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74
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 98 (9 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
The Concept of Comonotonicity in Actuarial Science and Finance: Applications
 Mathematics & Economics
, 2002
"... In an insurance c o text,o ne isoB interested in thedistributio functio o a sum o rando variables. Such a sum appears when coBBthe aggregate claimso f an insurance po rtfo o ver a certain reference perio d. Italso appears when coBdisco ted payments related to a single po licyo a poBat di#erent futur ..."
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Cited by 83 (38 self)
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In an insurance c o text,o ne isoB interested in thedistributio functio o a sum o rando variables. Such a sum appears when coBBthe aggregate claimso f an insurance po rtfo o ver a certain reference perio d. Italso appears when coBdisco ted payments related to a single po licyo a poBat di#erent future po ints in time. Theassumptio o mutual independence between the coB oB ts o the sum is very co venient fro a coB po int o view, butsoB no a realistico ne. In The Concept of Comonotonicity in Actuarial Science and Finance: Theory, we determined appro ximatio fo sumso f rando variables, when thedistributio o f the coB oB ts are kno wn, but thesto chastic dependence structure between them isunkno wn oto o cumberso to wo rk with. Practical applicatio o f thistheo will becoBin this paper. o. papers are to a large extent ano verviewo recent research resultsos tained by theautho butalso newtheoBand practical results are presented. 1
Robust Numerical Methods for PDE Models of Asian Options
 Journal of Computational Finance
, 1998
"... We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this p ..."
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Cited by 46 (14 self)
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We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this problem. In particular, the usual methods generally produce solutions containing spurious oscillations. We adapt flux limiting techniques originally developed in the field of computational fluid dynamics in order to rapidly obtain accurate solutions. We show that flux limiting methods are total variation diminishing (and hence free of spurious oscillations) for nonconservative PDEs such as those typically encountered in finance, for fully explicit, and fully and partially implicit schemes. We also modify the van Leer flux limiter so that the secondorder total variation diminishing property is preserved for nonuniform grid spacing. 1 Introduction Asian options are securities with payoffs...
Pricing and Hedging Spread Options
 SIAM Review
, 2003
"... Abstract. We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present ..."
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Cited by 24 (6 self)
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Abstract. We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools. We describe the mathematical framework used to model them, and we review the numerical algorithms actually used to price and hedge them. There is already extensive literature on the pricing of spread options in the equity and fixed income markets, and our contribution is mostly to put together material scattered across a wide spectrum of recent textbooks and journal articles. On the other hand, information about the various numerical procedures that can be used to price and hedge spread options on physical commodities is more difficult to find. For this reason, we make a systematic effort to choose examples from the energy markets in order to illustrate the numerical challenges associated with these instruments. This gives us a chance to discuss an interesting application of spread options to an asset valuation problem after it is recast in the framework of real options. This approach is currently the object of intense mathematical research. In this spirit, we review the two major avenues to modeling energy price dynamics. We explain how the pricing and hedging algorithms can be implemented in the framework of models for both the spot price dynamics and the forward curve dynamics.
Discrete Asian Barrier Options
, 1998
"... . A partial differential equation method based on using auxiliary variables is described for pricing discretely monitored Asian options with or without barrier features. The barrier provisions can be applied to either the underlying asset or to the average. They may also be of either instantaneous o ..."
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Cited by 14 (4 self)
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. A partial differential equation method based on using auxiliary variables is described for pricing discretely monitored Asian options with or without barrier features. The barrier provisions can be applied to either the underlying asset or to the average. They may also be of either instantaneous or delayed effect (i.e. Parisian style). Numerical examples demonstrate that this method can be used for pricing floating strike, fixed strike, American, or European options. In addition, examples are provided which indicate that an upstream biased quadratic interpolation is superior to linear interpolation for handling the jump conditions at observation dates. Moreover, it is shown that defining the auxiliary variable as the average rather than the running sum is more rapidly convergent for AmericanAsian options. Keywords: Asian options, Barrier options, Parisian options, PDE option pricing Running Title: Discrete Asian Barrier Options Acknowledgment: This work was supported by the Nation...
A new PDE approach for pricing arithmetic average Asian options
, 2000
"... . In this paper, arithmetic average Asian options are studied. It is observed that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple onedimensional partial dierential equation which could be applied to both continuous an ..."
