A Simple Proof of the Fundamental Theorem of Asset Pricing
aa r X i v : . [ q -f i n . M F ] D ec A SIMPLE PROOF OF THE FUNDAMENTAL THEOREM OFASSET PRICING
KEITH A. LEWIS
Abstract.
A simple statement and accessible proof of a version of the Fun-damental Theorem of Asset Pricing in discrete time is provided. Careful dis-tinction is made between prices and cash flows in order to provide uniformtreatment of all instruments. There is no need for a “real-world” measure inorder to specify a model for derivative securities, one simply specifies an ar-bitrage free model, tunes it to market data, and gets down to the business ofpricing, hedging, and managing the risk of derivative securities. Introduction
It is difficult to write a paper about the Fundamental Theorem of Asset Pricingthat is longer than the bibliography required to do justice to the excellent workthat has been done elucidating the key insight Fischer Black, Myron Scholes, andRobert Merton had in the early ’70’s. At that time, the Capital Asset PricingModel and equilibrium reasoning dominated the theory of security valuation so thenotion that the relatively weak assumption of no arbitrage could have such detailedimplications about possible prices resulted in well deserved Nobel prizes.One aspect of the development of the FTAP has been the technical difficultiesinvolved in providing rigorous proofs and the the increasingly convoluted statementsof the theorem. The primary contribution of this paper is a statement of thefundamental theorem of asset pricing that is comprehensible to traders and riskmanagers and a proof that is accessible to students at graduate level courses inderivative securities. Emphasis is placed on distinguishing between prices and cashflows in order to give a unified treatment of all instruments. No artificial “realworld” measures which are then changed to risk-neutral measures needed. (Seealso Biagini and Cont [4].) One simply finds appropriate price deflators.Section 2 gives a brief review of the history of the FTAP with an eye to demon-strating the increasingly esoteric mathematics involved. Section 3 states and provesthe one period version and introduces a definition of arbitrage more closely suitedto what practitioners would recognize. Several examples are presented to illustratethe usefulness of the theorem. In section 4 the general result for discrete time mod-els is presented together with more examples. The last section finishes with somegeneral remarks and a summary of the methodology proposed in this paper. Theappendix is an attempt to clairify attribution of early results.
Date : December 4, 2019.Peter Carr is entirely responsible for many enjoyable and instructive discussions on this topic.Andrew Kalotay provided background on Edward Thorpe and his contributions. Alex Mayusprovided practitioner insights. Robert Merton graciously straightened me out on the early history.Walter Schachermeyer provided background on the technical aspects of the state of the art proofs.I am entirely responsible for any omissions and errors. Review
From Merton’s 1973 [25] paper, “The manifest characteristic of (21) is the num-ber of variables that it does not depend on” where (21) refers to the Black-Scholes1973 [5] option pricing formula for a call having strike E and expiration τf ( S, τ ; E ) = S Φ( d ) − Ee − rt Φ( d − σ √ τ ) . Here, Φ is the cumulative standard normal distribution, σ is the instantaneousvariance of the return on the stock and d = [log( S/E ) + ( r + σ ) τ ] /σ √ τ . Inparticular, the return on the stock does not make a showing, unlike in the CapitalAsset Pricing Model where it shares center stage with covariance. This was the keyinsight in the connection between arbitrage-free models and martingales.In the section immediately following Merton’s claim he calls into question therigor of Black and Scholes’ proof and provides his own. His proof requires the bondprocess to have nonzero quadratic variation. Merton 1974 [26] provides what is nowconsidered to be the standard derivation.A special case of the valuation formula that European option prices are thediscounted expected value of the option payoff under the risk neutral measure makesits first appearance in the Cox and Ross 1976 [7] paper. The first version of theFTAP in a form we would recognize today occurs in a Ross 1978 [31] where