Antonie Kotze

Financial Chaos Theory · Financial Engineering and Quantitative Analyst

The Radial Basis Functions (RBF) interpolation is a popular approximation technique used to smooth scattered data in various dimensions. This study uses RBF interpolation to interpolate the volatility skew of the S&P500 index options. The interpolated skews are used to construct the risk-neutral densities of the index and its local volatility surfa...

Bilateral over the counter derivatives were clearly in the eye of the storm during the 2008 financial crisis. In contrast, the listed derivative markets managed by central counter-parties (CCPs), were much more stable. After 2008 regulators started forcing risk away from banks to central counterparties. A consequence is that CCPs can increase syste...

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 y...

Modeling is important because scientists investigate the world around us by building models that simulate real-world problems. Modeling is neither science nor mathematics; it is the craft that builds bridges between the two. Progress in modeling dynamics has always been closely associated with advances in computing. Monte Carlo simulation/modeling...

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 exce...

Monte Carlo simulation or probability simulation is a technique used to understand the impact of risk and uncertainty in financial and other forecasting models. It is very useful when complex financial instruments need to be priced. Exotic options are listed on the JSE on its Can-Do platform. Most listed exotic options are marked-to-model and the J...

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...

Volatility has been in decline since it spiked during the 2008 financial crisis. Is volatility eventually returning to the financial markets?

Recently, the Johannesburg Stock of 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 to list some of their exotic options on the exchange. Cando’s are market to model. These products are all modeled with the celebrated Black-Scholes (Black & Scholes,...

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 Organi...

Talk on implied and local volatility surfaces and pricing exotic options. I give a bit of history on heat diffusion and Joseph Fourier and the origination of the Black-Scholes parabolic partial differential equation.

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 a...

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 dif...

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 YieldX. These products give investors the advantages of listed derivatives with the exibility of \over the counter" (OTC) contracts. Investors can negotiate the terms for all...

Exposure-at-default is one of the most interesting and most difficult parameters to estimate in counterparty credit risk. Basel I offered only the non-internal Current Exposure Method for estimating this quantity whilst Basel II further introduced the Standardized Method and an Internal Model Method. Under new Basel III rules a central counterparty...

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 s...

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's lifetime, and a pre-specified observation frequency. We implement Vorst's method in valuing these options and give...

Exposure-at-default is one of the most interesting and most difficult parameters to estimate in counterparty credit risk. Basel I offered only the non-internal Current Exposure Method for estimating this quantity whilst Basel II further introduced the Standardized Method and an Internal Model Method. Under new Basel III rules a central counterparty...

The current method employed by the Johannesburg Stock Exchange (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 fut...

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 C...

If you have traded a few options but are relatively new to trading them, you are probably battling to understand why some of your trades aren't profitable. You start to realise that trying to predict what will happen to the price of a single option or a position involving multiple options as the market changes can be a difficult undertaking. You ex...

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 da...

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...

In 2007, the SAVI was launched as an index designed to measure the market’s expectation of the 3-month market volatility. The SAVI soon became the benchmark for measuring the market sentiment, and in this light can be thought of as a market “fear” index. Two years later, in 2009 the Johannesburg Stock Exchange updated the SAVI to reflect a new way...

According to the classical theory based on the work by Black, Scholes and Merton, the implied volatility of an option should be independent of its strike and expiration date. However, traded option data sketch a different picture. The structure of volatilities for different strikes for a given maturity tends to have the shape ofsmile'' orskew''. Ev...

In this paper we show how to generate the implied volatility surface by fitting a quadratic deterministic function to implied volatility data from Alsi index options traded on Safex. This market is mostly driven by structured spread trades, and very few at-the-money options ever trade. It is thus difficult to obtain the correct at-the-money volatil...

In this note we discuss and summarize the valuation methodology for Double barrier Cash or Nothing Options. We start off by briefly defining vanilla binary options and ordinary and double barrier options. We then move on to the valuation and price dynamics of the option at hand. After that we list the formulas for the Greeks and discuss their dynam...

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...

