In this paper, we show how the ambiguities in derivation of the BSE can be eliminated. We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.

1. 1
BLACK SCHOLES PRICING CONCEPT.
Ilya I. Gikhman
6077 Ivy Woods Court Mason,
OH 45040, USA
Ph. 513-573-9348
Email: ilyagikhman@yahoo.com
Classification code
Key words. Black Scholes, option, derivatives, pricing, hedging.
Abstract. In some papers it have been remarked that derivation of the Black Scholes Equation (BSE)
contains mathematical ambiguities. In particular in [2,3 ] there are two problems which can be raise by
accepting Black Scholes (BS) pricing concept. One is technical derivation of the BSE and other the
pricing definition of the option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on
market risk weighting. In such approach, we define random market price for each market scenario. The
spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market
participants.
BLACK SCHOLES WORLD.
We highlight two popular derivations of the BSE. One is the original derivation [1] and other is a popular
derivation represented in [5]. Following [1] let us first recall original derivation of the BSE. Next, we will
present original derivation in stochastic processes form.
Let w ( x , t ) denote the value of the call option which is a function of the stock price x and time t. The
hedge position is defined by the number
w 1 ( x , t ) =
x
)t,x(w


(1.1)

2. 2
of the options that would be sold short against one share of long stock. First order approximation of the
ratio of the change in the option value to the change in the stock price is w 1 ( x , t ). Indeed, if the stock
price changes by an amount x , the option price will change by an amount w 1 ( x , t ) x , and the
number of options given by expression (1.1) will be change by an amount of x . Thus, the change in the
value of long position in the stock will be approximately offset by the change in value of a short position
in 1 / w 1 options. The hedged position that contains one share of stock long and 1 / w 1 options short is
definrd by the formula
x – w / w 1 (1.2)
The change in the value of the hedged position over a short interval time period t is
x – w / w 1 (1.3)
Using stochastic calculus we expend note that
w = w ( x + x , t + t ) – w ( x , t ) = w 1 x +
2
1
w 11 v 2
x 2
t + w 2 t (1.4)
Here
w 11 = 2
2
x
)t,x(w


, w 2 =
t
)t,x(w


and v 2
is the variance of the return on stock. Substituting (1.4) into expression (1.3), we find that the
change in the value of the equity in hedged position is:
– (
2
1
w 11 v 2
x 2
+ w 2 ) t / w 1 (1.5)
Since return on the equity in the hedged position is certain, the return must be equal to r t . Thus the
change in the hedge position (1.5) must equal the value of the equity times r t
– (
2
1
w 11 v 2
x 2
+ w 2 ) t / w 1 = ( x – w / w 1 ) r t (1.6)
From (1.6) we arrive at Black Scholes equation
w 2 = r w – r x w 1 –
2
1
v 2
x 2
w 11 (1.7)
Boundary condition to equation (1.7) is defined by the call option payoff, which is specified at the
maturity of the option date T
w ( x , T ) = ( x – c ) χ ( x  c ) (1.8)
Here χ ( x ) denotes indicator function. This formula must be the option valuation formula.

3. 3
Remark. Using modern stochastic calculus we can represent Black Scholes derivation in the next form.
Let S ( t ) denote a security price at the moment t ≥ 0 and suppose that
dS ( t ) =  S ( t ) dt + σ S ( t ) dw ( t ) (1.9)
European call option written on security S is a contract, which grants buyer of the option the right to buy
a security for a known price K at a maturity T of the contract. The price K is known as the strike price of
the option. According to call option contract the payoff of the European call option is
max { S ( T , ω ) – K , 0 }
In order to buy the option contract at t buyer of the option should pay option premium at t. The option
premium is also called option price. The pricing problem is the problem of finding option price at any
moment t prior to maturity. Following [5] consider a hedged position, consisting of a long position in the
stock and short position in the number Δ ( t )
Δ ( t ) = [
S
))t(S,t(C


] – 1
of the options. Hence, hedge position (1.2) can be represented as
Π ( t ) = x – w / w 1 = S ( t ) – [
S
))t(S,t(C


