scholarly journals Robust option replication for a Black-Scholes model extended with nondeterministic trends

1999 ◽  
Vol 12 (2) ◽  
pp. 113-120 ◽  
Author(s):  
John G. M. Schoenmakers ◽  
Peter E. Kloeden
Keyword(s):  
Statistical Analysis ◽  
Stock Prices ◽  
Integration Theory ◽  
Long Range Dependence ◽  
Stochastic Integration ◽  
Classical Concept ◽  
European Option ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Replicating Portfolio

Statistical analysis on various stocks reveals long range dependence behavior of the stock prices that is not consistent with the classical Black and Scholes model. This memory or nondeterministic trend behavior is often seen as a reflection of market sentiments and causes that the historical volatility estimator becomes unreliable in practice. We propose an extension of the Black and Scholes model by adding a term to the original Wiener term involving a smoother process which accounts for these effects. The problem of arbitrage will be discussed. Using a generalized stochastic integration theory [8], we show that it is possible to construct a self financing replicating portfolio for a European option without any further knowledge of the extension and that, as a consequence, the classical concept of volatility needs to be re-interpreted.

scholarly journals FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE

2003 ◽  
Vol 06 (01) ◽  
pp. 1-32 ◽  
Author(s):  
YAOZHONG HU ◽  
BERNT ØKSENDAL
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Standard Brownian Motion ◽  
Stochastic Integration ◽  
European Option ◽  
No Arbitrage ◽  
Black Scholes ◽  
White Noise Calculus ◽  
Replicating Portfolio

The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Itô type of stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1. We show that if we use an Itô type of stochastic integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Itô fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our Itô fractional Black–Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t).

SIMULTANEOUS OPTION AND STOCK PRICES: ANOTHER LOOK AT THE BLACK-SCHOLES MODEL

1982 ◽  
Vol 17 (2) ◽  
pp. 68-68
Author(s):  
Thomas J. O'Brien ◽  
William F. Kennedy
Keyword(s):  
Stock Prices ◽  
Black Scholes Model ◽  
Black Scholes

A Binary Option Pricing Based on Fuzziness

2014 ◽  
Vol 13 (06) ◽  
pp. 1211-1227 ◽  
Author(s):  
Masatoshi Miyake ◽  
Hiroshi Inoue ◽  
Jianming Shi ◽  
Tetsuya Shimokawa
Keyword(s):  
Monte Carlo ◽  
Option Pricing ◽  
Primary Role ◽  
Pricing Model ◽  
European Option ◽  
Over The Counter ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Fuzzy Boundary

In pricing for European option Black–Scholes model has been widely used in various fields in which the model can be applied under appropriate conditions. In this paper, we discuss a binary option, which is popular in OTC (Over the Counter) market for hedging and speculation. In particular, asset-or-nothing option is basic for any other options but gives essential implications for constructing more complex option products. In addition to the primary role of the asset-or-nothing option, another availability of the option is considered by introducing fuzzy concept. Therefore, the uncertainty which an investor and intermediary usually have in their minds is incorporated in the pricing model. Thus, the model is described with fuzzy boundary conditions and applied to the conventional binary option, proposing more useful and actual pricing way of the option. This methodology with the analysis is examined, comparing with Monte Carlo simulations.

scholarly journals A EUROPEAN OPTION GENERAL FIRST-ORDER ERROR FORMULA

2013 ◽  
Vol 54 (4) ◽  
pp. 248-272 ◽  
Author(s):  
GUILLAUME LEDUC
Keyword(s):  
Random Walks ◽  
European Security ◽  
European Option ◽  
Order Error ◽  
First Order ◽  
Error Formula ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Option Values ◽  
Payoff Functions

AbstractWe study the value of European security derivatives in the Black–Scholes model when the underlying asset $\xi $ is approximated by random walks ${\xi }^{(n)} $. We obtain an explicit error formula, up to a term of order $ \mathcal{O} ({n}^{- 3/ 2} )$, which is valid for general approximating schemes and general payoff functions. We show how this error formula can be used to find random walks ${\xi }^{(n)} $ for which option values converge at a speed of $ \mathcal{O} ({n}^{- 3/ 2} )$.

AN APPLICATION OF MELLIN TRANSFORM TECHNIQUES TO A BLACK–SCHOLES EQUATION PROBLEM

2007 ◽  
Vol 05 (01) ◽  
pp. 51-66 ◽  
Author(s):  
MARIANITO R. RODRIGO ◽  
ROGEMAR S. MAMON
Keyword(s):  
Existence And Uniqueness ◽  
Mellin Transform ◽  
Formal Solution ◽  
Option Price ◽  
Contingent Claims ◽  
Time Dependent ◽  
European Option ◽  
Equation Problem ◽  
Black Scholes Model ◽  
Black Scholes

In this article, we use a Mellin transform approach to prove the existence and uniqueness of the price of a European option under the framework of a Black–Scholes model with time-dependent coefficients. The formal solution is rigorously shown to be a classical solution under quite general European contingent claims. Specifically, these include claims that are bounded and continuous, and claims whose difference with some given but arbitrary polynomial is bounded and continuous. We derive a maximum principle and use it to prove uniqueness of the option price. An extension of the put-call parity which relates the aforementioned two classes of claims is also given.

scholarly journals VALUATION OF EUROPEAN PUT OPTION BY USING THE QUADRATURE METHOD UNDER THE VARIANCE GAMMA PROCESS

2020 ◽  
Vol 4 (5) ◽  
pp. 1-5
Author(s):  
Akash Singh ◽  
Ravi Gor Gor ◽  
Rinku Patel
Keyword(s):  
Quadrature Method ◽  
European Option ◽  
Brownian Motion Process ◽  
Asset Pricing Model ◽  
Pricing Options ◽  
Put Option ◽  
Variance Gamma Process ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Dynamic Asset Pricing

Dynamic asset pricing model uses the Geometric Brownian Motion process. The Black-Scholes model known as standard model to price European option based on the assumption that underlying asset prices dynamic follows that log returns of asset is normally distributed. In this paper, we introduce a new stochastic process called levy process for pricing options. In this paper, we use the quadrature method to solve a numerical example for pricing options in the Indian context. The illustrations used in this paper for pricing the European style option.  We also try to develop the pricing formula for European put option by using put-call parity and check its relevancy on actual market data and observe some underlying phenomenon.

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