scholarly journals In Honor of the Nobel Laureates Robert C. Merton and Myron S. Scholes: A Partial Differential Equation That Changed the World

1999 ◽  
Vol 13 (4) ◽  
pp. 229-248 ◽  
Author(s):  
Robert A Jarrow
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Option Pricing ◽  
Nobel Prize ◽  
New Method ◽  
Financial Industry ◽  
The World ◽  
Economics Profession ◽  
History Of ◽  
Pricing Formula

The Nobel Prize was given to Robert C. Merton and Myron S. Scholes for discovering a new method for determining the value of an option. This is known as the Black-Merton-Scholes option pricing formula. The purpose of this essay is to explain why the Black-Merton-Scholes option pricing formula is so important to the finance profession, the economics profession, the financial industry, and society at large. This is done by studying the history of the formula's development, the economic logic underlying the formula's derivation, and the formula's ramifications for the various professions.

scholarly journals Fractional Order Stochastic Differential Equation with Application in European Option Pricing

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qing Li ◽  
Yanli Zhou ◽  
Xinquan Zhao ◽  
Xiangyu Ge
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Option Pricing ◽  
Memory Effect ◽  
Fractional Order ◽  
Mean Value ◽  
European Option ◽  
European Option Pricing ◽  
Proposed Model ◽  
Pricing Formula

Memory effect is an important phenomenon in financial systems, and a number of research works have been carried out to study the long memory in the financial markets. In recent years, fractional order ordinary differential equation is used as an effective instrument for describing the memory effect in complex systems. In this paper, we establish a fractional order stochastic differential equation (FSDE) model to describe the effect of trend memory in financial pricing. We, then, derive a European option pricing formula based on the FSDE model and prove the existence of the trend memory (i.e., the mean value function) in the option pricing formula when the Hurst index is between 0.5 and 1. In addition, we make a comparison analysis between our proposed model, the classic Black-Scholes model, and the stochastic model with fractional Brownian motion. Numerical results suggest that our model leads to more accurate and lower standard deviation in the empirical study.

GENERATING SHAPE INVARIANT POTENTIALS

2008 ◽  
Vol 23 (31) ◽  
pp. 4959-4978 ◽  
Author(s):  
ASIM GANGOPADHYAYA ◽  
JEFFRY V. MALLOW
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Quantum Mechanics ◽  
Supersymmetric Quantum Mechanics ◽  
New Method ◽  
Invariance Condition ◽  
Partial Differential ◽  
Shape Invariance ◽  
Shape Invariant

We transform the shape invariance condition, a difference-differential equation of supersymmetric quantum mechanics, into a local partial differential equation. We develop a new method for generating translationally shape invariant potentials from this equation. We generate precisely all the known shape invariant potentials, and argue that there are unlikely to be others.

scholarly journals Analisis Lanjut Metode Beda Hingga Eksplisit Untuk Menentukan Harga Opsi

2019 ◽  
Vol 5 (01) ◽  
pp. 41-46
Author(s):  
Wahyudi Sastro
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Option Pricing ◽  
Difference Method ◽  
A Value ◽  
Black Scholes ◽  
Explicit Finite Difference ◽  
Explicit Finite Difference Method

Abstract. Explicit finite difference method is used to approximate a partial differential equation that is applied to determine the option pricing. The results of this study note that the calculation of option pricing using explicit finite difference method is negative when partition N ≥ 25 with a value of -2.21. Thus, the results of the calculation of option pricing are not convergent and away from the results of analyzing the option pricirng (Black-Scholes) solution. This is because one of the three probabilities Bj = 1- σ2j2Δt  is negative, namely (-0.12) when j ≥ 12 with S ≥ 16.25  (in units). So this explicit finite difference method cannot be used to determine the option pricing. Keywords: Option Pricing, Explicit Finite Difference Method   Abstrak. Metode beda hingga eksplisit digunakan untuk mengaproksimasi suatu persamaan diferensial pasial yang aplikasikan untuk menentukan harga opsi. Hasil penelitian ini diketahui bahwa perhitungan harga opsi dengan menggunakan metode beda hingga eksplisit bernilai negatif pada saat partisi N ≥ 25  dengan nilai -2,21. Dengan demikian, hasil perhitungan harga opsi tidak konvergen dan menjauhi hasil solusi analitik perhitungan harga opsi (Black-Scholes). Hal ini disebabkan karena salah satu ketiga probabilitas Bj = 1- σ2j2Δt yaitu  bernilai negatif yaitu (-0.12) saat j ≥ 12 dengan S ≥ 16.25 (dalam satuan). Sehingga metode beda hingga eksplisit ini tidak dapat digunakan untuk menentukan harga opsi.  Kata Kunci: Harga Opsi, Metode Beda Hingga Eksplisit.

