Extended Wiener Measure by Nonstandard Analysis for Financial Time Series

We propose a new approach to construct an extended Wiener measure using nonstandard analysis by E. Nelson. For the new definition we construct non-standardized convolution of probability measure for independent random variables. As an application, we consider a simple calculation of financial time series.


Introduction
As for option pricing in financial mathematics we investigate the expectation of a functional of the Wiener process which can be represented as the integral with respect to the Wiener measure. Unfortunately it is difficult to calculate the Wiener integral using standard methods directly. Therefore Black and Scholes [1] investigated the PDE (5) for the option pricing. Then we choose another extended method, i.e. nonstandard analysis, to calculate the Wiener integral. It links the Wiener measure to the Black-Sholes model, which is of the Heat equations. It is known that their solutions are constructed by the Bessel's special functions. As they are described by the Fourier integral, we need to use the convolution of the non-standard version. We give some sufficient conditions to make it possible to this procedure. In 1988, M. Kionoshita introduced non-standard analysis for distributions [2]. It is one of the former investigations.

Definition of Standard Wiener Measure
1 , e . 2π

Nonstandard Convolution by Hyper Function
Let  be the space of all locally integrable functions on R . Define the space of all rapidly decreasing C ∞ functions  by for any , m k + ∈ Z . Furthermore define the space of all slowly increasing functions  by for any k + ∈ Z there exist some positive constants k C and holds for each , k m Z + ∈ , where Z + means the space of all positive integers.
From the above we can easily obtain that the convolution of f and g is defined by The definition of the convolution can be extended to the k-th convolution by From the above definitions we can easily obtain the next result.
Let H  be the set of rapidly decreasing functions satisfying 1) Proof.
1) can be shown easily. We next prove 2). Since f g S * ∈ from Proposition If a distribution U is a linear form and continuous on S, then U is called slowly increasing.
The Dirac distribution δ is defined by Let R * be the set of hyper real numbers. The hyper function : is a family of functions. Let *  , * ∞  , *  and *  be the set of hyper integrable functions, the set of hyper ∞  functions, the set of slowly decreasing hyper ∞  functions and the set of slowly increasing hyper ∞  functions, respectively.
In the above definitions, "it is a prediction ( ) P t " is equivalent to " ( ) In other words, at each neighborhood of the zero on ( ] Next we define the hyper inner product by Furthermore the convolution of hyper functions is defined by For the convolution of hyper functions, analogous results of Propositions 1 and 2 hold.
The hyper operator T * is also defined by

Extended Wiener Measure
Put , 0,1, , where N is a hyper natural number in the sense of nonstandard analysis.
Let V be the space of simple random walks defined by where the simple random walk is defined by Combining (2) and (3)

S t t T ≤ ≤
be a stock price, where T is a maturity. The well known Black-Scholes model [1], which describes the price evolution of some stock options, is the Itô stochastic differential equation where  ( ) W t is a standard Wiener process,  the volatility σ and the drift, or trend, μ are assumed to be constant. This model and its numerous generalizations are playing a major role in financial mathematics since more than thirty years although Equation (4) is often severely criticized (see, e.g., [1] [3] and the references therein).

The solution of Equation (4) is the geometric Brownian motion which reads
is the initial condition. It seems most natural to consider the mean ( ) 0 e t S µ of ( ) S t as the trend of ( ) S t . This choice unfortunately does not agree with the following fact: − is almost surely not a quickly fluctuating function around 0, i.e., the probability that is not "small", when   is "small",  T is neither "small" nor "large".

Option Pricing for One-Step Binomial Model
Let K be an exercise price. A derivative with a payoff is called a call option. On the other hand a put option is a derivative with a payoff ( ) ( ) We first consider the option premium by a one-step binomial tree model de- where d r u ≤ ≤ are returns on the stock and r is a risk-free rate. We next define a risk-neutral probability by We next consider the two-steps binomial tree model defined by

Option Pricing for Multi-Steps Binomial Model
We next consider an extension of the multi-steps (nT steps) binomial tree model defined by

Non-Standardization for Option Pricing
We next consider an extension of the multi-steps (nT steps) binomial model to the infinitesimal scale model from the view point of the non-standard analysis.
The binomial distribution is calculated by the convolution of sums of i.i.d. random variables. Therefore we should apply the notion of the convolution in the non-standard sense.
Consider a stock price defined by the Black-Sholes model Then the price of the European call option with a strike price K is given by  (5) where ( ) , V V S t = is the option price at time t. This investigation using the standard-analysis is so complicated.
Since it is impossible to calculate ( ) , V S t with respect to the Wiener measure W directly by the usual standard method, Black and Scholes [1] showed their famous formula for the option pricing to solve the PDE (5).
where the boundary condition: