Financial Derivative Uncertainty Claim Pricing Using Meixner Distribution Skewed Operators

In this paper, consideration is given to the problem of independent/uncertain claim pricing using distribution operators. This method was earlier studied in Insurance Pricing where the original operator was described in relation to normal distribution. We apply the Meixner process to financial pricing since the normal distribution is a very poor model to fit log-returns of financial assets like stocks or indices. For us to realize a better fit, we substitute the normal distribution by Meixner process. Hence, we generalize this approach by using an operator based on the density Meixner distribution. We further show how Meixner operator function can be used to derive the formular for asset pricing of independent/uncertain future returns of a risky asset.


Introduction
Of recent, Mathematics of Finance has gained huge respect on the bases of its effect on the financial industry.The foundation of financial mathematics as it is known today has its origin in the seminal papers by Black and Scholes (1973) and by Merton (1973), where the Ito's formula has been used for deriving a compact pricing formula for a standard European call option by formulating explicitly the model on the risk-neutral measure, under a set of assumptions.
Option valuation is one of the most important topics in financial mathematics.The exact modeling of financial price series is significant for the pricing and hedging of financial derivatives such as option.To price and hedge derivatives securities, it is vital to have a good modeling of the probability distribution of the underlying product.
The most well-known continuous time model used is the calibrated Black-Scholes model.It uses the normal distribution to fit the log-returns of the underlying; the price process of the underlying is given by the geometric Brownian motion Where {B t , t ≥ 0} is a standard Brownian motion, i.e.B t follow a Normal distribution with mean 0 and variance t.
It is known however that the log-returns of most financial assets have an actual kurtosis that is higher than that of the Normal distribution.
Empirical evidence has shown that the normal distribution is a very poor model to fit log-returns of financial asset such as stocks.In order to achieve a better fit we replace the Brownian motion by a special Levy process called the Meixner process.Several authors have proposed similar process models.Notable contributions were made, for instance, Luscher (2005) used the Normal Inverse Gaussian (NIG) to price synthetic Collateralized Debt Obligations (CDO), Barndorff-Nielsen (1995) proposed the NIG Levy process, Eberling and Keller (1995) proposed the Hyperbolic models and their generalizations, Wang (2000) proposed pricing based on a normal-based distribution operator for a form of insurance risk, Osu and Achi (2012) price contingent claim using distortion operators by Cauchy distribution under a simple transformation.
In this paper, we show how Meixner operator function can be used to derive the formula for asset pricing of uncertain future returns of a risky asset.More so, we generalize this approach by using an operator based on the density Meixner distribution.

Meixner Process
The Meixner process is a Levy process {X(t)} t≥0 that has Meixner distributed increments.Specifically, X(t) has a Meixner (x : a, b, dt, mt)-distributed with parameters a > 0, −π < b < π, d > 0 and m ∈ R. The density of the Meixner distribution Meixner (a, b, d, m) is given by where Γ denotes the Gamma function.
The characteristic function of Meixner (a, b, d, m) is given by A Meixner (x: a, b, dt, mt)-distributed random variable has the following stylized features (and compared to the Normal distribution): See (Schoutens, 2002) on more information about Meixner process.

Time Changed Levy Process
Let (Ω, f, p) be a probability space in this work; a random variable representing net income or profit after some time and h be a constant.We presume that for any random variable described on the probability space, its characteristic function exists.That is, for any random variable For the cumulative distribution, (cdf), F X (•) with differential dF X (•), corresponding to a given random variable X, we define by where the characteristic function of X under , is defined by (2.4) The Levy-Khintchine formula for the characteristic function of X t is; Where the characteristic exponent, ϕ X (u) , u ∈ R d is given by (2.6) The characteristic function of the time changed Levy-process Y t ≡ X t is That is the Laplace transform of the stochastic time T t = t 0 (v(s−)ds evaluated at the characteristic exponent of X.

The General Case of Correlated Time Changed
More generally, the general Fourier transform, CF, of Y t ≡ X T t under measure can be represented as the "Laplace Transform" of T t under a new complex-valued measure ∅(u), calculated at the characteristic-exponent ϕ x (u) of X t . (2.8) For each u ∈ D, ∅(u) is absolutely with respect to p and is defined by which is a complex valued p-Martingale with respect to f T t t≥0 for each u ∈ D.
The familiar Wald Martingale defined on a Levy process Time changed (ie replacing t by T t ) preserves the Martingality.((u) .While the validity of mathematical operation is constant, the multi-rated measure loses whatever could be its merit.

Existence Theorem of Esscher Martingale Measure (ESSMM) for Geometric Levy Process
We believe in the existence of what that follows, since the Martingale condition for an Esscher changed expectation measure This state equals the situation as follows and this equals also the following statement where ∅ (u) is the characteristic function of Z t .
To develop the existence theorem, setting (3.4)

Then we have
Theorem 1 Existence Condition For ESSMM.

