Antonie Kotzé

University of Johannesburg, South Africa, Faculty of Economic and Financial Sciences, Senior research associate

University of Pretoria, Mathematics and Applied Mathematics, Extraordinary Lecturer

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Antonie is an independent derivatives pricing, valuation and model validation expert and financial engineer. His expertise includes all financial derivatives (OTC and on exchange): FX, swaps, commodities, credit and employee share incentive schemes. Antonie holds a Ph.D. in Theoretical/Mathematical Physics from the University of the Witwatersrand (South Africa) where Quantum Chaos Theory was his field of interest. With more than 20 years’ experience as a quantitative analyst in the South African and African financial and derivatives market. Antonie is a former Rand Afrikaans University (currently University of Johannesburg) faculty member but is still associated via his research interests. He is still an active academic researcher and has published some cutting-edge research in international peer-reviewed journals. He has BCBS (Basel) and IFRS 9 credit risk experience.

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Papers by Antonie Kotzé

JSE Local Volatility Finite Difference Model

Inhouse, 2014

Recently, the Johannesburg Stock Exchange (JSE) launched a new range of exotic products, namely t... more Recently, the Johannesburg Stock Exchange (JSE) launched a new range of exotic products, namely the Can-Do. The main idea behind the Can-Do products is to help financial institutions list some of their exotic options on the exchange. Cando’s are market to model. These products are all modelled with the celebrated Black-Scholes (Black & Scholes, 1973) partial differential equation (PDE) but with exotic features. Due to the inherent complexity of these features in the modeling equations, one can rarely find closed-form analytical solutions to these options. Therefore one has to resort to numerical methods. In the literature, four main families of methods where developed and are used extensively to solve option pricing problems. These include lattice methods (Breen, 1991), (Cox, et al., 1979), Monte Carlo simulations (Barraquand & Martineau, 1995), (Broadie & Glasserman, 1997), finite difference (FD) methods (Brennan & Schwartz, 1977), (Brennan & Schwartz, 1978) and analytical approximations (Carr & Faguet, 1994), (Geske & Johnson, 1984).

The use of finite difference methods for solving PDEs on a computer goes back almost to the 1950s since its invention. In finance, these methods were introduced in the 1970’s after the derivation of the Black-Scholes model. The sophistication of these methods in finance has become a field of extensive research. These include alternating finite difference (ADI) methods (Hout & Foulon, 2010), adaptive grids (Persson & Sydow, 2010), grids stretching (Oosterlee et al., 2005) and compact finite differences (Tangman et al., 2008), to name but a few. The interested reader is referred to two important textbooks (Daniel J, 2006), (Tavella & Randall, 2000) in the literature that cover financial derivatives pricing with finite difference methods exclusively.

This document provides a base for the valuation methods used in the Can-Do product space. Thus, we will not re-document already well-documented models and methodologies. This document will rather focus on the application of these methods to the JSE Can-Do products.

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Current Exposure Method for CCP's Under Basel III

Social Science Research Network, 2012

Exposure-at-default (EAD) is one of the most interesting and most difficult parameters to estimat... more Exposure-at-default (EAD) is one of the most interesting and most difficult parameters to estimate in counterparty credit risk (CCR). Basel I offered only the non-internal Current Exposure Method (CEM) for estimating this quantity whilst Basel II further introduced the Standardized Method (SM) and an Internal Model Method (IMM). The Current Exposure Method relies on the Value at Risk (VaR) methodology and its characteristics are discussed in this short note [Tu 10].

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Reset Options on South African Equities

Social Science Research Network, 2003

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

The North American Journal of Economics and Finance, Dec 1, 2013

The current method employed by the Johannesburg Stock Exchange 1 (JSE) to determine implied volat... more The current method employed by the Johannesburg Stock Exchange 1 (JSE) to determine implied volatility is based on trade data and a linear deterministic approach. The aim of this paper is to construct a market-related arbitrage-free implied volatility surface, by using a quadratic deterministic function, for two stock indices and ten single stock futures (SSFs). Actual traded data is used and we show practically how all no-arbitrage conditions are implemented and tested.

