A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.

1. 1
PRICING and HEDGING OPTIONS.
Ilya I. Gikhman
6077 Ivy Woods Court Mason,
OH 45040, USA
Ph. 513-573-9348
Email: ilyagikhman@yahoo.com
Classification code G12, G13.
Key words. Option, pricing, hedging.
Abstract. A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic
calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The
perfect hedging on a finite time interval is a separate problem that does not change option pricing. The
cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative
approach to option pricing. It defines option price as a settlement between counterparties and in contrast
to BS approach presents the market risk of the option premium.
I. Black Scholes Pricing.
Black Scholes Equation, BSE is usually derived by using differential form, which informally is based on
equality
dw ( t ) dw ( t ) = dt
Such equality is heuristically true but it does not a formal. Differential form of presentation of the
stochastic calculus makes sense only in integral form. We are going to use integral form of stochastic
calculus to present derivation of the BSE. Assume that underlying asset is a stock, which price S ( t ),
t ≥ 0 follows a Geometric Brownian Motion equation
dS ( t ) = µ ( t ) S ( t ) dt + σ ( t ) S ( t ) dw ( t ) (1)
Here { w ( t ) , F t , t  0 } is a Wiener process, coefficients µ ( t ), σ ( t ) are known deterministic
functions, and S ( 0 ) > 0. Equation (1) should be interpreted in integral form as

2. 2
S ( t ) = S ( 0 ) + 
t
0
µ ( s ) S ( s ) ds + 
t
0
σ ( t ) S ( s ) dw ( s )
Let C ( t , S ) = C ( t , S ; T , K ) denote call option price at time t , t  [ 0 , T ]. Here T is an option
maturity given that S ( 0 ) = S. Let T , T < + ∞ and [ s , t ]  [ 0 , T ]. Consider a partition
s = t 0 < t 1 < … < t m = t , and λ =
mk1
max

( t k – t k – 1 ). Denote C 1, 2
{ [ 0 , T ] × ( 0 , + ∞ ) } the space
of continuously differentiable one in t and twice in S nonrandom functions of two variables ( t , S ). Let
C ( t , S ) be an arbitrary function from the space C 1, 2
{ [ 0 , T ] × ( 0 , + ∞ ) }. The hedged portfolio
Π ( t ) = C ( t , S ( t )) – C
/
S ( t , S ( t )) S ( t )
is usually used to define the value of the option. The change of the value of the portfolio that is implied by
the latter formula on [ s , t ] can be presented in the form
П ( t ) – П ( s ) = 
m
1k
П ( t k ) – П ( t k – 1 ) = 
m
1k
[ C ( t k , S ( t k ) ) – C ( t k – 1 , S ( t k – 1 )) ] –
– [ C
/
S ( t k , S ( t k )) S ( t k ) – C
/
S ( t k – 1 , S ( t k – 1 )) S ( t k – 1 )
This approach obviously fails to justify BSE. Other way to introduce hedged portfolio is presented in [2].
In this case the value of the hedged portfolio Π ( u , t ) is defined for any moment t  ( 0 , T ] and
u  [ t , T ] by the formula
Π ( u , t ) = C ( u , S ( u ) ) – C
/
S ( t , S ( t )) S ( u )
It represents the value of the one long call option and a portion of  ( t ) = C
/
S ( t , S ( t )) short shares
of the stocks. Then
П ( t , t m – 1 ) – П ( t 1 , s ) = 


1m
1k
П ( t k + 1 , t k ) – П ( t k , t k – 1 )
Then
П ( t , t m – 1 ) – П ( t 1 , s ) = 


1m
1k
[ C /
t ( t k – 1 , S ( t k – 1 ) ) +
+
2
1
C
//
SS ( t k – 1 , S ( t k – 1 ) ) σ 2
( t k – 1 ) S 2
( t k – 1 ) ]  t k + o ( λ )
where
0λ
l.i.m
 λ
)λ(o
= 0. Denote П ( t ) =
0t
lim

