Wiener measure W on C [ 0, T ] is defined by

W { ω ∈ C [ 0, T ] | ω ( t 1 ) ∈ A 1 , ⋯ , ω ( t k ) ∈ A k } = ∫ A 1 ⋯ ∫ A k p ( t 1 , x 1 ) p ( t 2 − t 1 , x 2 − x 1 ) ⋯ p ( t k − t k − 1 , x k − x k − 1 ) d x 1 ⋯ d x k

for any 0 ≤ t 1 ≤ ⋯ ≤ t n ≤ T and any Borel sets A 1 , ⋯ , A k , where

p ( t , x ) = 1 2 π t e − x 2 2 t .

Let F be the space of all locally integrable functions on R . Define the space of all rapidly decreasing C ∞ functions S by

sup x ∈ R | x m f ( k ) ( x ) | < ∞

for any m , k ∈ Z + . Furthermore define the space of all slowly increasing functions O by for any k ∈ Z + there exist some positive constants C k and M k ∈ Z + such that

| f ( k ) ( x ) | < C k ( 1 + | x | ) M k .

f ∈ F is Lebesgue integrable on any compact set K such that

− ∞ < ∫ K   f ( x ) d x < ∞

For f ∈ C ∞ there exists continuous k-th derivative f ( k ) ( x ) for each k ≥ 1 .

For f ∈ S ⊂ C ∞

sup x ∈ R | x m f ( k ) ( x ) | < ∞

holds for each k , m ∈ Z + , where Z + means the space of all positive integers.

From the above we can easily obtain that

S ⊂ O ⊂ C ∞ ⊂ F

For f , g ∈ S , the convolution of f and g is defined by

f ∗ g ( x ) = ∫ − ∞ ∞   f ( s ) g ( x − s ) d s

The definition of the convolution can be extended to the k-th convolution by

f 1 ∗ f 2 ∗ ⋯ ∗ f k ( x ) = ∫ − ∞ ∞ ( f 1 ∗ ⋯ ∗ f k − 1 ) ( s ) f k ( x − s ) d s

for each k ∈ Z + .

From the above definitions we can easily obtain the next result.

Proposition 1.

1) f ∗ g ( x ) = g ∗ f ( x ) .

2) f ∗ ( g + h ) ( x ) = g ∗ f ( x ) + g ∗ h ( x ) .

3) supp ( f ∗ g ) ⊆ supp ( f ) ∪ supp ( g ) .

Proposition 2. For φ , ψ ∈ S

1) φ ψ ∈ S .

2) φ ∗ ψ ∈ S .

3) ( φ ∗ ψ ) ( k ) ( x ) = φ ( k ) ∗ ψ ( x ) = φ ∗ ψ ( k ) ( x ) .

Let H ˜ be the set of rapidly decreasing functions satisfying

1) f ( − x ) = f ( x ) .

2) ∫ − ∞ ∞ f ( x ) d x = 1 .

Define a function f ¯ ( t ) by

f ¯ ( t ) ( x ) = 1 t f ( x t ) ,   t ∈ ( 0 , 1 ]

Proposition 3.

1) f ∈ H ˜ ⇒ f ¯ ( t ) ∈ H ˜ .

2) f , g ∈ H ˜ ⇒ f ∗ g ∈ H ˜ .

Proof.

1) can be shown easily. We next prove 2). Since f ∗ g ∈ S from Proposition 2 2), we have

1) f ∗ g ( − x ) = ∫ − ∞ ∞ f ( s ) g ( − x − s ) d s = ∫ − ∞ ∞ f ( − s ) g ( x + s ) d s = ∫ ∞ − ∞ f ( u ) g ( x − u ) ( − d u ) = ∫ − ∞ ∞ f ( u ) g ( x − u ) d u = f ∗ g (x)

2) ∫ − ∞ ∞ f ∗ g ( x ) d x = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( s ) g ( x − s ) d s d x = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( u ) g ( v ) d u d v = ∫ − ∞ ∞ f ( u ) d u ∫ − ∞ ∞ g ( v ) d v = 1

Remark 1. 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

δ ( ϕ ) = ϕ ( 0 ) ,   ϕ ∈ S

Let V be a set of the slowly increasing distributions and let L f ∈ V be defined by

L f ( φ ) = 〈 f , ϕ 〉 ,   ϕ ∈ S .

