Computational Methods in Finance by Ali Hirsa

By Ali Hirsa

As today’s monetary items became extra complicated, quantitative analysts, monetary engineers, and others within the monetary now require strong ideas for numerical research. protecting complicated quantitative options, Computational equipment in Finance explains tips on how to clear up complicated useful equations via numerical equipment.

The first a part of the ebook describes pricing tools for various derivatives less than various types. The e-book stories universal techniques for modeling resources in several markets. It then examines many computational methods for pricing derivatives. those comprise rework ideas, similar to the quick Fourier rework, the fractional quick Fourier rework, the Fourier-cosine procedure, and saddlepoint procedure; the finite distinction process for fixing PDEs within the diffusion framework and PIDEs within the natural leap framework; and Monte Carlo simulation.

The subsequent half makes a speciality of crucial steps in real-world by-product pricing. the writer discusses the best way to calibrate version parameters in order that version costs fit with marketplace costs. He additionally covers quite a few filtering strategies and their implementations and provides examples of filtering and parameter estimation.

Developed from the author’s classes at Columbia college and the Courant Institute of recent York collage, this self-contained textual content is designed for graduate scholars in monetary engineering and mathematical finance in addition to practitioners within the monetary undefined. it's going to aid readers effectively rate an unlimited array of derivatives.

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Its characteristic function is given by 1 E(eiuZV GSA (t) ) = φ(−iΨV G (u), t, , κ, η, λ) ν where ΨV G is the log characteristic function of the variance gamma process at unit time, namely, 1 ΨV G (u) = − log 1 − iuθν + σ 2 νu2 /2 ν We define the asset price process at time t as follows: S(t) = S(0) e(r−q)t+Z(t) E[eZ(t) ] Stochastic Processes and Risk-Neutral Pricing 27 We note that 1 E[eZ(t) ] = φ(−iΨV G (−i), t, , κ, η, λ) ν Therefore the characteristic function of the log of the asset price at time t is given by E[eiu log St ] = exp(iu(log S0 + (r − q)t)) × φ(−iΨV G (u), t, ν1 , κ, η, λ) φ(−iΨV G (−i), t, ν1 , κ, η, λ)iu Thus we have a closed form for the VGSA characteristic function for the log asset price.

Under the variance gamma model the unit period continuously compounded return is normally distributed conditional on the realization of a random process — a random time with a gamma density. The resulting process and associated pricing model provide us with a robust three parameter generalization of the standard Brownian motion model. The log of the asset price process under the variance gamma model is given by ln St = ln S0 + (r − q + ω)t + X(t; σ, ν, θ) or equivalently St = S0 e(r−q+ω)t+X(t;σ,ν,θ) ω is determined so that E(St ) = S0 e(r−q)t Stochastic Processes and Risk-Neutral Pricing 23 The density of the log asset price under the variance gamma model at time t can be expressed conditional on the realization of gamma time change g as a normal density function.

Stochastic Processes and Risk-Neutral Pricing 33 If the process is Markov but either there is no characteristic function or the derivative price has path dependency, we can use numerical solutions to PDEs/PIDEs for pricing. In case the process is non-Markov or high dimensional, or the derivative price and the payoff has very complex path dependency, then we must use Monte Carlo simulation methods. Problems 1. 48. 2. Derive the characteristic function of a normal inverse Gaussian (NIG) process using a similar approach used to derive the characteristic function of the variance gamma process.

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