scholarly journals Asian Option Pricing with Transaction Costs and Dividends under the Fractional Brownian Motion Model

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yan Zhang ◽  
Di Pan ◽  
Sheng-Wu Zhou ◽  
Miao Han
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Transaction Costs ◽  
Fractional Brownian Motion ◽  
Asian Option ◽  
Partial Differential ◽  
No Arbitrage ◽  
Geometric Average ◽  
Brownian Motion Model

The pricing problem of geometric average Asian option under fractional Brownian motion is studied in this paper. The partial differential equation satisfied by the option’s value is presented on the basis of no-arbitrage principle and fractional formula. Then by solving the partial differential equation, the pricing formula and call-put parity of the geometric average Asian option with dividend payment and transaction costs are obtained. At last, the influences of Hurst index and maturity on option value are discussed by numerical examples.

PRICING DIGITAL OUTPERFORMANCE OPTIONS WITH UNCERTAIN CORRELATION

2011 ◽  
Vol 14 (05) ◽  
pp. 709-722 ◽  
Author(s):  
JACINTO MARABEL
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Geometric Brownian Motion ◽  
Correlation Model ◽  
Partial Differential ◽  
Analytic Expression

Multi-asset options exhibit sensitivity to the correlations between the underlying assets and these correlations are notoriously unstable. Moreover, some of these options such as the digital outperformance options, have a cross-gamma that changes sign depending on the relative evolution of the underlying assets. In this paper, I present a model to price digital outperformance options when there is uncertainty about correlation, but it is assumed to lie within a certain range. Under the assumption that assets prices follow a Geometric Brownian motion with constant instantaneous volatilities I present an analytic expression for the price of the digital outperformance option under the constant correlation assumption, as well as the partial differential equation corresponding to the uncertain correlation model. The comparison of the prices obtained using both models shows that there is no constant correlation which allows attaining the price obtained under the uncertain correlation model. This fact shows that it can be dangerous to assume a constant instantaneous correlation for products with a cross-gamma that changes sign.

SOME RESULTS ON PARTIAL DIFFERENTIAL EQUATIONS AND ASIAN OPTIONS

2001 ◽  
Vol 11 (03) ◽  
pp. 475-497 ◽  
Author(s):  
E. BARUCCI ◽  
S. POLIDORO ◽  
V. VESPRI
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Three Dimensions ◽  
Asian Options ◽  
Partial Differential ◽  
No Arbitrage ◽  
Strongly Degenerate ◽  
Degenerate Partial Differential Equations

We analyze partial differential equations arising in the evaluation of Asian options. The equations are strongly degenerate partial differential equations in three dimensions. We show that the solution of the no-arbitrage partial differential equation is sufficiently regular and standard numerical methods can be employed to approximate it.

ASIAN OPTIONS WITH THE AMERICAN EARLY EXERCISE FEATURE

1999 ◽  
Vol 02 (01) ◽  
pp. 101-111 ◽  
Author(s):  
LIXIN WU ◽  
YUE KUEN KWOK ◽  
HONG YU
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Integral Representation ◽  
Asian Option ◽  
Asian Options ◽  
Partial Differential ◽  
Early Exercise ◽  
Equation Formulation ◽  
Option Model ◽  
Early Exercise Premium

By appropriate scaling of the variables, the reduction in the dimensionality of the partial differential equation formulation of an American-style Asian option model is achieved. The integral representation of the early exercise premium can be obtained in a succinct manner. The exercise policy of Asian options with the early exercise provision can then be examined.

scholarly journals A LOGISTIC NONLINEAR BLACK-SCHOLES-MERTON PARTIAL DIFFERENTIAL EQUATION: EUROPEAN OPTION

2018 ◽  
Vol 6 (6) ◽  
pp. 480-487
Author(s):  
Joseph Otula Nyakinda
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Stock Price ◽  
Option Price ◽  
Logistic Models ◽  
Geometric Brownian Motion ◽  
Partial Differential ◽  
Illiquid Markets ◽  
Black Scholes

Nonlinear Black-Scholes equations provide more accurate values by taking into account more realistic assumptions, such as transaction costs, illiquid markets, risks from an unprotected portfolio or large investor's preferences, which may have an impact on the stock price, the volatility, the drift and the option price itself. Most modern models are represented by nonlinear variations of the well-known Black-Scholes Equation. On the other hand, asset security prices may naturally not shoot up indefinitely (exponentially) leading to the use of Verhulst’s Logistic equation. The objective of this study was to derive a Logistic Nonlinear Black Scholes Merton Partial Differential equation by incorporating the Logistic geometric Brownian motion. The methodology involves, analysis of the geometric Brownian motion, review of logistic models, process and lemma, stochastic volatility models and the derivation of the linear and nonlinear Black-Scholes-Merton partial differential equation. Illiquid markets have also been analyzed alongside stochastic differential equations. The result of this study may enhance reliable decision making based on a rational prediction of the future asset prices given that in reality the stock market may depict a nonlinear pattern.

scholarly journals Pricing Currency Option in a Mixed Fractional Brownian Motion with Jumps Environment

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Foad Shokrollahi ◽  
Adem Kılıçman
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Exchange Rate ◽  
Fractional Brownian Motion ◽  
Fractional Partial Differential Equation ◽  
Currency Option ◽  
Mixed Fractional Brownian Motion ◽  
Spot Exchange Rate ◽  
New Framework

A new framework for pricing the European currency option is developed in the case where the spot exchange rate fellows a mixed fractional Brownian motion with jumps. The jump mixed fractional partial differential equation is obtained. Some Greeks and properties volatility are discussed. Finally the numerical simulations illustrate that our model is flexible and easy to implement.

scholarly journals Asian Option Pricing with Monotonous Transaction Costs under Fractional Brownian Motion

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Di Pan ◽  
Shengwu Zhou ◽  
Yan Zhang ◽  
Miao Han
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Transaction Cost ◽  
Asian Option ◽  
Option Value ◽  
Cost Rate ◽  
Geometric Average ◽  
Option Pricing Model ◽  
Analytical Expressions

Geometric-average Asian option pricing model with monotonous transaction cost rate under fractional Brownian motion was established. The method of partial differential equations was used to solve this model and the analytical expressions of the Asian option value were obtained. The numerical experiments show that Hurst exponent of the fractional Brownian motion and transaction cost rate have a significant impact on the option value.

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