Then 〈 f , ϕ 〉 is the inner product and it is well defined when

〈 f , ϕ 〉 = ∫ − ∞ ∞ f ( x ) ϕ ( x ) d x

is integrable.

Let R ∗ be the set of hyper real numbers. The hyper function f δ : R ∗ → R ∗ is defined by

f δ ( a δ ) = b δ ,   b ( t ) = f t ( a ( t ) ) ,   t ∈ ( 0 , 1 ]

for a representative a ( t ) of a δ ∈ R ∗ , where { f t ( ⋅ ) | t ∈ ( 0,1 ] } is a family of functions. Let F ∗ , C ∗ ∞ , S ∗ and O ∗ be the set of hyper integrable functions, the set of hyper C ∞ functions, the set of slowly decreasing hyper C ∞ functions and the set of slowly increasing hyper C ∞ functions, respectively.

In the above definitions, “it is a prediction P ( t ) ” is equivalent to “ P ( t ) for any t ∈ J , J ∈ U on ( 0,1 ] , where U is an ω-incomplete ultra filter, is prediction”. In other words, at each neighborhood of the zero on ( 0,1 ] P ( t ) is established almost everywhere.

Next we define the hyper inner product by

〈 f δ , g δ 〉 ∗ = a δ ∈ R ∗ ,   f δ , g δ ∈ F ,   a ( t ) = 〈 f ( t ) , g ( t ) 〉 ∈ R     ( t ∈ ( 0 , 1 ] )

Furthermore the convolution of hyper functions is defined by

f δ ∗ g δ = h δ ,   h ( t ) = f ( t ) ∗ g ( t )   ( t ∈ ( 0 , 1 ] )

For the convolution of hyper functions, analogous results of Propositions 1 and 2 hold.

The hyper operator T ∗ is also defined by

T ∗ ( ϕ s ) = a δ ∈ R ∗ ,   a ( t ) = T ( ϕ s )   for   T ∈ T ,   ϕ s ∈ S ∗

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.

Put

t k = T k N ,   k = 0 , 1 , ⋯ , N

where N is a hyper natural number in the sense of nonstandard analysis.

Let V be the space of simple random walks defined by

V = { ω ¯ ∈ C [ 0 , T ] |   ω ¯   is   a   simple   random   walk }

where the simple random walk is defined by

P { ω ¯ ( t k ) = x k − 1 + 1 / N | ω ¯ ( t k − 1 ) = x k − 1 } = 1 2

P { ω ¯ ( t k ) = x k − 1 − 1 / N | ω ¯ ( t k − 1 ) = x k − 1 } = 1 2

and

ω ¯ ( 0 ) = x 0 = 0

with probability 1. Then V is a support on the extended Wiener measure in C [ 0, T ] . The fact is based on the central limit theorem.

For any set A in the Borel σ-field B , the extended Wiener measure of A is defined by

W ( A ) = # ( A ∩ V ) 2 N

where # ( A ) the cardinality of A ∩ V .

Theorem 2.1. For any Borel set A ∈ B ( C [ 0, T ] ) , the Wiener measure W can be represented by

W ( A ) = ∑ ω ¯ ∈ A ∩ V W ( ω ¯ ) (1)

where B ( C [ 0, T ] ) is the Borel field of the space of continuous functions on [ 0, T ] .

Proof. Consider a calculus of the integral with respect to W in the sense of nonstandard analysis. For ω ¯ , ω ˜ ∈ A ∩ V with ω ¯ ≠ ω ˜ ,

ω ¯ ∩ ω ˜ = ∅ (2)

Since V is dense in C [ 0, T ] , we have

∪ ω ¯ ∈ A ∩ V ω ¯ = A (3)

Combining (2) and (3) we prove the theorem. □

Theorem 2.2. For a measurable function h on C [ 0, T ] we have

∫ C [ 0, T ] h ( ω ) W ( d ω ) = ∑ ω ¯ ∈ V h ( ω ¯ ) W ( ω ¯ )

Let K be an exercise price. A derivative with a payoff

D ( T ) = f ( S ( T ) ) = max { S ( T ) − K ,   0 }

is called a call option. On the other hand a put option is a derivative with a payoff

D ( T ) = f ( S ( T ) ) = max { K − S ( T ) ,   0 } .

