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Barrier Options
  • Advanced Monte Carlo methods for barrier and related exotic options In this work, we present advanced Monte Carlo techniques applied to the pricing of barrier options and other related exotic contracts. It covers in particular the Brown- ian bridge approaches, the barrier shifting techniques (BAST) and their extensions as well. We leverage the link between discrete and continuous monitoring to de- sign efficient schemes, which can be applied to the Black-Scholes model but also to stochastic volatility or Merton?s jump models. This is supported by theoretical results and numerical experiments. , E. Gobet (2008)
  • Valuing double barrier options with time-dependent parameters by Fourier series expansion Based upon the Fourier series expansion, we propose a simple and easy-to-use approach for computing accurate estimates of Black-Scholes double barrier option prices with time-dependent parameters. This new approach is also able to provide tight upper and lower bounds of the exact barrier option prices. Furthermore, this approach can be straightforwardly extended to the valuation of standard European options with specified moving boundaries as well , C.F. Lo, C. H. Hui (2007)
  • Pricing double barrier Parisian Options using Laplace transforms In this work, we study a double barrier version of the standard Parisian options. We give closed formulae for the Laplace transforms of their prices with respect to the maturity time. We explain how to invert them numerically and prove a result on the accuracy of the numerical inversion. , C. Labart, J. Lelong (2006)
  • Barrier option pricing for assets with Markov-modulated dividends We present a simple methodology to price single and double barrier options when the dividend process of the underlying is a Markov-modulated log-Brownian motion, and the stock is priced in equilibrium by a CRRA representative agent. In particular, we show how to derive the Laplace transform (in time) of the barrier price, by solving a system of ODEs. The method proposed is extremely simple to implement but also extremely e?ective. Pricing of double barrier option in the classical Black and Scholes framework arises as a special case of the model presented in the paper , G. Di Graziano, L.C.G. Rogers (2005)
  • Close Form Pricing of Plain and Partial Outside Double Barrier Outside Double Barrier Options are two-asset options where the payoff is defined on one asset and the barrier is defined on another asset. This paper gives the formulas for Outside Double Barrier Options where the barrier is either plain or partially monitored at the front, rear and middle. Since the corresponding Outside Single Barrier Options prices can be written down by taking the corresponding upper (lower) barrier to infinity (zero), the formulas in this paper can be also used as a reference for Outside Single Barrier Options , P. Banerjee (2003)
  • Hedging Complex Barrier Options We show how several complex barrier options can be hedged using a portfolio of standard European options. These hedging strategies only involve trading at a few times during the option?s life. Since rolling, ratchet, and lookback options can be decomposed into a portfolio of barrier options, our hedging results also apply , P. Carr, A. Chou (2002)
  • Analytic Method for Pricing Double Barrier Options in the Presence of Stochastic Volatility While there exist closed-form solutions for vanilla options in the presence of stochastic volatility for nearly a decade [Heston, 1993], practitioners still depend on numerical methods ? in particular the Finite Difference and Monte Carlo methods ? in the case of double barrier options. It was only recently that Lipton [2001] proposed (semi-)analytical solutions for this special class of path-dependent options. Although he presents two different approaches to derive these solutions, he restricts himself in both cases to a less general model, namely one where the correlation and the interest rate differential are assumed to be zero. Naturally the question arises, if these methods are still applicable for the general stochastic volatility model without these restrictions. In this paper we show that such a generalization fails for both methods. We will explain why this is the case and discuss the consequences of our results , O. Faulhaber (2002)
  • Static Replication of Barrier Options: Some General Results This paper presents a number of new theoretical results for replication of barrier options through a static portfolio of European put and call options. Our results are valid for options with completely general knock-out/knock-in sets, and allow for time- and state-dependent volatility as well as discontinuous asset dynamics. We illustrate the theory with numerical examples and discuss the practical implementation. , L.Andersen, J.Andreasen, D.A.Eliezer (2000)
  • Closed Form Valuation of American Barrier Options Closed form formulae for European barrier options are well known from the literature. This is not the case for American barrier options, for which no closed form formulae have been published. One has therefore had to resort to numerical methods. Using lattice models like a binomial or a trinomial tree for valuation of barrier options is known to converge extremely slowly, compared to plain vanilla options. Methods for improving the algorithms have been described by several authors. However, these are still numerical methods that are quite computer intensive. In this paper we show how American barrier options can be valued analytically in a very simple way. This speeds up the valuation dramatically as well as give new insight into barrier option valuation , E.G.Haug (1999)
  • Robust Hedging of Barrier Options , H. Brown, D. Hobson, L.C.G. Rogers (1998)
  • A Continuity Correction For Discrete Barrier Options Explanation of shift barrier correction for continuous barrier , M.Broadie, P.Glasserman, S.Kou (1997)
  • One-Touch Double Barrier Binary Option Values The valuation and applications of one-touch double barrier binary options that include features of knock-out, knock-in, European and American style are described. Using a conventional Black-Scholes option-pricing environment, analytical solutions of the options are derived. The relationships among different types of one-touch double barrier binary options are discussed. An investor having a particular view on values of foreign exchanges, equities or commodities can use the options as directional trades or structured products in financial market. , C. H. Hui (1996)
  • Two Extensions to Barrier Option Valuation We first present a brief but essentially complete survey of the literature on barrier option pricing. We then present two extensions of European up-and-out call option valuation. The first allows for an initial protection period during which the option cannot be knocked out. The second considers an option which is only knocked out if a second asset touches an upper barrier. Closed form solutions, detailed derivations, and the economic rationale for both types of options are provided. , P. Carr (1995)
  • Enhanced Numerical Methods for Options with Barriers In this paper we analyze the biases implicit in valuing options with barriers on a lattice. We then suggest a method for enhancing the numerical solution of boundary value problems on a lattice that helps to correct these biases. It seems to work well in practice. , E.Derman, I.Kani, D.Ergener, I.Bardhan (1995)

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