1. The Black-Scholes option pricing model assumes a perfect hedge using a dynamic trading strategy, but this requires frequent rebalancing of the hedge portfolio which incurs transaction costs that are not accounted for in the model. 2. When the hedge portfolio is rebalanced over discrete time intervals, an adjustment cost arises at each interval that affects the expected present value of the cash flows and thus the derived option price. 3. For the Black-Scholes model to accurately price options, it must account for the expected costs of dynamically rebalancing the hedge portfolio over the life of the option.

1. 1
BLACK SCHOLES PRICING and DYNAMIC HEDGING.
Ilya I. Gikhman
6077 Ivy Woods Court
Mason, OH 45040, USA
Ph. 513-573-9348
Email: ilya_gikhman@yahoo.com
Classification code
Key words. Black Scholes, option, derivatives, pricing, dynamic hedging.
Abstract. In this remark, we present structure of the perfect hedging. Closed form formulas clarify the
fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding
portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing
scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS
option price.
Perfect or dynamic hedging is the underlying idea of the Black-Scholes (BS) option pricing concept. The
price of the options is associated with the price of the BS-portfolio introduced in [1]. In other words there
is no other price of the options in market if no hedged portfolio is established. As far as sellers of the
options do not sell options incorporated in BS-portfolios it is not clear whether BS price coincides with
market pricing. It is possible to interpret deviations between theoretical and realized option values by
underlying theoretical assumptions for example such as (GBM) Geometric Brownian Motion of the
underlying asset distribution. For quite popular models in which underlying assets incorporates jump
terms the BS portfolio does not guarantee risk free return. Therefore, the option price notion can not be
defined.
Let us briefly recall BS pricing construction. In order to present BS pricing equation they [1] defined a
hedged position that lately was called BS hedged portfolio. The formula
Π ( t ) = C ( t , S ( t )) – C
/
S ( t , S ( t )) S ( t )
specifies portfolio value [2] at any moment during lifetime of the call option C ( t , S ). Here the term
C /
S ( t , S ( t )) defines number of short stocks S ( t ) in the portfolio. It is clear that the latter formula not
accurately represents definition of the portfolio. Indeed the variable t in the formula is variable in stock
price term S ( t ) and at the same time it is fixed in number of shares term C
/
S ( t , S ( t )). Following
[3, formula (1.12)] let us define BS-portfolio

2. 2
Π ( u , t ) = C ( u , S ( u )) – C
/
S ( t , S ( t )) S ( u ) (1)
where t, t  0 is a fixed parameter and variable u, u  t. In [3] we followed hedging scheme introduced in
[1] where underlying one share of stock S ( t ) is hedged by Δ shares of call option C ( t , S ( t )). Portfolio
(1) is a modern reduction of the hedging concept in which one option share C ( t , S ) = C ( t , S ; T , K )
is hedged by C
/
S ( t , S ( t )) shares of underlying stocks. Parameters T and K are option maturity and
strike price of the option. Assume that underlying asset follows GBM equation
dS ( t ) =  S ( t ) dt + σ S ( t ) dw ( t ) (2)
where  , σ are known deterministic continuous functions in t, t  0. Let 0 ≤ t 0 ≤ t 1 ≤ … ≤ t n = T
denote a partition of the time interval [ t 0 , T ]. Dates t j , j = 1, 2, … n – 1 we associate with dates of
adjustments of the BS-portfolio and for simplicity we can put t j + 1 = t j + 1. Following BS-concept
number of shares C
/
S ( t j , S ( t j )) of underlying stocks in portfolio does not changed over period
[ t j , t j + 1 ] and at the moment t j + 1 it should be adjusted by buying or selling
C
/
S ( t j + 1 , S ( t j + 1 )) – C
/
S ( t j , S ( t j ))
underlying shares. Hence, the value of the adjustment of the BS-portfolio at the date t j + 1 is equal to
Δ Π ( t j + 1 ) = – [ C
/
S ( t j + 1 , S ( t j + 1 )) – C
/
S ( t j , S ( t j )) ] S ( t j + 1 )
Bearing in mind latter adjustment equality the cash flow to the buyer can be represented by the formula
CF A = – [ C ( t 0 , S ( t 0 )) – C
/
S ( t 0 , S ( t 0 )) S ( t 0 ) ] χ ( t = t 0 ) –
– 


1n
1j
[ C
/
S ( t j , S ( t j )) – C
/
S ( t j – 1 , S ( t j – 1 )) ] S ( t j ) χ ( t = t j ) + (3)
+ [ max ( S ( T ) – K , 0 ) + C
/
S ( t n – 1 , S ( t j )) ] S ( t n – 1 ) χ ( t = T )
The expected present value (EPV) of the cash flow to counterparty is commonly associated with the price.
It follows then
EPV A = C
/
S ( t 0 , S ( t 0 )) S ( t 0 ) – 


1n
1j
B ( t 0 , t j ) E [ C
/
S ( t j , S ( t j )) –
– C
/
S ( t j – 1 , S ( t j – 1 )) ] S ( t j ) + (4)
+ B ( t 0 , T ) E [ max ( S ( T ) – K , 0 ) – C
/
S ( t n – 1 , S ( t j )) ] S ( t n – 1 )