П ( t +  t , t ). Taking limit when λ tends to zero we
arrive at the formula

3. 3
П ( t ) – П ( s ) = 
t
s
Π ( du , u )
where
Π ( du , u ) = Π /
u ( u , u ) du = [ C /
t ( u , S ( u )) +
2
1
C
//
SS ( u , S ( u )) σ 2
( u ) S 2
( u ) ] du
Note that if a function C ( u , S )  C 1, 2
{ [ 0 , T ] × ( 0 , + ∞ ) } the integrand on the right hand side of
(2) is a continuous function. Given S = S ( t ) the interest rate corresponding to the rate of return
)t(Π
)t(Πd
= Π ( dt , t ) =
S)S,t(C)S,t(C
)S,t(CS)t(σ
2
1
)S,t(C
/
S
//
SS
22/
t


dt
of the hedged portfolio is a deterministic continuous function. Let C ( t , S ) is a solution of the Black
Scholes equation, BSE.
C /
t ( t , S ) + S C
/
S ( t , S ) +
2
1
σ 2
( t ) S 2
C
//
SS ( t , S ) – r C ( t , S ) = 0 (BSE)
Then Π ( dt , t ) does not depend on space parameter S and
)t(Π
)t(Πd
= r dt (2)
On the other hand, in order to avoid arbitrage opportunity between hedged portfolio and risk free bond we
put we should assume equality (2) from which it follows (BSE). The boundary condition of the (BSE)
follows from the definition of the call option contract
C ( T , S ) = max { S ( T ) – K , 0 }
Note that Black Scholes option price C ( t , S ) = C BS ( t , S ) that is defined by the solution of the (BSE)
does not represent a settlement price between buyer and seller. It is a price, which is implied by no
arbitrage principle. In stochastic market, Black Scholes pricing can be interpreted as a price which is
implied by no arbitrage trading strategy. Such price can be sometimes close and sometimes not to
historical data depending on market conditions. In case when theoretical pricing does not close to
observed data theory developed adjustment tools like calibration, local volatility, jumps, and etc. Some of
the adjustments make sense some not.
II. Alternative Approach to Option Pricing.
Here we present other point of view on option pricing. This approach was introduced in [4-6]. Let us first
define the pricing equality for two or more risky investments. The pricing equality used in BS no
arbitrage pricing takes place only for non-risky instruments. Every investment is risky in sense that
buying an instrument investors have a chance to lose their investments. Besides no arbitrage, pricing idea

4. 4
there is other way, which is popular for the pricing risky instruments. In this case, the pricing equality is
interpreted as the equality of the expected present values, EPVs of the two cash flows. This pricing
method could not represent the ‘perfect’ hedging
*) Assume that a stock, a call option on stock, and the risk free bond represent trading instruments in a
market. Our goal is to define option price on stock. One can admit that there are many assets on the
market but an assumption is other assets do not effect on option pricing. Let  denote admissible set of
market scenarios and   . For the fixed market scenario there are two investment opportunities for an
investor: 1) to buy stock for known price S ( t ,  ) or 2) to buy its option at the same moment t.
If for given market scenario  =  ( u ), u  [ t , T ) the stock value at maturity T is bellow strike price
K then there is no sense to buy option and therefore C ( t , S ( t ) ;  ) = 0, for   {  : S ( t ,  ) < K }.
If for a chosen scenario  the value of stock larger than K at T then in order to avoid arbitrage for such
scenario we should assume that the rate of return on stock and its option are equal. Hence the formula that
eliminate arbitrage opportunity between stock and option for each market scenario can be written as
)t(S
)ω,T(S
 { S ( T ) > K } =
)ω;)t(S,t(C
)ω;)T(S,T(C
(4)
Formula (4) defines ‘fair’ price of the call option
C ( t , S ( t ) ;  ) =
)ω,T(S
)t(S
max { S ( T ,  ) – K , 0 } (5)
for each market scenario . In [5] stochastic pricing for different classes of options is considered in
details.
The second part of the option pricing problem is the spot option price c ( t , S ( t )), which is applied for
buying and selling options in the market. As far as the spot price of the assets are the settlement price
between buyers and sellers a reasonable estimate of the spot price is one, which does not promise an
advantage in corresponding deal. Let investor buys a stock and write option on the stock. When a buyer
buys the option paying premium to the seller. At maturity date T option seller holds the stock if
S ( T ,  ) ≤ K . Buyer of the option pays premium at initiation and exercises the option at maturity for
the scenarios for which S ( T ,  ) > K . We can also interpret situation for buyer as a choice to by option
or its underlying asset itself or for the seller is a choice to sell stock or write option. Corresponding cash
flows that formally define the deal are stochastic and can be written in the form
CF seller ( u ; 0 , T ) = [ – S ( 0 ) + c ( 0 , S ( 0 )) ]  { u = 0 } + [ ( K – S ( T ) )  { S ( T ,  ) > K } +
+ S ( T ,  )  { S ( T ,  ) ≤ K } ]  { u = T } (6)
CF buyer ( u ; 0 , T ) = – c ( 0 , S ( 0 ))  { u = 0 } + ( S ( T ,  ) – K )  { S ( T ,  ) > K }  { u = T }
There is no universal market law, which defines the spot price c ( 0 , S ( 0 )). No arbitrage BS option price
is a theoretical price implied by no arbitrage strategy of the trading. It can be either close to market
premium or not. The fact that it does not depend on drift of the underlying asset but depends on risk free

