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ON CONSTRUCTION OF THE RISKLESS DERIVATIVES PORTFOLIOS.

Ilya Gikhman

6077 Ivy Woods Court

Mason, OH 45040, USA

ph. 513-573-9348

email: ilya_gikhman@yahoo.com

Abstract. In this article we discuss a construction of riskless derivatives portfolios. First, following [1],

one examines the riskless portfolio introduced by Black and Scholes. Then Black Scholes approach is

extended on a portfolio of two derivatives. These derivatives are assumed to be solutions of linear

stochastic differential equations.

I. Construction a risk-free portfolio represents a fundamental problem for the financial theory. Derivatives

market was developed a long time ago and has played an important role in the world economy. In 1973,

Black and Scholes introduced a derivation of the option pricing formula. Options are a complex class of

derivatives. The main motivation of the derivatives development is the belief in the possibility of reducing

risks by including in risky portfolios derivatives instruments. This idea stems from the construction of

Black and Scholes option pricing. A modern version of the construction of the option price is given in [1].

In this report, we briefly present a critical point of view on the idea of the risk elimination which is a

primary part of the modern theory of derivatives.

Let {Ω, F, P} denote arbitrary complete probability space and assume that the stock price can be

approximated by the solution of the stochastic differential equation

d S ( t ) = μ S ( t ) d t + σ S ( t ) d w ( t ) (1)

t [ 0 , T ]. Denote f ( t , S ) the price of the option. Following Black and Scholes [1, p. 287] define the

portfolio ( t , S ) , S = S ( t ) which contains one short option and a portion S

)S,t(f

of long stocks,

i.e.

( t , S ) = – f ( t , S ) + S

)S,t(f

S (2)

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t [ 0 , T ]. Function ( t , S ) is continuous in ( t , S ) [ 0 , T ] ( 0 , + ). Price of the portfolio at

time ( t + Δt ) - 0 is determined by the formula

( t + Δt – 0, S ( t + Δt – 0 )) = – f ( t + Δt – 0 , S ( t + Δt – 0 )) + S

))t(S,t(f

S ( t + Δt – 0 ) (3)

Here, the notation t - 0 corresponds to the left hand side limit of the corresponding function at t. Using

Ito's formula we find that

d ( t , S ( t )) = – d f ( t , S ( t )) + S

))t(S,t(f

d S ( t ) =

= – [2

222

S

))t(S,t(f

2

)t(Sσ

t

))t(S,t(f

] d t

The right side of the latter equation does not contain terms containing dw (t). Therefore, to avoid arbitrage

opportunities relative income portfolio on the time interval [t, t + dt) should be proportional to the interest

rate of the risk-free bonds

d ( t , S ( t )) = r ( t , S ( t )) d t (4)

Bearing in mind (4) we arrive at the equation of the Black – Scholes

t

))t(S,t(f

+ r

S

))t(S,t(f

S ( t ) +

2

222

S

))t(S,t(f

2

)t(Sσ

– r f ( t , S ( t )) = 0 (5)

Note that equation (3) determines the price of the portfolio (t + Δt, S (t + Δt)) at time t + Δt does not

have the same structure as one that represented by the formula (2). According to formula (2) the price of

the portfolio at the time t + Δt must have the form

( t + Δt , S ( t + Δt )) = – f ( t + Δt , S ( t + Δt )) + S

))t + t ( S,t +t (f

S ( t + Δt ) (6)

Prices of the portfolios at time t + Δt defined by equalities (3) and (6) are different. This observation

justifies that formulas (2), (3) established at some moment t will not be guaranteed in future moments. As

far as these formulas represent sufficient conditions for the existence of the Black Scholes equation (5)

this equation in general does not take place in the future moments. In order to eliminate the contradiction

in the Black - Scholes model it is necessary to reconstruct the portfolio adding assets to the portfolio (3)

to bring it to the form (6). Recall that only formula (6) that corresponds to the portfolio (2) at the time

t + Δt . The value of reconstruction at time t + Δt of the portfolio at t + Δt is equal to

δ ( t + Δt ) = ( t + Δt , S ( t + Δt )) – ( t + Δt – 0 ,̀ S ( t + Δt – 0 ))

Taking into account the dependence of the solution on the initial data, we note that

S ( t + Δt ) = S ( t + Δt ; 0 , S ( 0 )) = S ( t + Δt ; t , S ( t ))

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Consequently, the price change of the portfolio has the form

δ ( t + Δt ) = [ S

))t + t ( S,t +t (f

–

S

))t(S,t(f

] S ( t + Δt ) =

= [ S

))t + t ( S,t +t (f

–

S

))t(S,t(f

] [ 1 + μ Δ t + σ Δ w ( t ) ] S ( t )

This equation implies that the Black and Scholes price of the portfolio has a risk on [ t , t + Δt ]. More

precisely, portfolio has no risk only half-open time interval [ t , t + Δt ) and gets the risk in the final

moment t + Δt . Now let 0 ≤ t 0 < t 1 < ... < t n = t + Δt = T be a partition of the time interval [ t 0 , T ].

