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

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.

Dynamic Properties of Foreign Exchange Complex Network

“…Centrality analysis is employed to measure the key nodes in the network. In practice, we choose four commonly used centrality measures, i.e., node degree, node strength, betweenness centrality and closeness centrality measures, which is calculated by Formulas (4), (5), (7) and (8). In addition, the results are shown in Figure 10.…”

“…It identifies the exchange rates of global trade and also determines the relative wealth of a country [4][5][6]. The volatility of the FX market is affected by many factors, and its occurrence, formation and evolution show the typical feature of complex system [7][8][9]. To date, networks and dynamics modeling have attracted great attention in natural, social science and engineering technology [10][11][12][13][14][15][16][17][18][19][20][21][22].…”