scholarly journals On the discrepancy between the objective and risk neutral densities in the pricing of European options

ORiON ◽  
2019 ◽  
Vol 35 (1) ◽  
pp. 33-56
Author(s):  
IJH Visagie ◽  
GL Grobler
Keyword(s):  
Option Price ◽  
Calibration Procedure ◽  
Process Models ◽  
Option Prices ◽  
Underlying Asset ◽  
Discrepancy Measure ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Geometric Levy Process ◽  
Normal Inverse Gaussian

A technique known as calibration is often used when a given option pricing model is fitted to observed financial data. This entails choosing the parameters of the model so as to minimise some discrepancy measure between the observed option prices and the prices calculated under the model in question. This procedure does not take the historical values of the underlying asset into account. In this paper, the density function of the log-returns obtained using the calibration procedure is compared to a density estimate of the observed historical log-returns. Three models within the class of geometric Lévy process models are fitted to observed data; the Black-Scholes model as well as the geometric normal inverse Gaussian and Meixner process models. The numerical results obtained show a surprisingly large discrepancy between the resulting densities when using the latter two models. An adaptation of the calibration methodology is also proposed based on both option price data and the observed historical log-returns of the underlying asset. The implementation of this methodology limits the discrepancy between the densities in question.

scholarly journals Valuing Coca-Cola And PepsiCo Options Using The Black-Scholes Option Pricing Model And Data Downloads From The Internet

2012 ◽  
Vol 8 (6) ◽  
pp. 559-564
Author(s):  
John C. Gardner ◽  
Carl B. McGowan Jr
Keyword(s):  
Option Pricing ◽  
Stock Price ◽  
Option Price ◽  
Option Prices ◽  
Pricing Model ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Risk Free Rate ◽  
Daily Returns ◽  
Free Rate

In this paper, we demonstrate how to collect the data and compute the actual value of Black-Scholes Option Pricing Model call option prices for Coca-Cola and PepsiCo.The data for the current stock price and option price are taken from Yahoo Finance and the daily returns variance is computed from daily prices.The time to maturity is computed as the number of days remaining for the stock option.The risk-free rate is obtained from the U.S. Treasury website.

scholarly journals OPTIONS WRITTEN ON STOCKS WITH KNOWN DIVIDENDS

2004 ◽  
Vol 07 (07) ◽  
pp. 901-907
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Stock Price ◽  
Option Price ◽  
Option Prices ◽  
Strike Price ◽  
Call Options ◽  
Alternative Approach ◽  
Black Scholes ◽  
Market Practice ◽  
The Difference ◽  
Risk Free Rate

There are two common methods for pricing European call options on a stock with known dividends. The market practice is to use the Black–Scholes formula with the stock price reduced by the present value of the dividends. An alternative approach is to increase the strike price with the dividends compounded to expiry at the risk-free rate. These methods correspond to different stock price models and thus in general give different option prices. In the present paper we generalize these methods to time- and level-dependent volatilities and to arbitrary contract functions. We show, for convex contract functions and under very general conditions on the volatility, that the method which is market practice gives the lower option price. For call options and some other common contracts we find bounds for the difference between the two prices in the case of constant volatility.

scholarly journals Nonparametric Estimation of Fractional Option Pricing Model

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Qing Li ◽  
Songlin Liu ◽  
Misi Zhou
Keyword(s):  
Option Pricing ◽  
Stock Exchange ◽  
Option Price ◽  
Integral Function ◽  
The State ◽  
Pricing Model ◽  
State Price ◽  
Black Scholes ◽  
Option Pricing Model ◽  
State Price Density

The establishment of the fractional Black–Scholes option pricing model is under a major condition with the normal distribution for the state price density (SPD) function. However, the fractional Brownian motion is deemed to not be martingale with a long memory effect of the underlying asset, so that the estimation of the state price density (SPD) function is far from simple. This paper proposes a convenient approach to get the fractional option pricing model by changing variables. Further, the option price is transformed as the integral function of the cumulative density function (CDF), so it is not necessary to estimate the distribution function individually by complex approaches. Finally, it encourages to estimate the fractional option pricing model by the way of nonparametric regression and makes empirical analysis with the traded 50 ETF option data in Shanghai Stock Exchange (SSE).

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 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 VOLATILITY SMILE AT THE RUSSIAN OPTION MARKET

2006 ◽  
Vol 7 (1) ◽  
pp. 9-15
Author(s):  
D. Golembiovsky ◽  
I. Baryshnikov
Keyword(s):  
Normal Distribution ◽  
Option Price ◽  
Underlying Asset ◽  
Basic Model ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Basic Purpose ◽  
Log Normal ◽  
In The Beginning ◽  
Log Normal Distribution

The main derivative exchange in Russia is FORTS (Futures and Options in RTS) which is a division of Russian Trade System (RTS). The underlying assets of option contracts are futures on Russian companies’ shares: OJSC “EES"1, OJPC “Lukoil"2 and OJSC “Gazprom"3. A basic model for estimation of fair option price is Black‐Scholes model, developed in the beginning of 70‐s’ years of the last century. This model defines the option premium as a cost of its hedging by underlying asset. It uses a number of assumptions: prices of underlying assets follow log‐normal distribution; hedging is accomplished continuously; an underlying asset is infinitely divisible; a volatility is constant on all period of option life. However, according to practice, prices of shares and futures do not follow normal or log‐normal distribution, a volatility can change during a life of option, and hedging is a discrete process. Thus, Black‐Scholes model can yield inexact results in real markets, especially it concerns deeply “in the money” or deeply “out of the money” options. The basic purpose of the paper is to investigate opportunities to apply Black‐Scholes model for an estimation of option premiums in the Russian market.