] – 1
C ( t , S ( t ) ) (1.2′)
The change in the value of the hedged position in a short interval t is equal to
S ( t + t ) – [
S
))t(S,t(C


] – 1
C ( S ( t + t ) , t + t ) (1.3′)
Note that in latter formula number of options at the next moment t + t does not change and equal to
[
S
))t(S,t(C


] – 1
Taking into account Ito formula (1.4) can be rewritten as
Δ C = C ( t + t , x + x ) – C ( t , x ) = C ( t + t , S ( t + t ; t , x ) ) – C ( t , S ( t + t ; t , x ) ) +
+ C ( t , S ( t + t ; t , x )) – C ( t , x ) = C ( t + t , x ) – C ( t , x ) + C ( t , S ( t + t ; t , x ) ) –
– C ( t , x ) + o ( t ) = C /
x ( t , x ) S +
2
1
C
//
xx ( t , x ) σ 2
x 2
t + C /
t ( t , x ) t + o ( t )
where o ( t ) is the random variable defined by Taylor formula taking in the integral form and
0t
l.i.m

( t ) – 1
o ( t ) = 0. Then the formula (1.5) representing the change in the value of the hedged
portfolio can be rewritten as

4. 4
S – C [ C /
x ] – 1
= S – [ C /
x ] – 1
[ C /
x ( t , x ) S +
2
1
C
//
xx ( t , x ) σ 2
x 2
+ C /
t ( t , x ) ] t =
= – [ C /
x ] – 1
[
2
1
C
//
xx ( t , x ) σ 2
x 2
+ C /
t ( t , x ) ] t (1.5′)
The rate of return of the portfolio at t does not contain risky term of the ’white’ noise type. To avoid
arbitrage opportunity the rate of return of the portfolio at t should be proportional to risk free bond rate r.
Hence, we arrive at the BSE (1.7′) which can be represented in the form
C /
t ( t , x ) + r x C /
x ( t , x ) +
2
1
C
//
xx ( t , x ) σ 2
x 2
– r C ( t , x ) = 0 (BSE)
with boundary condition C ( T , x ) = max { x – K , 0 }.
In modern handbooks [5] one usually consider derivation of the BSE by construction hedged position by
using one option long and a portion of stocks short. This derivation is similar to original derivation [1].
The only difference between two derivations is the value
Δ ( t ) = [ C /
S ( t , S ( t ) ) ] – 1
of options in hedged portfolio in original derivation and the number of stocks
N ( t , S ( t ) ) = C /
S ( t , S ( t ) ) (1.10)
in hedged portfolio
Π ( t , S ( t ) ) = − C ( t , S ( t ) ) + N ( t , S ( t ) ) S ( t ) (1.11)
in alternative derivation [5].
Comment. In some papers, authors expressed a confusion raised from the use value of the hedged
position (portfolio) and its difference, differential, or financial change in the value in definition of the
hedged portfolio. These notions are in general similar to each other. Misunderstanding comes from the
use one time parameter tin definition of the portfolio value differently in dynamics of the portfolio. Such
drawback can be easily corrected by introducing hedged portfolio by the function
Π ( u , t ) = S ( u ) – [ C /
S ( t , S ( t ) ) ] – 1
C ( u , S ( u )) (1.12)
of the variable u, u  t where t, t  0 is a fixed parameter. Formula (1.12) defines value of the portfolio at
u , u, u  t constructed at t, t  0. Then differential of the function Π ( u , t ) with respect to variable u is
defined by the formula
d Π ( u , t ) = d S ( u ) – [ C /
S ( t , S ( t ) ) ] – 1
d C ( u , S ( u ))
We arrive at the hedged position by putting variable u = t