We first consider the option premium by a one-step binomial tree model defined by

S ( T ) = { S ( 0 ) ( 1 + u ) S ( 0 ) ( 1 + d )

where d ≤ r ≤ u are returns on the stock and r is a risk-free rate. We next define a risk-neutral probability by

{ p u * = r − d u − d p d * = u − r u − d

and E * is the expectation under the risk-neutral probability { p u * ,   p d * } . Since the option premium can be obtained to discount the expectation of the payoff function D ( t ) , we have the following the call option premium C 0

C 0 = 1 1 + r E * [ D ( T ) ] = 1 1 + r [ p d * max { ( 1 + d ) S 0 − K , 0 } + p u * max { ( 1 + u ) S 0 − K , 0 } ]

P 0 = 1 1 + r E * [ G ( T ) ] = 1 1 + r [ p d * max { K − ( 1 + d ) S 0 , 0 } + p u * max { K − ( 1 + u ) S 0 , 0 } ]

We next consider the two-steps binomial tree model defined by

{ P * { S ( T 2 ) = ( 1 + u 2 ) S ( 0 ) } = p u * P * { S ( T 2 ) = ( 1 + d 2 ) S ( 0 ) } = p d *

and

{ P * { S ( T ) = ( 1 + u 2 ) S ( T 2 ) } = p u * P * { S ( T ) = ( 1 + d 2 ) S ( T 2 ) } = p d *

Then we have

{ P * { S ( T ) = ( 1 + u 2 ) 2 S ( 0 ) } = ( p u * ) 2 P * { S ( T ) = ( 1 + d 2 ) ( 1 + u 2 ) S ( 0 ) } = 2 p u * p d * P * { S ( T ) = ( 1 + d 2 ) 2 S ( 0 ) } = ( p d * ) 2

We next consider an extension of the multi-steps (nT steps) binomial tree model defined by

{ P * { S ( k n ) = ( 1 + u n ) S ( k − 1 n ) } = p u * P * { S ( k n ) = ( 1 + d n ) S ( k − 1 n ) } = p d *

for k = 1 , 2 , ⋯ , n T . Then

P * { S ( T ) = ( 1 + d n ) k ( 1 + u n ) n T − k S 0 } = ( n T k ) ( p u * ) k ( p d * ) n T − k ,

for k = 0 , 1 , 2 , ⋯ , n T . The call option premium for the multi-steps binomial model is the following

C 0 = 1 ( 1 + r n ) n T E * [ D ( T ) ] = ∑ k = 0 n T ( p d * ) k ( p u * ) n T − k max { ( 1 + d n ) k ( 1 + u n ) n T − k S ( 0 ) − K , 0 } ( 1 + r n ) n T

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

S ( t ) = S ( 0 ) exp ( ( μ − σ 2 2 ) t + σ W ( t ) ) ,   0 ≤ t ≤ T .

Then the price of the European call option with a strike price K is given by

V ( S , T ) = E * [ max { S ( t ) − K , 0 } ] ,

where E * is the expectation with respect to the risk neutral probability. To investigate the call option pricing Black and Scholes (1973) solve the following PDE

∂ V ∂ t + 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 + r S ∂ V ∂ S − r V = 0 , (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).

For a pay-off function h ( s ( W ) ) , the option price with respect to h ( s ( W ) ) is represented by

V ( T ) = ∫ C [ 0 , T ] h ( s ( ω ) ) W ( d ω ) (6)

where for the risk neutral interest rate r ≥ 0

s ( ω ) = { S ( 0 ) exp [ ( r − σ 2 2 ) t + σ ω ( t ) ] ,   0 ≤ t ≤ T } (7)

Putting ω ( t ) = W ( t ) , s ( W ) is the Black-Scholes model defined by

s ( W ) = { S ( 0 ) exp [ ( r − σ 2 2 ) t + σ W ( t ) ] ,   0 ≤ t ≤ T } . (8)

A typical example of the pay off function D ( S ( W ) ) for the strike price K of the European call option is

D ( s ( W ) ) = max { S T − K , 0 } . (9)