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Table Of Integrals Series And Products

Encyclopedia of Quantitative Finance

80 IE E E S P E C T R U M • Ju n e 20 02 T echnology is a language-generating machine. With letters, phonemes, prefixes, suffixes, and words as its raw materials, technology constantly manufactures shiny new acronyms, words, and phrases to describe its onrush of new ideas, processes, and products. While technology has always stamped out new words at an impressive clip, most of these terms remain warehoused within the narrow tech communities that defined them. Some of them, however, are shipped out on the linguistic equivalents of planes, trains, and automobiles and get distributed far and wide. A few even morph into general-purpose words and phrases. For example, nuclear technology provided us with such terms as ground zero, fallout, and meltdown; radio spun off flip side, fine-tune, and stereo; aviation contributed push the envelope, automatic pilot, bail out, and gremlin; the car industry donated spark plug, bypass, blow a gasket, and rev up; and even earlier, the railroads gave us derail, sidetrack, streamline, and pick up steam. But though technology has always been a kind of new-word assembly line, what’s different these days is that technology is cranking out fresh terms at a rate that is downright exponential. That’s not because people are making up new words in their spare time, but because we now have more technology than ever. We don’t just have telephones, we have mobile phones, pagers, satellite phones, and wireless devices. We don’t just have computers, we have desktops, servers, notebooks, palmtops, PDAs, and Internet appliances. New gadgets, new technologies, new services, and new ideas stride purposefully down Technology Road, each one pulling a bright red wagon full of newly minted words and phrases. These now have a super-efficient method of propagation: the electronic byways of e-mail, chat rooms, instant messaging, and the Web. In the pre-Internet world, new words would tend to stay within the cultural tributary that coined them; only a few would get swept into the mainstream. Today there is a subculture—the Internet and its adjunct technologies—of hundreds of millions of people, that by definition is part of the mainstream. This means it doesn’t take much for new words and phrases to catch on. Keeping up with this deluge of newfangled technical terms will be the future focus of the Technically Speaking column. I’ll examine new words and phrases that have jumped down from their technological niches and are poised to set up shop in the broader piazza of general language use. This column introduces Paul McFedries as the new author of Technically Speaking, which appears every other month. Readers are invited to correspond with him at techspkg@ieee.org. P E T E R H O R V A T H TECHNICALLY SPEAKING

Filling the gaps

Potential theory and its applications to basic problems of mathematical physics

Introduction: A Primer on Risk Sensitive Control Diffusion Models: The Complete Market Case Asset Management in Incomplete Markets Managing Against a Benchmark Introducing Constraints Asset and Liability Managements Jump-Diffusion Models: Jumps in Asset Prices Affine Processes General Jump-Diffusion Setting Extensions to Benchmarks and ALM Implementation: Securities Modeling Factor Selection Parameter Estimation and Partial Observations Numerical Methods Empirical Analysis.

Risk-Sensitive Investment Management

The authors continue a series of articles where prices of the barrier options written on the underlying, whose dynamics follow a one-factor stochastic model with time-dependent coefficients and the barrier, are obtained in semi-closed form; see Carr and Itkin (2020) and Itkin and Muravey (2020). This article extends this methodology to the Cox–Ingersoll–Ross model for zero-coupon bonds and to the constant elasticity of variance model for stocks, which are used as the corresponding underlying for the barrier options. The authors describe two approaches. One is a generalization of the method of heat potentials (for the heat equation) to the Bessel process, so it is named “the method of Bessel potentials.” The authors also propose a general scheme for constructing the potential method for any linear differential operator with time-independent coefficients. The second approach is the method of generalized integral transform, which is also extended to the Bessel process. In all cases, a semi-closed solution means that first, we need to numerically solve a linear Volterra equation of the second kind, and then the option price is represented as a one-dimensional integral. The authors demonstrate that their method is computationally more efficient than both the backward and forward finite difference methods, while providing better accuracy and stability. Also, it is shown that these methods do not duplicate, but instead they complement each other, as one provides very accurate results at shorter maturities and the other provides such results at longer maturities. TOPICS:Options, statistical methods Key Findings • This article extends this methodology to the CIR model for zero-coupon bonds and to the CEV model for stocks, which are used as the corresponding underlying for the barrier options. • One approach is a generalization of the method of heat potentials (for the heat equation) to the Bessel process, so we call it “the method of Bessel potentials.” The second approach is the method of generalized integral transform, which is also extended to the Bessel process. • These methods do not duplicate, but instead they complement each other, as one provides very accurate results at shorter maturities and the other provides such results at longer maturities.

