scholarly journals Approximation Formula for Option Prices under Rough Heston Model and Short-Time Implied Volatility Behavior

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1878
Author(s):  
Siow Woon Jeng ◽  
Adem Kilicman
Keyword(s):  
Term Structure ◽  
Taylor Expansion ◽  
Implied Volatility ◽  
Order Approximation ◽  
Heston Model ◽  
Approximation Formula ◽  
Option Prices ◽  
Second Order Approximation ◽  
Time Term ◽  
Short Time

Rough Heston model possesses some stylized facts that can be used to describe the stock market, i.e., markets are highly endogenous, no statistical arbitrage mechanism, liquidity asymmetry for buy and sell order, and the presence of metaorders. This paper presents an efficient alternative to compute option prices under the rough Heston model. Through the decomposition formula of the option price under the rough Heston model, we manage to obtain an approximation formula for option prices that is simpler to compute and requires less computational effort than the Fourier inversion method. In addition, we establish finite error bounds of approximation formula of option prices under the rough Heston model for 0.1≤H<0.5 under a simple assumption. Then, the second part of the work focuses on the short-time implied volatility behavior where we use a second-order approximation on the implied volatility to match the terms of Taylor expansion of call option prices. One of the key results that we manage to obtain is that the second-order approximation for implied volatility (derived by matching coefficients of the Taylor expansion) possesses explosive behavior for the short-time term structure of at-the-money implied volatility skew, which is also present in the short-time option prices under rough Heston dynamics. Numerical experiments were conducted to verify the effectiveness of the approximation formula of option prices and the formulas for the short-time term structure of at-the-money implied volatility skew.

CALIBRATION OF STOCHASTIC VOLATILITY MODELS VIA SECOND-ORDER APPROXIMATION: THE HESTON CASE

2015 ◽  
Vol 18 (06) ◽  
pp. 1550036 ◽  
Author(s):  
ELISA ALÒS ◽  
RAFAEL DE SANTIAGO ◽  
JOSEP VIVES
Keyword(s):  
Term Structure ◽  
Implied Volatility ◽  
Order Approximation ◽  
Calibration Procedure ◽  
Computational Effort ◽  
Heston Model ◽  
Option Prices ◽  
Volatility Models ◽  
Short And Long Term ◽  
Long Term Behavior

In this paper, we present a new, simple and efficient calibration procedure that uses both the short and long-term behavior of the Heston model in a coherent fashion. Using a suitable Hull and White-type formula, we develop a methodology to obtain an approximation to the implied volatility. Using this approximation, we calibrate the full set of parameters of the Heston model. One of the reasons that makes our calibration for short times to maturity so accurate is that we take into account the term structure for large times to maturity: We may thus say that calibration is not "memoryless," in the sense that the option's behavior far away from maturity does influence calibration when the option gets close to expiration. Our results provide a way to perform a quick calibration of a closed-form approximation to vanilla option prices, which may then be used to price exotic derivatives. The methodology is simple, accurate, fast and it requires a minimal computational effort.

A sharp approximation for ATM-forward option prices and implied volatilites

2016 ◽  
Vol 03 (01) ◽  
pp. 1650002 ◽  
Author(s):  
Dan Stefanica ◽  
Radoš Radoičić
Keyword(s):  
Implied Volatility ◽  
Standard Normal Distribution ◽  
Approximation Formula ◽  
Option Prices ◽  
First Results ◽  
Approximation Formulas ◽  
Sharp Approximation ◽  
Standard Normal ◽  
Relative Errors ◽  
Uniformly Bounded

In this paper, we provide an approximation formula for at-the-money forward options based on a Pólya approximation of the cumulative density function of the standard normal distribution, and prove that the relative error of this approximation is uniformly bounded for options with arbitrarily large (or small) maturities and implied volatilities. This approximation is viable in practice: for options with implied volatility less than 95% and maturity less than three years, which includes the large majority of traded options, the values given by the approximation formula fall within the tightest typical implied vol bid–ask spreads. The relative errors of the corresponding approximate option values are also uniformly bounded for all maturities and implied volatilities. The error bounds established here are the first results in the literature holding for all integrated volatilities, and are vastly superior to those of two other approximation formulas analyzed in this paper, including the Brenner–Subrahmanyam formula.