BOUNDS ON PRICES FOR ASIAN OPTIONS VIA FOURIER METHODS

2016 ◽  
Vol 57 (3) ◽  
pp. 299-318
Author(s):  
SCOTT ALEXANDER ◽  
ALEXANDER NOVIKOV ◽  
NINO KORDZAKHIA
Keyword(s):  
Asian Options ◽  
Option Prices ◽  
Underlying Asset ◽  
Fourier Methods ◽  
The Core ◽  
Numerical Studies ◽  
Highly Correlated ◽  
Joint Characteristic ◽  
Core Idea ◽  
Geometric Levy Process

The problem of pricing arithmetic Asian options is nontrivial, and has attracted much interest over the last two decades. This paper provides a method for calculating bounds on option prices and approximations to option deltas in a market where the underlying asset follows a geometric Lévy process. The core idea is to find a highly correlated, yet more tractable proxy to the event that the option finishes in-the-money. The paper provides a means for calculating the joint characteristic function of the underlying asset and proxy processes, and relies on Fourier methods to compute prices and deltas. Numerical studies show that the lower bound provides accurate approximations to prices and deltas, while the upper bound provides good though less accurate results.

scholarly journals Hedging and pricing illiquid options with market impacts

2017 ◽  
Vol 04 (02n03) ◽  
pp. 1750030
Author(s):  
Taiga Saito
Keyword(s):  
Option Price ◽  
Critical Issue ◽  
Option Prices ◽  
Risk Minimization ◽  
Hedging Strategy ◽  
Time Model ◽  
Underlying Asset ◽  
Market Impacts ◽  
Local Risk ◽  
Liquidity Costs

In this paper, we consider hedging and pricing of illiquid options on an untradable underlying asset, where an alternative asset is used as a hedging instrument. Particularly, we consider the situation where the trade price of the hedging instrument is subject to market impacts caused by the hedger and the liquidity costs paid as a spread from the mid price. Pricing illiquid options, which often appears in trading of structured products, is a critical issue in practice because of its difficulties in hedging mainly due to untradability of the underlying asset as well as the liquidity costs and market impacts of the hedging instrument. First, by setting the problem under a discrete time model, where the optimal hedging strategy is defined by the local risk-minimization, we present algorithms to obtain the option price along with the hedging strategy by an asymptotic expansion. Moreover, we provide numerical examples. This model enables the estimation of the effect of both the market impacts and the liquidity costs on option prices, which is important in practice.

scholarly journals DUPIRE'S EQUATION FOR BUBBLES

2012 ◽  
Vol 15 (06) ◽  
pp. 1250041 ◽  
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Option Price ◽  
Local Martingale ◽  
Option Prices ◽  
Uniqueness Of Solutions ◽  
Underlying Asset ◽  
Put Options ◽  
Price Process ◽  
Usual Form ◽  
Volatility Models ◽  
Strict Local Martingale

We study Dupire's equation for local volatility models with bubbles, i.e. for models in which the discounted underlying asset follows a strict local martingale. If option prices are given by risk-neutral valuation, then the discounted option price process is a true martingale, and we show that the Dupire equation for call options contains extra terms compared to the usual equation. However, the Dupire equation for put options takes the usual form. Moreover, uniqueness of solutions to the Dupire equation is lost in general, and we show how to single out the option price among all possible solutions. The Dupire equation for models in which the discounted derivative price process is merely a local martingale is also studied.

scholarly journals The Performance of Skewness and Kurtosis Adjusted Option Pricing Model in Emerging Markets

2016 ◽  
Vol 5 (3) ◽  
pp. 70-84
Author(s):  
Özge Sezgin Alp
Keyword(s):  
Emerging Markets ◽  
Option Pricing ◽  
Option Prices ◽  
Pricing Model ◽  
European Options ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Skewness And Kurtosis ◽  
Pricing Formula

In this study, the option pricing performance of the adjusted Black-Scholes model proposed by Corrado and Su (1996) and corrected by Brown and Robinson (2002), is investigated and compared with original Black Scholes pricing model for the Turkish derivatives market. The data consist of the European options written on BIST 30 index extends from January 02, 2015 to April 24, 2015 for given exercise prices with maturity April 30, 2015. In this period, the strike prices are ranging from 86 to 124. To compare the models, the implied parameters are derived by minimizing the sum of squared deviations between the observed and theoretical option prices. The implied distribution of BIST 30 index does not significantly deviate from normal distribution. In addition, pricing performance of Black Scholes model performs better in most of the time. Black Scholes pricing Formula, Carrado-Su pricing Formula, Implied Parameters

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