scholarly journals PDE Surface-Represented Facial Blendshapes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2905
Author(s):  
Haibin Fu ◽  
Shaojun Bian ◽  
Ehtzaz Chaudhry ◽  
Shuangbu Wang ◽  
Lihua You ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fourier Series ◽  
Mathematical Expression ◽  
Second Order ◽  
New Method ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Boundary Curves ◽  
Pde Surfaces

Partial differential equation (PDE)-based geometric modelling and computer animation has been extensively investigated in the last three decades. However, the PDE surface-represented facial blendshapes have not been investigated. In this paper, we propose a new method of facial blendshapes by using curve-defined and Fourier series-represented PDE surfaces. In order to develop this new method, first, we design a curve template and use it to extract curves from polygon facial models. Then, we propose a second-order partial differential equation and combine it with the constraints of the extracted curves as boundary curves to develop a mathematical model of curve-defined PDE surfaces. After that, we introduce a generalized Fourier series representation to solve the second-order partial differential equation subjected to the constraints of the extracted boundary curves and obtain an analytical mathematical expression of curve-defined and Fourier series-represented PDE surfaces. The mathematical expression is used to develop a new PDE surface-based interpolation method of creating new facial models from one source facial model and one target facial model and a new PDE surface-based blending method of creating more new facial models from one source facial model and many target facial models. Some examples are presented to demonstrate the effectiveness and applications of the proposed method in 3D facial blendshapes.

scholarly journals Rereading and Studying on The Woman from Sarajevo (Gospodjica)

2021 ◽  
Vol 12 (2) ◽  
pp. 281
Author(s):  
Li Pan
Keyword(s):  
Nobel Prize ◽  
Life Experience ◽  
Mental States ◽  
Single Woman ◽  
World War ◽  
Psychological Study ◽  
Philosophical Reflection ◽  
Artistic Style ◽  
The World ◽  
History Of

The book The Woman from Sarajevo (Gospodjica) is one of the important novels written by Serbian writer Ivo Andric who once lived through the world war and worked in significant department of the country. This novel is not only the product of that period of time but also his only long psychological one which represents his interest in describing the mental states of the main characters. It is a purely psychological study of greed from the point of the pathology and obsession. It also shows his greatness in writing which helps him win the Nobel Prize in literature for his epic force of tracing themes and depicting human destinies drawn from the history of his country. This novel describes the real experience of a single woman named Raica Radakovic from a unique perspective, unfolding the ordinary people’s life and fate in historical tide. It depicts Raica’s life experience objectively, showing the author’s philosophical reflection on people’s life and fate, which makes this novel demonstrate its objective and profound artistic style.

Uncertain Stochastic Option Pricing in the Presence of Uncertain Jumps

2019 ◽  
Vol 27 (04) ◽  
pp. 613-635
Author(s):  
Justin Chirima ◽  
Eriyoti Chikodza ◽  
Senelani Dorothy Hove-Musekwa
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Option Pricing ◽  
Stock Price ◽  
Call Option ◽  
Option Prices ◽  
Jump Size ◽  
Canonical Process ◽  
Option Pricing Model ◽  
Pricing Formula

In this paper, a new differential equation, driven by aleatory and epistemic forms of uncertainty, is introduced and applied to describe the dynamics of a stock price process. This novel class of differential equations is called uncertain stochastic differential equations(USDES) with uncertain jumps. The existence and uniqueness theorem for this class of differential equations is proposed and proved. An appropriate version of the chain rule is derived and applied to solve some examples of USDES with uncertain jumps. The differential equation discussed is applied in an American call option pricing problem. In this problem, it is assumed that the evolution of the stock price is driven by a Brownian motion, the Liu canonical process and an uncertain renewal process. MATLAB is employed for implementing the derived option pricing model. Results show that option prices from the proposed call option pricing formula increase as the jump size increases. As compared to the proposed call option pricing formula, the Black-Scholes overprices options for a certain range of strike prices and under-prices the same options for another range of exercise prices when the jump size is zero.

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