An investigation into the reasons for the pricing differences between a warrant and an option on the same stock in the South African derivatives market

This study set out to draw a pricing comparison between two similar contracts in the South African derivatives market. These contracts, a normal option and a warrant on the same underlying stock are considered. The research shows that although the two derivatives are the same in all respects, the premiums differ substantially when priced with the Black-Scholes-Merton model. It is clear that pricing has to take place over the same calendar period due to market changes when comparing the instruments. The Black-Scholes-Merton model was the proposed model to be used. However, due to certain limitations the Modified Black model was used as the best suited model. It was shown that warrant contracts always have a higher implied volatility and a higher premium than a comparable normal option per share of the same stock. These results werecompared with similar studies conducted in the European markets

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ASYMPTOTIC APPROXIMATIONS FOR PRICING DERIVATIVES UNDER MEAN-REVERTING PROCESSES

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024916500308 ◽

2016 ◽

Vol 19 (05) ◽

pp. 1650030 ◽

Cited By ~ 1

Author(s):

RICHARD JORDAN ◽

CHARLES TIER

Keyword(s):

Interest Rates ◽

Implied Volatility ◽

Asymptotic Approximations ◽

Ray Method ◽

Interest Rate Derivatives ◽

Put Options ◽

Merton Model ◽

Volatility Model ◽

Black Scholes ◽

Efficient Alternative

The problem of fast pricing, hedging, and calibrating of derivatives is considered when the underlying does not follow the standard Black–Scholes–Merton model but rather a mean-reverting and deterministic volatility model. Mean-reverting models are often used for volatility, commodities, and interest-rate derivatives, while the deterministic volatility accounts for the nonconstant implied volatility. Trading desks often use numerical methods for real-time pricing, hedging, and calibration when implementing such models. A more efficient alternative is to use an analytic formula, even if only approximate. A systematic approach is presented, based on the WKB or ray method, to derive asymptotic approximations to the density function that can be used to derive simple formulas for pricing derivatives. Such approximations are usually only valid away from any boundaries, yet for some derivatives the values of the underlying near the boundaries are needed such as when interest rates are very low or for pricing put options. Hence, the ray approximation may not yield acceptable results. A new asymptotic approximation near boundaries is derived, which is shown to be of value for pricing certain derivatives. The results are illustrated by deriving new analytic approximations for European derivatives and their high accuracy is demonstrated numerically.

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Modeling of implied volatility surfaces of nifty index options

International Journal of Financial Engineering ◽

10.1142/s2424786319500282 ◽

2019 ◽

Vol 06 (03) ◽

pp. 1950028 ◽

Cited By ~ 1

Author(s):

Mihir Dash

Keyword(s):

Implied Volatility ◽

Strike Price ◽

Index Options ◽

Put Options ◽

Call Options ◽

Merton Model ◽

Implied Volatility Surface ◽

Black Scholes ◽

Volatility Surface ◽

Implied Volatility Surfaces

The implied volatility of an option contract is the value of the volatility of the underlying instrument which equates the theoretical option value from an option pricing model (typically, the Black–Scholes[Formula: see text]Merton model) to the current market price of the option. The concept of implied volatility has gained in importance over historical volatility as a forward-looking measure, reflecting expectations of volatility (Dumas et al., 1998). Several studies have shown that the volatilities implied by observed market prices exhibit a pattern very different from that assumed by the Black–Scholes[Formula: see text]Merton model, varying with strike price and time to expiration. This variation of implied volatilities across strike price and time to expiration is referred to as the volatility surface. Empirically, volatility surfaces for global indices have been characterized by the volatility skew. For a given expiration date, options far out-of-the-money are found to have higher implied volatility than those with an exercise price at-the-money. For short-dated expirations, the cross-section of implied volatilities as a function of strike is roughly V-shaped, but has a rounded vertex and is slightly tilted. Generally, this V-shape softens and becomes flatter for longer dated expirations, but the vertex itself may rise or fall depending on whether the term structure of at-the-money volatility is upward or downward sloping. The objective of this study is to model the implied volatility surfaces of index options on the National Stock Exchange (NSE), India. The study employs the parametric models presented in Dumas et al. (1998); Peña et al. (1999), and several subsequent studies to model the volatility surfaces across moneyness and time to expiration. The present study contributes to the literature by studying the nature of the stationary point of the implied volatility surface and by separating the in-the-money and out-of-the-money components of the implied volatility surface. The results of the study suggest that an important difference between the implied volatility surface of index call and put options: the implied volatility surface of index call options was found to have a minimum point, while that of index put options was found to have a saddlepoint. The results of the study also indicate the presence of a “volatility smile” across strike prices, with a minimum point in the range of 2.3–9.0% in-the-money for index call options and of 10.7–29.3% in-the-money for index put options; further, there was a jump in implied volatility in the transition from out-of-the-moneyness to in-the-moneyness, by 10.0% for index call options and about 1.9% for index put options.