3. 3
Formula (4) shows that BS setting of the option pricing problem is incomplete. The BS pricing does not
specify who will pay adjustment cost: buyer, seller, or both of them. Ignoring the cost of adjustments
leads unexpected losses. At the date-t 0 the expectation of the cash value adjustments (cva) is equal to
C
/
S ( t 0 , S ( t 0 )) S ( t 0 ) – 


1n
1j
B ( t 0 , t j ) E [ C
/
S ( t j , S ( t j )) –
– C
/
S ( t j – 1 , S ( t j – 1 )) ] S ( t j ) + B ( t 0 , T ) E C
/
S ( t n – 1 , S ( t j )) S ( t n – 1 ) =
= – 


1n
1j
E C
/
S ( t j – 1 , S ( t j – 1 )) [ B ( t 0 , t j ) S ( t j ) – B ( t 0 , t j – 1 ) S ( t j – 1 ) ] –
– B ( t 0 , T ) E C
/
S ( t n – 1 , S ( t j )) S ( t n – 1 ) = – 


1n
1j
E C
/
S ( t j – 1 , S ( t j – 1 )) 
 { [ B ( t 0 , t j ) – B ( t 0 , t j – 1 ) ] S ( t j ) + B ( t 0 , t j – 1 ) [ S ( t j ) – S ( t j – 1 ) ] } –
– B ( t 0 , T ) E C
/
S ( t n – 1 , S ( t n – 1 )) S ( t n – 1 )
Therefore
cva δ = – 


1n
1j
E C
/
S ( t j – 1 , S ( t j – 1 )) { B ( t 0 , t j – 1 ) [ FRA ( t j – 1 , t j ; t 0 ) – 1 ] S ( t j ) +
(5)
+ B ( t 0 , t j – 1 ) [ S ( t j ) – S ( t j – 1 ) ] } – B ( t 0 , T ) E C
/
S ( t n – 1 , S ( t n – 1 )) S ( t n – 1 )
Here subscript δ on the left hand side (5) denotes discrete time scale applied for the adjustments (5) and
FRA ( t j – 1 , t j ; t 0 ) denotes date-t 0 implied forward discount rate over [ t j – 1 , t j ] period. In continuous
time formula (5) can be rewritten in integral form
cva = – 
T
t 0
E C /
S ( t , S ( t )) { B ( t 0 , t ) S ( t ) fd ( t ; t 0 ) dt + B ( t 0 , t ) dS ( t ) } –
(5)
– B ( t 0 , T ) E C
/
S ( T , S ( T )) S ( T )
Here the function fd ( t ; t 0 ) dt denotes date-t 0 implied instantaneous forward discount factor at t and
differential dS ( t ) on the right hand side (5) is defined by (2). In [1] the possibility of adjustments was

4. 4
announced but did not develop. Formulas (5), (6) suggest that adjustments of the portfolio should be
taking into account in BS pricing concept, which can be specified as following.
Buyer purchases BS portfolio (1) for Π ( t 0 , t 0 ) at t 0 . The price of the portfolio can be lower or
larger than the call option price C ( t 0 , S ( t 0 )). It depends on sign of the term C
/
S ( t 0 , S ( t 0 )). During
next period [ t 0 , t 1 ) BS price of the option approximately is defined by risk free value of the portfolio.
Next at each date t j , j = 1, 2, … n – 1 buyer should buy or sell
C
/
S ( t j , S ( t j )) – C
/
S ( t j – 1 , S ( t j – 1 ))
of underlying shares. It is also a possibility to share adjustment expenses between two parties of the deal.
In other words buying BS portfolio at t which guarantees risk free rate of return at the moment t does not
guarantee that portfolio will preserve risk free rate of return over any future period. At the maturity date
t n = T investor exercises call option for
max { S ( T ) – K , 0 }
and also owner of the equity portion C
/
S ( t n – 1 , S ( t n – 1 )) S ( T ) can sell it or hold it for valuable
period of time.
The perfect dynamic BS portfolio adjustment admits also an alternative interpretation. One can interpret
hedged portfolio with the help of floating-floating swap. Buyer of the option receives option
C ( t 0 , S ( t 0 )) from option seller at t 0 and send C
/
S ( t 0 , S ( t 0 )) S ( t 0 ) shares of underlying asset
to option seller at t 0 . During life time of the option holder should send
C /
S ( t j , S ( t j )) S ( t j )
shares of stock to option seller and option seller should send
C
/
S ( t j – 1 , S ( t j – 1 )) S ( t j )
shares of stock to option buyer at the dates t j , j = 1, 2, … n – 1. At maturity date t n = T option holder
exercises option for max { S ( T ) – K , 0 } and asset portion of C
/
S ( t n – 1 , S ( t n – 1 )) S ( T ) will be
sold either option buyer or seller. If C /
S ( t n – 1 , S ( t n – 1 )) < 0 the owner of the equity portion is option
holder otherwise if C
/
S ( t n – 1 , S ( t n – 1 )) > 0 equity owner is seller of the option.

5. 5
References.
1. Black, F., Scholes, M. The Pricing of Options and Corporate Liabilities. The Journal of Political
Economy, May 1973.
2. Hull J., Options, Futures and other Derivatives. Pearson Education International, 7ed. 814 p.
3. I. Gikhman. Black Scholes pricing concept. http://www.slideshare.net/list2do ,
http://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=365639