This document discusses issues with the derivation of the Black-Scholes equation and option pricing model. It highlights two popular derivations of the Black-Scholes equation, noting ambiguities in the original derivation. It proposes defining the hedged portfolio over a variable time interval to address these ambiguities. The document also notes drawbacks of the Black-Scholes price, including that it only guarantees a risk-free return over an infinitesimal time period and does not reflect market prices which may incorporate other strategies.

1. 1
COMMENTS ON OPTION PRICING.
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, hedging, local volatility.
Abstract. In some papers we 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 use options as hedging instruments. We develop a new approach to option
price based on market risk. We define random market price of the option for each market scenario. The
option premium is interpreted as the settlement between profit – loss exapectations.
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)
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

2. 2
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.
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)

3. 3
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 value of the hedged position (portfolio) at t + 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 over [ t , t + t ) can be rewritten as
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′)

4. 4
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
i Π ( t ) =
)t(Π
)t(Π)tΔt(Π
lim
0t


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 considers a derivation of the BSE by construction hedged position
by using one option long and a portion of stocks in short. This derivation is similar to original Black-
Scholes derivation [1]. The only difference is that the value
Δ ( t ) = [ C /
S ( t , S ( t ) ) ] – 1
of options in hedged ratio in original derivation and the number of stocks
N ( t , S ( t ) ) = C /
S ( t , S ( t ) ) (1.10)
short in hedged portfolio
Π ( t , S ( t ) ) = C ( t , S ( t ) ) + N ( t , S ( t ) ) S ( t ) (1.11)
in an alternative derivation [5].
Comment. Some authors have expressed a confusion defining Black Scholes hedged position (portfolio).
Misunderstanding comes from the definition of the portfolio value. The same parameter time t in the
equation (1.2′) is used in two different meanings. It is a constant in the term [
S
))t(S,t(C


] – 1
and it is
a variable in other terms in (1.2′). Such drawback can be easily corrected by introducing hedged portfolio
for u  t by the function
Π ( u , t ) = S ( u ) – [ C /
S ( t , S ( t ) ) ] – 1
C ( u , S ( u )) (1.12)
Here u, u  t is a variableand t, t  0 is a fixed parameter. Formula (1.12) defines the value of the
portfolio at u , u, u  t constructed at t, t  0. The differential of the portfolio value 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 ))

5. 5
and the differential of the hedged position at t is defined by putting u = t. Thus Black Scholes portfolio is
specified by equalities
Π BS ( t ) = Π ( u , t ) | u = t = Π ( t , t ) , d Π BS ( t ) = d Π ( u , t ) | u = t = d u Π ( t , t )
Therefore equation (1.12) covers two equations value of the portfolio and its change in the value, which
are used in Black Scholes equation derivation. 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.
We can summarize the Black-Scholes pricing concept. For each moment of time t  [ 0 , T ] there exist
option price C ( t , S ( t )) and a portfolio that contains one stock in long position and the portion of
options in short position for which the change in the value of the portfolio at t is riskless.
The BS price is synthetically defined by the BS’s hedged portfolio. At any moment t  [ 0 , T ) if the
initial value Π ( t , t ) < 0 investor lends or borrows if Π ( t , t ) > 0 to or from counterparty at risk free
interest rate r. These action is provided by the risk free interest r on infinitesimal small interval
[ t , t + dt ).
Drawbacks of the BS price.
On the other hand, there is no evidence that the date-t hedge price would sbe used by the market. Such
price could satisfy a hedger who buys options to get a risk free return at the small interval of time. On the
other hand it is difficult to expect that investors that are looking for higher return than risk free rate or use
other combinations of assets than Π BS ( t ) can be satisfied by the Black-Scholes price. Different groups
of investors should provide additional effects on option demand and therefore we can not completely rely
on interpretation of the option price represented by BSE solution. BS price is one thay guarantees a
realization of the particular strategy.
Market prices of the options can be close to Black-Scholes prices or not and there is no evidence or
justification that market actually uses Black Scholes price. Indeed market participants can buy option
contracts based on its own speculative strategies. This remark should take into account when we discuss
reasons of existence such phenomena as implied and local volatilitiy.
Another observation is the perfect hedging that actually covers only initial moment of time. Black and
Scholes remarked “As the variables x = S ( t ) , 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].
This remark highlights the fact that holding risk free position by maintained continuously hedged position
implies additional risky ( uncertain ) cash flow generated by the process of buying – selling options in
short position to maintain risk free hedged portfolio. The Black Scoles pricing does not imply future
maintance of the short position of the portfolio. Otherwise future cash flow should definitely adjust the
BS option price. Bearing in mind coupon bond pricing one can expect that adjustment of continuously
maintance of the hedge position should be represented in spot price as expected present value.