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Who predicts dollar-rupee volatility better? A tale of two options markets

Managerial Finance ◽

10.1108/mf-09-2018-0416 ◽

2019 ◽

Vol 45 (9) ◽

pp. 1292-1308

Author(s):

Aparna Prasad Bhat

Keyword(s):

Implied Volatility ◽

Realized Volatility ◽

Informational Efficiency ◽

Significant Information ◽

Content Type ◽

Merton Model ◽

Model Free ◽

Ex Post ◽

Black Scholes ◽

Practical Implications

Purpose The purpose of this paper is to examine whether volatility implied from dollar-rupee options is an unbiased and efficient predictor of ex post volatility, and to determine which options market is a better predictor of future realized volatility and to ascertain whether the model-free measure of implied volatility outperforms the traditional measure derived from the Black–Scholes–Merton model. Design/methodology/approach The information content of exchange-traded implied volatility and that of quoted implied volatility for OTC options is compared with that of historical volatility and a GARCH(1, 1)-based volatility. Ordinary least squares regression is used to examine the unbiasedness and informational efficiency of implied volatility. Robustness of the results is tested by using two specifications of implied volatility and realized volatility and comparison across two markets. Findings Implied volatility from both OTC and exchange-traded options is found to contain significant information for predicting ex post volatility, but is neither unbiased nor informationally efficient. The implied volatility of at-the-money options derived using the Black–Scholes–Merton model is found to outperform the model-free implied volatility (MFIV) across both markets. MFIV from OTC options is found to be a better predictor of realized volatility than MFIV from exchange-traded options. Practical implications This study throws light on the predictive power of currency options in India and has strong practical implications for market practitioners. Efficient currency option markets can serve as effective vehicles both for hedging and speculation and can convey useful information to the regulators regarding the market participants’ expectations of future volatility. Originality/value This study is a comprehensive study of the informational efficiency of options on an emerging currency such as the Indian rupee. To the author’s knowledge, this is one of the first studies to compare the predictive ability of the exchange-traded and OTC markets and also to compare traditional model-dependent volatility with MFIV.

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A proposed model for construction project management communication in the South African construction industry

Acta Structilia ◽

10.18820/1023-0564/as23i1.1 ◽

2016 ◽

Vol 23 (1) ◽

Keyword(s):

Project Management ◽

Construction Industry ◽

South African ◽

Construction Project ◽

Construction Project Management ◽

The South ◽

Management Communication ◽

Proposed Model

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The robustness and accuracy of Box-Jenkins ARIMA in modeling and forecasting household debt in South Africa

Journal of Economics and Behavioral Studies ◽

10.22610/jebs.v6i9.534 ◽

2014 ◽

Vol 6 (9) ◽

pp. 748-759 ◽

Cited By ~ 3

Author(s):

Ntebogang Dinah Moroke

Keyword(s):