5. 5
Π ( u , t ) | u = t = Π ( t , t ) , d Π ( u , t ) | u = t = d u Π ( t , t )
Hence, instead of defining two separate equations for portfolio value and its dynamics we present one
equation, which covers two equations, which are used in Black Scholes pricing concept. In doing such
correction we expand original coordinate space ( t , Π ) to ( t , u , Π ) , 0 ≤ t ≤ u. Correction makes enable
to present an accurate derivation of the BS pricing concept.
Black-Scholes pricing concept. For a fixed moment of time t  [ 0 , T ] there exist option price
C ( t , S ( t )) and portfolio that contains one stock in long position and a portion of options in short
position for which the change in the value of the portfolio at t is riskless.
The BS pricing concept is specified by the BS’s hedged portfolio
1) borrows S ( t ) at risk free interest rate r and
2) sell immediately Δ ( t ) call options for BS price
These transactions provide investor risk free interest r on infinitesimal interval [ t , t + dt ).
On the other hand, point wise BS price can do not satisfy all market participants. Such price could satisfy
only hedger. Counterparty who buys options to get high return is a speculator and he can be either
satisfied or not by the Black-Scholes price.
In general, price is interpreted as a settlement between buyer and seller. Hence, Black-Scholes pricing
does not a price in general. It likes a date-t strategy. If we use Black-Scholes pricing then we only
guarantee instantaneous risk free return on BS’s portfolio at t. Market prices of the options can be close to
Black-Scholes model prices or not and there is no evidence or justification that market actually use Black
Scholes price.
Recall that perfect hedge provided by the option covers only one moment of time. Black and Scholes
remarked “As the variables x , t ( here x = S ( t )) change, the number of options to be sold short to create
hedged position with one share of stock changes. If the hedged position is maintained continuously, then
the approximations mentioned above become exact, and the return on the hedged position is completely
independent of the change in the value of the stock. In fact, the return on the hedged position becomes
certain. (This was pointed out to us by Robert Merton)” [1].
Hence, seller of the option is subject to market risk at the next moment t + t , t > 0 and can get either
loss or profit and selling option for BS price. Let us consider the cash flow generated by continuously
maintained hedged portfolio. The date-t value of the hedged portfolio is defined by the formula (1.3′) and
equal to Π ( t + Δt , t ). On the other date-(t + Δt) value of the BS’s portfolio is equal to
Π ( t + Δt , t + Δt ) = S ( t + Δt ) – [ C /
S ( t + Δt , S ( t + Δt )) ] – 1
C ( t + Δt , S ( t + Δt ) )
Thus in order to maintain hedged portfolio one should add
Π ( t + Δt , t + Δt ) – Π ( t + Δt , t ) = (1.13)
= { [ C /
S ( t , S ( t ) ) ] – 1
– [ C /
S ( t + Δt , S ( t + Δt )) ] – 1
} C ( t + Δt , S ( t + Δt ) )

6. 6
to Π ( t + Δt , t ) the adjustment sum at the moment t + Δt. Denote f ( t , S ( t ) ) = [ C /
S ( t , S ( t ) ) ] – 1
.
Bearing in mind relationships
f /
t ( t , S ( t ) ) = – 2/
S ]))t(S,t(C[
1
C
//
tS ( t , S ( t ) )
f /
S ( t , S ( t ) ) = – 2/
S ]))t(S,t(C[
1
C
//
SS ( t , S ( t ) )
f
//
SS ( t , S ( t ) ) = – 2/
S ]))t(S,t(C[
1
C
///
SSS ( t , S ( t ) )
C ( t + Δt , S ( t + Δt ) ) = C ( t , S ( t )) + [ C ( S ( t + Δt ) , t + Δt ) – C ( t , S ( t )) ]
one can apply Ito formula. It follows from (1.13) that date-( t + Δt) adjustment is equal to
Π ( t + Δt , t + Δt ) – Π ( t + Δt , t ) =
= – 2/
S ]))t(S,t(C[
1
{ [ C
//
tS ( t , S ( t )) + C
//
SS ( t , S ( t )) μ S ( t ) +
+
2
1
C
///
SSS ( t , S ( t ) ) σ 2
S 2
( t ) ] Δt + C
//
SS ( t , S ( t ) ) σ S ( t ) Δw ( t ) } [ C ( t , S ( t )) +
+ C /
t ( t , S ( t )) + C /
S ( t , S ( t )) μ S ( t ) +
2
1
C
//
SS ( t , S ( t ) ) σ 2
S 2
( t ) ] Δt +
+ C /
S ( t , S ( t ) ) σ S ( t ) Δw ( t ) ] = (1.14)
= – 2/
S ]))t(S,t(C[
1
{ [ C
//
tS ( t , S ( t )) + C
//
SS ( t , S ( t )) μ S ( t ) +
+
2
1
C
///
SSS ( t , S ( t ) ) σ 2
S 2
( t ) +
))(tS,t(C
))(tS,t(C))(tS,t(C //
SS
/
S
σ 2
S 2
( t ) ] Δt +
+ C
//
SS ( t , S ( t ) ) σ S ( t ) Δw ( t ) } C ( t , S ( t ))
If the value Π ( t + Δt , t + Δt ) > Π ( t + Δt , t ) then portfolio adjustment is the amount which should
be added at t + Δt. Otherwise corresponding sum should be withdrawn. Such adjustment represents mark-
to-market transactions. Using formula (1.14) we represent cash flow that corresponds to maintenance of
the hedged portfolio during lifetime of the option.
Let t = t 0 < t 1 < … < t n = T be a part ion of the lifetime period of the option. Then applying formula
(1.14) the maintenance of the hedged position can be represented by sum