6. 6
Thus the Black Scholes call option price C ( t , S ) is a price on the interval on which there exisis the
hedged portfolio, i.e. at any moment on the interval [ 0 , T ) on the infitisimal interval [ t , t + dt ) and
C ( T , S ) = max { S – K , 0 }.
Hence the existence of the BS option price does not its lifetime. It is a period on which the perfect hedged
portfolio exists. This statement can be expressed by using financial terms as following. At a moment t
over lifetime of the option investor borrows the sum Π BS ( t ) equal to
C ( t , S ( t ) ) + N ( t , S ( t ) ) S ( t )
at risk free interest rate. If C ( t , S ) is a solution of the BSE then at the moment t + dt investor returns this
sum and risk free interest rate on this sum Π BS ( t ) [ 1 + r dt ] to the bank. Using synthetic pricing, i.e.
the assumption that N ( t , S ( t ) ) shares of stock can be sold for S ( t + dt ) at t + dt we arrive at the fact
that option price is a solution of the BSE on [ t , t + dt ) and not on [ 0 , T ]. If we ignore the fact of the
existence of the hedged portfolio then we arrive at theclassical solution of the BSE equation on [ 0 , T ].
Thus, if we take into account that option price is defined on interval on which hedged portfolio is defined
then the BSE solution is defined on [ t , t + dt ) and not on [ 0 , T ].
There is also a puzzle that connected to global solution of the BSE. On the one hand it follows from
mathematics that if C ( t , S ) is a smooth and bounded local in t solution of the BSE then it can be
considered as a solution of the BSE on [ 0 , T ) which satisfies the boundary condtion at T. On the other
hand the price of the option that is a solution of the BSE is defined as far as there existis BS hedged
portfolio on [ t , t +  t ). This definition implies that stock portion in portfolio can be sold and we arrive
at the option price. At the moment t +  t the portfolio should be adjusted. In the limit when  t tends to
zero hedged portfolio should be adjusted buying or selling its components. This non zero cash flow
provides maintence of the BS hedged portfolio which provides existence of the global in time hedged
portfolio. On the other hand option price is already defined on [ 0 , T ]. Hence there is no affect of the
portfolio maintence on BS option price.
Some wrong applications of the Black Scholes pricing.
1. We do not have general definition of the option pricing. Indeed if underlying stock does not
follow Geometric Browning Motion (GBM) equation (1.9) the hedged position defined by Black and
Scholes does not exist and therefore notion of the option price is undefined. Even if underlying stock is a
random process which distribution are closed to corresponding distributions of the a GBM we can not
formally justify that corresponding BS price is closed to the real option as it does not be defined.
Therefore for example a large number of research that are devoted to BS pricing for an asset having
random jumps, see for example [6] are formally incorrect. The BS hedged portfolio for such underlyings
does not exist and therefore the use of the BS formular is informal or even incorrect.
We are already highlighted that seller and buyer of the options are subject to market risk at the
moment t + t , t > 0 and they can either get loss or profit by applying ‘no-arbitrage’ BS price. Let us

7. 7
consider a cash flow generated by continuously maintained hedged portfolio. The date-t value of the
hedged portfolio is defined by the formula (1.3′) and it is equal to Π ( t + Δt , t ). On the other hand the
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 ) )
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 ) +

8. 8
+
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 and denote
 =
ni1
max

( t i – t i – 1 ). Then applying formula (1.14) the maintenance of the hedged position can be
represented by sum
H ( t , T ) =
0λ
lim


n
1j
[ Π ( t j , t j ) – Π ( t j , t j – 1 ) ] = –
0λ
lim


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 the hedged position over lifetime can be represented by a risky cash
flow. It is clear in general that it might be costly to maintain hedged position over the lifetime of the
option. On the other hand additional cash flow reflects additional cost for keeping hedge of the portfolio.
Expected PV (EPV) of the future cash flow should adjust the original BS price similar as periodic coupon

9. 9
payments adjusts zero coupon bond. On the other hand why investors that are not going to hold hedged
position over lifetime should think that the BS price is a reasonable option price.
II. It is important and quite popular concept called local volatility that somewhat is interpreted as an
adjustment of the BS pricing. We are outline the primary idea and briefly highlight its drawbacks. We
follow [7].
a) Let us briefly remind solution of the option pricing problem following [8]. The hedged portfolio
is usually defined by the formula
Π ( t , S ( t ) ) = C ( t , S ( t ) ) +
S
))t(S,t(C


S ( t )
In this formula the same latter t is fixed for the term
S
))t(S,t(C


and the variable for other terms. It is
incorrect. Define BS hedged portfolio Π by putting
Π ( u , S ( u ) ) = C ( u , S ( u ) ) +  ( t , S ( t ) ) S ( u ) (2.1)
where
 ( t , S ( t ) ) =
S
))t(S,t(C


u  [ t , T ]. Then
d Π ( u , S ( u ) ) | u = t = d C ( t , S ( t ) ) +  ( t , S ( t ) ) d S ( t ) (2.2)
Bearing in mind Ito formula the change in the value of the portfolio at u = t can be represented by the
formula
d Π ( u , S ( u ) ) | u = t = [ 2
222
S
))t(S,t(C
2
σ)t(S
t
))t(S,t(C





] d t
This formula does not contain a term proportional to d w ( t ). In order to exclude arbitrage one needs to
assume that
d P ( t , S ( t ) ) = r P ( t , S ( t ) ) d t (2.3)
Here r denotes the constant risk free interest rate. In other words rates of return of the riskless portfolio
and riskless borrowing rate should be equal. This is the essence of the no arbitrage pricing. Indeed one
can borrow Π ( t , S ( t ) ) at t and return this amount plus interest interest Π ( t , S ( t ) ) [ 1 + r dt ] at the
next moment t + dt. This is no arbitrage pricing applied to the BS hedged portfolio. From ( 2.3) it follows
the Black-Scholes equation

10. 10
2
222
x
)x,t(C
2
σx
t
)x,t(C





+ r
x
)x,t(C


– r C ( t , x ) = 0 (BSE)
defined in the area t  [ 0, T ), x > 0 with the backward boundary condition
C ( T , x ) = max { x – K , 0 }
The solution of the (BSE) equation can be represented in closed form
C ( t , x ) = E exp – r ( T – t ) max ( Sr ( T ; t , x ) – K , 0 ) (2.4) 5)
where
d S r ( u ) = r S r ( u ) d u + σ S r ( u ) d w ( u ) (2.5) (6)
u > t and Sr ( t ) = x .
Remark. Pricing formula (2.4) shows that actual underlying of the call option is the random process
S r ( u ) on original probability space{ Ω , F , P } while according to the general definition of the
derivative underlying should be the random process (1.9). The risk-neutral world was presented as a
solution of the confusion. It does not eliminate the fact that underlying of the Black-Scholes pricing
formula is the random process (2.5). First note that the random process S ( t ) is always defined on
original probability space { Ω , F , P } regardless of the whether options on this stock exist or not.
Risk neutral valuation concept suggests to consider equation (1.9) on the risk-neutral probability space
{ Ω , F , Q } where the risk-neutral probability measure Q is defined by the formula
Q ( A ) = A
{ exp 
T
0
[
σ
rμ 
d w Q ( t ) –
2
1

T
0
(
σ
rμ 
) 2
d t ] } P ( d ω )
for arbitrary set A  F. Here w Q ( t ) denote a Wiener process on { Ω , F , Q }. Then the random process
S ( t ) = S ( t , ω ) is a solution of the risk-neutral equation (2.5) on probability space { Ω , F , P } with a .
Wiener process
w ( t ) = w Q ( t ) + 
t
0
σ
rμ 
d u
on { Ω , F , P }. Thus the essence of the risk neutral valuations is to place real stock diffusion equation on
{ Ω , F , Q }. Recall that stock equation (1.9) regardless of option is defined on original probabability
space{ Ω , F , P }. We can omit controversial riskneutral valuations by starting with the equation (2.5).
Using explicit formula for the solution of the equation (2.5)
S r ( t ; 0 , x ) = x exp 
t
0
( r –
2
1
 2
) d l + 
t
0
 d w ( l )
the formula (2.4) can be rewritten in the well-known form

