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

2017 ◽  
Vol 20 (1) ◽  
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.

scholarly journals HISTORICAL BACKTESTING OF LOCAL VOLATILITY MODEL USING AUD/USD VANILLA OPTIONS

2016 ◽  
Vol 57 (3) ◽  
pp. 319-338
Author(s):  
T. G. LING ◽  
P. V. SHEVCHENKO
Keyword(s):  
Implied Volatility ◽  
Underlying Asset ◽  
Local Volatility ◽  
Market Data ◽  
Black Scholes Model ◽  
Local Volatility Model ◽  
Volatility Model ◽  
Wide Range ◽  
Black Scholes ◽  
Delta Hedging

The local volatility model is a well-known extension of the Black–Scholes constant volatility model, whereby the volatility is dependent on both time and the underlying asset. This model can be calibrated to provide a perfect fit to a wide range of implied volatility surfaces. The model is easy to calibrate and still very popular in foreign exchange option trading. In this paper, we address a question of validation of the local volatility model. Different stochastic models for the underlying asset can be calibrated to provide a good fit to the current market data, which should be recalibrated every trading date. A good fit to the current market data does not imply that the model is appropriate, and historical backtesting should be performed for validation purposes. We study delta hedging errors under the local volatility model using historical data from 2005 to 2011 for the AUD/USD implied volatility. We performed backtests for a range of option maturities and strikes using sticky delta and theoretically correct delta hedging. The results show that delta hedging errors under the standard Black–Scholes model are no worse than those of the local volatility model. Moreover, for the case of in- and at-the-money options, the hedging error for the Black–Scholes model is significantly better.

scholarly journals Robustness of Delta Hedging for Path-Dependent Options in Local Volatility Models

2007 ◽  
Vol 44 (04) ◽  
pp. 865-879 ◽  
Author(s):  
Alexander Schied ◽  
Mitja Stadje
Keyword(s):  
Payoff Function ◽  
Hedging Strategy ◽  
Local Volatility ◽  
Directional Convexity ◽  
Volatility Models ◽  
Local Volatility Model ◽  
Volatility Model ◽  
Black Scholes ◽  
Delta Hedging ◽  
Path Dependent

We consider the performance of the delta hedging strategy obtained from a local volatility model when using as input the physical prices instead of the model price process. This hedging strategy is called robust if it yields a superhedge as soon as the local volatility model overestimates the market volatility. We show that robustness holds for a standard Black-Scholes model whenever we hedge a path-dependent derivative with a convex payoff function. In a genuine local volatility model the situation is shown to be less stable: robustness can break down for many relevant convex payoffs including average-strike Asian options, lookback puts, floating-strike forward starts, and their aggregated cliquets. Furthermore, we prove that a sufficient condition for the robustness in every local volatility model is the directional convexity of the payoff function.

AN IMPLIED VOLATILITY MODEL DETERMINED BY CREDIT DEFAULT SWAPS

2012 ◽  
Vol 15 (07) ◽  
pp. 1250049
Author(s):  
PASCAL HEIDER
Keyword(s):  
Capital Structure ◽  
Stock Price ◽  
Implied Volatility ◽  
Contingent Claims ◽  
Stock Volatility ◽  
Numerical Examples ◽  
Volatility Model ◽  
Credit Default ◽  
Spread Dynamics ◽  
A Company

In this paper we propose a diffusion model relating the stock price dynamics to the CDS spread dynamics of a company by assuming a linear relationship between instantaneous stock volatility and CDS spread. To value contingent claims under this model we apply a finite elements discretization to the associated pricing partial differential equation. A robust calibration strategy is presented and numerical examples are studied to validate the model assumptions. Besides option pricing, we discuss further applications which are e.g. the identification of market situations allowing volatility and capital structure arbitrage.

Arbitrage Pricing

2019 ◽  
pp. 95-118
Author(s):  
Tomas Björk
Keyword(s):  
Continuous Time ◽  
Implied Volatility ◽  
Futures Contracts ◽  
Martingale Measures ◽  
Bank Account ◽  
No Arbitrage ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Financial Derivative ◽  
Delta Hedging

The chapter starts with a detailed discussion of the bank account in discrete and continuous time. The Black–Scholes model is then introduced, and using the principle of no arbitrage we study the problem of pricing an arbitrary financial derivative within this model. Using the classical delta hedging approach we derive the Black–Scholes PDE for the pricing problem and using Feynman–Kač we also derive the corresponding risk neutral valuation formula and discuss the connection to martingale measures. Some concrete examples are studied in detail and the Black–Scholes formula is derived. We also discuss forward and futures contracts, and we derive the Black-76 futures option formula. We finally discuss the concepts and roles of historic and implied volatility.