7. 7
H ( t , T ) = 
n
1j
[ Π ( t j , t j ) – Π ( t j , t j – 1 ) ] = – 
n
1j
2
1-j1-j
/
S
1-j1-j
]))t(S,t(C[
))t(S,t(C

 { [ C
//
tS ( t j – 1 , S ( t j – 1 )) + C
//
SS ( t j – 1 , S ( t j – 1 )) μ S ( t j – 1 ) +
+
2
1
C
///
SSS ( t j – 1 , S ( t j – 1 ) ) σ 2
S 2
( t j – 1 ) +
+
))t(S,t(C
))t(S,t(C))t(S,t(C
1-j1-j
1-j1-j
//
SS1-j1-j
/
S
σ 2
S 2
( t j – 1 ) ] Δt j – 1 +
+ C
//
SS ( t j – 1 , S ( t j – 1 ) ) σ S ( t j – 1 ) Δw ( t j – 1 ) } = (1.15)
= 
T
t
2/
S ]))u(S,u(C[
))u(S,u(C
[ C
//
tS ( u , S ( u )) + C
//
SS ( u , S ( u )) μ S ( u ) +
+
2
1
C
///
SSS ( u , S ( u ) ) σ 2
S 2
( u ) +
))u(S,u(C
))u(S,u(C))u(S,u(C //
SS
/
S
σ 2
S 2
( u ) ] du +
+ 
T
t
2/
S ]))u(S,u(C[
))u(S,u(C
C
//
SS ( u , S ( u )) σ S ( u ) dw ( u )
Formula (1.15) shows that keeping hedged position over lifetime of the option is represented by a risky
cash flow. It is clear that it is costly to maintain the hedged position over the lifetime of the option. On the
other hand additional cash flow reflects additional cost for keeping hedge. Expected PV of the future cash
flow, which will adjust hedged portfolio, should be considered as collateral option price similar to CVA.
Now let us look at the alternative option pricing. First recall that option price at t is looking as
deterministic smooth function. Such preliminary condition implies that market risk which actually
undefined by BS’world has no effect on option price. On the other hand profit-loss analysis shows that
no=arbitrage BS’s option price admits either loss or profit. Our approach starts with the similar
observation and prescribes admitted option values at maturity a probability distribution that is specified
by underlying stock.
ALTERNATIVE APPROACH.
In this section, we represent an alternative approach to option pricing. We have discussed some
drawbacks of the BS option-pricing concept. The alternative approach to derivatives pricing was
introduced in [2-4]. We call two cash flows equal over a time interval [ 0 , T ] if they have equal
instantaneous rates of return at any moment during [ 0 , T ].
Introduce financial equality principle. Two investments S i ( t ) , i = 1, 2 we call to be equal at moment t
if their instantaneous rates of return are equal at this moment. If two investments are equal for any

