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

scholarly journals On the role of Föllmer-Schweizer minimal martingale measure in risk-sensitive control asset management

2015 ◽  
Vol 52 (3) ◽  
pp. 703-717
Author(s):  
Amogh Deshpande
Keyword(s):  
Asset Management ◽  
Optimal Strategy ◽  
Stock Price ◽  
Short Selling ◽  
Martingale Measure ◽  
Minimal Martingale Measure ◽  
Risk Sensitive ◽  
Risk Sensitive Control ◽  
Riskless Asset

Kuroda and Nagai (2002) stated that the factor process in risk-sensitive control asset management is stable under the Föllmer-Schweizer minimal martingale measure. Fleming and Sheu (2002) and, more recently, Föllmer and Schweizer (2010) observed that the role of the minimal martingale measure in this portfolio optimization is yet to be established. In this paper we aim to address this question by explicitly connecting the optimal wealth allocation to the minimal martingale measure. We achieve this by using a ‘trick’ of observing this problem in the context of model uncertainty via a two person zero sum stochastic differential game between the investor and an antagonistic market that provides a probability measure. We obtain some startling insights. Firstly, if short selling is not permitted and the factor process evolves under the minimal martingale measure, then the investor's optimal strategy can only be to invest in the riskless asset (i.e. the no-regret strategy). Secondly, if the factor process and the stock price process have independent noise, then, even if the market allows short-selling, the optimal strategy for the investor must be the no-regret strategy while the factor process will evolve under the minimal martingale measure.

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.

The Esscher Transform and the Minimal Martingale Measure

2019 ◽  
pp. 405-412
Author(s):  
Tomas Björk
Keyword(s):  
Incomplete Market ◽  
Martingale Measure ◽  
Esscher Transform ◽  
Extended Version ◽  
Martingale Measures ◽  
Minimal Martingale Measure ◽  
Economic Significance

In this chapter we present two ways to choose a unique martingale measure in an incomplete market. The first way is to use an extended version of the Esscher transform, which implies that we restrict the class of martingale measures. The second way is to use the minimal martingale measure, that is, the measure which minimizes the norm of the associated Girsanov kernel. We exemplify the two methods and discuss the economic significance.

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