it iscalled the Basic Valuation Theorem. The use of the Hahn-Banach theorem in theproof also makes its first appearance here, although it is not clear precisely whattopological vector space is under consideration. The statement of the result is alsocouched in terms of market equilibrium, but that is not used in the proof. Onlythe lack of arbitrage in the model is required.Harrison and Kreps [13] provide the first rigorous proof of the one period FTAP(Theorem 1) in a Hilbert space setting. They are also the first to prove results forgeneral diffusion processes with continuous, nonsingular coefficients and make thepremonitory statement “Theorem 3 can easily be extended to this larger class ofprocesses, but one then needs quite a lot of measure theoretic notation to make arigorous statement of the result.”The 1981 paper of Harrison and Pliska [14] is primarily concerned with modelsin which markets are complete (Question 1.16), however they make the key obser-vation, “Thus the parts of probability theory most relevant to the general question(1.16) are those results, usually abstract in appearance and French in origin, whichare invariant under substitution of an equivalent measure.” This observation ap-plies equally to incomplete market models and seems to have its genesis in themuch earlier work of Kemeny 1955 [19] and Shimony 1955 [33] as pointed out byW. Schachermeyer.D. Kreps 1981 [21] was the first to replace the assumption of no arbitrage withthat of no free lunch : “The financial market defined by ( X, τ ), M , and π admits afree lunch if there are nets ( m α ) α ∈ I ∈ M and ( h α ) α ∈ I ∈ X + such that lim α ∈ I ( m α − h α ) = x for some x ∈ X + \{ } .” It is safe to say the set of traders and risk managersthat are able to comprehend this differs little from the empty set. It was a brillianttechnical innovation in the theory but the problem with first assuming a measurefor the paths instrument prices follow was that it made it difficult to apply theHahn-Banach theorem. The dual of L ∞ ( τ ) under the norm topology is intractable.The dual of L ∞ ( τ ) under the weak-star topology is L ( τ ), which by the Radon-Nikodym theorem can be identified with the set of measures that are absolutely SIMPLE PROOF OF THE FUNDAMENTAL THEOREM OF ASSET PRICING 3 continuous with respect to τ . This is what one wants when hunting for equivalentmartingale measures, however one obstruction to the proof is that the positivefunctions in L ∞ ( τ ) do not form a weak-star open set. Krep’s highly technical freelunch definition allowed him to use the full plate of open sets available in the normtopology that is required for a rigorous application of the Hahn-Banach theorem.The escalation of technical machinery continues in Dalang, Morton and Will-inger 1990 [8]. This paper gives a rigorous proof of the FTAP in discrete time foran arbitrary probability space and is closest to this paper in subject matter. Theycorrectly point out an integrability condition on the price process is not econom-ically meaningful since it is not invariant under change of measure. They give aproof that does not assume such a condition by invoking a nontrivial measurableselection theorem. They also mention, “However, if in addition the process wereassumed to be bounded, ...” and point out how this assumption could simplifytheir proof. The robust arbitrage definition and the assumption of bounded pricesis also used the original paper, Long Jr. 1990 [17], on numeraire portfolios.The pinnacle of abstraction comes in Delbaen and Schachermeyer 1994 [9] wherethey state and prove the FTAP in the continuous time case. Theorem 1.1 states anequivalent martingale measure exists if and only if there is no free lunch with van-ishing risk : “There should be no sequence of final payoffs of admissible integrands, f n = ( H n · S ) ∞ , such that the negative parts f − n tend to zero uniformly and suchthat f n tends almost surely to a [0 , ∞ ]-valued function f satisfying P [ f > > The One Period Model
The one period model is described by a vector, x ∈ R m , representing the pricesof m instruments at the beginning of the period, a set Ω of all possible outcomesover the period, and a bounded function X : Ω → R m , representing the prices ofthe m instruments at the end of the period depending on the outcome, ω ∈ Ω. Definition 3.1.
