scholarly journals Occupation time distributions for the telegraph process

2011 ◽  
Vol 121 (8) ◽  
pp. 1816-1844 ◽  
Author(s):  
Leonid Bogachev ◽  
Nikita Ratanov
Keyword(s):  
Occupation Time ◽  
Telegraph Process

THE FEYNMAN–KAC FORMULA AND PRICING OCCUPATION TIME DERIVATIVES

1999 ◽  
Vol 02 (02) ◽  
pp. 153-178 ◽  
Author(s):  
JULIEN-N. HUGONNIER
Keyword(s):  
Brownian Motion ◽  
Time Derivative ◽  
Occupation Time ◽  
Barrier Options ◽  
Occupation Times

In this paper, we undertake a study of occupation time derivatives that is derivatives for which the pay-off is contingent on both the terminal asset's price and one of its occupation times. To this end we use a formula of M. Kac to compute the joint law of Brownian motion and one of its occupation times. General pricing formulas for occupation time derivatives are established and it is shown that any occupation time derivative can be continuously hedged by a controlled portfolio of the basic securities. We further study some examples of interest including cumulative barrier options and discuss some numerical implementations.

scholarly journals Option Pricing Driven by a Telegraph Process with Random Jumps

2012 ◽  
Vol 49 (3) ◽  
pp. 838-849 ◽  
Author(s):  
Oscar López ◽  
Nikita Ratanov
Keyword(s):  
Option Pricing ◽  
Financial Market ◽  
Option Prices ◽  
Martingale Measures ◽  
Market Models ◽  
Telegraph Process ◽  
Equivalent Martingale Measures ◽  
Hedging Strategies ◽  
Random Jumps ◽  
Equivalent Martingale

In this paper we propose a class of financial market models which are based on telegraph processes with alternating tendencies and jumps. It is assumed that the jumps have random sizes and that they occur when the tendencies are switching. These models are typically incomplete, but the set of equivalent martingale measures can be described in detail. We provide additional suggestions which permit arbitrage-free option prices as well as hedging strategies to be obtained.

Bose-Einstein-type statistics, order statistics and planar random motions with three directions

2004 ◽  
Vol 36 (3) ◽  
pp. 937-970 ◽  
Author(s):  
S. Leorato ◽  
E. Orsingher
Keyword(s):  
Partial Differential Equations ◽  
Order Statistics ◽  
Constant Velocity ◽  
Uniform Motion ◽  
Change Of Direction ◽  
Telegraph Process ◽  
Cyclic Motion ◽  
Different Types ◽  
Random Motions ◽  
Bose Einstein

In this paper we study different types of planar random motions (performed with constant velocity) with three directions, defined by the vectors dj = (cos(2πj/3), sin(2πj/3)) for j = 0, 1, 2, changing at Poisson-paced times. We examine the cyclic motion (where the change of direction is deterministic), the completely uniform motion (where at each Poisson event each direction can be taken with probability ) and the symmetrically deviating case (where the particle can choose all directions except that taken before the Poisson event). For each of the above random motions we derive the explicit distribution of the position of the particle, by using an approach based on order statistics. We prove that the densities obtained are solutions of the partial differential equations governing the processes. We are also able to give the explicit distributions on the boundary and, for the case of the symmetrically deviating motion, we can write it as the distribution of a telegraph process. For the symmetrically deviating motion we use a generalization of the Bose-Einstein statistics in order to determine the distribution of the triple (N0, N1, N2) (conditional on N(t) = k, with N0 + N1 + N2 = N(t) + 1, where N(t) is the number of Poisson events in [0, t]), where Nj denotes the number of times the direction dj (j = 0, 1, 2) is taken. Possible extensions to four directions or more are briefly considered.

scholarly journals Transformations of Telegraph Processes and Their Financial Applications

Risks ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 147
Author(s):  
Anatoliy A. Pogorui ◽  
Anatoliy Swishchuk ◽  
Ramón M. Rodríguez-Dagnino
Keyword(s):  
Differential Equation ◽  
Stock Prices ◽  
Option Prices ◽  
Linear Transformations ◽  
Financial Applications ◽  
Telegraph Process ◽  
Put Option ◽  
New Models ◽  
Non Linear ◽  
New Applications

In this paper, we consider non-linear transformations of classical telegraph process. The main results consist of deriving a general partial differential Equation (PDE) for the probability density (pdf) of the transformed telegraph process, and then presenting the limiting PDE under Kac’s conditions, which may be interpreted as the equation for a diffusion process on a circle. This general case includes, for example, classical cases, such as limiting diffusion and geometric Brownian motion under some specifications of non-linear transformations (i.e., linear, exponential, etc.). We also give three applications of non-linear transformed telegraph process in finance: (1) application of classical telegraph process in the case of balance, (2) application of classical telegraph process in the case of dis-balance, and (3) application of asymmetric telegraph process. For these three cases, we present European call and put option prices. The novelty of the paper consists of new results for non-linear transformed classical telegraph process, new models for stock prices based on transformed telegraph process, and new applications of these models to option pricing.

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