scholarly journals Derivatives pricing via p-optimal martingale measures: some extreme cases

2006 ◽  
Vol 43 (3) ◽  
pp. 634-651 ◽  
Author(s):  
Marina Santacroce
Keyword(s):  
Asset Price ◽  
Martingale Measure ◽  
Power Utility ◽  
Derivatives Pricing ◽  
Trade Off ◽  
Continuous Semimartingale ◽  
Backward Equation ◽  
The Mean ◽  
Necessary And Sufficient ◽  
Mean Variance

In an incomplete financial market in which the dynamics of the asset prices is driven by a d-dimensional continuous semimartingale X, we consider the problem of pricing European contingent claims embedded in a power utility framework. This problem reduces to identifying the p-optimal martingale measure, which can be given in terms of the solution to a semimartingale backward equation. We use this characterization to examine two extreme cases. In particular, we find a necessary and sufficient condition, written in terms of the mean-variance trade-off, for the p-optimal martingale measure to coincide with the minimal martingale measure. Moreover, if and only if an exponential function of the mean-variance trade-off is a martingale strongly orthogonal to the asset price process, the p-optimal martingale measure can be simply expressed in terms of a Doléans-Dade exponential involving X.

scholarly journals Derivatives pricing via p-optimal martingale measures: some extreme cases

2006 ◽  
Vol 43 (03) ◽  
pp. 634-651
Author(s):  
Marina Santacroce
Keyword(s):  
Asset Price ◽  
Martingale Measure ◽  
Power Utility ◽  
Derivatives Pricing ◽  
Trade Off ◽  
Continuous Semimartingale ◽  
Backward Equation ◽  
The Mean ◽  
Necessary And Sufficient ◽  
Mean Variance

In an incomplete financial market in which the dynamics of the asset prices is driven by a d-dimensional continuous semimartingale X, we consider the problem of pricing European contingent claims embedded in a power utility framework. This problem reduces to identifying the p-optimal martingale measure, which can be given in terms of the solution to a semimartingale backward equation. We use this characterization to examine two extreme cases. In particular, we find a necessary and sufficient condition, written in terms of the mean-variance trade-off, for the p-optimal martingale measure to coincide with the minimal martingale measure. Moreover, if and only if an exponential function of the mean-variance trade-off is a martingale strongly orthogonal to the asset price process, the p-optimal martingale measure can be simply expressed in terms of a Doléans-Dade exponential involving X.

SOME REMARKS ON MEAN-VARIANCE HEDGING FOR DISCONTINUOUS ASSET PRICE PROCESSES

2005 ◽  
Vol 08 (04) ◽  
pp. 425-443 ◽  
Author(s):  
TAKUJI ARAI
Keyword(s):  
Asset Price ◽  
Jump Diffusion ◽  
Diffusion Models ◽  
Continuous Case ◽  
Martingale Measure ◽  
Trade Off ◽  
One Dimensional ◽  
The Mean ◽  
Duality Approach ◽  
Mean Variance

Mean-variance hedging for the discontinuous semimartingale case is obtained under some assumptions related to the variance-optimal martingale measure. In the present paper, two remarks on it are discussed. One is an extension of Hou–Karatzas' duality approach from the continuous case to discontinuous. Another is to prove that there is the consistency with the case where the mean-variance trade-off process is continuous and deterministic. In particular, one-dimensional jump diffusion models are discussed as simple examples.

Characterization of stochastic equilibrium controls by the Malliavin calculus

2021 ◽  
pp. 2150054
Author(s):  
Jiang Yu Nguwi ◽  
Nicolas Privault
Keyword(s):  
Malliavin Calculus ◽  
Linear Quadratic ◽  
Merton Problem ◽  
Classical Duality ◽  
The Mean ◽  
Time Inconsistent ◽  
Necessary And Sufficient ◽  
Mean Variance ◽  
Quadratic Case

We derive a characterization of equilibrium controls in continuous-time, time-inconsistent control (TIC) problems using the Malliavin calculus. For this, the classical duality analysis of adjoint BSDEs is replaced with the Malliavin integration by parts. This results into a necessary and sufficient maximum principle which is applied to a linear-quadratic TIC problem, recovering previous results obtained by duality analysis in the mean-variance case, and extending them to the linear-quadratic setting. We also show that our results apply beyond the linear-quadratic case by treating the generalized Merton problem.

THE COMPATIBLE BOND-STOCK MARKET WITH JUMPS

2011 ◽  
Vol 14 (05) ◽  
pp. 723-755 ◽  
Author(s):  
DEWEN XIONG ◽  
MICHAEL KOHLMANN
Keyword(s):  
Stock Market ◽  
Marked Point ◽  
Quadratic Functions ◽  
Marked Point Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
No Arbitrage ◽  
The Mean ◽  
Necessary And Sufficient ◽  
Mean Variance

We construct a bond-stock market composed of d stocks and many bonds with jumps driven by general marked point process as well as by an ℝn-valued Wiener process. By composing these tools we introduce the concept of a compatible bond-stock market and give a necessary and sufficient condition for this property. We study no-arbitrage properties of the composed market where a compatible bond-stock market is arbitrage-free both for the bonds market and for the stocks market. We then turn to an incomplete compatible bond-stock market and give a necessary and sufficient condition for a compatible bond-stock market to be incomplete. In this market we consider the mean-variance hedging in the special situation where both B(u, T) and eG(u, y, T)-1 are quadratic functions of T - u. So, we need to extend the notion of a variance-optimal martingale (VOM) as in Xiong and Kohlmann (2009) to the more general market. By introducing two virtual stocks [Formula: see text], we prove that the VOM for the bond-stock market is the same as the VOM for the new stock market [Formula: see text]. The mean-variance hedging problem in this incomplete bond-stock market for a contingent claim [Formula: see text] is solved by deriving an explicit solution of the optimal measure-valued strategy and the optimal cost induced by the optimal strategy of MHV for the stocks [Formula: see text] is computed.

