We show that no-arbitrage principal does not sufficient for correct understanding derivatives pricing in stochastic setting.

1. A COMMENT ON NO-ARBITRAGE PRICING.
Ilya Gikhman
6077 Ivy Woods Court
Mason OH 45040 USA
Ph. 513-573-9348
Email: ilyagikhman@yahoo.com
Keywords: No-arbitrage pricing, cash and carry, forward contract
JEL Classification: G12
Abstract. In this short notice we present critical comments on no-arbitrage principle. We show that
no-arbitrage pricing is complete in a pricing theory which ignores market risk and is dealing with the
deterministic implied price of instruments. There is a unique price of a derivative in deterministic setting.
The no-arbitrage pricing approach picks risk free bond which used as upfront funding instrument for
financing deals. In such approach the underlying of the derivatives in deterministic setting becomes risk
free bond. In stochastic setting no-arbitrage pricing replace real underlying on a virtual underlying that
has risk free expected return and the original volatility.
From our point of view this interpretation of the price of a derivative is incorrect. Our approach to
derivatives pricing was presented in [1]. The derivatives pricing contains two steps. On the first step we
define the ‘market price’. This is the price for each admissible market scenario. On the second step we
define a spot derivatives price. In some cases spot price can be implied price. In more complex situations
for example such as options pricing construction of the spot price does not so simple. Given market and
spot derivative prices we arrive at the market risk. The market risk by definition is the probability of
scenarios that counterparty pays or loses more than it is implied by the spot price. Market and spot prices
along with correspondent market risk is what we call derivatives price. We illustrate this approach by
considering a forward contract pricing.
Recall the essence of a standard no-arbitrage forward pricing. It can be found in any major
financial handbooks. Denote S ( t ) an asset price at a date t , t ≥ 0. By definition forward contract at
date t is an agreement between two parties called seller and buyer. Counterparts are contracting to sell and
buy underlying asset S at a future date T, T > t. The Seller of the forward obliges to deliver the
underlying asset to the buyer at maturity of the contract T. The Buyer of the contract obliges to pay the
agreed at t amount F upon delivery. The forward pricing problem is a construction of the forward price F.
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2. A forward is a simple and popular derivatives contract and its valuation highlights general ideas of pricing
more complex derivative instruments. No-arbitrage pricing can be briefly outlined as following. At date t
a forward buyer is going short and receives cash S ( t ). The net value of the go short at t is 0. Cash S ( t )
is invested in risk free bond B ( t , T ) at t.
At maturity T forward buyer pays amount F and receives the asset which price is S ( T ) at T. Date-T
accumulated cash amount in the bank is B – 1 ( t , T ) S ( t ). The value S ( T ) is unknown at the date t. The
financial rule which solves the pricing problem can be formulated as the following no-arbitrage principal.
Since the portfolio value at t is zero then the value of the portfolio should be equal to zero at a
future moment T.
The buyer’s portfolio value at T is
V ( T , F ) = B–1 ( t , T ) S ( t ) - F
Applying no-arbitrage principal we arrive at the forward price F = B – 1 ( t , T ) S ( t ).
Comment. The no-arbitrage principal statement is formulated as a general law which can be applied for
price discovery regardless of the underlying distribution. Underlying can be either deterministic or
stochastic. Assume first that S is a deterministic function. Then the rate of return on stock and bond
should be equal.
Indeed, let B ( t , T ) , 0 ≤ t ≤ T be a deterministic bond price at t and assume that B ( T , T ) = $1.
Assume that the price of the asset is another deterministic function S ( t ). Then from no-arbitrage
principal it follows that the rates of return on asset and bond must be equal. Indeed, let us for assume for
example that the statement of the theorem does not true and let
S( T )  S( t ) 1 - B( t ,T )
<
S( t ) B( t ,T )
Then investor can sell a portion B ( t , T ) S – 1 ( t ) of stock at t which results total
[ B ( t , T ) S –1 ( t ) ] S ( t ) = B ( t , T )
and purchase the bond at date t. As far as the rate of return on bond is higher than on stock then
S ( T ) < 1. At date T investor receives $1 for bond , buys back stock for $S ( T ) and makes riskless profit
of 1 - S ( T ) > 0. Similarly we can consider the case when the rate of return on stock is higher than the
rate of return of the bond. Thus, the assumption that deterministic stock and bonds have different rates of
return leads us to arbitrage opportunity.