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Cited by 13 (1 self)
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. In this paper, arithmetic average Asian options are studied. It is observed that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple onedimensional partial dierential equation which could be applied to both continuous and discrete average Asian option. The article also provides numerical implementation of the pricing equation. The implementation is fast and accurate even for low volatility and/or short maturity cases. Key words: Asian options, Options on a traded account, Brownian motion, xed strike, oating strike. 1 Introduction Asian options are securities with payo which depends on the average of the underlying stock price over certain time interval. Since no general analytical solution for the price of the Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options. A number of approximations that produce closed form expressions have appeared, se...
Competitive Monte Carlo methods for the Pricing of Asian Options
 Journal of Computational Finance
, 2000
"... We explain how a carefully chosen scheme can lead to competitive Monte Carlo algorithm for the computation of the price of Asian options. We give evidence of the eciency of these algorithms with a mathematical study of the rate of convergence and a numerical comparison with some existing methods. K ..."
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Cited by 13 (1 self)
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We explain how a carefully chosen scheme can lead to competitive Monte Carlo algorithm for the computation of the price of Asian options. We give evidence of the eciency of these algorithms with a mathematical study of the rate of convergence and a numerical comparison with some existing methods. Key Words: Asian option, Monte Carlo methods, Numerical methods, Diusion process. 1 Introduction Monte Carlo methods are known to be useful when the state dimension is large. This is widely true but we will give here an example of a small dimension problem coming from nance where a Monte Carlo (helped by a variance reduction technique) can be more ecient than other known methods. This example is based on the price of an Asian option (see subsection 2.1). This problem is known to be computationally hard and a lot of literature deals with this problem: using either analytic methods ([10], [9]), numerical methods based on the partial dierential equation associated ([4], [7], [12], [16]) or M...
Pricing Asian and Basket Options Via Taylor Expansion of the Underlying Volatility." mimeo (2000
"... comments and suggestions. ..."
A Refined Binomial Lattice for Pricing American Asian Options
, 1998
"... . We present simple and fast algorithms for computing very tight upper and lower bounds on the prices of American Asian options in the binomial model. We introduce a new refined version of the CoxRossRubinstein [4] binomial lattice of stock prices. Each node in the lattice is partitioned into "node ..."
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Cited by 11 (0 self)
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. We present simple and fast algorithms for computing very tight upper and lower bounds on the prices of American Asian options in the binomial model. We introduce a new refined version of the CoxRossRubinstein [4] binomial lattice of stock prices. Each node in the lattice is partitioned into "nodelets", each of which represents all paths arriving at the node with a specific geometric stock price average. The upper bound uses an interpolation idea similar to the HullWhite [5] method. From the backwardrecursive upperbound computation, we estimate a good exercise rule that is consistent with the refined lattice. This exercise rule is used to obtain a lower bound on the option price using a modification of a conditionalexpectation based idea from RogersShi [11] and ChalasaniJhaVarikooty [3]. Our algorithms run in time proportional to the number of nodelets in the refined lattice, which is smaller than n 4 =20 for n periods. Keywords: American Options, Asian Options, Pathdepend...
Asians and cash dividends: exploiting symmetries in pricing theory
, 2000
"... In this article we present new results for the pricing of arithmetic Asian options within a BlackScholes context. To derive these results we make extensive use of the local scale invariance that exists in the theory of contingent claim pricing. This allows us to derive, in a natural way, a simple P ..."
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Cited by 9 (4 self)
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In this article we present new results for the pricing of arithmetic Asian options within a BlackScholes context. To derive these results we make extensive use of the local scale invariance that exists in the theory of contingent claim pricing. This allows us to derive, in a natural way, a simple PDE for the price of arithmetic Asians options. In the case of European average strike options, a proper choice of numeraire reduces the dimension of this PDE to one, leading to a PDE similar to the one derived by Rogers and Shi. We solve this PDE, finding a Laplacetransform representation for the price of average strike options, both seasoned and unseasoned. This extends the results of Geman and Yor, who discussed the case of average price options. Next we use symmetry arguments to show that prices of average strike and average price options can be expressed in terms of each other. Finally we show, again using symmetries, that plain vanilla options on stocks paying known cash dividends are closely related to arithmetic Asians, so that all the new techniques can be directly applied to this case.