Arbitrage exists if there is a vector γ ∈ R m such that γ · x < and γ · X ( ω ) ≥ for all ω ∈ Ω . The cost of setting up the position γ is γ · x = γ x + · · · + γ m x m . This beingnegative means money is made by putting on the position. When the position isliquidated at the end of the period, the proceeds are γ · X . This being non-negativemeans no money is lost. KEITH A. LEWIS
It is standard in the literature to introduce an arbitrary probability measure onΩ and use the conditions γ · x = 0 and γ · X ≥ E [ γ · X ] > strong arbitrage .Define the realized return for a position, γ , by R γ = γ · X/γ · x , whenever γ · x = 0.If there exists ζ ∈ R m with ζ · X ( ω ) = 1 for ω ∈ Ω (a zero coupon bond) then theprice is ζ · x = 1 /R ζ . Zero interest rates correspond to a realized return of 1.Note that arbitrage is equivalent to the condition R γ < γ ∈ R m .In particular, negative interest rates do not necessarily imply arbitrage.The set of all arbitrages form a cone since this set is closed under multiplicationby a positive scalar and addition. The following version of the FTAP shows how tocompute an arbitrage when it exists. Theorem 3.1. (One Period Fundamental Theorem of Asset Pricing) Arbitrageexists if and only if x does not belong to the smallest closed cone containing therange of X . If x ∗ is the nearest point in the cone to x , then γ = x ∗ − x is anarbitrage.Proof. If x belongs to the cone, it is arbitrarily close to a finite sum P j X ( ω j ) π j ,where ω j ∈ Ω and π j > j . If γ · X ( ω ) ≥ ω ∈ Ω then γ · P X ( ω j ) π j ≥
0, hence γ · x cannot be negative. The other direction is a consequence of thefollowing with C being the smallest closed cone containing X (Ω). (cid:3) Lemma 3.2. If C ⊂ R m is a closed cone and x
6∈ C , then there exists γ ∈ R m suchthat γ · x < and γ · y ≥ for all y ∈ C .Proof. This result is well known, but here is an elementary self-contained proof.Since C is closed and convex, there exists x ∗ ∈ C such that k x ∗ − x k ≤ k y − x k for all y ∈ C . We have k x ∗ − x k ≤ k tx ∗ − x k for t ≥
0, so 0 ≤ ( t − k x ∗ k − t − x ∗ · x = f ( t ). Because f ( t ) is quadratic in t and vanishes at t = 1, we have0 = f ′ (1) = 2 k x ∗ k − x ∗ · x , hence γ · x ∗ = 0. Now 0 < k γ k = γ · x ∗ − γ · x , so γ · x < k x ∗ − x k ≤ k ty + x ∗ − x k for t ≥ y ∈ C , we have 0 ≤ t k y k + 2 ty · ( x ∗ − x ). Dividing by t and setting t = 0 shows γ · y ≥ (cid:3) Let B (Ω) be the Banach algebra of bounded real-valued functions on Ω. Its dual, B (Ω) ∗ = ba (Ω), is the space of finitely additive measures on Ω, e.g., Dunford andSchwartz [11]. If P is the set of non-negative measures in ba (Ω), then {h X, Π i :Π ∈ P} is the smallest closed cone containing the range of X , where the anglebrackets indicate the dual pairing. There is no arbitrage if and only if there existsa non-negative finitely additive measure, Π, on Ω such that x = h X, Π i . We callsuch Π a price deflator .If V ∈ B (Ω) is the payoff function of an instrument and V = γ · X for some γ ∈ R m , then the cost of replicating the payoff is γ · x = h γ · X, Π i = h V, Π i . Ofcourse the dimension of such perfectly replicating payoff functions can be at most SIMPLE PROOF OF THE FUNDAMENTAL THEOREM OF ASSET PRICING 5 m . The second fundamental theorem of asset pricing states that when there arecomplete markets, the price is unique. But that never happens in the real world.If a zero coupon bond, ζ ∈ R m , exists then the riskless realized return is R = R ζ = 1 / Π(Ω). If we let P = Π R , then P is a probability measure and x = h X/R, P i = EX/R . With V as in the previous paragraph, the cost of the replicatingpayoff is v = EV /R , the expected discounted payoff.3.0.1.
