PRICING FOR GEOMETRIC MARKED POINT PROCESSES UNDER PARTIAL INFORMATION: ENTROPY APPROACH

2009 ◽  
Vol 12 (02) ◽  
pp. 179-207 ◽  
Author(s):  
CLAUDIA CECI ◽  
ANNA GERARDI
Keyword(s):  
Stock Price ◽  
Point Processes ◽  
Partial Information ◽  
Marked Point ◽  
Asset Price ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Martingale Measure ◽  
Minimal Entropy Martingale Measure ◽  
Risk Neutral Measures

The problem of the arbitrage-free pricing of a European contingent claim B is considered in a general model for intraday stock price movements in the case of partial information. The dynamics of the risky asset price is described through a marked point process Y, whose local characteristics depend on some unobservable jump diffusion process X. The processes Y and X may have common jump times, which means that the trading activity may affect the law of X and could be also related to the presence of catastrophic events. Risk-neutral measures are characterized and in particular, the minimal entropy martingale measure is studied. The problem of pricing under restricted information is discussed, and the arbitrage-free price of the claim B w.r.t. the minimal entropy martingale measure is computed by using filtering techniques.

RISK-NEUTRAL MEASURES AND PRICING FOR A PURE JUMP PRICE PROCESS

2009 ◽  
Vol 24 (1) ◽  
pp. 47-76 ◽  
Author(s):  
Anna Gerardi ◽  
Paola Tardelli
Keyword(s):  
Point Processes ◽  
Marked Point ◽  
Asset Price ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Martingale Measure ◽  
Price Process ◽  
Risk Neutral Measures

This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process, modeled again by a marked point process. Taking into account the presence of catastrophic events, the possibility of common jump times between the risky asset price process and the arrivals process is allowed. By setting and solving a suitable control problem, the characterization of the minimal entropy martingale measure is obtained. In a particular case, a pricing problem is also discussed.

scholarly journals Partially informed investors: hedging in an incomplete market with default

2015 ◽  
Vol 52 (03) ◽  
pp. 718-735 ◽  
Author(s):  
P. Tardelli
Keyword(s):  
Point Processes ◽  
Value Function ◽  
Partial Information ◽  
Marked Point ◽  
Asset Price ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Control Problems ◽  
The Value Function

In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.

scholarly journals Partially informed investors: hedging in an incomplete market with default

2015 ◽  
Vol 52 (3) ◽  
pp. 718-735
Author(s):  
P. Tardelli
Keyword(s):  
Point Processes ◽  
Value Function ◽  
Partial Information ◽  
Marked Point ◽  
Asset Price ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Control Problems ◽  
The Value Function

In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.

UTILITY MAXIMIZATION IN A PURE JUMP MODEL WITH PARTIAL OBSERVATION

2010 ◽  
Vol 25 (1) ◽  
pp. 29-54 ◽  
Author(s):  
Paola Tardelli
Keyword(s):  
Utility Maximization ◽  
Point Processes ◽  
Marked Point ◽  
Asset Price ◽  
Main Tool ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Underlying Event ◽  
Jump Model

This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process—a marked point process having common jump times with the risky asset price process. The problem of utility maximization of terminal wealth is dealt with when the underlying event arrivals process is assumed to be unobserved by the market agents using, as the main tool, backward stochastic differential equations. The dual problem is studied. Explicit solutions in a particular case are given.

A MODEL FOR HIGH FREQUENCY DATA UNDER PARTIAL INFORMATION: A FILTERING APPROACH

2006 ◽  
Vol 09 (04) ◽  
pp. 555-576 ◽  
Author(s):  
CLAUDIA CECI ◽  
ANNA GERARDI
Keyword(s):  
Point Process ◽  
Stock Price ◽  
Asset Prices ◽  
Partial Information ◽  
Marked Point ◽  
Recursive Algorithm ◽  
Asset Price ◽  
Linearization Method ◽  
Marked Point Process ◽  
Jump Intensity

A general model for intraday stock price movements is studied. The asset price dynamics is described by a marked point process Y, whose local characteristics (in particular the jump-intensity) depend on some unobservable hidden state variable X. The dynamics of Y and X may be strongly dependent. In particular the two processes may have common jump times, which means that the actual trading activity may affect the law of X and could be also related to the possibility of catastrophic events. The agents, in this model, are restricted to observing past asset prices. This leads to a filtering problem with marked point process observations. The conditional law of X given the past asset prices (the filter) is characterized as the unique weak solution of the Kushner–Stratonovich equation. An explicit representation of the filter is obtained by the Feyman–Kac formula using a linearization method. This representation allows us to provide a recursive algorithm for the filter computation.

Marked point processes as limits of Markovian arrival streams

1993 ◽  
Vol 30 (02) ◽  
pp. 365-372 ◽  
Author(s):  
Søren Asmussen ◽  
Ger Koole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
The State ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
State Transitions ◽  
Poisson Arrival ◽  
Convergence Of Moments ◽  
Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

Reliability analysis of complex repairable systems by means of marked point processes

1980 ◽  
Vol 17 (1) ◽  
pp. 154-167 ◽  
Author(s):  
Peter Franken ◽  
Arnfried Streller
Keyword(s):  
Reliability Analysis ◽  
Point Processes ◽  
Marked Point ◽  
Regenerative Process ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Repairable Systems ◽  
Interval Reliability ◽  
Repair Facility ◽  
Markovian Systems

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.

Marked point processes as limits of Markovian arrival streams

1993 ◽  
Vol 30 (2) ◽  
pp. 365-372 ◽  
Author(s):  
Søren Asmussen ◽  
Ger Koole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
The State ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
State Transitions ◽  
Poisson Arrival ◽  
Convergence Of Moments ◽  
Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

Power spectra of general shot noises and Hawkes point processes with a random excitation

2002 ◽  
Vol 34 (01) ◽  
pp. 205-222 ◽  
Author(s):  
P. Brémaud ◽  
L. Massoulié
Keyword(s):  
Point Process ◽  
Spectral Measure ◽  
Point Processes ◽  
Marked Point ◽  
Power Spectra ◽  
Random Excitation ◽  
Marked Point Process ◽  
Martingale Measure ◽  
Shot Noise Process ◽  
Power Spectral

We give (i) the Cramér power spectral measure of the general shot noise process with random excitation and non-Poisson stationary driving point processes and (ii) the Bartlett power spectral measure of the self-exciting Hawkes point process with random excitation, also called the Hawkes branching point process with random fertility rate. The latter is obtained via the isometry formula for integrals with respect to the canonical martingale measure associated with a marked point process.

Reliability analysis of complex repairable systems by means of marked point processes

1980 ◽  
Vol 17 (01) ◽  
pp. 154-167 ◽  
Author(s):  
Peter Franken ◽  
Arnfried Streller
Keyword(s):  
Reliability Analysis ◽  
Point Processes ◽  
Marked Point ◽  
Regenerative Process ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Repairable Systems ◽  
Interval Reliability ◽  
Repair Facility ◽  
Markovian Systems

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.

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