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Pricing Asian and Basket Options Via Taylor Expansion of the Underlying Volatility." mimeo (2000
"... comments and suggestions. ..."
Bessel processes, the integral of geometric Brownian motion, and Asian options
 Theor. Probab. Appl
, 2004
"... Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been stu ..."
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Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the HartmanWatson theory of [Y80]. Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the HartmanWatson approach itself: this approach is in general applicable without modifications only if it does not involve Bessel processes of negative indices. The main mathematical contribution of this paper is the developement of three principal ways to overcome these restrictions, in particular by merging stochastics and complex analysis in what seems a novel way, and the discussion of their consequences for the valuation of Asian options proper.
Bessel processes, the integral of geometric Brownian motion, and Asian options
"... This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of ..."
Abstract
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This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes us ing the HartmanWatson theory of [Y80]. Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the HartmanWatson approach itself: this approach is in general applicable without modifications only if it does not involve Bessel processes of negative indices. The main mathematical contribution of this paper is the developement of three principal ways to overcome these restrictions, in particular by merging stochastics and complex analysis in what seems a novel way, and the discussion of their consequences for the valuation of Asian options proper.
Applied Probability Trust (5 September 2001) ON THE EQUIVALENCE OF FLOATING AND FIXEDSTRIKE ASIAN OPTIONS
"... There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating and fixedstrike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very use ..."
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There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating and fixedstrike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation and in this case, the result allows the use of more established fixedstrike pricing methods to price floatingstrike Asian options.
ANALYTICAL RAMIFICATIONS OF DERIVATIVES VALUATION: ASIAN OPTIONS AND SPECIAL FUNCTIONS
, 2002
"... Averaging problems are ubiquitous in Finance with the valuation of the so–called Asian options on arithmetic averages as their most conspicuous form. There is an abundance of numerical work on them, and their stochastic structure has been extensively studied by Yor and his school. However, the analy ..."
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Averaging problems are ubiquitous in Finance with the valuation of the so–called Asian options on arithmetic averages as their most conspicuous form. There is an abundance of numerical work on them, and their stochastic structure has been extensively studied by Yor and his school. However, the analytical structure of these problems is largely unstudied. Our philosophy now is that such valuation problems should be considered as an extension of the theory of special functions: they lead to new problems about new classes of special functions which should be studied in terms of and using of the methods of special functions and their theory. This is exemplified by deriving integral representations for the Black–Scholes prices based on Yor’s Laplace transform ansatz to their valuation. They are obtained by analytic Laplace inversion using complex analytic methods. The analysis ultimately rests on the gamma function which in this sense is found to be at the base of Asian options. The results improve on those of Yor and have served us a as starting point for deriving first time benchmark prices for these options. 1. Introduction: This
Bessel and VolatilityStabilized Processes
"... The work in this thesis expands the study of volatilitystabilized processes introduced in [17]. Using their representation as timechanged Bessel processes and a multidimensional version of the skewproduct decomposition theorem, we derive the conclusion that the vector of market weights is a multid ..."
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The work in this thesis expands the study of volatilitystabilized processes introduced in [17]. Using their representation as timechanged Bessel processes and a multidimensional version of the skewproduct decomposition theorem, we derive the conclusion that the vector of market weights is a multidimensional Jacobi diffusion. The Dirichlet distribution is proved to be the invariant distribution of this diffusion. The fact that the marginals of this vector process are onedimensional Jacobi diffusions having the Beta distribution as an invariant distribution provides new proofs for limiting behavior results for the individual market weights already established in [17]. Using the spectral representation of the transition density of a onedimensional diffusion, we establish a series representation involving Jacobi polynomials for the transition density of the individual market weights, thus answering one of the open questions in [17]. Using techniques pioneered