11. 11
C ( t , x ) = x N ( d 1 ) – K exp – r ( T – t ) N ( d 2 )
where
d 1 =
tTσ
1

[ ln
)t-T(r-expK
x
+
2
)t-T(σ 2
] , d 2 = d 1 –  t-T
and N (  ) is the cumulative distribution function of the standard Gaussian random variable. We reffered
the risk neutral valuation been controversial based on the next arguments. Paramer  in above formulas
for Q ( A ) and
b) Now let us look at an adjustment of the Black Scholes pricing concept presented by the local
volatility. The basic of this development was presented in [9-11]. Let us shortly outline the problem
putting for simplicity that drift coefficient is equal to zero
d S ( u ) =  S ( u ) d w ( u )
u > t and S ( t ) = x. Consider call option value
C ( T , K ) = E max { S ( T ; t , x ) - K , 0 } (2.6)
as a function of T , K when ( t , x ) are fixed parameters. Denote p ( t , x ; T , y ) the probability density
of the stochastic process S ( T ) = S ( T ; t , x ). Then latter equality can be rewritten in the form
C ( T , K ) = 

K
( y - K ) p ( t , x ; T , y ) d y (2.7)
Twice differentiation in (2.7) with respect to K leads us to equality
2
2
K
)K,T(C


= p ( t , x ; T , K ) (2.8)
To present local volatility consider a diffusion process K ( T ) , T ≥ t given by equation
K ( T ) = k + 
T
t
b ( u , K ( u ) ) d W ( u ) (2.9)
Diffusion coefficient b ( u , K ) is unknown function which should be specified. Assume that there exists
density f ( t , k ; T , K ) of the solution of the equation (2.9). Then Kolmogorov’s first equation for the
density can be representes in the form
T
)K,T;k,t(f
])K,T;k,t(f)K,T(b[
K2
1 2
2
2






12. 12
Replacing function f ( t , k ; T , K ) by the function 2
2
K
)K,T(C


leads to
2
2
2
2
2
2
2
K
)K,T(C
T
]
K
)K,T(C
)K,T(b[
K2
1









Changing order of differentiations on the right hand side of the latter equation and integrating twice with
respect to variable K lead us to
T
)K,T(C
K
)K,T(C
)K,T(b
2
1
2
2
2





(2.10)
Solving equation (2.10) for b we arrive at the local volatility diffusion also known as the local volatility
surface
2
2
K
)K,T(C
T
)K,T(C
2
)K,T(b




 (2.11)
In original papers variables ( T , K ) in the equation (2.10) and in the formula (2.11) were replaced by the
variables ( t , S ). Then it follows that x  ( t , x ) = b ( t , x ). This transformations of the constant
diffusion coefficient  in the stock dynamics to the ‘local volatility surface’ b ( t , x ) in formula (2.11)
might explain the ‘paradox’ that observed real world option prices depend on implied volatility and time
to maturity and strike price. In later papers the variables ( T , K ) and ( t , S ) were not mixed and
therefore BS option price in the form (2.6) is defined for t  [ 0 , T ] when parameter maturity T is fixed
while heuristic (unobserved) underlying K ( T ) which is defined by (2.9) as well as the same BS option
price C ( t , S ( t ) ; T , K ( T ) ) are define for the fixed date t for T > t.
Comment. Sometines local volatility concept is using to get an adjustment to BS option prices C ( t , S ).
We should remark that such idea is icorrect. We should take into account the fact that final formulas
(2.10), (2.11) take place when we suppose that BS price C ( t , S ; T , K ) is the same regardless whether
one of the groups variables ( t , S ) or ( T , K ) are fixed and other variable. If local volatility present
other value than it is represented by BSE then formulas (2.10), (2.11) do not take place.
ALTERNATIVE APPROACH TO OPTIONS PRICING.
We present now an alternative point of view on option pricing. Recall that BS option price is defined as a
deterministic smooth function. It is proved that there exists a risk free portfolio which consisists from
long option and delta share of short stocks. Hence instantaneous market risk of the option is equal to delta
shares of underlying stocks which eliminates risky ‘white’ noise term . We begin with market risk of the
option. This approach was introduced in [2-4].