Managing Risk.
The current theoretical foundations of Risk Mangagment arelacking . The classical theory assumes complete markets and perfect hedging andfails to provide useful tools for quantitatively assessing how wishful this thinkingis. The main defect of most current risk measures is that they fail to take intoaccount active hedging. E.g., VaR[18] assumes trades will be held to some timehorizon and only considers a percentile loss. The only use to someone running abusiness that they might lose X in n days with probability p if they do nothing isto put a tick in a regulatory checkbox.Multi-period models will be considered below, but a first step is to measurethe least squared error in the one-period model. Given any measure Π and anypayoff V ∈ B (Ω), we can minimize h ( γ · X − V ) , Π i . The solution is γ = h XX T , Π i − h XV, Π i . The least squared error ismin γ h ( γ · X − V ) , Π i = h V , Π i − h XV, Π i T h XX T , Π i − h XV, Π i . In the case of a two instrument market X = ( R, S ) where R is the realized returnon a zero coupon bond we get γ = (( EV − nES ) /R, n ) where n = Cov( S, V ) / Var S and the expectation corresponds to the probability measure P = Π R . If we furtherassume x = (1 , s ) we have γ · x = EV /R − n ( ES/R − s ) and the least squared errorreduces to sin θ Var( V ) /R where cos θ is the correlation of S with V .If Π is a price deflator we get the same answer for the price as in the one-periodmodel without the need to involve the Hahn-Banach theorem.3.1. Examples.
This section illustrates consequences of the one period model.Standard results follow from rational application of mathematics instead of ad hocarguments.
Example 1. (Put-Call parity) Let
Ω = [0 , ∞ ) , x = (1 , s, c, p ) , and X ( ω ) =( R, ω, ( ω − k ) + , ( k − ω ) + ) . This models a bond with riskless realized return R , a stock that can take onany non-negative value, and a put and call with the same strike. Take γ =( − k/R, , − , γ · X ( ω ) = − k + ω − ( ω − k ) + + ( k − ω ) + = 0 it follows0 = x · γ = − k/R + s − c + p so s − k/R = c − p .This is the first thing traders check with any European option model. Put-callparity does not hold in general for American options because the optimal exercisetime for each option is not necessarily the same. Example 2. (Cost of Carry) Let
Ω = [0 , ∞ ) , x = (1 , s, , and X ( ω ) = ( R, ω, ω − f ) . As empirically verified in September 2008
KEITH A. LEWIS
This models a bond with riskless realized return R , a stock, and a forwardcontract on the stock with forward f . The smallest cone containing the rangeof X is spanned by X (0) = ( R, , − f ) and lim ω →∞ X ( ω ) /ω = (0 , , , s ) = a ( R,
0) + b (0 ,
1) gives a = 1 /R and b = s . This implies 0 = − f /R + s so f = Rs . Example 3. (Standard Binomial Model) Let
Ω = { d, u } , < d < u , x = (1 , s, v ) and X ( ω ) = ( R, sω, V ( sω )) , where V is any given function. This is the usual (MBA) parametrization for the one period binomial model witha risk-less bond having realized return R , and a stock having price s that can go toeither sd or su . The smallest cone containing the range of X is spanned by X ( d )and X ( u ). Solving (1 , s ) = aX ( d ) + bX ( u ) for a and b yields a = ( u − R ) /R ( u − d )and b = ( R − d ) /R ( u − d ). The condition that a and b are non-negative implies d ≤ R ≤ u . The no arbitrage condition on the third component implies v = 1 R (cid:18) u − Ru − d V ( sd ) + R − du − d V ( su ) (cid:19) . In a binomial model, the option is a linear combination of the bond and stock.This is obviously a serious defect in the model. Solving V ( sd ) = mR + nsd and V ( su ) = mR + nsu for n we see the number of shares of stock to purchase in orderto replicate the option is n = ( V ( su ) − V ( sd )) / ( su − sd ). Note that if V is a callspread consisting of long one call with strike slightly greater than sd and short onecall with strike slightly less than su , then ∂v/∂s = 0 since V ′ ( sd ) = 0 = V ′ ( su ). Example 4. (Binomial Model) Let
Ω = { S + , S − } , x = (1 , s, v ) , and X ( ω ) =( R, ω, V ( ω )) , where V is any given function. As above we find v = 1 R (cid:18) S + − RsS + − S − V ( S − ) + Rs − S − S + − S − V ( S + ) (cid:19) and the number of shares of stock required to replicate the option is n = ( V ( S + ) − V ( S − ) / ( S + − S − ). Note ∂v/∂s = n indicates the number of stock shares to buyin order to replicate the option. Example 5.
