Stochastic string models with continuous semimartingales

Bueno-Guerrero, Alberto; Moreno, Manuel; Navas, Javier F.

doi:10.1016/j.physa.2015.03.070

Cited by 20 publications

(8 citation statements)

References 38 publications

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“…In this section, we recall definitions and briefly summarize relevant facts about the GBS model, stochastic strings, and Lévy processes that we will use in this paper. For stochastic string processes, the main references are [18,19]. The GBS model was introduced in [17].…”

Section: Preliminariesmentioning

confidence: 99%

“…Lévy processes are semimartingales and it is proven in [19] that stochastic integrals with respect to stochastic string processes are continuous martingales. Thus, if X t is the sum of a stochastic string integral and a Lévy process with Lévy triple (γ, 0, ν) and finite first moment, then it is a semimartingale.…”

Section: Combining Stochastic Strings and Jump Processesmentioning

confidence: 99%

“…In a recent paper, Ref. [17] introduces a novel generalization of the Black-Scholes model incorporating stochastic interest rates and stochastic string shocks ( [18,19]). Termed the generalized Black and Scholes (GBS) model, this new framework yields a Black-Scholeslike formula for European call option prices in which volatility is a function of the expiration times of underlying options.…”

Section: Introductionmentioning

confidence: 99%

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Option Pricing under a Generalized Black–Scholes Model with Stochastic Interest Rates, Stochastic Strings, and Lévy Jumps

Bueno-Guerrero,

Clark

2023

Mathematics

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We introduce a novel option pricing model that features stochastic interest rates along with an underlying price process driven by stochastic string shocks combined with pure jump Lévy processes. Substituting the Brownian motion in the Black–Scholes model with a stochastic string leads to a class of option pricing models with expiration-dependent volatility. Further extending this Generalized Black–Scholes (GBS) model by adding Lévy jumps to the returns generating processes results in a new framework generalizing all exponential Lévy models. We derive four distinct versions of the model, with each case featuring a different jump process: the finite activity lognormal and double–exponential jump diffusions, as well as the infinite activity CGMY process and generalized hyperbolic Lévy motion. In each case, we obtain closed or semi-closed form expressions for European call option prices which generalize the results obtained for the original models. Empirically, we evaluate the performance of our model against the skews of S&P 500 call options, considering three distinct volatility regimes. Our findings indicate that: (a) model performance is enhanced with the inclusion of jumps; (b) the GBS plus jumps model outperform the alternative models with the same jumps; (c) the GBS-CGMY jump model offers the best fit across volatility regimes.

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Section: Preliminariesmentioning

confidence: 99%

Section: Combining Stochastic Strings and Jump Processesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Option Pricing under a Generalized Black–Scholes Model with Stochastic Interest Rates, Stochastic Strings, and Lévy Jumps

Bueno-Guerrero,

Clark

2023

Mathematics

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“…that is the call option value in the Gaussian stochastic string framework (Bueno-Guerrero et al, 2015a). Consider now discounted contingent claims of the form:…”

Section: Hedging Portfolios Without Bank Accountmentioning

confidence: 99%

“…Nevertheless, in our framework the source of uncertainty is not Brownian motion but the socalled stochastic string shock [Santa-Clara and Sornette (2001), Bueno-Guerrero et al (2015a)]. Fortunately, in Bueno-Guerrero et al (2017), a Malliavin calculus for stochastic string shocks has been developed.…”

Section: Introductionmentioning

confidence: 99%

Interest rate option hedging portfolios without bank account

Bueno-Guerrero¹

2019

SEF

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Purpose This paper aims to study the conditions for the hedging portfolio of any contingent claim on bonds to have no bank account part. Design/methodology/approach Hedging and Malliavin calculus techniques recently developed under a stochastic string framework are applied. Findings A necessary and sufficient condition for the hedging portfolio to have no bank account part is found. This condition is applied to a barrier option, and an example of a contingent claim whose hedging portfolio has a bank account part different from zero is provided. Originality/value To the best of the authors’ knowledge, this is the first time that this issue has been addressed in the literature.

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