13. 13
Define equality of two risky investments. BS no arbitrage option pricing also uses equality. The equality
makes sense for riskless instruments. The corresponding pricing is known as no arbitrage pricing. From
our point of view no arbitrage pricing is a possible price for a group of investors. It is not sufficient to
extend this price for the market which stochastic nature reflect different individual goals of market
participants. Our setting of the pricing problem admits that capital for purchase is available. Every
purchase is risky in sense that investors can either get or loose their money. Besides no arbitrage pricing
there is a different way which is popular for the pricing risky instruments which do not admit perfect
hedging. In this case equality is interpreted as equality expected present values of two sides of the deal.
One can easy verify that applying EPV pricing approach leads to a different option price
C ( t , S ( t ) ) = B ( t , T ) E max { S ( T ) – K , 0 ) }
This solution does not apbitrage free but in average it provides equal opportunity for option buyer and
seller.
Two cash flows is said to be 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. In option pricing we have
two primary assets underlying stock and the option written on this stock. Risk free bond we consider as an
accessory asset. 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
(3.1)
where 0  t  T and C ( t , S ( t )) = C ( t , S ( t ) ; T , K ) denotes option price at t with maturity T. K is
a strike price and C ( T , X ; T , K ) = max { X – K , 0 } is known as call option payoff. Solution of the
equation (3.1) is a random function C ( t , S ( t ), ω ) that promises the same rate of return as its
underlying S ( t ) for the set of scenarios 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 which depends on a
market scenario we call this this stochastic price the market price. Deterministic spot price also known as
option premium we denote as c ( t , S ( t )). It is the settlement price between sellers and buyers of the
option at t. 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 ( t , x ) > C ( t , S ( t ) ; T , K ) } (3.2′)
where C ( t , S ( t ) ; T , K ) is a solution of the equation (3.1). It is a measure of the chance that option
premium is overpriced market price. Similarly market risk of the option seller is measured by the chance
of the adjacent market event, i.e.
P { c ( t , x ) < C ( t , S ( t ) ; T , K ) } (3.2′′)
It represents the probability of the chance that the premium c ( t , x ) received by option seller at t is
underpriced, i.e. it is less than it is implied by the market scenarios. In contrast to BSE solution which
represents option price that guarantees risk free rate of return on BS portfolio c ( t , x ) as a settlement
between events

14. 14
{  : c ( t , x ) > C ( t , S ( t ) ; T , K ;  ) } , {  : c ( t , x ) < C ( t , S ( t ) ; T , K ;  ) }
can be realized in the form
E C ( t , S ( t ) ; T , K ) { c ( 0 , x ) > C ( t , S ( t ) ; T , K ;  ) } =
= E C ( t , S ( t ) ; T , K ) { c ( t , x ) < C ( t , S ( t ) ; T , K ;  ) }
This equation represents equality of expected profit and loss for the option buyer. Actually option
premium can be chosen based on particular interest of an investor. In particular, it can be equal to be the
BS price too.
Let S ( t ) be a solution of the equation (1.9). Bearing in mind that solution of the equation (3.3) 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 random 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 function
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 (3.3) can be represented by formulas

15. 15
p S , profit = P { S ( T )  B – 1
( t , T ) x } = P { exp [ ( μ –
2
σ 2
) ( T – t ) +
+ 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 (3.3 ) 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 } (3.3)
Next, first order adjustment to option premium value can be calculated by taking into account average
loss / profit ratios for stock and option
R S ( t , T ; x ) =
}x)T,t(B)ω,T(S{χ)ω,T(SE
}x)T,t(B)ω,T(S{χ)ω,T(SE
1-
1-



16. 16
R C ( t , T ; x ) =
})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 }.
If the value R C ( t , T ) is small then the use option price in the form (3.3) does not looks as a reasonable
approximation. In this case it looks mo reasonable to use the solution of the equation
R C ( t , T ; x ) = R S ( t , T ; x ) (3.4)
Right hand side of the equation (3.5) 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-


