Local risk minimization and numéraire

1999 ◽  
Vol 36 (4) ◽  
pp. 1126-1139 ◽  
Author(s):  
F. Biagini ◽  
M. Pratelli
Keyword(s):  
Stochastic Process ◽  
Stochastic Volatility ◽  
Contingent Claims ◽  
Continuous Case ◽  
Risk Minimization ◽  
Complete Markets ◽  
Local Risk ◽  
The Right

The ‘change of numéraire’ technique has been introduced by Geman, El Karoui and Rochet for pricing and hedging contingent claims in the case of complete markets. In this paper we study the ‘change of numéraire’ using the ‘locally risk-minimizing approach’, when the market is not complete. We prove that, if the stochastic process which represents the prices is continuous, the l.r.m. strategy is invariant by a change of numéraire (this result is false in the right-continuous case, as is shown by some counterexamples).We also give an extension of Merton's formula to the case of stochastic volatility.

Local risk minimization and numéraire

1999 ◽  
Vol 36 (04) ◽  
pp. 1126-1139 ◽  
Author(s):  
F. Biagini ◽  
M. Pratelli
Keyword(s):  
Stochastic Process ◽  
Stochastic Volatility ◽  
Contingent Claims ◽  
Continuous Case ◽  
Risk Minimization ◽  
Complete Markets ◽  
Local Risk ◽  
The Right

The ‘change of numéraire’ technique has been introduced by Geman, El Karoui and Rochet for pricing and hedging contingent claims in the case of complete markets. In this paper we study the ‘change of numéraire’ using the ‘locally risk-minimizing approach’, when the market is not complete. We prove that, if the stochastic process which represents the prices is continuous, the l.r.m. strategy is invariant by a change of numéraire (this result is false in the right-continuous case, as is shown by some counterexamples). We also give an extension of Merton's formula to the case of stochastic volatility.

NUMERICAL ANALYSIS ON LOCAL RISK-MINIMIZATION FOR EXPONENTIAL LÉVY MODELS

2016 ◽  
Vol 19 (02) ◽  
pp. 1650008 ◽  
Author(s):  
TAKUJI ARAI ◽  
YUTO IMAI ◽  
RYOICHI SUZUKI
Keyword(s):  
Incomplete Markets ◽  
Contingent Claims ◽  
Risk Minimization ◽  
Gamma Model ◽  
Variance Gamma ◽  
Call Options ◽  
Estimated Parameters ◽  
Exponential Lévy Models ◽  
Local Risk ◽  
Variance Gamma Model

We illustrate how to compute local risk minimization (LRM) of call options for exponential Lévy models. Here, LRM is a popular hedging method through a quadratic criterion for contingent claims in incomplete markets. Arai & Suzuki (2015) have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transform (FFT) method suggested by by Carr & Madan (1999). Considering Merton jump-diffusion models and variance gamma models as typical examples of exponential Lévy models, we provide the forms for the FFT explicitly; and compute the values of LRM numerically for given parameter sets. Furthermore, we illustrate numerical results for a variance gamma model with estimated parameters from the Nikkei 225 index.

scholarly journals Super-replication in stochastic volatility models under portfolio constraints

1999 ◽  
Vol 36 (02) ◽  
pp. 523-545 ◽  
Author(s):  
Jakša Cvitanić ◽  
Huyên Pham ◽  
Nizar Touzi
Keyword(s):  
Stochastic Volatility ◽  
Stock Price ◽  
Convex Set ◽  
Short Selling ◽  
Contingent Claims ◽  
Stochastic Volatility Models ◽  
Portfolio Constraints ◽  
Volatility Models ◽  
Closed Convex Set

We study a financial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints. The latter are given in terms of bounds imposed on the borrowing and short-selling of a ‘hedger’ in this market, and can be described by a closed convex set K. We find explicit characterizations of the minimal price needed to super-replicate European-type contingent claims in this framework. The results depend on whether the volatility is bounded away from zero and/or infinity, and also, on if we have linear dynamics for the stock price process, and whether volatility process depends on the stock price. We use a previously known representation of the minimal price as a supremum of the prices in the corresponding shadow markets, and we derive a PDE characterization of that representation.

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