5. 5
rate suggests that this price might be good when expected return on underlying is close to risk free rate.
Otherwise, deviation between theory and practice can be quite visible to ignore it. The primary motivation
in trading securities and derivatives are expected return and the corresponding risks. Reduction of the
trade primary motivation to buy or sell assets to no arbitrage looks like an oversimplified theoretical
assumption.
A simple example of the settlement pricing can be found by taking equality of the expected present
values, (EPVs) of the two cash flows (6). Hence
– S ( 0 ) + c ( 0 , S ( 0 )) + E [ ( K – S ( T ) )  { S ( T ,  ) > K } +
+ S ( T ,  )  { S ( T ,  ) ≤ K } ] B ( 0 , T ) = – c ( 0 , S ( 0 )) +
+ B ( 0 , T ) E ( S ( T ,  ) – K )  { S ( T ,  ) > K }
Function
c ( 0 , S ( 0 )) = (7)
= S ( 0 ) + B ( 0 , T ) E { [ S ( T ,  ) – K ]  { S ( T ) > K } –
2
1
S ( T )  { S ( T ) ≤ K }}
is a solution of the latter equation. It can be interpreted as an estimate of the settlement price implied by
the EPVs equality. Market price of the option can follow this principle or it can be adjusted by the
external risk factors. Nevertheless, this estimate of the option premium has more profound sense compare
with the no arbitrage pricing. It takes into account the settlement between buyer and seller.
We should highlight the fact that there is no the best definition of the estimate of the option premium as
far as any two different option premium have different risk characteristics, i.e. higher return always
implies higher risk, which by definition is a lower probability to reach it.
One can also use other estimate of the premium we can start with expected values of the equation (5).
Another models for settlement estimate can be expected value of the stochastic option price represented
by (6), i.e.
c 1 ( 0 , S ( 0 )) =
}K>)T(S{)ω,T(SE
)0(S
E max { S ( T ,  ) – K , 0 } (8)
c 2 ( 0 , S ( 0 )) = S ( 0 ) E
)ω,T(S
}0,K-)ω,T(S{max
(9)
In general any choice of option premium implies different market risk. Buyer market risk is defined by
the formula
P { c ( 0 , S ( 0 )) > C ( 0 , S ( 0 ) ;  ) }

6. 6
where C ( t , S ( t ) ;  ) is the stochastic price defined by (7). It presents a measure of the chance that
option premium is overpriced. The market risk of the option seller is defined by the adjacent event
P { c ( 0 , S ( 0 )) > C ( 0 , S ( 0 ) ;  ) }
It represents the probability of the chance that the premium received by the option seller is underpriced. In
contrast to BSE solution which represents option price that guarantees risk free rate of return on BS
portfolio the spot price of the option c ( t , x ) is a settlement price which is specified by the risk events
{  : c ( t , x ) > C ( t , S ( t ) ; T , K ;  ) } , {  : c ( t , x ) < C ( t , S ( t ) ; T , K ;  ) }
or say broadly by the equality of the primary risk characteristics of the counterparties.
**) For practical applications, we usually use a discrete space-time approximation of the continuous
pricing models. Consider a discrete approximation of the stochastic stock price at T