=
= [ 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
On the other hand
R S ( t , T ) = E S ( T ) χ { S ( T )  B– 1
( t , T ) x } =
=
)tT(σπ2
1
2



x)T,t(Bln-
v exp –
)tT(σ2
])tT()
2
σ
μ(xlnv[
2
2
2


dv
Hence the equality ( 3.5) can be rewritten as


 
)x,t(c)T,t(BlnK 1
1
( v – K ) exp –
)tT(σ2
])tT()
2
σ
μ(xlnv[
2
2
2


dv =
= 

x)T,t(Bln-
v exp –
)tT(σ2
])tT()
2
σ
μ(xlnv[
2
2
2


dv

17. 17
It is difficult to present solution of the equation Equation (3.4) admits numeric approach to calculate
option premium c 1 ( t , x ). Such situation suggests establishing option price. 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
It is difficult or even impossible to define a unique financial law for the option premium that rules the
total market. There are too many investment strategies  on the market. Bearing in mind stochastic nature
of the risk any option price is risky. ‘No arbitrage’ BS pricing is no arbitrage for the BS’s hedged
portfolio which implies that the price of the option to be similar as the delta shares of underlying stocks.
Let us illustrate alternative approach tooption pricing by using discrete space-time approximations of
continuous model (3.1). Consider a discrete approximation of the stoc price at T

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 in theory we can assume that p j for a particular j could be as close to 1 or to 0 as we wish. Let t
be a current moment of time. We eliminate arbitrage opportunity for each market scenario ω   j by
putting
)ω;x,t(C
)ω;x,t(C)KS(
x
)t(SS jj 


, if S j  K
and

18. 18
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 ] }
Then the option premium at t 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 underlying security is equal to 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 somewhat 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 , ω i ) > K }.
Our goal to present a reasonable way to estimate the price choice c ( t , x ) that represented in the market.
One possible estimate is BS price. Pricing equation (3.1) represents definition of the option price for each
market scenario. The reasonable first order approximation of the settlement between buyers and sellers is
the option price that represents market risk that equal to underlying stock. We use PV concept of equality
to represent settlement pricing concept .
The value of the stock x = S ( t ) at t is equal to the value B – 1
( t , T ) x at T. No arbitrage pricing
suggests next market risk valuation. Borrow x at risk free rate and buy stock at t. At T sell stock for
S ( T ) and arrive at the next the chance of profit / loss which specifies market risk of the buyer
P { S ( T , ω )  B – 1
( t , T ) x } , P { S ( T , ω ) < B – 1
( t , T ) x }
Correspondingly. The average profit / loss on stock at date T are defined then 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 }
Similarly the chances of underpriced and overpriced option are equal to
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 is based on market risk can be defined as following

19. 19
P { S ( T , ω )  B – 1
( t , T ) x } = P { C ( T , S ( T ) )  B – 1
( t , T ) c ( t , x ) }
This equation represents the value of the spot option price as equality of loss-chances on stock and the
option.
Conclusion. We are consideing Black Scholes derivatives pricing concept as oversimplified pricing. The
oversimplified pricing here means that definition of the derivatives price ignores market risk of the
option’s premium. Our definition of option price is based on weighted risk-reward or profit-loss ratios.
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, ISBN-3838366050,
2010, p.154.
5. Hull J., Options, Futures and other Derivatives. Pearson Education International, 7ed. p.814.
6. P.Tankov, E. Voltchkova., Jump-diffusion models: a practitioner’s guide.
http://www.proba.jussieu.fr/pageperso/tankov/tankov_voltchkova.pdf
7. Gikhman, Il., Failings of the Option Pricing.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2406585
8. Hull J., Options, Futures and other Derivatives. Pearson Education International, 7ed. 814 p.
9. Derman, E., and Kani,I., Riding on a smile, Risk, 7 (1994), p.32-33.
10. Dupire, B., “Pricing and Hedging with Smiles” , Research paper, 1993, p.9.
11. Dupire, B., “Pricing with a Smile”. Risk Magazine, 7, 1994, p.8-20.

20. 20