scholarly journals The Small-Time Smile and Term Structure of Implied Volatility under the Heston Model

2012 ◽  
Vol 3 (1) ◽  
pp. 690-708 ◽  
Author(s):  
Martin Forde ◽  
Antoine Jacquier ◽  
Roger Lee
Keyword(s):  
Term Structure ◽  
Implied Volatility ◽  
Small Time ◽  
Heston Model

SHORT-TIME IMPLIED VOLATILITY IN EXPONENTIAL LÉVY MODELS

2015 ◽  
Vol 18 (04) ◽  
pp. 1550025
Author(s):  
ERIK EKSTRÖM ◽  
BING LU
Keyword(s):  
Implied Volatility ◽  
Upper And Lower Bounds ◽  
Gaussian Component ◽  
Option Prices ◽  
Strike Price ◽  
Short Time Asymptotics ◽  
Black Scholes ◽  
Exponential Lévy Models ◽  
Necessary And Sufficient ◽  
Short Time

We show that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential Lévy models is the existence of jumps towards the strike price in the underlying process. When such jumps do not exist, the implied volatility converges to the volatility of the Gaussian component of the underlying Lévy process as the time to maturity tends to zero. These results are proved by comparing the short-time asymptotics of the Black–Scholes price with explicit formulas for upper and lower bounds of option prices in exponential Lévy models.

scholarly journals Option Pricing under the Jump Diffusion and Multifactor Stochastic Processes

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Shican Liu ◽  
Yanli Zhou ◽  
Yonghong Wu ◽  
Xiangyu Ge
Keyword(s):  
Diffusion Process ◽  
Option Pricing ◽  
Stochastic Volatility ◽  
Term Structure ◽  
Implied Volatility ◽  
Jump Diffusion ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Volatility Model ◽  
Jump Diffusion Process

In financial markets, there exists long-observed feature of the implied volatility surface such as volatility smile and skew. Stochastic volatility models are commonly used to model this financial phenomenon more accurately compared with the conventional Black-Scholes pricing models. However, one factor stochastic volatility model is not good enough to capture the term structure phenomenon of volatility smirk. In our paper, we extend the Heston model to be a hybrid option pricing model driven by multiscale stochastic volatility and jump diffusion process. In our model the correlation effects have been taken into consideration. For the reason that the combination of multiscale volatility processes and jump diffusion process results in a high dimensional differential equation (PIDE), an efficient finite element method is proposed and the integral term arising from the jump term is absorbed to simplify the problem. The numerical results show an efficient explanation for volatility smirks when we incorporate jumps into both the stock process and the volatility process.

scholarly journals GARCH Generated Volatility Indices of Bitcoin and CRIX

2020 ◽  
Vol 13 (6) ◽  
pp. 121 ◽  
Author(s):  
Pierre J. Venter ◽  
Eben Maré
Keyword(s):  
Option Pricing ◽  
Term Structure ◽  
Implied Volatility ◽  
Conditional Heteroskedasticity ◽  
Option Prices ◽  
Pricing Model ◽  
European Option ◽  
Short Term ◽  
Market Prices ◽  
Option Pricing Model

In this paper, the pricing performance of the generalised autoregressive conditional heteroskedasticity (GARCH) option pricing model is tested when applied to Bitcoin (BTCUSD). In addition, implied volatility indices (30, 60-and 90-days) of BTCUSD and the Cyptocurrency Index (CRIX) are generated by making use of the symmetric GARCH option pricing model. The results indicate that the GARCH option pricing model produces accurate European option prices when compared to market prices and that the BTCUSD and CRIX implied volatility indices are similar when compared, this is consistent with expectations because BTCUSD is highly weighted when calculating the CRIX. Furthermore, the term structure of volatility indices indicate that short-term volatility (30 days) is generally lower when compared to longer maturities. Furthermore, short-term volatility tends to increase to higher levels when compared to 60 and 90 day volatility when large jumps occur in the underlying asset.