8. 8
moment of time during [ 0 , T ] then we call these investments equal on [ 0 , T ]. Applying this definition
to a stock and European call option on this stock we arrive at the equation
)t(S
)T(S
 { S ( T ) > K } =
))t(S,t(C
))T(S,T(C
(2.1)
where C ( t , S ( t )) = C ( t , S ( t ) ; T , K ), 0  t  T denotes option price at t with maturity T and
strike price K and C ( T , X ; T , K ) = max { X – K , 0 }. Solution of the equation (2.1) is a random
function C ( t , S ( t ), ω ) that promises the same rate of return as its underlying S ( t ) for a scenario
ω  { ω : S ( T ) > K } and C ( t , S ( t ) ; ω ) = 0 for each scenario ω  { ω : S ( T ) ≤ K }.
Bearing in mind that this definition of the price depending on market scenario we call this price as the
market price. Spot price which we denote c ( t , S ( t )) is interpreted as the settlement price between
sellers and buyers of the option at t. It is deterministic function in t. Let S ( t ) = x and c ( t, x ) be a
spot call option price. The market risk of the buyer of the option is defined by the chance that buyer pays
more than it is implied by the market, i.e.
P { c ( 0 , x ) > C ( t , S ( t ) ; T , K ) } (2.2′)
On the other hand, option seller’s market risk is measured by the chance of the adjacent market event, i.e.
P { c ( 0 , x ) < C ( t , S ( t ) ; T , K ) } (2.2′′)
It represents the probability of the chance that the premium received by option seller is less than it is
implied by the market.
Let us illustrate alternative pricing by using a discrete space-time approximation of continuous model
(2.1). Consider a discrete approximation of the S ( T , ω ) in the form

n
1j
S j  { S ( T , ω )  [ S j – 1 , S j ) }
where 0 = S 0 < S 1 … < S n < +  and denote p j = P (  j ) = P { S ( T )  [ S j – 1 , S j ) }.
Note that p j for a particular j could be as close to 1 or to 0 as we wish. We eliminate arbitrage opportunity
for each scenario ω   j by putting
)ω;x,t(C
KS
x
S jj 
 , if S j  K
and
C ( t , x ; ω ) = 0 , if S j < K
The solution of the latter equation is
C ( t , x ; ω ) =
jS
x
( S j - K )  { S j  K }  { S ( T , ω )  ( S j – 1 , S j ] }

9. 9
Then the market price of the call option can be approximated by the random variable
C ( t , x ; ω ) = 
n
1j jS
x
( S j - K )  { S j  K }  { S ( T , ω )  ( S j – 1 , S j ] }
For the scenarios ω  ω j = { S j ≤ K } return on the underlying security is 0 < x – 1
S j ≤ K while
option return is equal to 0. Thus, the option premium c ( 0 , x ) for the scenarios ω i = { S i > K }
should compensate losses of the security return for the scenarios ω = { S ( t , ω ) ≤ K }. Investor will be
benefitted by the option c ( t , x ) < C ( t , x ; ω ) for the scenarios ω for which { S i ( T , ωi ) > K }.
Our goal to present a reasonable estimates of the spot price choice c ( t , x ) represented by the market.
One possible estimate is BS price. It is price is formed by no arbitrage principle. The drawback of the
no arbitrage pricing is the fact that in stochastic market there is no classical arbitrage that implied by the
BS’s model. We rather have a probability distribution that prescribes a particular probability for each
option value as well as a positive probability of losing the original premium including no arbitrage BS
price. As far as market, pricing equation (2.1) represents definition of the option market price for each
market scenario the reasonable first order approximation of the settlement between buyers and sellers is
the option price that represents equal market risk as underlying stock.
Let t, t  [ 0 , T ] denote a moment of time. Then the value of the stock x = S ( t ) at t is equal to the
value B – 1
( t , T ) x at T. From buyer perspective the chances of profit / loss
P { S ( T , ω )  B – 1
( t , T ) x } , P { S ( T , ω ) < B – 1
( t , T ) x }
and average profit / loss on stock at date T are defined as following
avg S , profit = E S ( T , ω ) χ { S ( T , ω )  B – 1
( t , T ) x } ,
avg S , loss = E S ( T , ω ) χ { S ( T , ω ) < B – 1
( t , T ) x }
correspondingly. Similarly define chances of underpriced and overpriced option
P { C ( T , S ( T ) )  B – 1
( t , T ) c ( t , x ) } , P { C ( T , S ( T ) )  B – 1
( t , T ) c ( t , x ) }
and average profit / loss on option at maturity
avg C , profit = E C ( T , S ( T ) ) χ { C ( T , S ( T ) )  B – 1
( t , T ) c ( t , x ) } ,
avg C , loss = E C ( T , S ( T ) ) χ { C ( T , S ( T ) ) < B – 1
( t , T ) c ( t , x ) } ,
The zero-order approximation of the option price based on market risk can be defined by the equality
P { S ( T , ω )  B – 1
( t , T ) x } = P { C ( T , S ( T ) )  B – 1
( t , T ) c ( t , x ) } (2.3)