Suppose that portfolio in the form (2) is constructed for each moment t k , k = 0, 1, ... n. Then

( T , S ( T ) ) – ( t , S ( t ) =

n

1j

( t j + 1 , S ( t j + 1 )) – ( t j , S ( t j )) =

=

n

1j

– [ f ( t j + 1 , S ( t j + 1 )) – f ( t j , S ( t j )) ] + [ S

))t(S,t(f 1j1j

S ( t j + 1 ) –

– S

))t(S,t(f jj

S ( t j ) ] = Δ BS ( t , T ) + (7)

+

n

1j

[ S

))t(S,t(f 1j1j

–

S

))t(S,t(f jj

] S ( t j + 1 )

Here, Δ BS ( t , T ) denotes the price changes of the Black and Scholes portfolio on the [ t , T ], and

Δ BS ( t , T ) =

=

n

1j

– [ f ( t j + 1 , S ( t j + 1 )) – f ( t j , S ( t j )) ] + S

))t(S,t(f jj

[ S ( t j + 1 ) – S ( t j ) ]

The last term on the right-hand side of (7) represents adjustment term in Black Scholes world. This

adjustment term specifies market risk implied by the Black and Scholes portfolio. Market risk of the

derivatives price is the attribute of the pricing [ ] and it does not take into account in the modern pricing

theory. To obtain the final expression for the market risk of Black and Scholes model it is necessary to

calculate the limit of the right-hand side of (7) when max ( t j + 1 – t j ) → 0. Taking this limit, we note

that the equation of the Black - Scholes does not take place in any point of the interval [ t , T ], and at

every moment of time Black and Scholes’ portfolio has the risk. Thus, the assertions about risk free

nature of the Black and Scholes price (2) are incorrect. Analogous statement takes place regarding perfect

hedging realized by Black Scholes portfolio.

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To illustrate the fallacy of the mathematical idea in the derivation of Black and Scholes let us consider an

example. Let us construct its piecewise approximation w Δ ( t ) of the Wiener process w ( t ) , t ≥ 0

w Δ ( t ) = w ( 0 ) χ { t [ 0 , t 1 ) } +

n

1j

w ( t j – 1 ) χ { t [ t j – 1 , t j ) }

where Δ = t j – t j – 1 , j = 1, 2, … n. Random function w Δ ( t ) is a constant on each time the interval

[ t j – 1 , t j ) and therefore is the solution of the equation f ' ( x ) = 0. However, the limit process w ( t ) is

not a solution of the equation f ' ( x ) = 0 and even it is not differentiable on any small interval. In the

same way the option price defined by Black and Scholes does hold market risk and is a solution of their

equation on each open interval ( t j – 1 , t j ). Nevertheless, the limit Black Scholes price of the option is

specified by the risky portfolio at any given time due to the reconstruction of the portfolio value at each

moment t j . Moreover, the option price, defined as the pointwise limit of the portfolio defined by

equalities (2), (3) when Δ → 0 does not satisfy the equation (5).

II. Technique for constructing a riskless portfolio proposed for pricing options was extended to portfolios

of derivatives [1, p. 617]. Consider for example a portfolio with two derivatives written on the same

underlying asset θ ( t ). Let dynamics of the underlying asset and derivatives correspondingly follow

stochastic equations

d θ ( t ) = m θ ( t ) d t + s θ ( t ) d w ( t )

d f j ( t ) = μ j f j ( t ) d t + σ j f j ( t ) d w ( t )

Define [1, p. 617] portfolio , which contains a portion of σ 2 f 2 of the first type of derivative and a

portion – σ 1 f 1 of the second type of derivative. Therefore

( t ) = [ σ 2 f 2 ( t ) ] f 1 ( t ) – [ σ 1 f 1 ( t ) ] f 2 ( t )

During Δt period the price of the portfolio will be changed for the value

Δ ( t ) = [ σ 2 f 2 ( t ) ] Δ f 1 ( t ) – [ σ 1 f 1 ( t ) ] Δ f 2 ( t ) = ( μ 1 σ 2 – μ 2 σ 1 ) f 1 ( t ) f 2 ( t ) Δ t

This equality shows that the portfolio ( t ) has no risk and therefore one can apply equality (4).

Substituting the values of the of the derivatives we arrive at the equalities

1

1

σ

rμ =

2

2

σ

rμ = λ

Latter equalities show that parameter λ has the same value for any derivative defined by the same

underlying security and it is known as the market price of θ ( t ). As in the case of portfolio Black and

Scholes equation (4) should be satisfied for the entire time period, not just for an initial moment.

Similarly as it was shown above, in order to provide extension of the portfolio upon the whole interval [ t

0 , T ] we need to carry out the reconstructions of the portfolio by adding a portion of assets which

represent market risk. Taking first the limit when the time between two interval between two successive

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moments of reconstruction Δ tend to 0, Δt 0 we note that ( t ) is a risky portfolio at each time.

Therefore, the statement that the portfolio ( t ) has no risk is incorrect.

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References.

1. Hull, J. Options, Futures and Other Derivatives, 7th ed. Pearson Education International, p. 814.