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DERIVATIVES PRICING.

Ilya I. Gikhman 6077 Ivy Woods Court Mason, OH 45040, USA Ph. 513-573-9348 Email: ilyagikhman@yahoo.com

JEL : G12, G13

Key words. Option, derivatives, pricing, dynamic hedging.

Abstract. This paper presents a fallacy of the Black and Scholes’ (BS) option pricing concept. The BS pricing is still the unique theoretical way for pricing derivatives though quite a large number of expert have found a lot of remarks concerning its theoretical and practical failings. We should note that implementation of BS methodology uses a real historical data and it is difficult to identify in practice whether deviations of the theoretical predictions of the option prices are due to incorrect interpretation of the BS option prices or deviations are implied by initial assumptions of the theoretical model regarding to say for example a stock model. In [4] one presented a critical point of view on BS pricing concept. After this paper was published BS methodology is continue to be the benchmark interpretation of the derivatives pricing. Our goal is to introduce more substantial formal arguments which demonstrate that Black and Scholes’ approach to derivative pricing is incorrect.

I. BLACK SCHOLES APPROACH.

We presented our critical point of view on Black-Scholes interpretation of the option price in [4]. In [5] we extended our alternative approach pricing to cover different derivative classes. This book does not present any critics of the benchmark interpretation of the derivatives pricing. Critics of the benchmark pricing on different can be found in some papers presented in [6] .

We begin with European call option pricing. Let S ( t ) denote a security price at the moment t ≥ 0. Security can be of any class such as stock or interest rate. A mathematical model usually interprets S ( t ) as a random process in which variable S ( t , ω ) = ω ( t ) is associated with a market scenario. European call option written on security S is a contract which grants a buyer of the option the right to buy

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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. Thus, based on a market scenario there are two possibilities. Buyer of the option can refuse to exercise option at T if the price of the underlying security less than strike price, i.e. S ( T ) ≤ K or he/she agrees to exercise option at T, if S ( T ) > K. 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 known as the option price. The pricing problem is the problem of finding option price at any moment prior to maturity. The benchmark construction of the option price was given by Black and Scholes in [1]. Purchasing call option contract at t buyer of the option can expect ether to get profit equal to the option payoff or lose the premium. In classical setting the call option seller grants the exercise of the option at maturity, i.e. there is no chance of default. Hence, the option price should be interpreted as a settlement price between buyer and seller of the option. This interpretation of the price is the cornerstone of the pricing in the Financial Market regardless whether we deal with securities or derivatives in theory or in practice.

The essence of the Black and Scholes definition of the option price is derived from construction that assumes that option buyer borrows funds to purchase option at the risk free interest at the initiation date t. Then Black and Scholes introduced a portfolio which contains a portion of long stocks and short option. It was showed that portfolio dynamics is riskless. Therefore the only one way to avoid arbitrage is to suppose that the rate of return of the portfolio is equal to risk free interest rate. By construction such pricing approach is called arbitrage free. The dynamics of the stock portion of such portfolio one calls the perfect dynamical hedging of the option. Black Scholes option pricing can also be represented by using self-financing concept. The recent point of view on the connection between benchmark dynamic hedging and the modern funding problems is presented in [2].

Consider the dynamic hedging introduced by Black and Scholes in [6]. The Black Scholes hedging problem is discussed in the V section of the paper [3]. We briefly introduce essence of the arguments. It was mentioned that in some published papers revealed the fact that the original derivation of the Black-Scholes equation is incorrect. Nevertheless, the authors of the papers were recognized that the final result is correct. In [3] it was noted that the proof used in these papers are ‘unsound’ and it was made an attempt to bring a new derivation of the Black Scholes equation. Consider the value of a portfolio Π consisting of one short option - C ( t , S ( t )) and N ( t , S ( t ) ) portion of underlying stocks held long. The value of the portfolio at time t can be written in the form

Π ( t , S ( t ) ) = − C ( t , S ( t ) ) + N ( t , S ( t ) ) S ( t ) (1.1)

It was remarked that following the Black Scholes derivation, textbooks suggested that differential of the portfolio represented by the formula

d Π ( t , S ( t ) ) = − d C ( t , S ( t ) ) + N ( t , S ( t ) ) d S ( t ) (1.2)

