Mercurio, F. Rapisarda, R. Scotti (2001)
  • Valuing Exotic Options by Approximating the SPD with Higher Moments The financial economic No Arbitrage assumption implies that in a complete market the price of any derivative security is the discounted value of its payoff function integrated against the appropriate state-price density (SPD). For most exotic path dependent payoffs, it is quite difficult to obtain a closed form expression for the state-price density, thus eliminating the prized possibility of an analytic expression for the option in question. In this article we value exotic options by matching moments to an approximating SPD from a very general, robust, and convenient class of density functions, known in the statistical literature as the Johnson (1949) family. The end result is an analytic expression for a wide variety of European style exotic options. Our formula is tested on Asian and Basket options and is found to be extremely robust when compared with Monte Carlo simulations and other numerical techniques , S.E.Posner, M.A.Milevsky (1998)





















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