South African ◽

Predictive Ability ◽

Information Criteria ◽

Household Debt ◽

The South ◽

Confidence Bounds ◽

Policy Makers ◽

Reserve Bank ◽

Univariate Time Series ◽

Proposed Model

Abstract: This paper adopted the Box-Jenkins methodology to estimate a univariate time series model. Quarterly data collected from the South African Reserve Bank covering the period 1994 to 2014 was used. The initial plot of the series revealed that household debt is explained by an irregular and non-seasonal component. Owing to the non stationarity of the series, first differencing was applied to induce stationarity. The ACFs and PACFs identified six models. Of the six identified models,ð´ð‘…ð¼ð‘€ð´ 3, 1, 0 was selected according to the standard error estimates and the information criteria. The proposed model passed all the diagnostic tests and was further used for producing ten period forecasts of household debt. The forecasted household debt rates obtained were above 75% and within confidence bounds of 95%. Insample and out-of-sampling forecasts moved together confirming the reliability of the model in forecasting household debt and vigour in predictive ability. The proposed model exhibited the best performance in terms of Max APE and Max AE and ascertained the robustness and accuracy of the BoxJenkins ARIMA in forecasting. Both a trend of the data captured and non-seasonal peaks were predicted by the model. These forecasts were proven to be realistic and a true reflection of economic reality in the country. The paper recommended a non-seasonalð´ð‘…ð¼ð‘€ð´ 3, 1, 0 be used by researchers, policy makers and decision makers of different countries to make forecasts of household debt. The South African authorities were also encouraged to use this model to produce further forecasts of the series when making long term planning.

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Fractional Black–Scholes option pricing, volatility calibration and implied Hurst exponents in South African context

South African Journal of Economic and Management Sciences ◽

10.4102/sajems.v20i1.1532 ◽

2017 ◽

Vol 20 (1) ◽

Cited By ~ 2

Author(s):

Emlyn Flint ◽

Eben Maré

Keyword(s):

South African ◽

Long Memory ◽

Hurst Exponent ◽

Implied Volatility ◽

Contingent Claims ◽

Deterministic Function ◽

Volatility Model ◽

Black Scholes ◽

Delta Hedging ◽

Equity Index

Background: Contingent claims on underlying assets are typically priced under a framework that assumes, inter alia, that the log returns of the underlying asset are normally distributed. However, many researchers have shown that this assumption is violated in practice. Such violations include the statistical properties of heavy tails, volatility clustering, leptokurtosis and long memory. This paper considers the pricing of contingent claims when the underlying is assumed to display long memory, an issue that has heretofore not received much attention.Aim: We address several theoretical and practical issues in option pricing and implied volatility calibration in a fractional Black–Scholes market. We introduce a novel eight-parameter fractional Black–Scholes-inspired (FBSI) model for the implied volatility surface, and consider in depth the issue of calibration. One of the main benefits of such a model is that it allows one to decompose implied volatility into an independent long-memory component – captured by an implied Hurst exponent – and a conditional implied volatility component. Such a decomposition has useful applications in the areas of derivatives trading, risk management, delta hedging and dynamic asset allocation.Setting: The proposed FBSI volatility model is calibrated to South African equity index options data as well as South African Rand/American Dollar currency options data. However, given the focus on the theoretical development of the model, the results in this paper are applicable across all financial markets.Methods: The FBSI model essentially combines a deterministic function form of the 1-year implied volatility skew with a separate deterministic function for the implied Hurst exponent, thus allowing one to model both observed implied volatility surfaces as well as decompose them into independent volatility and long-memory components respectively. Calibration of the model makes use of a quasi-explicit weighted least-squares optimisation routine.Results: It is shown that a fractional Black–Scholes model always admits a non-constant implied volatility term structure when the Hurst exponent is not 0.5, and that 1-year implied volatility is independent of the Hurst exponent and equivalent to fractional volatility. Furthermore, we show that the FBSI model fits the equity index implied volatility data very well but that a more flexible Hurst exponent parameterisation is required to fit accurately the currency implied volatility data.Conclusion: The FBSI model is an arbitrage-free deterministic volatility model that can accurately model equity index implied volatility. It also provides one with an estimate of the implied Hurst exponent, which could be very useful in derivatives trading and delta hedging.

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