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The standard textbook argument for not including the term S ( t ) × d N ( t , S ( t ) ) is that the number of shares held is “instantaneously constant”. To a mathematician, this argument must be perplexing since the total variation of the number of shares held in any finite time interval is in fact infinite. In fact, the number of shares is changing so fast that the ordinary rules of calculus do not apply. To resolve the mathematical problem was an idea according to which we do not need to compute the total derivative. Instead one should consider the ‘gain’ on the portfolio which is defined as

g Π ( t , S ( t ) ) = − d D ( t , S ( t ) ) + N ( t , S ( t ) ) d S ( t ) (1.3)

This formula similar to the correspondent formula in [5] if the dividend rate of the underlying security

δ = 0. Here D and d D are defined in [3] by the formulas (1), (2), and (4). It is not clear whether this function can be equal to call option price C we used in (1.1) and (1.2). If D does not equal to C then we deal with a different portfolio it is not clear a technology of the option valuation. The problem is that in [3] it was defined as a derivative security that is a linear function with respect to underlying security S and a plain vanilla option payoff does not possesses of this property. Assuming that D = C we should note that ’gain’ does not equal to total differential and therefore it does not make any sense to state that instantaneous rate of return of the risk free bond is equal to the ‘gain’ that represents the instantaneous rate of return of the part of the Black Scholes portfolio. It is easy to see by add and subtract a ‘gain’ term any portfolio contains a part which instantaneous rate of return is equal to risk free rate. To present Black-Scholes ideas more accurately we assume that at the initial moment t = 0 we construct portfolio in the form of (1) and during the lifetime of the option the infinitesimal change in value of the portfolio is defined by formula (1.2). It was exactly was stated by Black and Scholes. Putting in (1.1)

N ( t , S ( t ) ) = S

))t(S,t(C

∂∂

we remark that for any t ∈ [ 0 , T ] the right value of the new portfolio P at moment t > 0 is equal to

P ( t , S ( t ) ) = Π ( 0, S ( 0 ) ) − ∫t

0

d C ( v , S ( v ) ) + ∫t

0

N ( v , S ( v ) ) d S ( v ) =

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= − C ( t , S ( t ) ) + S

))0(S,0(C

∂∂

S ( 0 ) + ∫t

0 S

))v(S,v(C

∂∂

µ S ( v ) d v + (1.4)

+ ∫t

0 S

))v(S,v(C

∂∂

σ S ( v ) d w ( v )

Expression (1.4) does not actually equal to the right hand side of the (1), i.e.

P ( t , S ( t ) ) = Π ( 0, S ( 0 ) ) − ∫t

0

d C ( v , S ( v ) ) + ∫t

0

N ( v , S ( v ) ) d S ( v ) ≠ Π ( t , S ( t ))

Following Black Scholes derivation we can calculate differential of the portfolio P ( t , S ( t ) )

d P ( t , S ( t ) ) = [ 2

222

S

))t(S,t(C

2

σ)t(S

t

))t(S,t(C

∂∂+

∂∂

] d t (1.5)

Change in value of the portfolio is riskless. It does not contain a term proportional to dw ( t ). In order to exclude arbitrage we should to assume that

d P ( t , S ( t ) ) = r P ( t , S ( t ) ) d t

where r denotes the risk free interest rate. Assuming that underlying security price is govern by equation

d S ( t ) = µ S ( t ) d t + σ S ( t ) d w ( t ) (1.6)

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and bearing in mind formulas (1.4), (1.5), and (1.6) we arrive at the fact that correct Black Scholes equation should be written as

2

222

S

))t(S,t(C

2

σ)t(S

t

))t(S,t(C

∂∂+

∂∂

= r [ S

))0(S,0(C

∂∂

S ( 0 ) – C ( t , S ( t ) ) +

(7)

+ ∫t

0 S

))v(S,v(C

∂∂

µ S ( v ) d v + ∫t

0 S

))v(S,v(C

∂∂

σ S ( v ) d w ( v ) ]

This construction strictly follows the BS’s derivation of the benchmark equation without heuristic and unproved statement that states the portfolio has the form (1.1) at any time prior to maturity. It might be difficult to expect that equation (1.7) can be helpful in derivation of the Black Scholes price. It also highlight the erroneous of the perfect hedging strategy that was derived from BS concept. Also we need to highlight the fact that BS price likely can not be interpreted as a settlement price between buyer and seller of the option. This price possibly can satisfy only buyer of the option and probably a bank that might be take a risk to borrow money at risk free rate thinking that the buyer will return borrowing funds plus correspondent cumulative risk free interest.

