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.

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

Scalas, E.
;
Toaldo, B.
2021-01-01

Abstract

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.
2021
105
0
3
33
Differential geometry; algebraic geometry
Scalas, E.; Toaldo, B.
File in questo prodotto:
File Dimensione Formato  
option_published.pdf

Accesso riservato

Tipo di file: PDF EDITORIALE
Dimensione 378.97 kB
Formato Adobe PDF
378.97 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1837633
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 3
social impact