Existence of Unbiased Estimators of the Black/Scholes Option Price, Other Derivatives, and Hedge Ratios

1997 ◽  
Vol 13 (6) ◽  
pp. 791-807 ◽  
Author(s):  
John L Knight ◽  
Stephen E. Satchell
Keyword(s):  
Option Price ◽  
Statistical Bias ◽  
Unbiased Estimators ◽  
Black Scholes ◽  
Hedge Ratios

In this paper, we reexamine the question of statistical bias in the classic Black/Scholes option price where randomness is due to the use of the historical variance. We show that the only unbiased estimated option is an at the money option.

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 The Black-Scholes Model Guideline For Options Course As Taught At Notre Dame University - Lebanon

2011 ◽  
Vol 4 (1) ◽  
Author(s):  
Viviane Y. Naimy
Keyword(s):  
Option Price ◽  
Stochastic Calculus ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Calculus I ◽  
Relationship Of ◽  
The Relationship

This paper presents the methodology used for Notre Dame University’s finance students to explain and explore the Black-Scholes model without going through the complexity of mathematics to model random movements or through stochastic calculus. I will name and develop the steps that I follow in order to allow students to properly use the Black-Scholes model and to understand the relationship of the model’s inputs to the option price while monitoring the risk via delta and gamma hedging.

Distributed Parameter Systems Control and Its Applications to Financial Engineering

2018 ◽  
pp. 15-35
Author(s):  
Gerasimos Rigatos ◽  
Pierluigi Siano
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Option Price ◽  
Distributed Parameter ◽  
Specific Reference ◽  
Differential Flatness ◽  
Differential Flatness Theory ◽  
Financial Model ◽  
Performance Indexes ◽  
Black Scholes

The chapter analyzes differential flatness theory for the control of single asset and multi-asset option price dynamics, described by PDE models. Through these control methods, stabilization of distributed parameter (PDE modelled) financial systems is achieved and convergence to specific financial performance indexes is made possible. The main financial model used in the chapter is the Black-Scholes PDE. By applying semi-discretization and a finite differences scheme the single-asset (equivalently multi-asset) Black-Scholes PDE is transformed into a state-space model consisting of ordinary nonlinear differential equations. For this set of differential equations it is shown that differential flatness properties hold. This enables to solve the associated control problem and to stabilize the options' dynamics. By showing the feasibility of control of the single-asset (equivalently multi-asset) Black-Scholes PDE it is proven that through selected purchases and sales during the trading procedure, the price of options can be made to converge and stabilize at specific reference values.

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).

Valuation of Covered Warrant Subject to Default Risk

2003 ◽  
Vol 06 (01) ◽  
pp. 21-44 ◽  
Author(s):  
Shen-Yuan Chen
Keyword(s):  
Default Risk ◽  
Market Price ◽  
Option Price ◽  
Option Value ◽  
Valuation Model ◽  
Black Scholes ◽  
Covered Warrant ◽  
The Difference ◽  
Vulnerable Option ◽  
Settlement Mechanism

There is no margin settlement mechanism for existing covered warrants in Taiwan, thus the credit risk of the warrant issuer must be considered when investors evaluate the price of a covered warrant. This paper applies the vulnerable option valuation model to empirically study the difference in the theoretical value of a vulnerable warrant, Black–Scholes option price and the market price of warrant by using the Taiwan warrant data. Empirical results show that the theoretical value of a vulnerable warrant is lower than the Black–Scholes non-vulnerable option value and its market value.

Abstract: Option Valuation Models–Some Implications of Parameter Estimation

1977 ◽  
Vol 12 (4) ◽  
pp. 667-667
Author(s):  
P. P. Boyle ◽  
A. L. Ananthanarayan
Keyword(s):  
Parameter Estimation ◽  
Option Pricing ◽  
Stock Prices ◽  
Unbiased Estimate ◽  
Estimation Error ◽  
Option Price ◽  
Nonlinear Function ◽  
Option Valuation ◽  
Black Scholes ◽  
Parameter Values

The Black-Scholes option pricing formula assumes that the variance of the return on the underlying stock is known with certainty. In practice an estimate of the variance, based on a sample of historical stock prices, is used. The estimation error in the variance induces error in the option price. Since the option price is a nonlinear function of the variance, an unbiased estimate of the variance does not produce an unbiased estimate of the option price. For reasonable parameter values, it is shown that the magnitude of the bias is not large.

PENENTUAN HARGA OPSI DENGAN MODEL BLACK-SCHOLES MENGGUNALKAN METODE BEDA HINGGA FORWARD TIME CENTRAL SPACE

2018 ◽  
Vol 1 (1) ◽  
pp. 45
Author(s):  
Werry Febrianti
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Option Price ◽  
Call Option ◽  
Put Option ◽  
Daily Data ◽  
Black Scholes ◽  
One Year ◽  
The Right

Option can be defined as a contract between two sides/parties said party one and party two. Party one has the right to buy or sell of stock to party two. Party two can invest by observe the put option price or call option price on a time period in the option contract. Black-Scholes option solution using finite difference method based on forward time central space (FTCS) can be used as the reference for party two in the investment determining. Option price determining by using Black-Scholes was applied on Samsung stock (SSNLF) by using finite difference method FTCS. Daily data of Samsung stock in one year was processed to obtain the volatility of the stock. Then, the call option and put option are calculated by using FTCS method after discretization on the Black-Scholes model. The value of call option was obtained as $1.457695030014260 and the put option value was obtained as $1.476925604670225.

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