Conclusions. There are two primary errors In BS concept. One is financial type error. They presented incorrect interpretation of the option price. Instead to define option price as a settlement price between buyer and seller they define option price focusing on risk free borrowing. This construction has lead to incorrect understanding of the derivatives price. Second error is the pure mathematical one. Incorrect implementation of the original BS’ s idea has lead them to famous BS equation

2

222

S

)S,t(C

2

σS

t

)S,t(C

∂∂+

∂∂

+ r S

)S,t(C

∂∂

– r C ( t , S ) = 0 (BSE)

while the accurate derivation should lead them to the equation (1.7). The (BSE) explicitly suggests that the underlying of the BS price is the heuristic random process of the type (6) in which real return is replaced by the risk free interest rate. In order to present the close relationships between real and heuristic underlying it was developed risk neutral world concept. If the equation (1.6) which originally was defined on original probability space consider with respect to some other measure then its image on the original probability space will be equal to heuristic BS underlying. On the current stage of derivatives pricing all real world of underlings are replaced by its risk neutral counterparts regardless whether they are equities or variable stochastic interest rates. Note that originally defined on initial probability space real world processes are converted to its risk neutral versions in order to match to BSE. Beside the correct hedging

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strategy now should use equation (1.7). The derivation of the equation (1.7) shows the real way of the dynamic hedging in realization of the BS strategy.

At the end of this section I would recall that drawbacks of the Back Scholes pricing concept were not accepted by top professors primarily in USA and the primary USA financial institutions such as Federal Reserve, SEC, and Treasury Department. They did not paid attention or they did not understand explicit failure of the benchmark derivatives pricing.

II. ALTERNATIVE APPROACH.

In this section we represent an alternative approach to option pricing. We have discussed failing of the Black and Scholes’ (BS) option pricing concept. The alternative approach to derivatives pricing was introduced in [4 , 5]. Our approach is based on investment equality principle that represents the formal financial law of equality. That states that two cash flows are equal over an interval of time when and only when they have equal instantaneous rates of return at any moment during this time interval. This approach to equality for cash flows is more accurate then present value (PV) reduction of the cash flows which permanently has used in Finance.

Let us first introduce the financial equality principle. We call two investments S i ( t ) , i = 1, 2 are equal at moment t if their instantaneous rates of return are equal at this moment. If equality of two investments holds for any moment of time over [ t , T ] then we call these investments are equal on [ t , T ]. It is not difficult to note that this concept presents no arbitrage notion of the price for each scenario. Applying this definition to European call option at t we arrive at the equation

)t(S

)T(S χ { S ( T ) > K } =

)K,T;)t(S,t(C

)K,T;)T(S,T(C (2.1)

0 ≤ t ≤ T where C ( t , S ( t ) ; T , K ) is the call option price at t with maturity T and strike price K and C ( T , X ; T , K ) = max { X - K , 0 }. The solution of the equation (1) is a random function that promises the same rate of return as its real underlying (1) for any meaningful scenario, i.e.

ω ∈ { ω : S ( T ) > K } and C ( t , S ( t ) ; ω ) = 0 for each ω ∈ { ω : S ( T ) ≤ K }. Bearing in mind that this price of the option depends on a scenario we call such price as market price at t in contrast to the spot price. The spot price c ( t , S ( t )) at t we interpret as the value which represents settlement price between sellers and buyers of the option. Let S ( t ) = x and c ( t, x ) be a spot call option price. The call option buyer’s market risk is measured by the value of the chance that buyer pays more than it is implied by the market, i.e.

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P { c ( 0 , x ) > C ( t , S ( t ) ; T , K ) }

On the other hand the call option seller’s market risk is correspondingly equal to

P { c ( 0 , x ) < C ( t , S ( t ) ; T , K ) }

It represents the probability of the chance that seller receives less that would implies by the market.

Let us present a space discrete reduction of the above continuous model. It also can be applied for discrete time problem setting. Consider approximation of the S ( t ) in the form

∑=

n

1j

S j χ { S ( T ) ∈ [ S j – 1 , S j ) }

where 0 = S 0 < S 1 … < S n < + ∞ and denote p j = P ( ω j ) = P { S ( T ) ∈ [ S j – 1 , S j ) }. Note that p j for a particular j could be as close to 1 as we wish. We eliminate arbitrage opportunity for each scenario ω j by putting

)ω;x,t(C

KS

x

S

j

jj −= , if S j > K

and

C ( t , x ; ω j ) = 0 , if S j ≤ K

Then market price of the call option can be approximated by the discrete valued random variable

C ( t , x ; ω j ) = ∑=

n

1j jS

x( S j - K ) χ { S ( T ) ∈ [ S j – 1 , S j ) }

This random option price offers lower return than the underlying security. Indeed, the option return for the scenarios ω j = { S j ≤ K } is equal to 0 while stock return is x – 1 S j > 0. Thus, the option premium can be interpreted as a price that reimburses the risk of the loss of the security return

∑=

n

1j x

S j χ { S j ≤ K }

and on the other hand investor will be benefitted by the option to get underlying stock or its value for K for the scenarios ω ∈ { S j ( T , ω ) > K }.

