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.
Highlights
One of the central ideas of economic thought is that, in properly functioning markets, prices of traded goods contain valuable information that can be used to make a wide variety of economic decisions.In financial derivative markets, implied volatility is one such traded quantity
This led them to state that under risk neutrality, there was a unique diffusion process consistent with the risk neutral probability densities derived from the prices of European options
If we understand the concept of instantaneous volatility, we are much closer to understanding the statement by Gatheral [33]: local volatility is representing some kind of average over all possible instantaneous volatilities in a stochastic volatility world
Summary
One of the central ideas of economic thought is that, in properly functioning markets, prices of traded goods contain valuable information that can be used to make a wide variety of economic decisions. Solved this problem numerically by implementing binomial trees They showed that the local volatility can be extracted from vanilla options, priced using the implied volatility surface. Dupire [9] showed that the existence of a forward equation that describes the evolution of call option prices as functions of maturity time and strike price makes it possible to express the unknown volatility function directly in terms of known option prices It captures the implied volatility skew without introducing additional sources of risk. Options.aspx function with an exponential time decay to ALSI4 futures implied volatility trade data — the ALSI is the liquid future on the tradable FTSE/JSE Top 40 equity index5 This means that the ALSI implied volatility surface can be represented by an algebraic equation and implementing the Dupire mapping is straightforward.
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