scholarly journals DECOMPOSITION FORMULA FOR JUMP DIFFUSION MODELS

2018 ◽  
Vol 21 (08) ◽  
pp. 1850052
Author(s):  
R. MERINO ◽  
J. POSPÍŠIL ◽  
T. SOBOTKA ◽  
J. VIVES
Keyword(s):  
Option Pricing ◽  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heston Model ◽  
Approximation Formula ◽  
Option Prices ◽  
Currency Options ◽  
Decomposition Formula

In this paper, we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alòs [(2012) A decomposition formula for option prices in the Heston model and applications to option pricing approximation, Finance and Stochastics 16 (3), 403–422, doi: https://doi.org/10.1007/s00780-012-0177-0 ] for Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ] SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular class of SVJ models — models utilizing a variance process postulated by Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ]. In particular, we inspect in detail the approximation formula for the Bates [(1996), Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options, The Review of Financial Studies 9 (1), 69–107, doi: https://doi.org/10.1093/rfs/9.1.69 ] model with log-normal jump sizes and we provide a numerical comparison with the industry standard — Fourier transform pricing methodology. For this model, we also reformulate the approximation formula in terms of implied volatilities. The main advantages of the introduced pricing approximations are twofold. Firstly, we are able to significantly improve computation efficiency (while preserving reasonable approximation errors) and secondly, the formula can provide an intuition on the volatility smile behavior under a specific SVJ model.

JOINING THE HESTON AND A THREE-FACTOR SHORT RATE MODEL: A CLOSED-FORM APPROACH

2015 ◽  
Vol 18 (08) ◽  
pp. 1550056 ◽  
Author(s):  
ROMAN HORSKY ◽  
TILMAN SAYER
Keyword(s):  
Term Structure ◽  
Implied Volatility ◽  
Heston Model ◽  
Correlation Structure ◽  
Market Model ◽  
Characteristic Functions ◽  
Rate Model ◽  
Bond Price ◽  
Equity Options ◽  
Three Factor

In this paper, we present an innovative hybrid model for the valuation of equity options. Our approach includes stochastic volatility according to Heston (1993) [Review of Financial Studies 6 (2), 327–343] and features a stochastic interest rate that follows a three-factor short rate model based on Hull and White (1994) [Journal of Derivatives 2 (2), 37–48]. Our model is of affine structure, allows for correlations between the stock, the short rate and the volatility processes and can be fitted perfectly to the initial term structure. We determine the zero bond price formula and derive the analytic solution for European type options in terms of characteristic functions needed for fast calibration. We highlight the flexibility of our approach, by comparing the price and implied volatility surfaces of our model with the Heston model, where we in particular focus on the correlation structure. Our approach forms a comprehensive market model with an intuitive correlation structure that connects both the equity and interest market to the market volatility.

scholarly journals FAST COMPUTATION OF VANILLA PRICES IN TIME-CHANGED MODELS AND IMPLIED VOLATILITIES USING RATIONAL APPROXIMATIONS

2012 ◽  
Vol 15 (04) ◽  
pp. 1250031 ◽  
Author(s):  
MARTIJN PISTORIUS ◽  
JOHANNES STOLTE
Keyword(s):  
Rational Function ◽  
Error Estimates ◽  
Implied Volatility ◽  
Heston Model ◽  
Option Prices ◽  
Transform Method ◽  
Motion Models ◽  
Wide Range ◽  
Black Scholes ◽  
Function Approximations

We present a new numerical method to price vanilla options quickly in time-changed Brownian motion models. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier transform method with respect to accuracy and speed appears to favour the newly developed method in the cases considered. We present error estimates for the option prices. Additionally, we use this method to derive a procedure to compute, for a given set of arbitrage-free European call option prices, the corresponding Black-Scholes implied volatility surface. To achieve this, rational function approximations of the inverse of the Black-Scholes formula are used. We are thus able to work out implied volatilities more efficiently than one can by the use of other common methods. Error estimates are presented for a wide range of parameters.

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