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Instalment Warrant Mechanics

Social Science Research Network, 2010

Instalment warrants are very popular in Australia and these instruments have been listed by Nedba... more Instalment warrants are very popular in Australia and these instruments have been listed by Nedbank and Standard Bank in South Africa. Instalments are financial products, that allow investors to gain direct exposure to shares by making a part payment upfront and delaying an optional final payment (or second instalment) until a later date (expiry date). This allows an investor to buy shares, and other securities, for a fraction of the current share price whilst receiving the benefits of capital growth and ordinary dividends.

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The Dilemma of a Central Counterparty versus a Qualified Central Counterparty in a Developing Country

Procedia. Economics and finance, 2014

The Basel Committee on Banking Supervision (BCBS) has a policy framework for how clearing member ... more The Basel Committee on Banking Supervision (BCBS) has a policy framework for how clearing member banks should treat their exposures to central counterparties (CCPs). Default funds play a crucial role as a risk mitigant in this framework. Furthermore, the Committee on Payment and Settlement Systems and Technical Committee of the International Organization of Securities Commissions (together abbreviated as CPSS-IOSCO) produced a set of 26 principles for financial market infrastructures. For a CCP to be deemed a qualified CCP, it has to abide by these principles. Clearing member banks with trade exposures to a qualified CCP will get preferential capital treatment under the BCBS framework. None of the principles or the policy framework requires, or forces a CCP to have a default fund. However, regulators prefer CCPs to have default funds in place to enhance their credit risk management practices, as required by principles 4 and 6 of the CPSS-IOSCO, before they will be qualified. We consider the merits of the BCBS's requirements for a clearing house, in a developing country such as South Africa, to become a qualified clearing house and conclude that a CCP with prudent risk management processes and controls, should not be disqualified if a default fund is not established.

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FTSE/JSE Top 40 Index Long-Term Returns

Social Science Research Network, 2017

The FTSE/JSE Top 40 Index is the flagship index at the Johannesburg Stock Exchange (JSE). It capt... more The FTSE/JSE Top 40 Index is the flagship index at the Johannesburg Stock Exchange (JSE). It captures more than 80% of the total market capitalisation of all the shares listed on the JSE. It is tradable and the liquid ALSI future is listed on this index. A superficial view of the long-term return of this index points to massive returns. Over a 22 year period, the index grew by 900%. Investors are celebrating. However, should investors be rejoicing? A deeper look demonstrates that returns in local currency terms are not what they seem when compared to returns in hard currencies. The index in terms of gold is a good indicator of the growth in the Gross Domestic Product of South Africa. Investors should keep this in mind when investing in South Africa.

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Local Volatility Modeling of JSE Exotic Can-Do Options

Social Science Research Network, 2014

Can-Do Options are derivative products listed on the JSE's derivative exchanges-mostly equity der... more Can-Do Options are derivative products listed on the JSE's derivative exchanges-mostly equity derivative products listed on Safex and currency derivative products listed on Yield-X. These products give investors the advantages of listed derivatives with the flexibility of "over the counter" (OTC) contracts. Investors can negotiate the terms for all option contracts, choosing the type of option, underlying asset and the expiry date. Many exotic options and even exotic option structures are listed. Exotic options cannot be valued using closed-form solutions or even by numerical methods assuming constant volatility. Most exotic options on Safex and Yield-X are valued by local volatility models. 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 the volatility to be constant. In this document we discuss various topics that influence the successful construction of implied and local volatility surfaces in practice. We focus on arbitrage-free conditions, choice of calibrating functionals and selection of numerical algorithms to price options. We illustrate our methodologies by studying the local volatility surfaces of South African index and foreign exchange options. Numerical experiments are conducted using Excel and MATLAB.