10. 10
The same value of the spot option price can be defined by the use of equality chances of losses for stock
and option. Next order adjustment can be calculated by taking into account average loss / profit ratios for
stock and option
R S ( t , T ) =
}x)T,t(B)ω,T(S{χ)ω,T(SE
}x)T,t(B)ω,T(S{χ)ω,T(SE
1-
1-


R C ( t , T ) =
})x,t(c)T,t(B))T(S,T(C{χ))T(S,T(CE
})x,t(c)T,t(B))T(S,T(C{χ))T(S,T(CE
1-
1-


where C ( T , S ( T ) ) = max { S ( T ) – K , 0 }. Let S ( t ) be a solution of the equation (1.9). Bearing
in mind that solution of the equation (2.3) can be written in the form
S ( T ) = x exp 
T
t
( μ –
2
σ 2
) dv + 
T
t
σ dw ( v )
We note that for any q > 0
P { S ( T ) < q } = P { ln S ( T ) < ln q } = P { ln x + 
T
t
( μ –
2
σ 2
) dv + 
T
t
σ dw ( v ) < ln q }
Right hand side represents distribution of the normal distributed variable with mean and variance equal to
ln x + ( μ –
2
σ 2
) ( T – t ) , σ 2
( T – t )
correspondingly. Therefore
P { S ( T ) < q } =
)tT(σπ2
1
2


qln
-
exp –
)tT(σπ2
])tT()
2
σ
μ(xlnv[
2
2
2


dv
Differentiation of the right hand side with respect to q brings the density distribution ρ ( t , x ; T , q ) of
the random variable S ( T )
ρ ( t , x ; T , q ) =
)tT(qσπ2
1
22

exp –
)tT(σπ2
])tT()
2
σ
μ(
x
q
ln[
2
2
2


Left and right hand sides of the equation (2.3) can be represented by formulas
p S , profit = P { S ( T )  B – 1
( t , T ) x } = P { exp [ ( μ –
2
σ 2
) ( T – t ) +

11. 11
+ σ [ w ( T ) – w ( t ) ] ]  B – 1
( t , T ) } =
=
)tT(σπ2
1
2



)T,t(Bln-
exp –
)tT(σ2
])tT()
2
σ
μ(xlnv[
2
2
2


dv =
= N (
tTσ
)tT()
2
σ
μ(xln)T,t(Bln
2


 )
where N ( · ) is the standard normal distribution cumulative distribution function. Then
p C , profit = P { C ( T , S ( T ))  B – 1
( t , T ) c ( t , x ) } =
= P { max { S ( T ) – K , 0 }  B – 1
( t , T ) c ( t , x ) } = P { S ( T )  K + B – 1
( t , T ) c ( t , x ) } =
= N (
tTσ
)tT()
2
σ
μ(xln])x,t(c)T,t(BK[ln
2
1




)
Then equation (2.3 ) can be rewritten as
N (
tTσ
)tT()
2
σ
μ(xln])x,t(c)T,t(BK[ln
2
1