Let
Ω = [90 , , x = (1 , , , and X ( ω ) = (1 , ω, max { ω − , } ) . This corresponds to zero interest rate, a stock having price 100 that will certainlyend with a price in the range 90 to 110, and a call with strike 100. One might thinkthe call could have any price between 0 are 10 without entailing arbitrage, but thatis not the case.This model is not arbitrage free. The smallest cone containing the range of X isspanned by X (90), X (100), and X (110). It is easy to see that x does not belong tothis cone since it lies above the plane determined by the origin, X (90) and X (110).Using e b , e s , and e c as unit vectors in the bond, stock, and call directions, X (90) = e b + 90 e s and X (110) = e b + 110 e s + 10 e c . Grassmann algebra[28] yields X (110) ∧ X (90) = 90 e b ∧ e s + 110 e s ∧ e b + 10 e c ∧ e b + 900 e c ∧ e s = − e s ∧ e c +10 e c ∧ e b − e b ∧ e s . The vector perpendicular to this is − e b + 10 e s − e c .After dividing by 10, we can read off an arbitrage from this: borrow 90 usingthe bond, buy one share of stock, and sell two calls. The amount made by puttingon this position is − γ · x = 90 −
100 + 12 = 2. At expiration the position will be
SIMPLE PROOF OF THE FUNDAMENTAL THEOREM OF ASSET PRICING 7 liquidated to pays γ · X ( ω ) = −
90 + ω − { ω − , } = 10 − | − ω | ≥ ≤ ω ≤ Example 6.
Let
Ω = [90 , , x = (100 , . , and X ( ω ) = ( ω, max { ω − , } ) . Eliminating the bond does not imply the call can have any price between 0 and10 without arbitrage. The position γ = (1 , −
11) is an arbitrage.
Example 7. (Normal Model) Let
Ω = ( −∞ , ∞ ) , x = (1 , s ) , X = ( R, S ) with R scalar, and S normally distributed. This model was developed by Louis Bachelier in his 1900 PhD Thesis[1] with animplicit dependence on R . Choose the parameterization S = Rs (1 + σZ ) wherewhere Z is standard normal and the price deflator is Π = P/R where P is theprobability measure underlying Z . This model is arbitrage free for any value of σ ,however it does allow for negative stock values. As long as σ is much smaller than s the probability of negative prices is negligible. Every model has its limitations.A useful formula is Cov( N, f ( M )) = Cov( N, M ) Ef ′ ( M ) whenever M and N are jointly normal. This follows from Ee αN f ( M ) = Ee αN Ef ( M + α Cov(
M, N )),taking a derivative with respect to α , then setting α = 0.The price of a put option with strike k is p ( k ) = E ( k − S ) + /R = E ( k − S )1( S ≤ k ) /R = ( k/R ) P ( S ≤ k ) − ( ES/R )1( S ≤ k )= ( k/R − s ) P ( S ≤ k ) + (Var( S ) /R ) Eδ k ( S )since d k − s ) + /ds = − δ k ( s ), where δ k is a delta function with unit mass at k .Let φ ( z ) = e − z / / √ π be the standard normal density and Φ( z ) = R z −∞ φ ( z ) dz be the cumulative standard normal distribution. We have Eδ k ( S ) = Eδ k ( Rs (1 + σZ )) = φ ( z ) /Rsσ where z = ( k/Rs − /σ hence p ( k ) = ( k/R − s )Φ( z ) + sσφ ( z ).For an at-the-money option, k = Rs , this reduces to