This a short notice about what is the option price theory nowadays.

1. 1
CRITICAL THOUGHTS ABOUT MODERN OPTION PRICING.
Ilya Gikhman
6077 Ivy Woods Court
Mason OH 45040 USA
Ph. 513-573-9348
Email: ilyagikhman@mail.ru
JEL : G13.
Keywords: Black Scholes pricing, alternative options pricing.
I. What is right and what is wrong in Black Scholes (BS) pricing.
What is Right. On the one hand it looks legitimate to construct portfolio
P ( t ) = – f ( t ) + g ( t ) S ( t ) (1)
Here f ( t ) = f ( t , S ( t ) ) denote option price and – f ( t ) is the value of the short option,
g ( t ) =
S
))t(S,t(f


is a portion of stock in portfolio at date t. It is right that change in value of the
portfolio at t is indeed
CiV P ( t ) = – df ( t ) + g ( t ) dS ( t ) (2)
and therefore
– df ( t ) + g ( t ) dS ( t ) = r P ( t ) (3)
That leads us to BS equation BSE on [ t , t + dt ). Hence, borrowing the sum P ( t ) from the bank at risk
free interest rate r at t and constructing BS portfolio as it suggested by the formula (1) investor could
return borrowing sum at t + dt and there is no profit or loss of the investor position at t + dt. The last
observation coincides with definition of no arbitrage pricing.
What is Wrong.
Statement 1. The change in value of P ( t ) does not equal to dP ( t ).
Proof. Given explicit form of option price represented by BSE solution one can present function in the
explicit form. Calculating the differential of the function P ( t ) we arrive at the proof of the Statement 1.

2. 2
What does the effect of the difference between differential and change in value of P. BSE should be take
place in each point ( t , S ) of the area [ 0 , T ) × ( 0 , + ∞ ) otherwise BS option formula defines option
price not for all ( t , S ). Next, we note that
P ( t ) + change in value of P ( t ) does not equal to P ( t ) + dP ( t ) = P ( t + dt ) , i.e.
P ( t ) + change in value of P ( t ) does not equal to P ( t + dt )
In order to extend BSE forward in time one needs to make an adjustment adding the term S ( t ) dg ( t ) .
Otherwise, the BSE takes place only at the point t + 0. As far as t is interpreted as a current moment we
usually put S ( t ) = S. Hence
dg ( t ) =
S
))tdt(S,tdt(f


–
S
))t(S,t(f


is a risky term representing stochastic number of stocks which should added to portfolio at t + 0 if we
work with differential in time model. One can also use more correctly the finite-difference form of the
model. In this case, sign differential in above formulas should be replaced by the sign delta. In this case
adjustment S ( t ) dg ( t ) should be replaced by S ( t ) Δ g ( t ) and this adjustment should take place at the
moment immediately prior to the moment t + Δ t. To continue construction similar construction should be
developed on each time interval [ t + (k - 1) Δ t , t + k Δ t ) and adjustments provided at the end of the
period, i.e. at t + k Δ t , k = 1, 2, … m to arrive at the BS portfolio that consistent with formula (1). The
process is continuing up to the maturity date T. Besides the stochastic nature of the S ( t ) dg ( t ) which
implies that the term can be either positive or negative. At date t + 0 or ( t + Δ t – 0 ) in finite difference
form ) investor does not know whether to borrow or short this amount. This term destroy no arbitrage BS
pricing concept. On the other hand taking limit in finite difference form of the model when delta t tends to
zero, we arrive at the fact that the BSE does not take place on any small closed time interval.
PS. The deficiency of the BS’s logic in derivation of the BSE can be illustrated in a simple example. Let
Δt be a fixed step and consider a stepwise approximation P ( t ) of the function S ( t ) = t on the interval
( 0 , T ]. On each interval [ t + ( k - 1) Δt , t + k Δt ) function P ( t ) satisfies equation dP / dt = 0 while
the limit function S ( t ) satisfies equation dS / dt = 1. If we assume that S is measured in a currency units
then the equation which specifies approximation does not have any relationship to the equation to the real
pricing function. The Black Scholes world the situation is even worse. We do not know anything about
the limit function which actually does not defined. Approximation function f ( t , S ) = C Δt ( t , S ) is
defined by BSE on each infinitesimal interval [ t + ( k - 1) Δt , t + k Δt ) while the limit function C ( t , S )
does not. Actually we even do not have the definition of the function C ( t , S ). This phenomena comes to
the existence because at the end of the each k-th interval approximation P ( t + k Δt - 0 ) should be
adjusted in order to present value S ( t + k Δt ).
II. Here we present alternative point of view on option pricing.
We focus on a single period of time and let t , T denote the beginning and end moments of this period.
Suppose that asset price at initial moment is known S ( t ) = x while the final price S ( T ,  ) is a known
and is assumed to be a discrete random variable taking values S j , j = 1, 2 … n and

3. 3
0 < S 1 < … < S q ≤ K < S q + 1 < …< S n
with a known probability distribution P { S ( T ,  ) = S j } = p j , j = 1, 2 … n. Here the constant K is a
known strike price of the call option that is defined by its payoff at T
C ( T , S ( T )) = max { S ( T ) - K , 0 }
Let us define option price C ( t , x ; ω ) for each market scenario  which is specified by value of the
stock at T. Denote a set of market scenarios ω j for which S ( T ,  j ) = S j , j = 1, 2, … n. Then
C ( t , x ; ω j ) = 0 , j = 1, 2, … q (4)
Next, for each market event ω j for which S ( T ,  j ) > K the price of the call option is defined by
equation
Then
}K)T(S{χ}0,K-)T(S{max
)T(S
x
)ω;x,t(C 
and therefore
C ( t , x ; ω j ) =
jS
x
( S j - K ) , j = q + 1, … , n (5)
We call the option price defined by (4), (5) as stochastic market price. The distribution of the stochastic
call option is defined follows by the distribution of the underlying asset
P { C ( t , x ; ω ) = 0 } = P { S ( T ,  ) < K }
P { C ( t , x ; ω ) =
jS
x
( S j - K ) } = P { S ( T ,  ) = S j } , j = q + 1, … , n
If market uses the price c ( t , x ) at t for buying call option then this price implies market risk. Market risk
of the buyer / seller of the option is defined by probabilities
P { c ( t , x ) > C ( t , x ; ω ) } , P { c ( t , x ) < C ( t , x ; ω ) }
correspondingly. First probability in above lane estimates a chance that buyer of the option pays higher
price that it is implies by the market, i.e. the option price overpriced while the second probability
estimates a chance that seller of the option receives lower price that it is implies by the market, i.e. the
option price underpriced. A few more market risk quantitative characteristics are available to quantify
}K)T(S{χ
x
)T(S
)x,t(C
}0,K-)TS({max


4. 4
market risk of the deal. We do not represent them here. Note that the spot option price c ( t , x ) in
particular can be chosen as it is suggested by the BSE. However, it is not a derivatives theory axiom. The
real theoretical axiom of the derivatives pricing is the fact that a spot price of the option does not a
complete definition of the price. The complete definition of the option price is composed by the spot price
along with attached to it correspondent market risk.
One can think about market risk as a way of quantification of the spot price. In other words the spot price
is similar to as an admissible value of a random variable while market risk is cumulative distribution
function associated with this number. The set of admissible values of the random variable holds minimum
information about random variable and cumulative distribution function is complete information which
defines a random variable.