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24
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.
On the integral of geometric Brownian motion
 Adv. Appl. Prob
, 2003
"... Abstract. This paper studies the law of any power of the integral of geometric Brownian motion over any finite time interval. As its main results, two integral representations for this law are derived. This is by enhancing the Laplace transform ansatz of [Y] with complex analytic methods, which is t ..."
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Abstract. This paper studies the law of any power of the integral of geometric Brownian motion over any finite time interval. As its main results, two integral representations for this law are derived. This is by enhancing the Laplace transform ansatz of [Y] with complex analytic methods, which is the main methodological contribution of the paper. The one of our integrals has a similar structure to that obtained in [Y], while the other is in terms of Hermite functions as those of [Du01]. Performing or not performing a certain Girsanov transformation is identified as the source of these two forms of the laws. For exponents equal to 1 our results specialize to those obtained in [Y], but for exponents equal to minus 1 they give representations for the laws which are markedly different from those obtained in [Du01].
On collisions of Brownian particles
 Annals of Applied Probability
, 2010
"... Abstract. We examine the behavior of n Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence, and for the prese ..."
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Cited by 5 (2 self)
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Abstract. We examine the behavior of n Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence, and for the presence, of triple collisions among the particles. As an application to the Atlas model for equity markets, we study a special construction of such systems of diffusing particles using Brownian motions with reflection on polyhedral domains. 1.
Asymptotic behavior of the stock pricedistributiondensityandimpliedvolatilityinstochasticvolatilitymodels
 Appl. Math. Optim
"... Abstract We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Ste ..."
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Cited by 3 (2 self)
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Abstract We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the SteinStein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the implied volatility in the SteinStein and the Heston model are obtained. Keywords SteinStein model · Heston model · Mixing distribution density · Stock price · Bessel processes · OrnsteinUhlenbeck processes · CIR processes · Asymptotic formulas · Implied volatility 1
Investment hysteresis under stochastic interest rates
, 2005
"... Most decision making research in real options focuses on revenue uncertainty assuming discount rates remain constant. However for many decisions, revenue or cost streams are relatively static and investment is driven by interest rate uncertainty, for example the decision to invest in durable machine ..."
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Most decision making research in real options focuses on revenue uncertainty assuming discount rates remain constant. However for many decisions, revenue or cost streams are relatively static and investment is driven by interest rate uncertainty, for example the decision to invest in durable machinery and equipment. Using interest rate models from Cox et al. (1985b), we generalize the work of Ingersoll and Ross (1992) in two ways. Firstly we include real options on perpetuities (in addition to ”zero coupon ” cash flows). Secondly we incorporate abandonment or disinvestment as well as investment options and thus model interest rate hysteresis (parallel to revenue uncertainty, Dixit (1989a)). Under stochastic interest rates, economic hysteresis is found to be significant, even for small sunk costs.
THE CALCULATION OF EXPECTATIONS FOR CLASSES OF DIFFUSION PROCESSES BY LIE SYMMETRY METHODS
, 902
"... This paper uses Lie symmetry methods to calculate certain expectations for a large class of Itô diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals t of the form Ex(e −λXt− 0 g(Xs) ds) can be reduced to evaluating a single integral of known func ..."
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This paper uses Lie symmetry methods to calculate certain expectations for a large class of Itô diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals t of the form Ex(e −λXt− 0 g(Xs) ds) can be reduced to evaluating a single integral of known functions. Given a drift f we determine the functions g for which the corresponding functional can be calculated by symmetry. Conversely, given g, we can determine precisely those drifts f for which the transition density and the functional may be computed by symmetry. Many examples are presented to illustrate the method.
Managing Volatility Risk Innovation of Financial Derivatives, Stochastic Models and Their Analytical Implementation
, 2010
"... This dissertation investigates two timely topics in mathematical finance. In particular, we study the valuation, hedging and implementation of actively traded volatility derivatives including the recently introduced timer option and the CBOE (the Chicago Board Options Exchange) option on VIX (the Ch ..."
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This dissertation investigates two timely topics in mathematical finance. In particular, we study the valuation, hedging and implementation of actively traded volatility derivatives including the recently introduced timer option and the CBOE (the Chicago Board Options Exchange) option on VIX (the Chicago Board Options Exchange volatility index). In the first part of this dissertation, we investigate the pricing, hedging and implementation of timer options under Heston’s (1993) stochastic volatility model. The valuation problem is formulated as a firstpassagetime problem through a noarbitrage argument. By employing stochastic analysis and various analytical tools, such as partial differential equation, Laplace and Fourier transforms, we derive a BlackScholesMerton type formula for pricing timer options. This work motivates some theoretical study of Bessel processes and Feller diffusions as well as their numerical implementation. In the second part, we analyze the valuation of options on VIX under Gatheral’s double meanreverting stochastic volatility model, which is able to consistently price options on S&P 500 (the Standard and Poor’s 500 index), VIX and realized variance (also well known as historical variance calculated by thevariance of the asset’s daily return). We employ scaling, pathwise Taylor expansion
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.
The Skorokhod embedding problem and its offspring
, 2004
"... This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are studied in detail and their proofs are presented. A certain unification of proofs, based on onedi ..."
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This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are studied in detail and their proofs are presented. A certain unification of proofs, based on onedimensional potential theory, is made. Some new facts which appeared in a natural way when different solutions were crossexamined, are reported. Azéma and Yor’s and Root’s solutions are studied extensively. A possible use of the latter is suggested together with a conjecture.