A SHOT NOISE MODEL FOR FINANCIAL ASSETS

2008 ◽  
Vol 11 (01) ◽  
pp. 87-106 ◽  
Author(s):  
TIMO ALTMANN ◽  
THORSTEN SCHMIDT ◽  
WINFRIED STUTE
Keyword(s):  
Simulation Study ◽  
Shot Noise ◽  
Stock Prices ◽  
Continuous Time ◽  
Noise Model ◽  
Martingale Measure ◽  
Financial Assets ◽  
Hedging Strategy ◽  
Noise Effects ◽  
Abrupt Changes

In this article we propose and study a model for stock prices which allows for shot-noise effects. This means that abrupt changes caused by jumps may fade away as time goes by. This model is incomplete. We derive the minimal martingale measure in discrete and continuous time and discuss the associated hedging strategy. Finally, a simulation study is included to show that our model is able to produce smile effects.

Incomplete markets: convergence of options values under the minimal martingale measure

1999 ◽  
Vol 31 (04) ◽  
pp. 1058-1077 ◽  
Author(s):  
Jean-Luc Prigent
Keyword(s):  
Financial Market ◽  
Incomplete Markets ◽  
Stock Prices ◽  
Continuous Time ◽  
Martingale Measure ◽  
Minimal Martingale Measure ◽  
Mild Assumption ◽  
The Stability ◽  
Minimal Measure ◽  
Nikodym Derivative

In the setting of incomplete markets, this paper presents a general result of convergence for derivative assets prices. It is proved that the minimal martingale measure first introduced by Föllmer and Schweizer is a convenient tool for the stability under convergence. This extends previous well-known results when the markets are complete both in discrete time and continuous time. Taking into account the structure of stock prices, a mild assumption is made. It implies the joint convergence of the sequences of stock prices and of the Radon-Nikodym derivative of the minimal measure. The convergence of the derivatives prices follows.This property is illustrated in the main classes of financial market models.

Incomplete markets: convergence of options values under the minimal martingale measure

1999 ◽  
Vol 31 (4) ◽  
pp. 1058-1077 ◽  
Author(s):  
Jean-Luc Prigent
Keyword(s):  
Financial Market ◽  
Incomplete Markets ◽  
Stock Prices ◽  
Continuous Time ◽  
Martingale Measure ◽  
Minimal Martingale Measure ◽  
Mild Assumption ◽  
The Stability ◽  
Minimal Measure ◽  
Nikodym Derivative

In the setting of incomplete markets, this paper presents a general result of convergence for derivative assets prices. It is proved that the minimal martingale measure first introduced by Föllmer and Schweizer is a convenient tool for the stability under convergence. This extends previous well-known results when the markets are complete both in discrete time and continuous time. Taking into account the structure of stock prices, a mild assumption is made. It implies the joint convergence of the sequences of stock prices and of the Radon-Nikodym derivative of the minimal measure. The convergence of the derivatives prices follows.This property is illustrated in the main classes of financial market models.

Arbitrage Theory in Continuous Time

2019 ◽  
Author(s):  
Tomas Björk
Keyword(s):  
Incomplete Markets ◽  
Continuous Time ◽  
Dynamic Equilibrium ◽  
Factor Model ◽  
Good Deal ◽  
Financial Derivatives ◽  
Equilibrium Theory ◽  
Martingale Measure ◽  
Fourth Edition ◽  
Optimal Stopping Theory

The fourth edition of this textbook on pricing and hedging of financial derivatives, now also including dynamic equilibrium theory, continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous time arbitrage pricing of financial derivatives, including stochastic optimal control theory and optimal stopping theory, the book is designed for graduate students in economics and mathematics, and combines the necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. All concepts and ideas are discussed, not only from a mathematics point of view, but the mathematical theory is also always supplemented with lots of intuitive economic arguments. In the substantially extended fourth edition Tomas Björk has added completely new chapters on incomplete markets, treating such topics as the Esscher transform, the minimal martingale measure, f-divergences, optimal investment theory for incomplete markets, and good deal bounds. There is also an entirely new part of the book presenting dynamic equilibrium theory. This includes several chapters on unit net supply endowments models, and the Cox–Ingersoll–Ross equilibrium factor model (including the CIR equilibrium interest rate model). Providing two full treatments of arbitrage theory—the classical delta hedging approach and the modern martingale approach—the book is written in such a way that these approaches can be studied independently of each other, thus providing the less mathematically oriented reader with a self-contained introduction to arbitrage theory and equilibrium theory, while at the same time allowing the more advanced student to see the full theory in action.