If the equation
has a solution h * , then the ESSMM of X t , P ES S MM , exists and The process Z T same as a Levy process under P ES S MM and the generating triplet of Z t under P ES S MM , (δ Proof.The Equation (3.5) is equivalent to (3.3).Therefore, and this is equal to By simple calculation, we obtain; (3.12) This formula proves the result of the theorem.

Price of a European Call Option
Theorem 2 For a probability space (Ω, f, P), a random variable X T which is the net income or profit at after some time.And for any X T defined on the probability space, its characteristic function exits, i.e, for any random variable X: Where C is the complex plane and the characteristic function of X T has the form Proof.Assuming that the stock prices are log-normally distributed and that the stochastic process Let the stochastic process {X t } be a Weiner process with mean per unit time μ and variance per unit time δ 2 .Then Using Equation (3.13a) to find the value of x which determines the region of integration, δe (1+h)x e −ht(−iμθ+ 1 2 θ 2 ) ≥ 0. (3.13b) We have which is equivalent to the well-known Kolmogorov Backward Equation in (Goovaerts & Laeven, 2008) with the drift function modified for the change of probability measure as proved by the Esscher transform. Thus, From Equations (3.11) and (3.13b), we can find h * for a random variable normally distributed X 2 t , Replacing in (3.15) we obtain, Thus from the characteristic function of the time changed Levy process, we obtained the traditional Black-Scholes formula for pricing a European call option.Note that the expected rate of return iμ, which represents the preferences of investors, does not appear in the last formula.

Meixner and Option Pricing
Let ∅ m denotes the Meixner distribution function We define the Meixner operator as

Let z be a variable and let X = h (z) be a transformation through a continuous function h, then
By definition, we have If we apply now the operator function h (x,a,b,m,k) on S x , we have Applying the probability again we have that P z > h −1 (t) − λ m , normalizing Multiplying through by h, we have Where E n denotes the expectation under the density measure n, then This result shows that under a h x,a,b,m,d operator, a meixner random variable is translated by a factor λ m a.This generalizes the equivalent result found in NIG.
The reason for every m ∈ R, is that, stochastically, the market is changing over time by making business time more stochastic.The m incorporates the stochastic volatility effect and the resulting option prices can be determined very similarly to empirical prices.S T evolves like a risk neutral asset as λ m is calliberated to verify the risk neutral condition even as m tends to nor∞.
-Then the expression for H Here, under meixner operator with value λ m gives ∅ m [(u) + λ m ], the price at S T evolves like a risk neutral asset because λ m is calibrated to verify the risk neutral condition. Hence -More generally, the proposition can be extended to the case when the security price is a function of a symmetrically distributed random variable.
-If we apply the price E[z + λ m a], the capital required for the standard European call payoff is Having known the density function H (F (S T )) of the stock price at the expiring time, S T under the risk neutral measure λ m a we can easily prize European call and put option by simply calculating the expected value (Schoutens, 2003).

Contingent Claim Pricing
Let us look at the Meixner asset price model ∅ m [(U) + λ m ] and a standard European call option pay-off at maturity time T given by f (S T , K) = (S T , −K).This is purely a function of the ( f T , p)-random variable S T that becomes the change for the security offered at the time T .It can be stated as S T = h (z T ) for a function h (z) = (S 0 e z − K), the value at time zero is therefore given by the expectation of the payoff under the Martingale measure.
Where π is the probability under ∅ of the future prize of the risky asset If we set λ m to be the one that make S T evolve like a risk neutral asset i.e. λ m = π−r c a T .This shows that the price assessed with the pricing function related to the Meixner operator with parameter λ m is given by which is the prize at time zero i.e. the price at S t .Therefore Applying the function we have Therefor, the parameter λ m measure the obvious discount equals the for seen security charge to the substantive security charge considering the continual market price variations in order to make business time more stochastic.The option price result incorporates these stochastic volatility effects which is determined very similarly to empirical prices.If we consider the pay-off of an European call option (with maturity T and strike price k) we have

Conclusion
λ m is the risk free rate for every m ∈ R, which shows that as market changes over time, it becomes more stochastic and the parameter m incorporates these stochastic volatility effect making the option price determine very similarly to empirical prices as it fit log-returns of financial assets like stock or indices.
In other words, under meixner operator with value λ m = μ−r c a T ln∅ w [(k) + λ m ] the prize at S T evolves like a risk neutral asset because λ m is calibrated to verify the risk neutral condition.
Consequently, we assumed that the prize of return at S T follows a meixner distribution in place of normal distribution this is because for us to achieve a better fit we replace the normal distribution by meixner distribution.Hence, we generalize this approach by using an operator based on the density meixner distribution.The meixner family allows for heavier and skewed tail, as the kurtosis of normal distribution is always less than that of meixner distribution kurtosis.
Simplifiesto H (S T , −λ) = S 0 e −λ m a T (4.10) then the current price becomes e −λ m b T H (S T , λ m ) = e −λ m a T S 0 e −λ m a T = S 0 .(4.11)