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J. Risk Financial Manag. 2015, 8(1), 43-82; doi:10.3390/jrfm8010043

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

1
Department of Finance and Investment Management, University of Johannesburg, PO Box 524, Aucklandpark 2006, South Africa
2
The Johannesburg Stock Exchange (JSE), One Exchange Square, Gwen Lane, Sandown 2196, South Africa
3
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
*
Author to whom correspondence should be addressed.
Academic Editor: Michael McAleer
Received: 30 September 2014 / Accepted: 5 January 2015 / Published: 26 January 2015
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Abstract

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. View Full-Text
Keywords: exotic options; JSE; Can-Do options; implied volatility; local volatility; dupire transforms; gyöngy theorem; calibration; deterministic volatility function exotic options; JSE; Can-Do options; implied volatility; local volatility; dupire transforms; gyöngy theorem; calibration; deterministic volatility function
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Kotzé, A.; Oosthuizen, R.; Pindza, E. Implied and Local Volatility Surfaces for South African Index and Foreign Exchange Options. J. Risk Financial Manag. 2015, 8, 43-82.

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