Let us consider inverse problem. Assume that option premium c = c ( t , x ) > 0 is a known. It might differ from a theoretical one while it can be occurred that c ( t , x ) coincides with the BSE price. Recall that our interpretation of the option price can not reduce it to a particular number. Regardless on valuable background of the option price like providing perfect hedging incase when underlying is a stochastic process any valuable theoretical premium choice implies risk. Indeed, let c > 0 denote call option price at t then date-t return on call option over [ t , T ] is

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c – 1 [ S ( T ; t , x ) – K ] χ ( S ( T ; t , x ) > K )

Therefore, with any choice of the option premium c the option contract remains risky. Its return can be any non negative number and the option return does not perfectly replicate underlying return x – 1 S ( T ; t , x ). The market risk of the option price does not represented in the BS pricing concept. Nevertheless, it should be the inalienable part of any option price model.

Now let us calculate the stock price Y = Y ( T ) , Y > K at T which is implied by the known option price c = c ( t , x ). This is the inverse problem to option pricing. For a scenario ω ( • ) = S ( • , ω ) for which ω ( T ) = { S ( T , ω ) = Y } it follows from the equality (2.1) that we have the equality

x

Y =

c

K-Y

Solving it for Y we arrive at the implied stock value

Y = cx

xk

− .

Assume that underlying stock price is govern by the equation

d S ( t ) = µ S ( t ) d t + σ S ( t ) d w ( t ) (2.2)

where coefficients µσ are constants or known deterministic functions on time. Putting S ( T ) = S ( T ; t , x ) we know that for any given x > 0 the random function S ( T ; t , x ) > 0 for any T > t with probability 1.The solution of the equation (2.1) can represented in the form

C ( t , x ; T , K ; ω ) = )x,t;T(S

x [ S ( T ; t , x ) - K ] χ { S ( T ; t , x ) > K }

The function C ( t , x ; ω ) = C ( t , x ; T , K ) on the left hand side we call the market price of the call option. For each market scenario ω = ω ( l ) , l ∈ [ t , T ] there exists a unique value of the stock at T as well as there exists the unique value of the option at the moment t. Admitting that the spot option price at the moment t can be an unbiased estimate of the market prices one can put that

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c ( t , x ) = E )x,t;T(S

x [ S ( T ; t , x ) - K ] χ { S ( T ; t , x ) > K } (2.3)

Unbiased estimate implies stability of the prices of the underlying security. Other approximation of the call premium can be defined as the spot price c ( t , x ) which is the solution of the expected ‘risk/reward’ equality problem

E C ( t , x ; ω ) χ { C ( t , x ; ω ) > c ( t , x ) } = E C ( t , x ; ω ) χ { C ( t , x ; ω ) < c ( t , x ) }

The latter equation can be rewritten in a simpler form

E C ( t , x ; ω ) χ { C ( t , x ; ω ) > c ( t , x ) } = 0.5 E C ( t , x ; ω )

More generally c ( t , x ) is defined with the help of the profit-loss characteristics. For example, one can

define c = c ( t , x ) by the given market ratio γ = γ ( t , x ; T ) ∈ ( 0 , 1 )

E C ( t , x ; ω ) χ { C ( t , x ; ω ) > c } = γ E C ( t , x ; ω ) (2.4)

Let us assume that formula (2.3) is a sufficiently good for the representation of the spot price of the option. Given the value of the option at t = 0 we are interested in its valuation c ( t ; 0 , x ) at the future moments. Define a random process c ( t ) = c 0 , x ( t ; ω ) , t > 0 putting by definition

x – 1 c ( t ) = E { S – 1 ( T ; 0 , x ) [ S ( T ; 0 , x ) - K ] χ { S ( T ; 0 , x ) > K }} | F [ 0 , t ] } =