Open-loop optimal control of a class of continuous-time stochastic systems—A simulation study†

1974 ◽  
Vol 19 (6) ◽  
pp. 1165-1175 ◽  
Author(s):  
EDGAR C. TACKER ◽  
THOMAS D. LINTON ◽  
CHARLES W. SANDERS
Keyword(s):  
Optimal Control ◽  
Simulation Study ◽  
Continuous Time ◽  
Stochastic Systems ◽  
Open Loop

scholarly journals Pricing Basket Default Swaps in a Tractable Shot-Noise Model

2011 ◽  
Author(s):  
Alexander Herbertsson ◽  
Jiwook Jang ◽  
Thorsten Schmidt
Keyword(s):  
Shot Noise ◽  
Noise Model

scholarly journals Pricing Chinese Convertible Bonds with Default Intensity by Monte Carlo Method

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xin Luo ◽  
Jinlin Zhang
Keyword(s):  
Monte Carlo ◽  
Stock Prices ◽  
Default Risk ◽  
Emerging Market ◽  
Cash Flows ◽  
Convertible Bonds ◽  
Martingale Measure ◽  
Default Intensity ◽  
Long Run ◽  
Two Parameters

This article proposes a new way to price Chinese convertible bonds by the Longstaff-Schwartz Least Squares Monte Carlo simulation. The default intensity and the volatility are the two important parameters, which are difficultly obtained in the emerging market, in pricing convertible bonds. By developing the Merton theory, we find a new effective method to get the theoretical value of the two parameters. In the pricing method, the default risk is described by the default intensity, and a default on a bond is triggered by the bottom Q(T) (default probability) percentile of the simulated stock prices at the maturity date. In the present simulation, a risk-free interest rate is used to discount the cash flows. So, the new pricing model is considered to tally with the general pricing rule under martingale measure. The empirical results of the CEB and the XIG convertible bonds by the proposed method are compared with those obtained by the credit spreads method. It is also found that the theoretical prices calculated by the method proposed in the article fit the market prices well, especially, in the long run tendency.

On the GIX/G/∞ system

1990 ◽  
Vol 27 (3) ◽  
pp. 671-683 ◽  
Author(s):  
L. Liu ◽  
B. R. K. Kashyap ◽  
J. G. C. Templeton
Keyword(s):  
Steady State ◽  
Shot Noise ◽  
Continuous Time ◽  
Transient State ◽  
System Size ◽  
Noise Process ◽  
Shot Noise Process ◽  
Special Cases

By using a shot noise process, general results on system size in continuous time are given both in transient state and in steady state with discussion on some interesting results concerning special cases. System size before arrivals is also discussed.

STATE-SPACE SELF-TUNING CONTROL FOR NONLINEAR STOCHASTIC AND CHAOTIC HYBRID SYSTEMS

2001 ◽  
Vol 11 (04) ◽  
pp. 1079-1113 ◽  
Author(s):  
SHU-MEI GUO ◽  
LEANG-SAN SHIEH ◽  
CHING-FANG LIN ◽  
JAGDISH CHANDRA
Keyword(s):  
State Space ◽  
Hybrid Systems ◽  
Continuous Time ◽  
Digital Control ◽  
Stochastic Systems ◽  
Moving Average ◽  
Computation Time ◽  
Sampling Period ◽  
Noise Model ◽  
Self Tuning

This paper presents a new state-space self-tuning control scheme for adaptive digital control of continuous-time multivariable nonlinear stochastic and chaotic systems, which have unknown system parameters, system and measurement noises, and inaccessible system states. Instead of using the moving average (MA)-based noise model commonly used for adaptive digital control of linear discrete-time stochastic systems in the literature, an adjustable auto-regressive moving average (ARMA)-based noise model with estimated states is constructed for state-space self-tuning control of nonlinear continuous-time stochastic systems. By taking advantage of a digital redesign methodology, which converts a predesigned high-gain analog tracker/observer into a practically implementable low-gain digital tracker/observer, and by taking the non-negligible computation time delay and a relatively longer sampling period into consideration, a digitally redesigned predictive tracker/observer has been newly developed in this paper for adaptive chaotic orbit tracking. The proposed method enables the development of a digitally implementable advanced control algorithm for nonlinear stochastic and chaotic hybrid systems.

Shot-noise processes and the minimal martingale measure

2007 ◽  
Vol 77 (12) ◽  
pp. 1332-1338 ◽  
Author(s):  
Thorsten Schmidt ◽  
Winfried Stute
Keyword(s):  
Shot Noise ◽  
Martingale Measure ◽  
Minimal Martingale Measure
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