scholarly journals Learning agents in Black–Scholes financial markets

2020 ◽  
Vol 7 (10) ◽  
pp. 201188
Author(s):  
Tushar Vaidya ◽  
Carlos Murguia ◽  
Georgios Piliouras
Keyword(s):  
Financial Markets ◽  
Implied Volatility ◽  
Analytical Formula ◽  
Opinion Dynamics ◽  
Option Prices ◽  
Agent Based ◽  
European Option Pricing ◽  
Black Scholes ◽  
Market Practices ◽  
Natural Agent

Black–Scholes (BS) is a remarkable quotation model for European option pricing in financial markets. Option prices are calculated using an analytical formula whose main inputs are strike (at which price to exercise) and volatility. The BS framework assumes that volatility remains constant across all strikes; however, in practice, it varies. How do traders come to learn these parameters? We introduce natural agent-based models, in which traders update their beliefs about the true implied volatility based on the opinions of other agents. We prove exponentially fast convergence of these opinion dynamics, using techniques from control theory and leader-follower models, thus providing a resolution between theory and market practices. We allow for two different models, one with feedback and one with an unknown leader.

Option pricing in a subdiffusive constant elasticity of variance (CEV) model

2019 ◽  
Vol 06 (02) ◽  
pp. 1950018
Author(s):  
Kevin Z. Tong ◽  
Allen Liu
Keyword(s):  
Financial Markets ◽  
Analytical Formula ◽  
Planck Equation ◽  
Option Prices ◽  
Cev Model ◽  
Constant Elasticity ◽  
Constant Elasticity Of Variance ◽  
New Process ◽  
Stable Variable ◽  
Classical Constant

In this paper, we extend the classical constant elasticity of variance (CEV) model to a subdiffusive CEV model, where the underlying CEV process is time changed by an inverse [Formula: see text]-stable subordinator. The new model can capture the subdiffusive characteristics of financial markets. We find the corresponding fractional Fokker–Planck equation governing the PDF of the new process. We also derive the analytical formula for option prices in terms of eigenfunction expansion. This method avoids the evaluation of PDF of an inverse [Formula: see text]-stable variable and also eliminates the need for numerical integration to calculate the option prices. We numerically investigate the sensitivities of the option prices to the key parameters of the newly developed model.

An Empirical Research on the Informational Efficiency of Model Free Implied Volatility

2008 ◽  
Vol 16 (2) ◽  
pp. 67-94
Author(s):  
Byung Kun Rhee ◽  
Sang Won Hwang
Keyword(s):  
Implied Volatility ◽  
Realized Volatility ◽  
Informational Efficiency ◽  
Option Prices ◽  
Index Options ◽  
Model Free ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Korean Market ◽  
Better Than

Black-Scholes Imolied volatility (8SIV) has a few drawbacks. One is that the model Is not much successful in fitting the option prices. and It Is n야 guaranteed the model is correct one. Second. the usual tradition in using the BSIV is that only at-the-money Options are used. It is well-known that IV's of In-the-money or Qut-of-the-money ootions are much different from those estimated from near-the-money options. In this regard, a new model is confronted with Korean market data. Brittenxmes and Neuberger (2000) derive a formula for volatility which is a function of option prices‘ Since the formula is derived without using any option pricing model. volatility estimated from the formula is called model-tree implied volatillty (MFIV). MFIV overcomes the two drawbacks of BSIV. Jiang and Tian (2005) show that. with the S&P index Options (SPX), MFIV is suoerlor to historical volatility (HV) or BSIV in forecasting the future volatllity. In KOSPI 200 index options, when the forecasting performances are compared, MFIV is better than any other estimated volatilities. The hypothesis that MFIV contains all informations for realized volatility and the other volatilities are redundant is oot rejected in any cases.

scholarly journals Markov Chain Monte Carlo Method for Estimating Implied Volatility in Option Pricing

2018 ◽  
Vol 10 (6) ◽  
pp. 108
Author(s):  
Yao Elikem Ayekple ◽  
Charles Kofi Tetteh ◽  
Prince Kwaku Fefemwole
Keyword(s):  
Option Pricing ◽  
Implied Volatility ◽  
Call Option ◽  
Option Prices ◽  
Options Market ◽  
European Call Option ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Real Market ◽  
Independent Path

Using market covered European call option prices, the Independence Metropolis-Hastings Sampler algorithm for estimating Implied volatility in option pricing was proposed. This algorithm has an acceptance criteria which facilitate accurate approximation of this volatility from an independent path in the Black Scholes Model, from a set of finite data observation from the stock market. Assuming the underlying asset indeed follow the geometric brownian motion, inverted version of the Black Scholes model was used to approximate this Implied Volatility which was not directly seen in the real market: for which the BS model assumes the volatility to be a constant. Moreover, it is demonstrated that, the Implied Volatility from the options market tends to overstate or understate the actual expectation of the market. In addition, a 3-month market Covered European call option data, from 30 different stock companies was acquired from Optionistic.Com, which was used to estimate the Implied volatility. This accurately approximate the actual expectation of the market with low standard errors ranging between 0.0035 to 0.0275.

scholarly journals Correcting the Bias in the Practitioner Black-Scholes Method

2019 ◽  
Vol 12 (4) ◽  
pp. 157
Author(s):  
Yun Yin ◽  
Peter G. Moffatt
Keyword(s):  
Linear Regression ◽  
Implied Volatility ◽  
Option Price ◽  
Option Prices ◽  
Technical Problems ◽  
Black Scholes ◽  
Option Values ◽  
Non Linear ◽  
Alternative Means ◽  
Log Linear

