Geodesic
Study of the dynamics of a jump-like change in price in the generalized Black–Scholes model
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 1, Tome 190 (2021), pp. 88-92
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this paper, we examine the stability of the generalized Black–Scholes model. We assume that the dynamics of the option price depends on random spasmodic changes in the volatility. We prove that the study of the mean-square stability of the Black–Scholes stochastic system can be reduced to the study of the stability of a deterministic system of second-order moments for the underlying asset. Using the example of an option with two structural volatility states, we obtain conditions of mean-square stability.
Keywords: stochastic system, volatility, Black–Scholes model, stability. 
@article{INTO_2021_190_a7,
     author = {T. V. Zav'yalova and G. A. Timofeeva},
     title = {Study of the dynamics of a jump-like change in price in the generalized {Black{\textendash}Scholes} model},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {88--92},
     year = {2021},
     volume = {190},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_190_a7/}
}
TY  - JOUR
AU  - T. V. Zav'yalova
AU  - G. A. Timofeeva
TI  - Study of the dynamics of a jump-like change in price in the generalized Black–Scholes model
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2021
SP  - 88
EP  - 92
VL  - 190
UR  - http://geodesic.mathdoc.fr/item/INTO_2021_190_a7/
LA  - ru
ID  - INTO_2021_190_a7
ER  - 
%0 Journal Article
%A T. V. Zav'yalova
%A G. A. Timofeeva
%T Study of the dynamics of a jump-like change in price in the generalized Black–Scholes model
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2021
%P 88-92
%V 190
%U http://geodesic.mathdoc.fr/item/INTO_2021_190_a7/
%G ru
%F INTO_2021_190_a7
T. V. Zav'yalova; G. A. Timofeeva. Study of the dynamics of a jump-like change in price in the generalized Black–Scholes model. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 1, Tome 190 (2021), pp. 88-92. http://geodesic.mathdoc.fr/item/INTO_2021_190_a7/

[1] Zavyalova T. V., Kats I. Ya., Timofeeva G. A., “Ob ustoichivosti dvizheniya stokhasticheskikh sistem so sluchainym usloviem skachka fazovoi traektorii”, Avtomat. telemekh., 7 (2002), 33–43 | MR

[2] Kats I. Ya., Metod funktsii Lyapunova v zadachakh ustoichivosti i stabilizatsii sistem sluchainoi struktury, Izd-vo UrGUPS, Ekaterinburg, 1998

[3] Shiryaev A. N., Osnovy stokhasticheskoi finansovoi matematiki. I. Fakty. Modeli, Fazis, M., 1998

[4] Bates D., “Jump and stochastic volatility: Exchange rate processes implicit in Deutsche mark options”, Rev. Fin. Stud., 9 (1996), 69–-107 | DOI

[5] Heston S. L., “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Fin. Stud., 6 (1993), 327–344 | DOI | MR

[6] Jorion P., “On jump processes in the foreign exchange and stock markets”, Rev. Fin. Stud., 1 (1988), 427–445 | DOI

[7] Merton R., “Option pricing when underlying stock returns are discontinuous”, J. Fin. Econ., 3 (1976), 125–-144 | DOI | Zbl

[8] Scott L. O., “The information content of prices in derivative security markets”, IMF Staff Papers., 39 (1992), 596–625 | DOI