= [ E S – 1 ( T ; t , y ) [ S ( T ; t , y ) - K ] χ { S ( T ; t , y ) > K } ] y = S ( t ; 0 , x ) =

= [ y – 1 c ( t , y ) ] y = S ( t ; 0 , x ) = ∫∞+

K v

Kv − p ( t , y ; T , v ) d v | y = S ( t ; 0 , x )

where p ( t , y ; T , v ) is the density of the random variable S ( T ; t , y ) at the point v. Here,

S ( T ; t , y ) is the solution of the equation (2.2) and it can be presented in the form

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S ( T ; t , y ) = y exp { ( µ - 2

σ2

) ( T - t ) + σ [ w ( T ) - w ( t ) ] }

Therefore

p ( t , y ; T , v ) = vd

d P { S ( T ; t , y ) < v } =

vd

dP { σ [ w ( T ) – w ( t ) ] < h }

where

h = v

1 { ln

y

v - ( µ -

2

σ2

) ( T - t ) }

Then

p ( t , y ; T , v ) = )t-T(vσπ2

122

exp - )tT(σ2

)tT()2

σµ(

y

vln

2

2

−

−−−

Putting q ( t , y ) = y – 1 c ( t , y ) we note that

q ( t , y ) = E S – 1 ( T ; t , y ) [ S ( T ; t , y ) - K ] χ { S ( T ; t , y ) > K }

Here q ( t , y ) = q ( T , t , y ) and t ∈ [ 0 , T ]. it is easy to verify that the function q ( t , y ) defined for

( t , y ) ∈ [ 0 , T ] × ( 0 , + ∞ ) is a smooth solution of the backward Cauchy problem

t

)y,t(q

∂∂

+ µ y y

)y,t(q

∂∂

+ 2

1σ 2 y 2

2

2

y

)y,t(q

∂∂

= 0 (2.5)

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with the backward terminal condition

q ( T , y ) = y – 1 ( y - K ) χ ( y > K ) (2.5′)

Indeed,

q ( t - ∆ t , y ) - q ( t , y ) = q ( t , S ( t ; t - ∆ t , y ) ) - q ( t , y ) =

= q /y ( t , y ) [ S ( t ; t - ∆ t , y ) - y ] + 2

1 q //

yy ( t , y ) [ S ( t ; t - ∆ t , y ) - y ] 2 + o ( ∆ t , ω )

where 0t∆

lim→

E | o ( ∆ t , ω ) | 2 = 0 . Dividing both sides of the latter equality by ∆ t and taking limit

when ∆ t tends to 0 we arrive at the equation (2.5). From equality

x – 1 c ( t ) = q ( t , y ) | y = S ( t ; 0 , x ) = y

)y,t(c

| y = S ( t ; 0 , x )

and bearing in mind that q ( t , y ) is a smooth deterministic function we apply Ito formula

d q ( t , S ( t ) ) = [ t

))t(S,t(q

∂∂

+ µ S ( t ) y

))t(S,t(q

∂∂

+ 2

1σ 2 S 2 ( t )

2

2

y

))t(S,t(q

∂∂

] d t +

+ σ S ( t ) y

))t(S,t(q

∂∂

d w ( t )

From (7) it follows that the expression in brackets on the right hand side is equal to 0 and therefore

y

)y,t(c

| y = S ( t ; 0 , x ) =

x

)x,0(c + ∫

t

0

σ S ( u ; 0 , x ) y

))x,0;u(S,u(q

∂∂

d w ( u ) =

12

= x

)x,0(c + ∫

t

0

σ [ y y∂

∂y

)z,u(c ] z = S ( u ; 0 , x ) d w ( u )

References.

1. Black, F., Scholes, M. The Pricing of Options and Corporate Liabilities. The Journal of Political

Economy, May 1973. 2. D. Brigo, C. Buescu, A. Pallavicini, Q. Liu, Illustrating a problem in the self-financing condition

in two 2010-2011 papers on funding, collateral and discounting. 2012, http://ssrn.com/abstract=2103121.

3. Carr, P. Frequently Asked Questions in Option Pricing Theory. 2002, http://www.math.nyu.edu/research/carrp/papers/pdf/faq2.pdf

4. Gikhman, Il.I. On Black- Scholes Equation. J. Applied Finance (4), 2004, p. 47-74, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=500303.

5. Gikhman, Il.I. Alternative Derivatives Pricing: Formal Approach, LAP Lambert Academic Publishing 2010, p.164.

6. http://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=365639.