scholarly journals Recursive formula for the double-barrier Parisian stopping time

2018 ◽  
Vol 55 (1) ◽  
pp. 282-301 ◽  
Author(s):  
Angelos Dassios ◽  
Jia Wei Lim
Keyword(s):  
Stopping Time ◽  
Recursive Formula ◽  
Formal Proof ◽  
Computational Method ◽  
Double Barrier ◽  
Parisian Options ◽  
Probabilistic Proof

Abstract In this paper we obtain a recursive formula for the density of the double-barrier Parisian stopping time. We present a probabilistic proof of the formula for the first few steps of the recursion, and then a formal proof using explicit Laplace inversions. These results provide an efficient computational method for pricing double-barrier Parisian options.

scholarly journals Double-Barrier Parisian Options

2011 ◽  
Vol 48 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Angelos Dassios ◽  
Shanle Wu
Keyword(s):  
Brownian Motion ◽  
Markov Model ◽  
Laplace Transform ◽  
Mathematical Finance ◽  
Important Application ◽  
Double Barrier ◽  
Parisian Options ◽  
Brownian Motion With Drift ◽  
The Laplace Transform ◽  
Excursion Time

In this paper we study the excursion time of a Brownian motion with drift outside a corridor by using a four-state semi-Markov model. In mathematical finance, these results have an important application in the valuation of double-barrier Parisian options. We subsequently obtain an explicit expression for the Laplace transform of its price.

scholarly journals Stochastic Recursive Zero-Sum Differential Game and Mixed Zero-Sum Differential Game Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Lifeng Wei ◽  
Zhen Wu
Keyword(s):  
Saddle Point ◽  
Differential Game ◽  
Stopping Time ◽  
Optimal Solution ◽  
Game Problem ◽  
Existence Results ◽  
Double Barrier ◽  
Zero Sum ◽  
Reflected Bsdes ◽  
Theoretical Results

Under the notable Issacs's condition on the Hamiltonian, the existence results of a saddle point are obtained for the stochastic recursive zero-sum differential game and mixed differential game problem, that is, the agents can also decide the optimal stopping time. The main tools are backward stochastic differential equations (BSDEs) and double-barrier reflected BSDEs. As the motivation and application background, when loan interest rate is higher than the deposit one, the American game option pricing problem can be formulated to stochastic recursive mixed zero-sum differential game problem. One example with explicit optimal solution of the saddle point is also given to illustrate the theoretical results.

scholarly journals PRICING DOUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS

2009 ◽  
Vol 12 (01) ◽  
pp. 19-44 ◽  
Author(s):  
CÉLINE LABART ◽  
JÉRÔME LELONG
Keyword(s):  
Laplace Transforms ◽  
Numerical Inversion ◽  
Option Prices ◽  
Double Barrier ◽  
Parisian Options

In this article, we study a double barrier version of the standard Parisian options. We give closed formulas for the Laplace transforms of their prices with respect to the maturity time. We explain how to invert them numerically and prove a result on the accuracy of the numerical inversion when the function to be recovered is sufficiently smooth. Henceforth, we study the regularity of the Parisian option prices with respect to maturity time and prove that except for particular values of the barriers, the prices are of class [Formula: see text] (see Theorem 5.1). This study heavily relies on the existence of a density for the Parisian times, so we have deeply investigated the existence and the regularity of the density for the Parisian times (see Theorem 5.3).

scholarly journals Double-Barrier Parisian Options

2011 ◽  
Vol 48 (01) ◽  
pp. 1-20 ◽  
Author(s):  
Angelos Dassios ◽  
Shanle Wu
Keyword(s):  
Brownian Motion ◽  
Markov Model ◽  
Laplace Transform ◽  
Mathematical Finance ◽  
Important Application ◽  
Double Barrier ◽  
Parisian Options ◽  
Brownian Motion With Drift ◽  
The Laplace Transform ◽  
Excursion Time

In this paper we study the excursion time of a Brownian motion with drift outside a corridor by using a four-state semi-Markov model. In mathematical finance, these results have an important application in the valuation of double-barrier Parisian options. We subsequently obtain an explicit expression for the Laplace transform of its price.

scholarly journals The complexity of the four colour theorem

2010 ◽  
Vol 13 ◽  
pp. 414-425 ◽  
Author(s):  
Cristian S. Calude ◽  
Elena Calude
Keyword(s):  
Complex Systems ◽  
Planar Graph ◽  
Riemann Hypothesis ◽  
Logic Program ◽  
Formal Proof ◽  
Complexity Class ◽  
Computational Method ◽  
Equational Logic ◽  
Four Color Theorem ◽  
Mathematical Problems

AbstractThe four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq [G. Gonthier, ‘Formal proof–the four color theorem’,Notices of Amer. Math. Soc.55 (2008) no. 11, 1382–1393]. In this paper we describe an implementation of the computational method introduced by C. S. Calude and co-workers [Evaluating the complexity of mathematical problems. Part 1’,Complex Systems18 (2009) 267–285; A new measure of the difficulty of problems’,J. Mult. Valued Logic Soft Comput.12 (2006) 285–307] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem is in the complexity class ℭU,4. For comparison, the Riemann hypothesis is in class ℭU,3while Fermat’s last theorem is in class ℭU,1.

Products of 2 × 2 stochastic matrices with random entries

1986 ◽  
Vol 23 (04) ◽  
pp. 1019-1024
Author(s):  
Walter Van Assche
Keyword(s):  
Rate Of Convergence ◽  
Beta Distribution ◽  
Stopping Time ◽  
Stochastic Matrices

The limit of a product of independent 2 × 2 stochastic matrices is given when the entries of the first column are independent and have the same symmetric beta distribution. The rate of convergence is considered by introducing a stopping time for which asymptotics are given.

Type Theory and Formal Proof

2009 ◽  
Author(s):  
Rob Nederpelt ◽  
Herman Geuvers
Keyword(s):  
Type Theory ◽  
Formal Proof
Sign in / Sign up

Export Citation Format

Share Document