Now let us assume that stock is a random process S ( t ,  ). Show that the no-arbitrage principal
does not correct pricing approach consider similar portfolio without forward contract. The value of the
portfolio at initiation date t is also zero but portfolio without forward contract has value at T
V ( T , 0 ) = B –1 ( t , T ) S ( t ) > 0
Note this value is provided by the market and it does not depend on whether derivatives exist itself or
not.
Our point of view on derivatives pricing was presented in [1] and its application to forwards pricing was
presented in [2]. We first define market price of the contract for each admissible market scenario. The
spot price we interpret as a constant which developed by the market participants based on market risk of
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3. the contract. For equilibrium market one can assume that spot price of a derivative is expected value of F,
i.e. < F >. For non equilibrium market the spot price can be a biased statistics of the F. We do not
interpret derivatives pricing as a game with zero cost. Such assumption immediately leads us to no-
arbitrage pricing that incorrectly interpret pricing in stochastic setting.
We interpret the ‘price’ as a settlement between forward buyer and seller. The price should be looked
equally from buyer and seller perspectives. The seller of the forward contract should bring the asset to the
forward buyer. To complete such obligation according to no-arbitrage principle the seller borrows funds
at risk free interest and buys the stock for S ( t ). The value of these transactions is zero and therefore it
can not determine the price of the forward. At maturity T the seller of the forward receives the price F and
should return to the bank the sum $B – 1 ( t , T ) S ( t ). The value of the seller position immediately after
settlement of the forward contract at T is equal to
F - B –1 ( t , T ) S ( t )
The forward buyer position immediately after settlement at T is S ( T ) - F. The settlement pricing
implies that buyer of the contract can be considered as a seller of the contract. This interpretation of the
forward price leads us to equation
F - B –1 ( t , T ) S ( t ) = S ( T ,  ) - F (1)
which brings the solution
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F(t,T;) = [ S ( T ,  ) + B –1 ( t , T ) S ( t ) ] (2)
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If S ( T ) is a random variable then F also depends on a market scenario . This is the market price of the
contract. The spot price < F > can be either expected value of F or not. It is a deterministic number
defined by the market participants based on the risk of the forward. The market risk implied by the spot
price we define as the risk factor F - < F > . Buyer risk value is P { F - < F > < 0 } the probability of
scenarios that buyer payment < F > is more than implied by market scenario F ( t , T ;  ). While seller
risk value is P { F - < F > > 0 } that is the measure of scenarios for which seller receives less than
implied by scenario.
We have defined settlement forward price based on definition expressed by equality (1). On the
other hand we can consider other definition of the market price of the forward contract then in turn
implies other market risk of the observed spot price of the forward.
We call two investment opportunities equal if they provide equal rates of return. Buyer of the forward
contract has a choice to buy stock or its forward. We used this definition in [1] for pricing options and
other types of derivatives. Denote forward contract market price as f. Note that equality of the rates of
return on forward and its underlying asset leads to the equation
S( T )  S( t ) [S( T ) - f ] - B( t ,T ) f
=
S( t ) B( t ,T )f
The left hand side of the latter equation defines rate of return on stock over [ t , T ]. Buying forward
contract buyer should invest B ( t , T ) f in bank at date t. At T forward buyer receives $f from the bank at
T and exchanges it immediately for stock S ( T ). The value of the transaction at T is S ( T ) - f. These
transactions justify right hand side of the latter equation. Assume that either stock or its forward admis a
portion of investment. Then we can ignore inequality
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4. B ( t , T ) f ≠ S ( t ) and assume that the same amount of money can be invested in sock or its forward.
The solution of the latter equation brings the market price of the forward contract in the form
S( T , ω) S( t )
f(t,T;) =
S( T ,ω) B( t ,T )  S( t )
The date-t spot price of the forward < f > implies market risk defined by the quantity f ( t , T ;  ) - < f >.
Note that spot forward price presented by the market and it does not depends on a model and therefore
< f > = < F >.
Finally, looking back at the underlying idea of the no-arbitrage price we conclude that it could be more
likely interpreted as a fair spot price of the forward buyer.
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5. References.
1. Gikhman I. Alternative Derivatives Pricing: Formal Approach. LAP LAMBERT Academic
Publishing, 2010, p. 164.
2. Gikhman I. Forward Contract Pricing. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1538944.
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