scholarly journals Mean-Variance Hedging Under Additional Market Information

2003 ◽  
Vol 06 (06) ◽  
pp. 613-636 ◽  
Author(s):  
F. Thierbach
Keyword(s):  
Martingale Measure ◽  
Option Prices ◽  
Market Information ◽  
Hedging Strategy ◽  
No Arbitrage ◽  
Finite Set ◽  
The Mean ◽  
Hedging Problem ◽  
Equivalent Martingale ◽  
Mean Variance

In this paper we analyze the mean-variance hedging approach in an incomplete market under the assumption of additional market information, which is represented by a given, finite set of observed prices of non-attainable contingent claims. Due to no-arbitrage arguments, our set of investment opportunities increases and the set of possible equivalent martingale measures shrinks. Therefore, we obtain a modified mean-variance hedging problem, which takes into account the observed additional market information. Solving this we obtain an explicit description of the optimal hedging strategy and an admissible, constrained variance-optimal signed martingale measure, that generates both the approximation price and the observed option prices.

QUADRATIC HEDGING FOR THE BATES MODEL

2007 ◽  
Vol 10 (05) ◽  
pp. 873-885 ◽  
Author(s):  
FRIEDRICH HUBALEK ◽  
CARLO SGARRA
Keyword(s):  
Option Pricing ◽  
Explicit Expression ◽  
Martingale Measure ◽  
Risk Minimization ◽  
Hedging Strategy ◽  
Preliminary Results ◽  
Quadratic Hedging ◽  
The Mean ◽  
Mean Variance ◽  
Martingale Case

In the present paper we give some preliminary results for option pricing and hedging in the framework of the Bates model based on quadratic risk minimization. We provide an explicit expression of the mean-variance hedging strategy in the martingale case and study the Minimal Martingale measure in the general case.

scholarly journals A Note on Asymptotic Exponential Arbitrage with Exponentially Decaying Failure Probability

2013 ◽  
Vol 50 (03) ◽  
pp. 801-809 ◽  
Author(s):  
Kai Du ◽  
Ariel David Neufeld
Keyword(s):  
Large Deviations ◽  
Growth Condition ◽  
Failure Probability ◽  
Quantitative Form ◽  
Continuous Semimartingale ◽  
The Mean ◽  
Mean Variance

The goal of this paper is to prove a result conjectured in Föllmer and Schachermayer (2007) in a slightly more general form. Suppose that S is a continuous semimartingale and satisfies a large deviations estimate; this is a particular growth condition on the mean-variance tradeoff process of S. We show that S then allows asymptotic exponential arbitrage with exponentially decaying failure probability, which is a strong and quantitative form of long-term arbitrage. In contrast to Föllmer and Schachermayer (2007), our result does not assume that S is a diffusion, nor does it need any ergodicity assumption.

scholarly journals A Note on Asymptotic Exponential Arbitrage with Exponentially Decaying Failure Probability

2013 ◽  
Vol 50 (3) ◽  
pp. 801-809 ◽  
Author(s):  
Kai Du ◽  
Ariel David Neufeld
Keyword(s):  
Large Deviations ◽  
Growth Condition ◽  
Failure Probability ◽  
Quantitative Form ◽  
Continuous Semimartingale ◽  
The Mean ◽  
Mean Variance

The goal of this paper is to prove a result conjectured in Föllmer and Schachermayer (2007) in a slightly more general form. Suppose that S is a continuous semimartingale and satisfies a large deviations estimate; this is a particular growth condition on the mean-variance tradeoff process of S. We show that S then allows asymptotic exponential arbitrage with exponentially decaying failure probability, which is a strong and quantitative form of long-term arbitrage. In contrast to Föllmer and Schachermayer (2007), our result does not assume that S is a diffusion, nor does it need any ergodicity assumption.

MEAN VARIANCE HEDGING IN A GENERAL JUMP MARKET

2010 ◽  
Vol 13 (05) ◽  
pp. 789-820
Author(s):  
DEWEN XIONG ◽  
MICHAEL KOHLMANN
Keyword(s):  
Financial Market ◽  
Martingale Measure ◽  
Contingent Claim ◽  
Predictable Process ◽  
Price Process ◽  
The Mean ◽  
Hedging Problem ◽  
Mean Variance ◽  
Intuitive Interpretation ◽  
Density Process

We consider a financial market in which the discounted price process S is an ℝd-valued semimartingale with bounded jumps, and the variance-optimal martingale measure (VOMM) Q opt is only known to be a signed measure. We give a backward semimartingale equation (BSE) and show that the density process Z opt of Q opt with respect to P is a possibly non-positive stochastic exponential if and only if this BSE has a solution. For a general contingent claim H, we consider the following generalized version of the classical mean-variance hedging problem [Formula: see text] where [Formula: see text]. We represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward martingale equation (BME) and an appropriate predictable process δ both with a straightforward intuitive interpretation.

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