n
1j
S j  { S ( T , ω )  [ S j , S j + 1 ) }
where 0 = S 0 < S 1 … < S n < +  and put p j = P (  j ) = P { S ( T )  [ S j , S j + 1 ) }.
Here we assume that S n + 1 = +  . In theory, one can assume that one of the probabilities p j could be
close to 1 or to 0. Let t be a current moment of time. Then we can eliminate arbitrage opportunity for each
market scenario ω   j by putting
)ω;x,t(C
)KS(
x
S jj 
 , if S j  K
C ( t , x ; ω ) = 0 , if S j < K
Solution of the latter equation is a stochastic process that can be written in the form
C ( t , x ; ω ) = 
n
1j jS
x
max { S j - K , 0 }  { S ( T , ω )  ( S j – 1 , S j ] } (10)
Premium c ( t , x ) which corresponds to stochastic price (10) can be presented in one of the forms (7-9).
III. Hedging.
BS model is based on perfect hedge of the call option only at initial point of time. This idea suggests
using BS portfolio for hedging during a finite time interval. We begin with a discrete time
approximation. Let us s = t 0 < t 1 < … < t m = t and λ =
mk1
max

( t k – t k – 1 ) be a partition of an
interval [ s , t ]. For holding hedged portfolio over [ s , t ] an investor should buy BS portfolio at the
date s and then should make an adjustment at the end of each subinterval [ t k , t k + 1 ] by changes its
-value

7. 7
 ( t k ) = C
/
S ( t k , S ( t k ))
which represents the number of the short stocks in the hedged portfolio. At the date t k the number of
shares C
/
S ( t k – 1 , S ( t k – 1 )) in short stocks in the hedged portfolio should be replaced by C
/
S ( t k ,
S ( t k )). The cash flow, which specifies the value of the hedged portfolio on [ s , t ] can be written as
CF hp ( u ; t 0 , t m ) = – [ C ( t 0 , S ( t 0 ) ) – C
/
S ( t 0 , S ( t 0 ) ) S ( t 0 ) ]  { u = t 0 } +
+ 
m
1k
[ C
/
S ( t k , S ( t k )) – C
/
S ( t k – 1 , S ( t k – 1 )) ] S ( t k )  { u = t k }
The EPV of the cash flow CF hp ( u ; s , t ) presents the date-s estimate of cost of the hedged
portfolio over [ s , t ]
P λ ( s , t ) = EPV { CF hp ( u ; s , t ) } = – [ C ( t 0 , S ( t 0 ) ) – C
/
S ( t 0 , S ( t 0 ) ) S ( t 0 ) ] +
+ 
m
1k
B ( s , t k ) E [ C
/
S ( t k , S ( t k )) – C
/
S ( t k – 1 , S ( t k – 1 )) ] S ( t k )
The estimate (8) implies market risk, which is stipulated by the random deviation of the real world
PV realization and its estimate EPV of the portfolio value
PV { CF hp ( u ; s , t ) } – EPV { CF hp ( u ; s , t ) }
We can present formula (8) in continuous time as following
P ( s , t ) =
0λ
lim

P λ ( s , t ) = – [ C ( s , S ( s ) ) – C
/
S ( s , S ( s ) ) S ( s ) ] +
+ 
t
s
B ( s , u ) E [
t

C
/
S ( u , S ( u )) + μ ( u ) S ( u )
S

C
/
S ( u , S ( u )) + (11)
+
2
1
 2
( u ) S 2
( u ) 2
2
S

C
/
S ( u , S ( u )) ] S ( u ) du
In original paper [1] call options was used as a hedge for a single stock. For each moment t during
lifetime of the option [ 0 , T ] the BS hedged portfolio is defined by the formula
П C ( u, t ) = S ( u ) – [
S
))t(S,t(C


] – 1
C ( u , S ( u ))
Hedged portfolio represents the value of the one long stock and the number

8. 8
 ( t ) = [
S
))t(S,t(C


] – 1
of short shares of the call option. Then
П C ( t , t m – 1 ) – П C ( t 1 , s ) = 