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Current exposure method for CCP’s under Basel III

Risk Governance and Control: Financial Markets & Institutions, 2013

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Valuing Asian Options Using Vorst's Approximation

Social Science Research Network, 2003

ABSTRACT Asian options are options based on some average of the underlying asset price. Generally... more ABSTRACT Asian options are options based on some average of the underlying asset price. Generally, an Asian option is an option whose payoff depends on the average price of the underlying asset during a pre-specified period within the option&#39;s lifetime, and a pre-specified observation frequency. We implement Vorst&#39;s method in valuing these options and give the relevant formulas for the Delta and Gamma. We also list some pseudo VBA code. Discrete Fixed Strike Arithmetic Average Options A fixed strike arithmetic average option 1 is an option where the strike is set on the deal date and the average is taken of the underlying asset to determine the payoff. There is currently no known closed form solution to the arithmetic average option problem. Many approximation methods 2 have been proposed [Hu 06; Ha 07]. Vorst proposed an approximation where he adjusts the strike price of the option using the geometric average [Vo 92]. We define the option as follows: let 0 t be the deal date (annualised time, usually zero) and we have n averaging dates over which the average will be taken. We also let m be the number of averaging dates already passed if we have entered an averaging period. We can then define the times on which the averaging is done as

Comparing Futures and Forwards for Managing Currency Exposures in a South African Context

Social Science Research Network, 2010

For internationally oriented firms or individuals that choose to eliminate the effects of fluctua... more For internationally oriented firms or individuals that choose to eliminate the effects of fluctuating exchange rates, either currency forward contracts or currency futures can be used to fulfil this requirement. Both tools essentially lock in prospective exchange rates, thereby eliminating both risk and opportunity, and thus eliminate currency risk completely. Though similar in their result, futures and forwards have a number of institutional differences that may foster different preferences among different users. This research paper strives to highlight those differences, allowing the selection between these two alternatives to be made on a rational basis. CURRENCY FUTURES DISPENSATION IN SOUTH AFRICA Currency futures were launched predominately as a retail product. The initial dispensation granted by the Minister of Finance in 2007 allows individuals to trade over and above their foreign allocation allowance stipulated by the South African Reserve Bank. Individuals, in other words, have no limits to the value traded in the currency futures market. The Minister of Finance in his 2008 budget speech extended the currency futures qualifying audience to include all South African corporate entities. Corporate entities, including limited or unlimited companies, private and public companies, close corporations, partnerships, trusts, hedge funds and banks are authorised to trade currency futures with no restrictions on the value traded. Corporate entities do not need to apply to Reserve Bank for approval to trade the currency futures nor do they have to report their trades. Unfortunately, pension funds and long term insurance companies are subject to their 15 % foreign allocation limits while asset managers and registered collective investment schemes are subject to their 25% foreign allocation limits.

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Refund Options on South African Equities

Social Science Research Network, 2002

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Debt, Defaults and Crises: A Historical Perspective

Social Science Research Network, 2015

The financial crisis that has been wreaking havoc in markets across the world since August 2007 h... more The financial crisis that has been wreaking havoc in markets across the world since August 2007 had its origins in an asset price bubble that interacted with new kinds of financial innovations that masked risk; with companies that failed to follow their own risk management procedures; and with regulators and supervisors that failed to restrain excessive risk taking [BLJ 08]. A bubble formed in the U.S. housing markets as home prices across the country increased each year from the mid 1990s to 2006, moving out of line with fundamentals like household income. Like traditional asset price bubbles, expectations of future price increases developed and were a significant factor in inflating house prices. As individuals witnessed rising prices in their neighborhood and across the country, they began to expect those prices to continue to rise, even in the late years of the bubble when it had nearly peaked. When the 2008 crisis hit, governments, corporations and individuals defaulted on interest payments. However, government debt defaults are a recurring feature of public finance. These defaults have typically involved low-income and emerging-market economies, although recent cases include advanced-economy sovereigns. Sovereign states have borrowed money for hundreds of years. Sovereign debt was one of the first financial assets ever traded, and continues to comprise a significant fraction of global financial assets. Unlike private debt, sovereign debt is especially difficult to enforce. For centuries, the legal doctrine of sovereign immunity limited suit against defaulting sovereigns, while few government assets are available for attachment in foreign jurisdictions [TW 13].

