Abstract Cauchy problems for the generalized fractional calculus

Ascione, Giacomo

doi:10.1016/j.na.2021.112339

Cited by 25 publications

(23 citation statements)

References 25 publications

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“…With a similar argument it is possible to show (V.13). See [3,22] for a thorough discussion on these properties.…”

Section: Discussionmentioning

confidence: 99%

Limit theorems for prices of options written on semi-Markov processes

Scalas¹,

Toaldo²

2021

Preprint

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We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator's Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

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“…With a similar argument it is possible to show (V.13). See [3,22] for a thorough discussion on these properties.…”

Section: Discussionmentioning

confidence: 99%

Limit theorems for prices of options written on semi-Markov processes

Scalas¹,

Toaldo²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…It can be shown, by Laplace transform arguments (see, for instance [6,38]) or by Green functions arguments (see [39]), that the Cauchy problem…”

Section: Inverse Subordinators and Non-local Convolution Derivativesmentioning

confidence: 99%

“…For the definition of strong solution, we still refer to Definition 4. To do this, we first need to show the following preliminary result, whose proof can be omitted since it follows by direct calculations and by using Pearson equation (6).…”

Section: The Time-non-local Forward Kolmogorov Equationmentioning

confidence: 99%

Time-Non-Local Pearson Diffusions

2021

Self Cite

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In this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker–Planck operator of a Pearson diffusion, covering a large class of important stochastic processes. Such kind of time-non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusions to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffusions. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence.

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“…With a similar argument it is possible to show (5.13). See [3,22] for a thorough discussion on these properties.…”

Section: Fractional-type Black-scholes Equationmentioning

confidence: 99%

“…(5.13) (alternatively, in order to justify(5.38) one can apply the general result[3, Lemma 3.1]). By substituting (5.38) into (4.16) we obtainq (x, y, z) = (x − K) + ν(y + z) ν(y) + z 0 (x − K) + + I z−τ x 2 ∂ 2 x q (x, 0, z − τ ) ν(y + dτ ) ν(y) = (x − K) + + ∂ 2 x q (x, y, z − τ − s)ν (s)ds ν(y + dτ ) ν∂ 2 x q (x, 0, z − s − τ ) ν(y + dτ ) ν(y) ν (s) ds = (x − K) + + I z z ∂ 2 x q (x, 0, z − τ ) ν(y + dτ ) ν(y) .…”

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Limit theorems for prices of options written on semi-Markov processes

Scalas

Toaldo

2021

Theor. Probability and Math. Statist.

View full text Add to dashboard Cite

We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

show abstract

Abstract Cauchy problems for the generalized fractional calculus

Cited by 25 publications

References 25 publications

Limit theorems for prices of options written on semi-Markov processes

Limit theorems for prices of options written on semi-Markov processes

Time-Non-Local Pearson Diffusions

Limit theorems for prices of options written on semi-Markov processes

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