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 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.

scholarly journals Option Pricing under Risk-Minimization Criterion in an Incomplete Market with the Finite Difference Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Xinfeng Ruan ◽  
Wenli Zhu ◽  
Shuang Li ◽  
Jiexiang Huang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Option Pricing ◽  
Difference Method ◽  
Incomplete Market ◽  
Martingale Measure ◽  
Risk Minimization ◽  
European Option ◽  
The Finite Difference Method ◽  
Nikodym Derivative

We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset is governed by a jump diffusion equation with stochastic volatility. We obtain the Radon-Nikodym derivative for the minimal martingale measure and a partial integro-differential equation (PIDE) of European option. The finite difference method is employed to compute the European option valuation of PIDE.

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.

scholarly journals MARKET VALUE MARGIN VIA MEAN–VARIANCE HEDGING

2013 ◽  
Vol 43 (3) ◽  
pp. 301-322 ◽  
Author(s):  
Andreas Tsanakas ◽  
Mario V. Wüthrich ◽  
Aleš Černý
Keyword(s):  
Market Value ◽  
Hedging Strategy ◽  
Market Prices ◽  
Regulatory Cost ◽  
Regulatory Approach ◽  
Consistent Extension ◽  
Capital Efficiency ◽  
The Mean ◽  
Mean Variance ◽  
Development Result

AbstractWe use mean–variance hedging in discrete time in order to value an insurance liability. The prediction of the insurance liability is decomposed into claims development results, that is, yearly deteriorations in its conditional expected values until the liability is finally settled. We assume the existence of a tradeable derivative with binary pay-off written on the claims development result and available in each development period. General valuation formulas are stated and, under additional assumptions, these valuation formulas simplify to resemble familiar regulatory cost-of-capital-based formulas. However, adoption of the mean–variance framework improves upon the regulatory approach by allowing for potential calibration to observed market prices, inclusion of other tradeable assets, and consistent extension to multiple periods. Furthermore, it is shown that the hedging strategy can also lead to increased capital efficiency.

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.

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.

scholarly journals Continuous-Time Portfolio Selection and Option Pricing under Risk-Minimization Criterion in an Incomplete Market

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Xinfeng Ruan ◽  
Wenli Zhu ◽  
Jiexiang Huang ◽  
Shuang Li
Keyword(s):  
Option Pricing ◽  
Portfolio Selection ◽  
Incomplete Market ◽  
Martingale Measure ◽  
Call Option ◽  
Risk Minimization ◽  
European Call Option ◽  
Optimal Portfolio Selection ◽  
Transformation Methods ◽  
Special Case

We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset are governed by a jump diffusion equation. We obtain the Radon-Nikodym derivative in the minimal martingale measure and a partial integrodifferential equation (PIDE) of European call option. In a special case, we get the exact solution for European call option by Fourier transformation methods. Finally, we employ the pricing kernel to calculate the optimal portfolio selection by martingale methods.

scholarly journals Locally Risk-minimizing Hedging of Insurance Payment Streams

2007 ◽  
Vol 37 (1) ◽  
pp. 67-91 ◽  
Author(s):  
Martin Riesner
Keyword(s):  
Financial Market ◽  
Life Insurance ◽  
Levy Process ◽  
Contingent Claim ◽  
Risk Minimization ◽  
Hedging Strategy ◽  
Insurance Contracts ◽  
Insurance Payment ◽  
Local Risk ◽  
Martingale Case

For the martingale case Föllmer and Sondermann (1986) introduced a unique admissible risk-minimizing hedging strategy for any square-integrable contingent claim H. Schweizer (1991) developed their theory further to the semimartingale case introducing the notion of local risk-minimization. Møller (2001) extended the theory of Föllmer and Sondermann (1986) to hedge general payment processes occurring mainly in insurance. We expand local risk-minimization to the theory of hedging general payment processes and derive such a hedging strategy for general unit-linked life insurance contracts in a general Lévy process financial market.

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.

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