Option Pricing in a Fractional Brownian Motion Environment

Necula, Ciprian

doi:10.2139/ssrn.1286833

Cited by 81 publications

(64 citation statements)

References 6 publications

(2 reference statements)

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“…Lemma 13 (see [29], Theorem 4.3). The price of a derivative on the stock price with a bounded payoff ( ) is given by ( , ), where ( , ) is the solution of the PDE:…”

Section: Resultsmentioning

confidence: 99%

Optimal Exercise Boundary of American Fractional Lookback Option in a Mixed Jump-Diffusion Fractional Brownian Motion Environment

Yang

2017

Mathematical Problems in Engineering

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A new framework for pricing the American fractional lookback option is developed in the case where the stock price follows a mixed jump-diffusion fraction Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.

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“…Lemma 13 (see [29], Theorem 4.3). The price of a derivative on the stock price with a bounded payoff ( ) is given by ( , ), where ( , ) is the solution of the PDE:…”

Section: Resultsmentioning

confidence: 99%

Optimal Exercise Boundary of American Fractional Lookback Option in a Mixed Jump-Diffusion Fractional Brownian Motion Environment

Yang

2017

Mathematical Problems in Engineering

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“…Lemma 3.1.1 (see Necula (2002), Theorem 4.3). The price of a derivative on the stock price with a bounded payoff f ðS t Þ is given by Dðt, S t Þ , where Dðt, SÞ is the solution of the PDE:…”

Section: Fundamental Solution Derivation Of the General Equationsmentioning

confidence: 99%

“…One way to a more realistic modeling is to change the geometric Brownian motion to a geometric fractional Brownian motion: the dependence of the log-return increments can now be modeled with the Hurst parameter of the fractional Brownian motion. It can be said that the properties of financial return series are non-normal, non-independent and nonlinear, self-similar, with heavytails, in both auto-correlations and cross-correlations, and volatility clustering (see Necula (2002), Hu and Øksendal (2003) and David (2004)). Since fractional Brownian motion has two substantial features such as self-similarity and longrange dependence, thus using it is more applicable to capture behavior from the financial asset (Biagini et al 2008).…”

Section: Introductionmentioning

confidence: 99%

Efficient valuation and exercise boundary of American fractional lookback option in a mixed jump-diffusion model

Yang

2017

J. Finan. Eng.

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This study presents an efficient method for pricing the American fractional lookback option in the case where the stock price follows a mixed jump diffusion fraction Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral, a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.

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“…Cajueiro and Tabak (2004), as well as Rejichi and Aloui (2012), use the Hurst exponent to test the evolving efficiency of emerging equity markets. Hu and Øksendal (2003) derived closed-form solutions for contingent claim valuation in a fractional Black-Scholes market, where the standard Brownian motion in the asset price process is replaced with an fBm (see also Necula 2002). Their work was extended by Elliott and Van der Hoek (2003).…”

Section: Introductionmentioning

confidence: 99%

Fractional Black–Scholes option pricing, volatility calibration and implied Hurst exponents in South African context

Flint

Maré

2017

SAJEMS

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Background: Contingent claims on underlying assets are typically priced under a framework that assumes, inter alia, that the log returns of the underlying asset are normally distributed. However, many researchers have shown that this assumption is violated in practice. Such violations include the statistical properties of heavy tails, volatility clustering, leptokurtosis and long memory. This paper considers the pricing of contingent claims when the underlying is assumed to display long memory, an issue that has heretofore not received much attention.Aim: We address several theoretical and practical issues in option pricing and implied volatility calibration in a fractional Black–Scholes market. We introduce a novel eight-parameter fractional Black–Scholes-inspired (FBSI) model for the implied volatility surface, and consider in depth the issue of calibration. One of the main benefits of such a model is that it allows one to decompose implied volatility into an independent long-memory component – captured by an implied Hurst exponent – and a conditional implied volatility component. Such a decomposition has useful applications in the areas of derivatives trading, risk management, delta hedging and dynamic asset allocation.Setting: The proposed FBSI volatility model is calibrated to South African equity index options data as well as South African Rand/American Dollar currency options data. However, given the focus on the theoretical development of the model, the results in this paper are applicable across all financial markets.Methods: The FBSI model essentially combines a deterministic function form of the 1-year implied volatility skew with a separate deterministic function for the implied Hurst exponent, thus allowing one to model both observed implied volatility surfaces as well as decompose them into independent volatility and long-memory components respectively. Calibration of the model makes use of a quasi-explicit weighted least-squares optimisation routine.Results: It is shown that a fractional Black–Scholes model always admits a non-constant implied volatility term structure when the Hurst exponent is not 0.5, and that 1-year implied volatility is independent of the Hurst exponent and equivalent to fractional volatility. Furthermore, we show that the FBSI model fits the equity index implied volatility data very well but that a more flexible Hurst exponent parameterisation is required to fit accurately the currency implied volatility data.Conclusion: The FBSI model is an arbitrage-free deterministic volatility model that can accurately model equity index implied volatility. It also provides one with an estimate of the implied Hurst exponent, which could be very useful in derivatives trading and delta hedging.

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Option Pricing in a Fractional Brownian Motion Environment

Cited by 81 publications

References 6 publications

Optimal Exercise Boundary of American Fractional Lookback Option in a Mixed Jump-Diffusion Fractional Brownian Motion Environment

Optimal Exercise Boundary of American Fractional Lookback Option in a Mixed Jump-Diffusion Fractional Brownian Motion Environment

Efficient valuation and exercise boundary of American fractional lookback option in a mixed jump-diffusion model

Fractional Black–Scholes option pricing, volatility calibration and implied Hurst exponents in South African context

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