scholarly journals Aspects of volatility targeting for South African equity investors

2014 ◽  
Vol 17 (5) ◽  
pp. 691-699 ◽  
Author(s):  
Bhekinkosi Khuzwayo ◽  
Eben Mare
Keyword(s):  
South African ◽  
Asset Allocation ◽  
Implied Volatility ◽  
Equity Market ◽  
Risky Asset ◽  
Risk And Return ◽  
Asset Classes ◽  
The Absolute ◽  
Equity Index ◽  
Targeting Strategy

We consider so-called volatility targeting strategies in the South African equity market. These strategies are aimed at keeping the volatility of a portfolio consisting of a risky asset, typically an equity index, and cash fixed. This is done by changing the allocation of the assets based on an indicator of the future volatility of the risky asset. We use the three month rolling implied volatility as an indicator of future volatility to influence our asset allocation. We compare investments based on different volatility targets to the performance of bonds, equities, property as well as the Absolute Return peer mean. We examine risk and return characteristics of the volatility targeting strategy as compared to different asset classes.

ASYMPTOTIC APPROXIMATIONS FOR PRICING DERIVATIVES UNDER MEAN-REVERTING PROCESSES

2016 ◽  
Vol 19 (05) ◽  
pp. 1650030 ◽  
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.

A Simple Model for Option Pricing with Jumping Stochastic Volatility

1998 ◽  
Vol 01 (04) ◽  
pp. 487-505 ◽  
Author(s):  
Stefano Herzel
Keyword(s):  
Implied Volatility ◽  
Market Price ◽  
Volatility Risk ◽  
Simple Modification ◽  
Closed Form Solutions ◽  
Black Scholes Model ◽  
A Value ◽  
Black Scholes ◽  
Delta Hedging ◽  
Actual Price

This paper proposes a simple modification of the Black–Scholes model by assuming that the volatility of the stock may jump at a random time τ from a value σa to a value σb. It shows that, if the market price of volatility risk is unknown, but constant, all contingent claims can be valued from the actual price C0, of some arbitrarily chosen "basis" option. Closed form solutions for the prices of European options as well as explicit formulas for vega and delta hedging are given. All such solutions only depend on σa, σb and C0. The prices generated by the model produce a "smile"-shaped curve of the implied volatility.

scholarly journals On the Solution of Fractional Option Pricing Model by Convolution Theorem

2019 ◽  
pp. 143-157
Author(s):  
A. I. Chukwunezu ◽  
B. O. Osu ◽  
C. Olunkwa ◽  
C. N. Obi
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Long Memory ◽  
Hurst Exponent ◽  
Convolution Theorem ◽  
Pricing Model ◽  
One Dimensional ◽  
Black Scholes ◽  
Option Pricing Model

The classical Black-Scholes equation driven by Brownian motion has no memory, therefore it is proper to replace the Brownian motion with fractional Brownian motion (FBM) which has long-memory due to the presence of the Hurst exponent. In this paper, the option pricing equation modeled by fractional Brownian motion is obtained. It is further reduced to a one-dimensional heat equation using Fourier transform and then a solution is obtained by applying the convolution theorem.

scholarly journals 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

2010 ◽  
Vol 8 (1) ◽  
pp. 379-389
Author(s):  
F.Y. Jordaan ◽  
J.H. van Rooyen
Keyword(s):  
South African ◽  
Implied Volatility ◽  
Calendar Period ◽  
The South ◽  
Merton Model ◽  
Proposed Model ◽  
Black Scholes ◽  
European Markets ◽  
Market Changes

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

FROM THE IMPLIED VOLATILITY SKEW TO A ROBUST CORRECTION TO BLACK-SCHOLES AMERICAN OPTION PRICES

2001 ◽  
Vol 04 (04) ◽  
pp. 651-675 ◽  
Author(s):  
JEAN-PIERRE FOUQUE ◽  
GEORGE PAPANICOLAOU ◽  
K. RONNIE SIRCAR
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Stochastic Volatility ◽  
Free Boundary Problem ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
One Dimensional ◽  
Volatility Model ◽  
Black Scholes

We describe a robust correction to Black-Scholes American derivatives prices that accounts for uncertain and changing market volatility. It exploits the tendency of volatility to cluster, or fast mean-reversion, and is simply calibrated from the observed implied volatility skew. The two-dimensional free-boundary problem for the derivative pricing function under a stochastic volatility model is reduced to a one-dimensional free-boundary problem (the Black-Scholes price) plus the solution of a fixed boundary-value problem. The formal asymptotic calculation that achieves this is presented here. We discuss numerical implementation and analyze the effect of the volatility skew.

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