Arbitrage-free implied volatility surfaces for options on single stock futures

Kotze, Antonie; Labuschagne, Coenraad C.A.; Nair, Merell L.; Padayachi, Nadine

doi:10.1016/j.najef.2013.02.012

Cited by 17 publications

(9 citation statements)

References 19 publications

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“…In the Dumas parametric model the IV surface is modeled as a quadratic function of the so-called normalized strike (rather than the strike price). Later this approach was further extended by Tompkins (2001), Kotzé et al (2013), Carr et al (2013). Th normalized strike is defined…”

Section: Overviewmentioning

confidence: 99%

To sigmoid-based functional description of the volatility smile

Itkin

2015

The North American Journal of Economics and Finance

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We propose a new static parameterization of the implied volatility surface which is constructed by using polynomials of sigmoid functions combined with some other terms. This parameterization is flexible enough to fit market implied volatilities which demonstrate smile or skew. An arbitrage-free calibration algorithm is considered that constructs the implied volatility surface as a grid in the strike-expiration space and guarantees a lack of arbitrage at every node of this grid. We also demonstrate how to construct an arbitrage-free interpolation and extrapolation in time, as well as build a local volatility and implied pdf surfaces. Asymptotic behavior of this parameterization is discussed, as well as results on stability of the calibrated parameters are presented. Numerical examples show robustness of the proposed approach in building all these surfaces as well as demonstrate a better quality of the fit as compared with some known models.

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Section: Overviewmentioning

confidence: 99%

To sigmoid-based functional description of the volatility smile

Itkin

2015

The North American Journal of Economics and Finance

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“…To ensure the positivity of the numerator, we note that no-arbitrage arguments state that calendar spreads should have positive values. Kotzé and Joseph [44] and Kotzé et al [18] discusses no-arbitrage arguments in detail. We can turn these statements around: An implied volatility surface is arbitrage free if the local volatility is a positive real number (not imaginary) where σ loc (K, T ) ∈ R + 0 .…”

Section: Dupire Local Volatilitymentioning

confidence: 99%

“…In order to obtain the local volatility surface, the JSE first constructs the implied volatility surface. Kotzé et al [18] discussed the current method employed by the JSE to determine implied volatility surfaces. This method is based on trade data and a linear deterministic approach.…”

Section: Introductionmentioning

confidence: 99%

Implied and Local Volatility Surfaces for South African Index and Foreign Exchange Options

Kotze

Oosthuizen²,

Pindza

2015

JRFM

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Certain exotic options cannot be valued using closed-form solutions or even by numerical methods assuming constant volatility. Many exotics are priced in a local volatility framework. Pricing under local volatility has become a field of extensive research in finance, and various models are proposed in order to overcome the shortcomings of the Black-Scholes model that assumes a constant volatility. The Johannesburg Stock Exchange (JSE) lists exotic options on its Can-Do platform. Most exotic options listed on the JSE's derivative exchanges are valued by local volatility models. These models needs a local volatility surface. Dupire derived a mapping from implied volatilities to local volatilities. The JSE uses this mapping in generating the relevant local volatility surfaces and further uses Monte Carlo and Finite Difference methods when pricing exotic options. In this document we discuss various practical issues that influence the successful construction of implied and local volatility surfaces such that pricing engines can be implemented successfully. We focus on arbitrage-free conditions and the choice of calibrating functionals. We illustrate our methodologies by studying the implied and local volatility surfaces of South African equity index and foreign exchange options.

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“…Furthermore, they give evidence that this functional form is a viable model for each β i parameter over time. In this manner, Kotzé et al (2013) showed that one can fully characterise an implied volatility surface using only six parameters:…”

Section: 42mentioning

confidence: 99%

“…5.4.4 Figure 13 displays the long-term implied volatility surface calculated from equation (21) Kotzé et al (2013) for full implementation details. The 50-year at-the-money implied volatility is 27,54%, the 50-year volatility curve ranging between 28,27% and 26,82%.…”

Section: 42mentioning

confidence: 99%

Estimating long-term volatility parameters for market-consistent models

Flint¹,

Ochse²,

Polakow³

2015

S. A. Act. J.

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Contemporary actuarial and accounting practices (APN 110 in the South African context) require the use of market-consistent models for the valuation of embedded investment derivatives. These models have to be calibrated with accurate and up-to-date market data. Arguably, the most important variable in the valuation of embedded equity derivatives is implied volatility. However, accurate long-term volatility estimation is difficult because of a general lack of tradable, liquid medium-and long-term derivative instruments, be they exchange-traded or over the counter. In South Africa, given the relatively short-term nature of the local derivatives market, this is of particular concern. This paper attempts to address this concern by: -providing a comprehensive, critical evaluation of the long-term volatility models most commonly used in practice, encompassing simple historical volatility estimation and econometric, deterministic and stochastic volatility models; and -introducing several fairly recent nonparametric alternative methods for estimating long-term volatility, namely break-even volatility and canonical option valuation.The authors apply these various models and methodologies to South African market data, thus providing practical, long-term volatility estimates under each modelling framework whilst accounting for real-world difficulties and constraints. In so doing, they identify those models and methodologies they consider to be most suited to long-term volatility estimation and propose best estimation practices within each identified area. Thus, while application is restricted to the South African market, the general discussion, as well as the suggestion of best practice, in each of the evaluated modelling areas remains relevant for all long-term volatility estimation.

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Arbitrage-free implied volatility surfaces for options on single stock futures

Cited by 17 publications

References 19 publications

To sigmoid-based functional description of the volatility smile

To sigmoid-based functional description of the volatility smile

Implied and Local Volatility Surfaces for South African Index and Foreign Exchange Options

Estimating long-term volatility parameters for market-consistent models

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