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  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  . It is one of the former investigations.
2. Construction of Extended Wiener Measure
2.1. Definition of Standard Wiener Measure
Wiener measure on is defined by
for any and any Borel sets , where
2.2. Nonstandard Convolution by Hyper Function
Let be the space of all locally integrable functions on . Define the space of all rapidly decreasing functions by
for any . Furthermore define the space of all slowly increasing functions by for any there exist some positive constants and such that
is Lebesgue integrable on any compact set K such that
For there exists continuous k-th derivative for each .
holds for each , where means the space of all positive integers.
From the above we can easily obtain that
For , the convolution of f and g is defined by
The definition of the convolution can be extended to the k-th convolution by
for each .
From the above definitions we can easily obtain the next result.
Proposition 2. For
Let be the set of rapidly decreasing functions satisfying
Define a function by
1) can be shown easily. We next prove 2). Since from Proposition 2 2), we have
Remark 1. If a distribution U is a linear form and continuous on S, then U is called slowly increasing.
The Dirac distribution is defined by
Let V be a set of the slowly increasing distributions and let be defined by
Then is the inner product and it is well defined when
Let be the set of hyper real numbers. The hyper function is defined by
for a representative of , where 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 ” is equivalent to “ for any , on , where is an ω-incomplete ultra filter, is prediction”. In other words, at each neighborhood of the zero on is established almost everywhere.
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 is also defined by
Remark 2. Similarly to the definition of the convolution in hyper functions, we can well define the convolution of hyper functions and hyper distributions, the convolution of hyper distributions and hyper functions and the convolution in distributions, respectively.
2.3. Extended Wiener Measure
where 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
with probability 1. Then V is a support on the extended Wiener measure in . The fact is based on the central limit theorem.
For any set A in the Borel σ-field , the extended Wiener measure of A is defined by
where the cardinality of .
Theorem 2.1. For any Borel set , the Wiener measure can be represented by
where is the Borel field of the space of continuous functions on .
Proof. Consider a calculus of the integral with respect to W in the sense of nonstandard analysis. For with ,
Since V is dense in , we have
Combining (2) and (3) we prove the theorem. □
Theorem 2.2. For a measurable function h on we have
3. Black-Scholes Model
Let be a stock price, where T is a maturity. The well known Black-Scholes model  , which describes the price evolution of some stock options, is the Itô stochastic differential equation
• 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.,   and the references therein).
The solution of Equation (4) is the geometric Brownian motion which reads
where is the initial condition. It seems most natural to consider the mean of as the trend of . 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”.
A rigorous treatment, which would agree with nonstandard analysis (see, e.g.,    ), may be deduced from some infinitesimal time-sampling of Equation (4).
3.1. 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 defined by
where are returns on the stock and r is a risk-free rate. We next define a risk-neutral probability by
and is the expectation under the risk-neutral probability . Since the option premium can be obtained to discount the expectation of the payoff function , we have the following the call option premium
We next consider the two-steps binomial tree model defined by
Then we have
3.2. Option Pricing for Multi-Steps Binomial Model
We next consider an extension of the multi-steps (nT steps) binomial tree model defined by
for . Then
for . The call option premium for the multi-steps binomial model is the following
3.3. 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
where is the expectation with respect to the risk neutral probability. To investigate the call option pricing Black and Scholes (1973) solve the following PDE
where is the option price at time t. This investigation using the standard-analysis is so complicated.
Since it is impossible to calculate with respect to the Wiener measure directly by the usual standard method, Black and Scholes  showed their famous formula for the option pricing to solve the PDE (5).
For a pay-off function , the option price with respect to is represented by
where for the risk neutral interest rate
Putting , is the Black-Scholes model defined by
A typical example of the pay off function for the strike price K of the European call option is
4. Heat Equations
Consider a thin circular plate whose faces are impervious to heat flow and whose circular edge is kept zero.
Initial temperature of the plate at is a function of the distance from the center. Let the radius of the plate be a. It is obvious because of symmetry that the temperature v must be a function of r and t only.
Using cylindrical coordinates,
where the boundary condition:
The initial condition:
We easily find that the Black-Sholes Equation (5) can be implied by the Heat Equation (10).
Put . Then the heat equation is transformed to
Furthermore we have
which is a well-known Bessel equation.
It is known that the solution of the Bessel’s equation is described by the first and the second Bessel function. They are typical special functions and described by the Fourier series. The Fourier series is calculated by the Fourier integral. When calculating Fourier integral, the convolution integral is applied. Here, let us consider the extension of the Bessel differential equation to the form in non-standard analysis directly. In fact, changing the standard real valuables to the non-standard valuables in a sense of Nelson. “Transfer Principle”  is effective to solve the equation. Then, if the convolution integral is extended to the nonstandard form, see the Section 2, it is sufficient to get the solutions.
Taking into non-standardization   for option pricing, that is, discretizing Wiener process in a sense of the nonstandard, we confirm that it is possible to discretize all the system. If the convolution is extended by using non-standard analysis, and if the hyper functions are satisfied with the conditions in the Section 2, we can obtain the solution for the Black-Sholes equation, which is described as the Fourier integral. Notice that they are of the Bessel special functions.