No arbitrage and randomization of derivatives pricing


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

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No arbitrage and randomization of derivatives pricing

  1. 1. A COMMENT ON NO-ARBITRAGE PRICING.Ilya Gikhman6077 Ivy Woods CourtMason OH 45040 USAPh. 513-573-9348Email: ilyagikhman@yahoo.comKeywords: No-arbitrage pricing, cash and carry, forward contractJEL Classification: G12Abstract. In this short notice we present critical comments on no-arbitrage principle. We show thatno-arbitrage pricing is complete in a pricing theory which ignores market risk and is dealing with thedeterministic 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 forfinancing deals. In such approach the underlying of the derivatives in deterministic setting becomes riskfree bond. In stochastic setting no-arbitrage pricing replace real underlying on a virtual underlying thathas 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 toderivatives pricing was presented in [1]. The derivatives pricing contains two steps. On the first step wedefine the ‘market price’. This is the price for each admissible market scenario. On the second step wedefine a spot derivatives price. In some cases spot price can be implied price. In more complex situationsfor example such as options pricing construction of the spot price does not so simple. Given market andspot derivative prices we arrive at the market risk. The market risk by definition is the probability ofscenarios that counterparty pays or loses more than it is implied by the spot price. Market and spot pricesalong with correspondent market risk is what we call derivatives price. We illustrate this approach byconsidering a forward contract pricing. Recall the essence of a standard no-arbitrage forward pricing. It can be found in any majorfinancial handbooks. Denote S ( t ) an asset price at a date t , t ≥ 0. By definition forward contract atdate t is an agreement between two parties called seller and buyer. Counterparts are contracting to sell andbuy underlying asset S at a future date T, T > t. The Seller of the forward obliges to deliver theunderlying asset to the buyer at maturity of the contract T. The Buyer of the contract obliges to pay theagreed at t amount F upon delivery. The forward pricing problem is a construction of the forward price F. 1
  2. 2. A forward is a simple and popular derivatives contract and its valuation highlights general ideas of pricingmore complex derivative instruments. No-arbitrage pricing can be briefly outlined as following. At date ta 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-Taccumulated cash amount in the bank is B – 1 ( t , T ) S ( t ). The value S ( T ) is unknown at the date t. Thefinancial 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 afuture moment T.The buyer’s portfolio value at T is V ( T , F ) = B–1 ( t , T ) S ( t ) - FApplying 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 forprice discovery regardless of the underlying distribution. Underlying can be either deterministic orstochastic. Assume first that S is a deterministic function. Then the rate of return on stock and bondshould 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-arbitrageprincipal it follows that the rates of return on asset and bond must be equal. Indeed, let us for assume forexample 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 thenS ( T ) < 1. At date T investor receives $1 for bond , buys back stock for $S ( T ) and makes riskless profitof 1 - S ( T ) > 0. Similarly we can consider the case when the rate of return on stock is higher than therate of return of the bond. Thus, the assumption that deterministic stock and bonds have different rates ofreturn leads us to arbitrage opportunity. Now let us assume that stock is a random process S ( t ,  ). Show that the no-arbitrage principaldoes not correct pricing approach consider similar portfolio without forward contract. The value of theportfolio 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 ornot.Our point of view on derivatives pricing was presented in [1] and its application to forwards pricing waspresented in [2]. We first define market price of the contract for each admissible market scenario. Thespot price we interpret as a constant which developed by the market participants based on market risk of 2
  3. 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 notinterpret 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 lookedequally from buyer and seller perspectives. The seller of the forward contract should bring the asset to theforward buyer. To complete such obligation according to no-arbitrage principle the seller borrows fundsat risk free interest and buys the stock for S ( t ). The value of these transactions is zero and therefore itcan not determine the price of the forward. At maturity T the seller of the forward receives the price F andshould return to the bank the sum $B – 1 ( t , T ) S ( t ). The value of the seller position immediately aftersettlement 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 pricingimplies that buyer of the contract can be considered as a seller of the contract. This interpretation of theforward price leads us to equation F - B –1 ( t , T ) S ( t ) = S ( T ,  ) - F (1)which brings the solution 1 F(t,T;) = [ S ( T ,  ) + B –1 ( t , T ) S ( t ) ] (2) 2If S ( T ) is a random variable then F also depends on a market scenario . This is the market price of thecontract. The spot price < F > can be either expected value of F or not. It is a deterministic numberdefined by the market participants based on the risk of the forward. The market risk implied by the spotprice we define as the risk factor F - < F > . Buyer risk value is P { F - < F > < 0 } the probability ofscenarios that buyer payment < F > is more than implied by market scenario F ( t , T ;  ). While sellerrisk value is P { F - < F > > 0 } that is the measure of scenarios for which seller receives less thanimplied by scenario. We have defined settlement forward price based on definition expressed by equality (1). On theother hand we can consider other definition of the market price of the forward contract then in turnimplies 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 forwardcontract has a choice to buy stock or its forward. We used this definition in [1] for pricing options andother types of derivatives. Denote forward contract market price as f. Note that equality of the rates ofreturn 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 )fThe left hand side of the latter equation defines rate of return on stock over [ t , T ]. Buying forwardcontract buyer should invest B ( t , T ) f in bank at date t. At T forward buyer receives $f from the bank atT and exchanges it immediately for stock S ( T ). The value of the transaction at T is S ( T ) - f. Thesetransactions justify right hand side of the latter equation. Assume that either stock or its forward admis aportion of investment. Then we can ignore inequality 3
  4. 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 morelikely interpreted as a fair spot price of the forward buyer. 4
  5. 5. References.1. Gikhman I. Alternative Derivatives Pricing: Formal Approach. LAP LAMBERT Academic Publishing, 2010, p. 164.2. Gikhman I. Forward Contract Pricing. 5