We address a number of technical problems with the popular Practitioner Black-Scholes (PBS) method for valuing options. The method amounts to a two-stage procedure in which fitted values of implied volatilities (IV) from a linear regression are plugged into the Black-Scholes formula to obtain predicted option prices. Firstly we ensure that the prediction from stage one is positive by using log-linear regression. Secondly, we correct the bias that results from the transformation applied to the fitted values (i.e., the Black-Scholes formula) being a highly non-linear function of implied volatility. We apply the smearing technique in order to correct this bias. An alternative means of implementing the PBS approach is to use the market option price as the dependent variable and estimate the parameters of the IV equation by the method of non-linear least squares (NLLS). A problem we identify with this method is one of model incoherency: the IV equation that is estimated does not correspond to the set of option prices used to estimate it. We use the Monte Carlo method to verify that (1) standard PBS gives biased option values, both in-sample and out-of-sample; (2) using standard (log-linear) PBS with smearing almost completely eliminates the bias; (3) NLLS gives biased option values, but the bias is less severe than with standard PBS. We are led to conclude that, of the range of possible approaches to implementing PBS, log-linear PBS with smearing is preferred on the basis that it is the only approach that results in valuations with negligible bias.

scholarly journals Pricing Various Types of Power Options under Stochastic Volatility

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1911
Author(s):  
Youngrok Lee ◽  
Yehun Kim ◽  
Jaesung Lee
Keyword(s):  
Financial Markets ◽  
Stochastic Volatility ◽  
Asset Price ◽  
Stochastic Volatility Model ◽  
Exotic Options ◽  
Option Prices ◽  
Black Scholes Model ◽  
Volatility Model ◽  
Black Scholes ◽  
Pricing Power

The exotic options with curved nonlinear payoffs have been traded in financial markets, which offer great flexibility to participants in the market. Among them, power options with the payoff depending on a certain power of the underlying asset price are widely used in markets in order to provide high leverage strategy. In pricing power options, the classical Black–Scholes model which assumes a constant volatility is simple and easy to handle, but it has a limit in reflecting movements of real financial markets. As the alternatives of constant volatility, we focus on the stochastic volatility, finding more exact prices for power options. In this paper, we use the stochastic volatility model introduced by Schöbel and Zhu to drive the closed-form expressions for the prices of various power options including soft strike options. We also show the sensitivity of power option prices under changes in the values of each parameter by calculating the resulting values obtained from the formulas.

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.

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.

ON THE ASYMPTOTICS OF FAST MEAN-REVERSION STOCHASTIC VOLATILITY MODELS

2007 ◽  
Vol 10 (05) ◽  
pp. 817-835 ◽  
Author(s):  
MAX O. SOUZA ◽  
JORGE P. ZUBELLI
Keyword(s):  
Asymptotic Expansion ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Correction Term ◽  
Option Prices ◽  
Stochastic Volatility Models ◽  
Black Scholes Model ◽  
Volatility Models ◽  
Asymptotic Formulae ◽  
Black Scholes

We consider the asymptotic behavior of options under stochastic volatility models for which the volatility process fluctuates on a much faster time scale than that defined by the riskless interest rate. We identify the distinguished asymptotic limits and, in contrast with previous studies, we deal with small volatility-variance (vol-vol) regimes. We derive the corresponding asymptotic formulae for option prices, and find that the first order correction displays a dependence on the hidden state and a non-diffusive terminal layer. Furthermore, this correction cannot be obtained as the small variance limit of the previous calculations. Our analysis also includes the behavior of the asymptotic expansion, when the hidden state is far from the mean. In this case, under suitable hypothesis, we show that the solution behaves as a constant volatility Black–Scholes model to all orders. In addition, we derive an asymptotic expansion for the implied volatility that is uniform in time. It turns out that the fast scale plays an important role in such uniformity. The theory thus obtained yields a more complete picture of the different asymptotics involved under stochastic volatility. It also clarifies the remarkable independence on the state of the volatility in the correction term obtained by previous authors.

Arbitrage-free interpolation of call option prices

2020 ◽  
Vol 37 (1-2) ◽  
pp. 55-78
Author(s):  
Christian Bender ◽  
Matthias Thiel
Keyword(s):  
Implied Volatility ◽  
Reference Model ◽  
Interpolation Method ◽  
Call Option ◽  
Option Prices ◽  
One Dimensional ◽  
Black Scholes Model ◽  
Free Interpolation ◽  
Black Scholes ◽  
Special Case

AbstractIn this paper, we introduce a new interpolation method for call option prices and implied volatilities with respect to the strike, which first generates, for fixed maturity, an implied volatility curve that is smooth and free of static arbitrage. Our interpolation method is based on a distortion of the call price function of an arbitrage-free financial “reference” model of one’s choice. It reproduces the call prices of the reference model if the market data is compatible with the model. Given a set of call prices for different strikes and maturities, we can construct a call price surface by using this one-dimensional interpolation method on every input maturity and interpolating the generated curves in the maturity dimension. We obtain the algorithm of N. Kahalé [An arbitrage-free interpolation of volatilities, Risk 17 2004, 5, 102–106] as a special case, when applying the Black–Scholes model as reference model.

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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