) = p S , profit
The solution of the equation can be represented in closed form
c ( t , x ) = B ( t , T ) { exp – [ σ tT  N – 1
( p S , profit ) +
+ ln x + ( μ –
2
σ 2
) ( T – t ) ] – K } (2.4)
Given c ( t , x ) one can calculate the value account average loss / profit ratio of the option. If the value
R C ( t , T ) is small then the use option price in the form (2.4) does not looks reasonable. In this case one
can start with numeric solution of the equation
R C ( t , T ) = R S ( t , T ) (2.5)
Right hand side of the equation (2.5) is known number and left hand side is equal to
R C ( t , T ) =
})x,t(c)T,t(BK)T(S{χ}0,K-)T(S{maxE
})x,t(c)T,t(BK)T(S{χ}0,K-)T(S{maxE
1
1-
1
1-


=

12. 12
= [ E max { S ( T ) – K , 0 } χ { S ( T )  K + B – 1
( t , T ) c 1 ( t . x ) } ] – 1
– 1
Here
E max { S ( T ) – K , 0 } χ { S ( T )  K + B – 1
( t , T ) c 1 ( t . x ) } =
=
)tT(σπ2
1
2



 
)x,t(c)T,t(BlnK 1
1
( v – K) exp –
)tT(σ2
])tT()
2
σ
μ(xlnv[
2
2
2


dv
Hence, equation (2.5) admits numeric approach for solution it with respect to c 1 ( t , x ). In this case it
might be that left hand side of the inequality
P { S ( T , ω )  B – 1
( t , T ) x } > P { C ( T , S ( T ) )  B – 1
( t , T ) c 1 ( t , x ) }
remarkably exceeds right hand side, i.e. chance to get profit is too small with respect to similar
characteristic on stock investment. Such situation suggests establishing option price, which is a
combination of two estimates represented by equations (2.3) and (2.5). One can define variance of the
loss and profit of the option
V 2
C , loss = E { C ( T , S ( T ) ) χ [ C ( T , S ( T ) ) < B – 1
( t , T ) c ( t , x ) ] –
– E C ( T , S ( T ) ) χ [ C ( T , S ( T ) ) < B – 1
( t , T ) c ( t , x ) ] } 2
V 2
C , profit = E { C ( T , S ( T ) ) χ [ C ( T , S ( T ) )  B – 1
( t , T ) c ( t , x ) ] –
– E C ( T , S ( T ) ) χ [ C ( T , S ( T ) )  B – 1
( t , T ) c ( t , x ) ] } 2
which will supplement to above risk characteristics of the latter estimates. Option loss and profit
variances can be compared with correspondent characteristics of the underlying asset
V 2
S , loss =
= E { S ( T , ω ) χ [ S ( T , ω ) < B – 1
( t , T ) x ] – E S ( T , ω ) χ [ S ( T , ω ) < B – 1
( t , T ) x ] } 2
V 2
S , profit =
= E { S ( T , ω ) χ [ S ( T , ω )  B – 1
( t , T ) x ] – E S ( T , ω ) χ [ S ( T , ω )  B – 1
( t , T ) x ] } 2
Conclusion. We consider Black Scholes derivatives pricing concept as oversimplified pricing. The
oversimplified pricing means that definition of the derivatives price ignores market risk of any spot option
price including no arbitrage pricing. Our definition of option price is based on weighted risk-reward or
profit-loss ratios.
I am grateful to P.Carr for his interest and useful discussions.

13. 13
References.
1. Black, F., Scholes, M. The Pricing of Options and Corporate Liabilities. The Journal of Political
Economy, May 1973.
2. Gikhman, Il., On Black- Scholes Equation. J. Applied Finance (4), 2004, p. 47-74,
3. Gikhman, Il., Derivativs Pricing. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=500303.
4. Gikhman, Il., Alternative Derivatives pricing. Lambert Academic Publishing, p.154.
5. Hull J., Options, Futures and other Derivatives. Pearson Education International, 7ed. p.814.