THE FORWARD PDE FOR EUROPEAN OPTIONS ON STOCKS WITH FIXED FRACTIONAL JUMPS

2005 ◽  
Vol 08 (02) ◽  
pp. 239-253 ◽  
Author(s):  
PETER CARR ◽  
ALIREZA JAVAHERI
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Stock Price ◽  
Arrival Rate ◽  
European Options ◽  
Standard Assumption ◽  
Integro Differential Equation ◽  
Local Variance ◽  
Calendar Dates

We derive a partial integro differential equation (PIDE) which relates the price of a calendar spread to the prices of butterfly spreads and the functions describing the evolution of the process. These evolution functions are the forward local variance rate and a new concept called the forward local default arrival rate. We then specialize to the case where the only jump which can occur reduces the underlying stock price by a fixed fraction of its pre-jump value. This is a standard assumption when valuing an option written on a stock which can default. We discuss novel strategies for calibrating to a term and strike structure of European options prices. In particular using a few calendar dates, we derive closed form expressions for both the local variance and the local default arrival rate.

scholarly journals Development and Retrospective Clinical Assessment of a Patient-Specific Closed-Form Integro-Differential Equation Model of Plasma Dilution

2017 ◽  
Vol 8 ◽  
pp. 117959721773030 ◽  
Author(s):  
Glen Atlas ◽  
John K-J Li ◽  
Shawn Amin ◽  
Robert G Hahn
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Human Study ◽  
Patient Specific ◽  
Sine Function ◽  
Integro Differential Equation ◽  
Hyperbolic Sine ◽  
Hyperbolic Cosine ◽  
Group 2 ◽  
Group 1

A closed-form integro-differential equation (IDE) model of plasma dilution (PD) has been derived which represents both the intravenous (IV) infusion of crystalloid and the postinfusion period. Specifically, PD is mathematically represented using a combination of constant ratio, differential, and integral components. Furthermore, this model has successfully been applied to preexisting data, from a prior human study, in which crystalloid was infused for a period of 30 minutes at the beginning of thyroid surgery. Using Euler’s formula and a Laplace transform solution to the IDE, patients could be divided into two distinct groups based on their response to PD during the infusion period. Explicitly, Group 1 patients had an infusion-based PD response which was modeled using an exponentially decaying hyperbolic sine function, whereas Group 2 patients had an infusion-based PD response which was modeled using an exponentially decaying trigonometric sine function. Both Group 1 and Group 2 patients had postinfusion PD responses which were modeled using the same combination of hyperbolic sine and hyperbolic cosine functions. Statistically significant differences, between Groups 1 and 2, were noted with respect to the area under their PD curves during both the infusion and postinfusion periods. Specifically, Group 2 patients exhibited a response to PD which was most likely consistent with a preoperative hypovolemia. Overall, this IDE model of PD appears to be highly “adaptable” and successfully fits clinically-obtained human data on a patient-specific basis, during both the infusion and postinfusion periods. In addition, patient-specific IDE modeling of PD may be a useful adjunct in perioperative fluid management and in assessing clinical volume kinetics, of crystalloid solutions, in real time.

Some Optimal Dividend Problems for a Surplus Process with Stochastic Interest

2010 ◽  
Vol 108-111 ◽  
pp. 1097-1102
Author(s):  
Wen Guang Yu
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Closed Form ◽  
Present Value ◽  
Integro Differential Equation ◽  
Surplus Process ◽  
Dividend Payments ◽  
Optimal Dividend ◽  
Stochastic Interest

In this paper, some results on the dividend payments prior to ruin in the classical surplus process with stochastic interest are derived. An integro-differential equation with a boundary conditions satisfied by the expected present value of dividend payments is derived and solved. Furthermore, closed-form expressions for exponential claims are given.

Abundant closed-form solitons for time-fractional integro–differential equation in fluid dynamics

2021 ◽  
Vol 53 (3) ◽  
Author(s):  
Emad A. Az-Zo’bi ◽  
Wael A. AlZoubi ◽  
Lanre Akinyemi ◽  
Mehmet Şenol ◽  
Islam W. Alsaraireh ◽  
...  
Keyword(s):  
Differential Equation ◽  
Fluid Dynamics ◽  
Closed Form ◽  
Integro Differential Equation

Empirical study for uncertain finance

2021 ◽  
pp. 1-8
Author(s):  
Han Tang ◽  
Wenfei Li
Keyword(s):  
Differential Equation ◽  
Interest Rate ◽  
Empirical Study ◽  
Method Of Moments ◽  
Stock Price ◽  
Uncertain Differential Equation ◽  
European Options ◽  
Financial Model ◽  
Uncertain Finance

Interest rate, stock and option are all important parts of finance. This paper introduces uncertain differential equation to study the evolution of interest rate and stock price separately. Based on actual observations, we estimate the parameters in uncertain differential equation with the method of moments. Using the introduced interest rate and stock models, we price European options and compare the results pricing with actual observations. Finally, a paradox of the stochastic financial model is stated.

Some numerical graphs with successive approximation method of fractional integro–differential equation

2019 ◽  
Vol 8 (4) ◽  
pp. 36
Author(s):  
Samir H. Abbas
Keyword(s):  
Differential Equation ◽  
Successive Approximation ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
Successive Approximation Method ◽  
Integro Differential Equation

This paper studies the existence and uniqueness solution of fractional integro-differential equation, by using some numerical graphs with successive approximation method of fractional integro –differential equation. The results of written new program in Mat-Lab show that the method is very interested and efficient. Also we extend the results of Butris [3].

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

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