scholarly journals A mean bound financial model and options pricing

2017 ◽  
Vol 04 (04) ◽  
pp. 1750047 ◽  
Author(s):  
Yu Li
Keyword(s):  
Implied Volatility ◽  
Heavy Tails ◽  
Asset Returns ◽  
Mathematical Explanation ◽  
Options Pricing ◽  
Fat Tails ◽  
Volatility Smile ◽  
Financial Model ◽  
Black Scholes Model ◽  
Black Scholes

Most of financial models, including the famous Black–Scholes–Merton options pricing model, rely upon the assumption that asset returns follow a normal distribution. However, this assumption is not justified by empirical data. To be more concrete, the empirical observations exhibit fat tails or heavy tails and implied volatilities against the strike prices demonstrate U-shaped curve resembling a smile, which is the famous volatility smile. In this paper we present a mean bound financial model and show that asset returns per time unit are Pareto distributed and assets are log Gamma distributed under this model. Based on this we study the sensitivity of the options prices to a change in underlying parameters, which are commonly called the Greeks, and derive options pricing formulas. Finally, we reveal the relation between correct volatility and implied volatility in Black–Scholes model and provide a mathematical explanation of volatility smile.

APPLYING THE GENETIC-BASED NEURAL NETWORKS TO VOLATILITY TRADING

2005 ◽  
Vol 01 (02) ◽  
pp. 285-293
Author(s):  
SHINN-WEN WANG
Keyword(s):  
Theoretical Model ◽  
Implied Volatility ◽  
Options Pricing ◽  
Pricing Model ◽  
Options Markets ◽  
Volatility Smile ◽  
Two Phase ◽  
Enterprise Value ◽  
Black Scholes ◽  
Value Estimation

The Black-Scholes options pricing model is widely applied in various options contracts, including contract design, trading, assets evaluation, and enterprise value estimation, etc. Unfortunately, this theoretical model limited by the influences of many unexpected real world phenomena due to six unreasonable assumptions. If we were to soundly take these phenomena into account, the opportunity to gain an excess return would be created. This research therefore combines both the remarkable effects caused by the implied volatility smile (or skew) and the tick-jump discrepancy between the underlying and derivative prices to establish a two-phase options arbitrage model using a genetic-based neural network (GNN). Using evidence from the warrant market in Taiwan, it is shown that the GNN model with arbitrage operations is superior in terms of performance to the original Black-Scholes-based arbitrage model. The GNN model is found to be suitable for application to various options markets as the valuation factors are modified. This paper helps to integrate the theoretical model with important practical considerations.

Option Pricing using Black Scholes Model (Case-Tata Motors and Reliance)

2015 ◽  
Vol 9 (1and2) ◽  
Author(s):  
Ms. Mamta Shah
Keyword(s):  
Stock Price ◽  
Implied Volatility ◽  
Safety Net ◽  
Volatility Smile ◽  
Model Case ◽  
Underlying Asset ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Option Strategies ◽  
Additional Income

The power of options lies in their versatility. It enables the investors to adjust position according to any situation that arises. Options can be speculative or conservative. This means investor can do everything from protecting a position from a decline to outright betting on the movement of a market or index. Options can enable the investor to buy a stock at a lower price, sell a stock at a higher price, or create additional income against a long or short stock position. One can also uses option strategies to profit from a movement in the price of the underlying asset regardless of market direction. the responsible act and safe thing to do. Options provide the same kind of safety net for trades and investments already committed, which is known as hedging. The research paper is based on Black Scholes Model. The study includes the Implied Volatility Test and Volatility Smile Test. This study also includes the solver available in MS Excel. This study is based on stock price of Reliance and Tata Motors.

TERM STRUCTURE OF VANILLA OPTIONS

2007 ◽  
Vol 10 (08) ◽  
pp. 1323-1337
Author(s):  
DORJE C. BRODY ◽  
IRENE C. CONSTANTINOU ◽  
BERNHARD K. MEISTER
Keyword(s):  
Term Structure ◽  
Implied Volatility ◽  
Volatility Smile ◽  
Discount Function ◽  
Forward Rates ◽  
Black Scholes Model ◽  
Initial Term ◽  
Complete Market ◽  
Black Scholes ◽  
Structure Density

Every maturity-dependent derivative contract entails a term structure. For example, when the value of the portfolio consisting of a long position in a stock and a short position in a vanilla option is expressed in units of its instantaneous exercise value, the resulting quantity defines a discount function. Thus, the derivative of the discount function with respect to the time left until maturity defines a term structure density function, and the "hazard rate" associated with the discount function determines the forward rates for the vanilla option portfolio. The dynamics associated with these quantities are obtained in the complete market setting. In particular, one can model vanilla options based on the associated forward rates. The formulation based on forward rates for options extends the approach based on modeling the implied volatility process. As an illustrative example, the initial term structure of the Black–Scholes model is considered. It is shown in this example that the implied volatility smile has the effect of making the option forward rates homogeneous across different strikes.

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.

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 Efficacy of Nifty Index Options through BSM Model

2019 ◽  
Vol 8 (3S2) ◽  
pp. 947-949
Keyword(s):  
Options Pricing ◽  
Price Determination ◽  
Strike Price ◽  
Index Options ◽  
Put Options ◽  
Black Scholes Model ◽  
Exercise Price ◽  
Black Scholes ◽  
Long Time ◽  
Optimum Price

Options are one of the products in financial derivatives, which gives the rights to buy and sell the product to an option holder in pre-fixed price which known as the strike price or exercise price at certain periods. Options contract was existed in various countries for long time, but it became very popular among the investors when the Fisher Black, Myron Scholes and Robert Merton were introduced the Black-Scholes Model in the year of 1973. This model was formerly developed by these three economists who were also receiving the Nobel prize for finding this innovative model. This model is mainly used to deal with the theoretical pricing challenge in options price determination. In India the trading in Index Options commenced on 4th June 2001 and Options on individual securities commenced on 2nd July 2001. There are many types in options contracts like stock options; Index options, weather options, real options and etc. This study has mainly been focusing on Nifty 50 index options which are effectively trade at NSE. This paper goes to describe about the importance of options pricing and how the BSM model has effectively used to find the optimum price of the theoretical value of call and put options.

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.

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