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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.

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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

)) td t(S, td t(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

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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{χx

) T ( S

)x,t(C

} 0 ,K- ) T S( { max

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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.