1m
1k
П C ( t k + 1 , t k ) – П C ( t k , t k – 1 ) =
= 


1m
1k
[
S
))t(S,t(C 1k1k

 
] – 1
[ C /
t ( t k – 1 , S ( t k – 1 )) +
+
2
1
C
//
SS ( t k – 1 , S ( t k – 1 )) σ 2
( t k – 1 ) S 2
( t k – 1 ) ]  t k + o ( λ )
where
0λ
l.i.m
 λ
)λ(o
= 0. Putting П ( t ) =
0t
lim

П ( t +  t , t ). Taking limit when λ tends to zero, we
arrive at the formula
П C ( t ) – П C ( s ) = 
t
s
Π C ( du , u ) = C /
t ( u , S ( u )) +
2
1
C
//
SS ( u , S ( u )) σ 2
( u ) S 2
( u )
The date-t instantaneous rate of return on the hedged portfolio is equal to
)t(Π
)t(Πd
= Π ( dt , t ) =
S)S,t(C)S,t(C
)S,t(CS)t(σ
2
1
)S,t(C
/
S
//
SS
22/
t


dt
It is a deterministic continuous function and therefore to avoid arbitrage opportunity we should assume
that
)t(Π
)t(Πd
= r dt
From this equality it follows that C ( t , S ) is a solution of the BSE. The cash flow, which specifies the
value of the hedged portfolio on [ s , t ] is
CF C, hp ( u ; t 0 , t m ) = S ( t 0 ) – [ C
/
S ( t 0 , S ( t 0 )) ] – 1
C ( t 0 , S ( t 0 )) ]  { u = t 0 } +
+ 
m
1k
{ [ C
/
S ( t k , S ( t k )) ] – 1
– [ C
/
S ( t k – 1 , S ( t k – 1 )) ] – 1
} C ( t k , S ( t k ))  { u = t k }
Therefore, the price of holding hedged portfolio in discrete time approximation is given by the
estimate

9. 9
EPV { CF C, hp ( u ; t 0 , t m ) } = S ( t 0 ) – [ C
/
S ( t 0 , S ( t 0 )) ] – 1
C ( t 0 , S ( t 0 )) ] +
+ 
m
1k
B ( s , t k ) { [ C
/
S ( t k , S ( t k )) ] – 1
– [ C
/
S ( t k – 1 , S ( t k – 1 )) ] – 1
} C ( t k , S ( t k ))
which in continuous time over [ s , t ] can be written in the form
P ( s , t ) =
0λ
lim

P λ ( s , t ) = S ( t 0 ) – [ C
/
S ( t 0 , S ( t 0 )) ] – 1
C ( t 0 , S ( t 0 )) ] +
+ 
t
s
B ( s , u ) E {
t

[ C
/
S ( u , S ( u )) ] – 1
+ μ ( u ) S ( u )
S

[ C
/
S ( u , S ( u )) ] – 1
+
+
2
1
 2
( u ) S 2
( u ) 2
2
S

[ C
/
S ( u , S ( u )) ] – 1
} C ( u , S ( u )) du
Comment.
It will be a good idea based on historical data observed over t k , k = 0, 1, 2, … to verify the rate of
return of the values of the BS hedged portfolio on [ t k , t k + 1 ] against the models of the risk free rate
on this interval. Other problem that might have a trading interest to use the swap of the rate of return
of the ’risk free’ specified by the BS hedged portfolio for the primary assets like DJI , S&P, oil, or
gold and LIBOR against the risk free rate specified by the risk free security itself.

10. 10
REFERENCES.
1. Black, F., Scholes, M. The Pricing of Options and Corporate Liabilities. The Journal of Political
Economy, May 1973.
2. Gikhman, Il., Comments on Option Pricing.
3. Gikhman, I., Critical Point on Stochastic Volatility Option Pricing.
https://www.slideshare.net/list2do/crit...g-80339422
https://papers.ssrn.com/sol3/papers.cfm ... id=3046261
4. Gikhman, Il., On Black- Scholes Equation. J. Applied Finance (4), 2004, p. 47-74,
5. Gikhman, Il., Derivativs Pricing. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=500303.
6. Gikhman, Il., Alternative Derivatives pricing. Lambert Academic Publishing, ISBN-3838366050,
2010, p.154.