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SAFCOM/JSE Derivatives: Quantification of the South African Default Fund

Social Science Research Network, 2012

ABSTRACT Exposure-at-default (EAD) is one of the most interesting and most difficult parameters t... more ABSTRACT Exposure-at-default (EAD) is one of the most interesting and most difficult parameters to estimate in counterparty credit risk (CCR). Basel I offered only the non-internal Current Exposure Method (CEM) for estimating this quantity whilst Basel II further introduced the Standardised Method (SM) and an Internal Model Method (IMM) [Tu 10]. The Basel Committee on Banking Supervision, however, forces Central Counterparties (CCPs) to use the CEM when calculating their exposures to counterparties. We suggest that the CEM can be used in estimating the size of the default fund.This discussion document explains how the CEM is used in quantifying the default fund. The soundness of the CEM answers is tested by implementing various other methods like Expected Shortfall (ES) and Value-at-Risk (VaR) in this regard. We use actual trading data as supplied by SAFCOM/JSE.

JSE Exotic Can-Do Options: Determining Initial Margins

Social Science Research Network, 2013

Can-do options are bespoke option structures listed on Safex and Yield-X. The JSE is the first ex... more Can-do options are bespoke option structures listed on Safex and Yield-X. The JSE is the first exchange in the world to list, trade and clear exotic options. The first exotic was listed on 8 January 2007 with the onset of the financial crisis that played out during 2008. The option was on the FTSE/JSE Top 40 index and was a discrete look-back put spread with an averaging feature thrown in for spice. Since then, the types of exotics traded has grown in leaps and bounds, with many of them being complex in nature. Can-Dos are now also listed on Yield-X, with various currency pairs as underlyers. Every product listed on either Safex or Yield-X needs to be marked-to-market, and risk managed on a daily basis. Initial margins need to be estimated for these complex products. This note will describe and explain the methodologies currently used to price these exotic Can-Dos, as well as the methodologies currently in use to determine initial margins for these instruments.

Volatility Smile Interpolation with Radial Basis Functions

International Journal of Theoretical and Applied Finance, Nov 1, 2022

Black-Scholes or Black-Holes?

The Nobel laureates Fischer Black, Myron Scholes and Robert Merton revolutionised financial econo... more The Nobel laureates Fischer Black, Myron Scholes and Robert Merton revolutionised financial economics with the publication of their option valuation formula in 1973. The model, however, was devised for an elementary, ideal and frictionless world. Understanding the framework and underlying simplifying assumptions behind their formulation will help to minimize the risks in trading and managing derivative securities.

Variance Futures listed on Safex: pricing

Volatility measures variability, or dispersion about a central tendency --- it is simply a measur... more Volatility measures variability, or dispersion about a central tendency --- it is simply a measure of the degree of price movement in a stock, futures contract or any other market. Volatility also has many subtleties that make it challenging to analyze and implement. The following questions immediately come to mind: is volatility a simple intuitive concept or is it complex in nature, what causes volatility, how do we estimate volatility and can it be managed? The management of volatility is currently a topical issue. Due to this, many institutional and individual investors have shown an increased interest in volatility as an investment vehicle. Variance swaps or variance futures (also called variance contracts) are equity derivative instruments offering pure exposure to daily realised future variance.

Reset Options on South African Equities

SSRN Electronic Journal, 2003

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A one dimensional model f�r quantum Chaos

Zeitschrift f�r Physik A Atomic Nuclei, 1990

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Antonie Kotzé

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