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In financial option pricing, the stable Lévy framework is a problematic issue because of its (theoretical) infinite invariance. This paper deals with the integration of these processes into option pricing by defining the minimal theoretical condition required for an optimal risk hedging based on a stable Lévy framework with an exponentially truncated distribution.

Stable Levy processes played a particular role in the evolution of financial knowledge since they were the first theoretical alternative to Gaussian framework for the statistical description of financial leptokurticity [

The second way of implementing stable Levy processes in finance is to associate them with a unique and unconditional distribution capturing empirical data. This methodological perspective has mainly been developed by econophysicists who often emphasised the advantage of dealing with historical data: studying financial data as they appeared in the past (and not as they should be according to a pre-existing theoretical framework) allows modellers to develop a less biased estimation of risk (McCauley, 2004; Famer and Geneakoplos, 2008). However, although empirical evidence suggesting the use of stable Levy processes for describing financial data, the theoretical properties of these processes imply an infinite variance which did not favour their empirical application. Indeed, an infinite variance property means it is not possible to value the major risk parameter (volatility) used in finance making obsolete all classical perspective of risk management. In the same vein, an infinite variance of financial prices leads to an impossibility to price financial assets since their variance would not converge toward a finite value. This situation makes stable Levy processes very interesting in knowledge because they required a specific statistical treatment to make them “applicable” in assets pricing. These treatments generated a lot of debates in the 1960s and 1970s [

Roughly speaking, the unconditional application of stable Levy processes in options pricing can be decomposed into through three categories of models which can be identified depending on the technical solution developed to escape from the infinite variance problem: i) time-changed processes; ii) models using a specific calibration of parameters and iii) models based on a particular calibration (truncation) of the distribution. The first category refers to time-changed models which introduce an “intrinsic time” (for options’ underlying) providing tail effects as observed in the market. Examples of these models are contained in Mandelbrot et al. [

to

The second category of models refers to a specific calibration of parameters in order to have finite statistical moments for the processes describing the options’ underlying distribution. Empirical distributions or time are not changed but authors give specific statistical conditions implying the existence of all statistical moments. In this framework, modellers defined specific conditions implying the possibility for variance to converge toward a finite value. Carr and Wu ( [

The third category of statistical solutions (within this article) developed to solve the problem of infinite variance, refers to all models based on a particular calibration of the whole distribution. That approach is well known by econophysicists who developed several truncation techniques in order to capture the fat-tailed distribution with an unconditional approach and a finite variance [

This paper aims to define the minimal condition required for an optimal risk hedging for a particular cut-off based on an exponentially truncated stable Lévy distribution. In other words, our contribution is to show that a (no-conditional) exponential stable Lévy description of a financial distribution (i.e. description of the whole distribution) can admit an admissible hedging strategy in a sense defined by Harrison and Kreps [

While the first section will present our generalized model in line with the Black and Scholes framework (based on a martingale measure), the second section will define a specific risk measure for a stable Levy version of this model that we will present in the third section. In line with [

Let’s consider a portfolio made up of a (call) option and a short position on _{i} the price for each stock. Initially the stock price is considered to be

where the first term is the value of the option at time T and the second term is the premium paid for the option at time

Due to the stochastic nature of the price process, risk is inherent to the financial evaluation of options and stocks. For the log-normal distribution it was shown by Black and Scholes [^{1}. A measure of risk that was also used in Bouchaud and Sornette [^{2}. Thus:

First of all let us note that for uncorrelated assets, we have the following expression:

our conceptual model defined in Equation (3) is in line with the generalized call option pricing formula defined by Tan [

This equation is valid for a martingale process

In the simplest case, it is straightforward to observe that in the case of the normal distribution with log-returns the optimal hedging strategy given in Equation (4) is the same as the hedging strategy from the Black and Scho-

les model, i.e.

Equation (3):

strategy, defined as

nette [

As previously mentioned, stable Levy framework generates infinite variance, therefore a no computable risk leading authors to find a solution: while Tan [

This section focuses on a particular cut-off based on an exponentially truncated stable Lévy process for option pricing by determining conditions for which the risk measure defined in the previous section can be viable in a hedging framework developed by Harrison and Kreps [

The number of parameters refers to the great diversification of models developed in the econophysical literature^{3}. In other words, all existing use of stable Levy processes (included the Gaussian one) in econophysics can be seen as a specific case of this generic equation.

^{3}Although, in the great majority of the models a_{1}, b_{1} = 1; b_{2} and d are equal to zero—See Bucsa et al. [

^{4}It is worth mentioning that this convergence of x to a Gausian regime is extremely slow due to the stable property of the Levy distribution [_{c} can be derived by using Berry-Esseen Theorem [

In line with the central limit theorem, a stable Levy regime will converge^{4} toward a Gaussian asymptotic distribution after a very high number of variables

as a specific case of the generic Formula (6) with the following parameters,

distribution density for the log returns of options underlying can take the following form,

where

Because stable Lévy processes generate infinite variance, we use an exponentially truncation implying an exponential decay of the distribution. This restriction means that the truncated distribution generates finite variations making possible the estimation of the variance (in the stable Levy regime) which is given by the following equation:

Using the general Equation (2) we calculate the option price for this model for the chosen portfolio, by considering the density distribution of stock returns:

Using the result

of squared volatility, yields:

Given this result, we can estimate the hedging strategy minimizing the risk by using Equation (4):

However, this optimal hedging can be implemented in the non-asymptotic regime implying that the variance

(Equation (7)) is finite only for x < l. By integrating

Although Tan [

This article defined conditions for which an option pricing model based on an exponentially truncated stable Lévy distribution can be viable in a Harrison and Kreps [