Infinite-Dimensional Black-Scholes Equation with Hereditary Structure

Chang, Mou-Hsiung; Youree, Roger K.

doi:10.1007/s00245-007-9003-z

Cited by 16 publications

(40 citation statements)

References 13 publications

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“…More generally, we will consider a class of semilinear versions of the parabolic Kolmogorov equation associated to the process X. This class includes as a very special case some infinitedimensional variants of the Black-Scholes equation for the fair price of an option, of great interest in mathematical finance and already considered in [5].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Equations with Delay: Optimal Control via BSDEs and Regular Solutions of Hamilton–Jacobi–Bellman Equations

Fuhrman¹,

Masiero²,

Tessitore³

2010

SIAM J. Control Optim.

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We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process X with values in a space of continuous functions C, with generator L. We then consider a backward stochastic differential equation depending on X, with unknown processes (Y, Z), and we study properties of the resulting system, in particular we identify the process Z as a deterministic functional of X. We next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space C driven by L, and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms the process X.

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Section: Introductionmentioning

confidence: 99%

Stochastic Equations with Delay: Optimal Control via BSDEs and Regular Solutions of Hamilton–Jacobi–Bellman Equations

Fuhrman¹,

Masiero²,

Tessitore³

2010

SIAM J. Control Optim.

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“…This procedures allows the MT model parameters to depend on an option's strike and maturity, leading to an increase in the number of effective model parameters, and escaping the assumptions/constraints of the underlying asset process (7). Therefore, the practitioner's MT method is the method of choice for those who seek to minimize errors leaving other considerations aside.…”

Section: Discussionmentioning

confidence: 99%

“…The MT model is one of several published models that account for memory in the underlying asset [6][7][8][9][10][11][12][13]. Within this set of models, there are two reasons why the MT model is the simplest: (i) the model requires the estimation of only three parameters; and (ii) the MT model is fundamentally a discrete-time model, allowing us to bypass the technical difficulties of estimation and filtering for stochastic delay/functional differential equations.…”

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confidence: 99%

Large-Scale Empirical Tests of the Markov Tree Model

Bhat

Kumar²

2015

IJFS

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The Markov Tree model is a discrete-time option pricing model that accounts for short-term memory of the underlying asset. In this work, we compare the empirical performance of the Markov Tree model against that of the Black-Scholes model and Heston's stochastic volatility model. Leveraging a total of five years of individual equity and index option data, and using three new methods for fitting the Markov Tree model, we find that the Markov Tree model makes smaller out-of-sample hedging errors than competing models. This comparison includes versions of Markov Tree and Black-Scholes models in which volatilities are strike-and maturity-dependent. Visualizing the errors over time, we find that the Markov Tree model yields more accurate and less risky single instrument hedges than Heston's stochastic volatility model. A statistical resampling method indicates that the Markov Tree model's superior hedging performance is due to its robustness with respect to noise in option data.

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“…This model has been considered in [7][8], and similar market models with hereditary structures or delayed responses have been considered by other researchers, e.g., Arriojas, et al [9] , Kazmerchuk, et al [10−12] , in which various versions of stochastic functional differential equations were introduced to describe the dynamics of riskless assets and stock prices.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

An approximation scheme for Black-Scholes equations with delays

Chang¹,

Pang

Pemy

2010

J Syst Sci Complex

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This paper addresses a finite difference approximation for an infinite dimensional BlackScholes equation obtained by Chang and Youree (2007). The equation arises from a consideration of an European option pricing problem in a market in which stock prices and the riskless asset prices have hereditary structures. Under a general condition on the payoff function of the option, it is shown that the pricing function is the unique viscosity solution of the infinite dimensional Black-Scholes equation. In addition, a finite difference approximation of the viscosity solution is provided and the convergence results are proved.

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Infinite-Dimensional Black-Scholes Equation with Hereditary Structure

Cited by 16 publications

References 13 publications

Stochastic Equations with Delay: Optimal Control via BSDEs and Regular Solutions of Hamilton–Jacobi–Bellman Equations

Stochastic Equations with Delay: Optimal Control via BSDEs and Regular Solutions of Hamilton–Jacobi–Bellman Equations

Large-Scale Empirical Tests of the Markov Tree Model

An approximation scheme for Black-Scholes equations with delays

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