Volatility is a measure of the risk or uncertainty and it has an important role in the financial markets. Volatility is defined as the variation of an asset's returns – it indicates the range of a return's movement. Large values of volatility mean that returns fluctuate in a wide range. Volatility measures variability, or dispersion about a central...

The classical Hamiltonian Hα(p, q) = 1/2(p12 + p22) + Uα(q) with 0 α 1 and Uα(q) = (1 – α)/12(q14 + q24) + 1/2q12q22 is integrable for α = 0. For α = 1 the motion is always irregular except for special orbits. Here we study the energy spectra of the quantized version and discuss the connection with "quantum chaos". For α = 0 the distribution of the...

Corporates enter the financial markets as natural hedgers for their interest rate and/or foreign exchange exposures. Few companies, however, utilise the equity derivative markets to their advantage - to hedge certain corporate action events. In this exposé we will discuss three case studies to understand how equity derivatives are utilised by corpo...

A price series or an economic indicator that changes a lot and swings wildly is said to be “volatile”. This simple and intuitive concept is the cause of many difficulties in finance. Unlike many other market parameters which can be directly observed, volatility has to be estimated. This is difficult, if not impossible, because we cannot say that vo...

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 t...

“Timing the markets” has been a topic of investigation for many decades. Every investor dreams of the perfect system that will tell him to buy right at the bottom or sell right at the top. The same holds for investors who want to hedge or gear their portfolio by using options. If an investor buys a call and the market turns down he always thinks th...

A cost-effective structure is sought whereby an institution can hedge its balance sheet against adverse market movements. Vanilla puts would suffice but it is generally very expensive if these puts are rolled on a continuous basis. An institution, however, might be willing to pay the intrinsic value of the put back to the writer thereof if the mark...

A cost-effective structure is sought whereby an institution can hedge its balance sheet against adverse market movements. Vanilla puts would suffice but it is generally very expensive if these puts are rolled on a continuous basis. An institution, however, might be willing to pay the intrinsic value of the put back to the writer thereof if the mark...

The intricate connection between the distribution of exceptional points and the spectral behaviour of Hamiltonian systems is investigated. A method to determine the distribution of exceptional points for a Hamilton operator with systematic degeneracies is described. Its implementation is demonstrated considering the hydrogen atom in a strong magnet...

An easy to implement numerical procedure is described to calculate the matrix representation of the position operator, if only the eigenvalues of the underlying Hamiltonian are known.

The coupling matrix elements of a chaotic Hamilton system play a crucial role in describing the chaotic behaviour where the coupling elements are related to the imaginary parts of the exceptional points. The relationship between the coupling matrix elements and the avoided level crossings is investigated. From this analysis a quantitative measure o...

Quantum chaos is associated with the phenomenon of avoided level crossings on a large scale which leads to a statistical behaviour similar to that of a Gaussian Orthogonal Ensemble (GOE) of matrices. The same behaviour is seen in a pure quantum one dimensional system consisting of a square well containing ±-function barriers on a Cantor set. For a...

Energy levels of a square well containing -function barriers on a Cantor set are analysed. According to the standard tests the spectrum has the characteristics of quantum chaotic behaviour. A discussion on the connection between the spectrum, the wave functions and the exceptional points in the case of quantum chaos is given.

It is argued that the eigenstates of a Hamiltonian which gives rise to a GOE-type spectrum are only weakly correlated and that the individual eigenstates are expected to be highly sensitive against a generic perturbation. A simple mathematical model is used for demonstration.

The variational approach was scrutinized within the Lipkin model using projected wave functions. It was found that projection before variation leads to spurious singularities in the transitional region of the phase transition, with the effect that the variational parameter shows a discontinuity. While the energy is slightly improved in comparison w...

Transitional regions are investigated of models describing finite Fermi systems which give rise to phase transitions. The emphasis lies on the connection between the pattern of the exceptional points of the Hamiltonian and fluctuations induced by multiple level crossings. It is found that the Lipkin model produces no fluctuations in the transitiona...

We study a Hamiltonian for a quartic potential where the motion is always irregular except for special orbits. We compare the classical and quantized versions where both show chaotic behaviour.