Semi-closed form prices of barrier options in the time-dependent CEV and CIR models

This paper studies a mean-variance investment-reinsurance problem under a new stochastic volatility model, namely the 4/2 stochastic volatility model. Solving this problem requires a deep understanding of a class of parabolic partial differential equations (PPDEs). By the parametrix method and the integral transform method, we derive explicit solutions to the PPDEs in several special cases. Through the Lie symmetry analysis, we obtain a four-parameter family of the 4/2 stochastic volatility models such that the corresponding PPDEs have closed-form solutions. The efficient strategy and the efficient frontier of the mean-variance problem are represented by using the closed-form solutions to PPDEs. Numerical examples for the obtained efficient frontier are provided by Monto Carlo method.

Optimal investment and reinsurance strategies under 4/2 stochastic volatility model

In this paper we consider a variation of the Merton's problem with added stochastic volatility and finite time horizon. It is known that the corresponding optimal control problem may be reduced to a linear parabolic boundary problem under some assumptions on the underlying process and the utility function. The resulting parabolic PDE is often quite difficult to solve, even when it is linear. The present paper contributes to the pool of explicit solutions for stochastic optimal control problems. Our main result is the exact solution for optimal investment in Heston model.

/pdf/an-explicit-solution-for-optimal-investment-in-heston-model-3sr1fqymcu.pdf

An explicit solution for optimal investment in Heston model

We continue a series of papers devoted to construction of semi-analytic solutions for barrier options. These options are written on underlying following some simple one-factor diffusion model, but all the parameters of the model as well as the barriers are time-dependent. We managed to show that these solutions are systematically more efficient for pricing and calibration than, e.g., the corresponding finite-difference solvers. In this paper we extend this technique to pricing double barrier options and present two approaches to solving it: the General Integral transform method and the Heat Potential method. Our results confirm that for double barrier options these semi-analytic techniques are also more efficient than the traditional numerical methods used to solve this type of problems.

Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

We continue a series of papers where prices of the barrier options written on the underlying, which dynamics follows some one factor stochastic model with time-dependent coefficients and the barrier, are obtained in semi-closed form, see (Carr and Itkin, 2020, Itkin and Muravey, 2020). This paper extends this methodology to the CIR model for zero-coupon bonds, and to the CEV model for stocks which are used as the corresponding underlying for the barrier options. We describe two approaches. One is generalization of the method of heat potentials for the heat equation to the Bessel process, so we call it the method of Bessel potentials. We also propose a general scheme how to construct the potential method for any linear differential operator with time-independent coefficients. The second one is the method of generalized integral transform, which is also extended to the Bessel process. In all cases, a semi-closed solution means that first, we need to solve numerically a linear Volterra equation of the second kind, and then the option price is represented as a one-dimensional integral. We demonstrate that computationally our method is more efficient than both the backward and forward finite difference methods while providing better accuracy and stability. Also, it is shown that both method don't duplicate but rather compliment each other, as one provides very accurate results at small maturities, and the other one - at high maturities.

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Semi-closed form prices of barrier options in the time-dependent CEV and CIR models

Optimal investment and reinsurance strategies under 4/2 stochastic volatility model

An explicit solution for optimal investment in Heston model

Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

Semi-closed form prices of barrier options in the time-dependent CEV and CIR models