p ( k ) = sσ/ √ π .The hedge position in the underlying is ∂p ( k ) /∂s = − ER S ≤ k ) /R = − Φ( z )so the at-the-money hedge is to short 1 / p we have the delta hedge is Cov( S, f ( S )) / Var( S ) = Ep ′ ( S ). If p is linear then we can find a perfect hedge so let’s estimate the leastsquared error for quadratic payoffs. Letting µ k = E ( S − f ) k be the k -th centralmoment, where f = Rs = ES , and using EZ = 1 and EZ = 3 we findVar( p ( S )) = µ p ′ ( f ) + ( µ − µ ) p ′′ ( f ) / f σ p ′ ( f ) + f σ p ′′ ( f ) / . Since Cov(
S, p ( S )) = Var( S ) Ep ′ ( S ) = Var( S ) p ′ ( f ) we havecorr( S, p ( S )) = 1 / p f σ p ′′ ( f ) / p ′ ( f ) ≈ − f σ p ′′ ( f ) / p ′ ( f ) if p ′ ( f ) > θ ≈ f σp ′′ ( f ) / p ′ ( f ) for small σ . The least squared error isVar( p ( S )) sin θ/R ≈ f σ p ′′ ( f ) / R which is second order in σ and does not de-pend (strongly) on p ′ ( f ).If p ′ ( f ) = 0 then the correlation is zero and the the best hedge is a cash positionequal to Ep ( S ). If p ′ ( f ) < − S, ( S − f ) + ) = 1 / p − /π ≈ .
856 independent of R , s , and σ . This fol-lows from Cov( S, p ( S )) = Var( S ) / p ( S ) = Var( S )(1 / − / π ) where p ( x ) = ( x − f ) + .One technique traders use to smooth out gamma for at-the-money options is toextend the option expiration by a day or two. This gives a quantitative estimateof how bad that hedge might be.3.2. An Alternate Proof.
The preceding proof of the fundamental theorem ofasset pricing does not generalized to multi-period models.Define A : R m → R ⊕ B (Ω) by Aξ = − γ · x ⊕ γ · X . This linear operatorrepresents the account statements that would result from putting on the position γ at the beginning of the period and taking it off at the end of the period. Define P to be the set of { p ⊕ P } where p > R and P ≥ B (Ω). Arbitrageexists if and only if ran A = { Aγ : γ ∈ R m } meets P . If the intersection is empty,then by the Hahn-Banach theorem [2] there exists a hyperplane H containing ran A that does not intersect P . Since we are working with the norm topology, clearly1 ⊕ P , so the theorem applies. Thehyperplane consist of all y ⊕ Y ∈ R ⊕ B (Ω) such that 0 = yπ + h Y, Π i for some π ⊕ Π ∈ R ⊕ ba (Ω).First note that hP , π ⊕ Π i cannot contain both positive and negative values. Ifit did, the convexity of P would imply there is a point at which the dual pairing iszero and thereby meets H . We may assume that the dual pairing is always positiveand that π = 1. Since 0 = h Aγ, π ⊕ Π i = h− γ · x, π i + h γ · X, Π i for all γ ∈ R m itfollows x = h X, Π i for the non-negative measure Π. This completes the alternateproof.This proof does not yield the arbitrage vector when it exists, however it can bemodified to do so. Define P + = { π ⊕ Π : h p ⊕ P, π ⊕ Π i > , p ⊕ P ∈ P} . TheHahn-Banach theorem implies ran A ∩ P 6 = ∅ if and only if ker A ∗ ∩ P + = ∅ , where A ∗ is the adjoint of A and ker A ∗ = { π ⊕ Π : A ∗ ( π ⊕ Π) = 0 } . If the later holdswe know 0 < inf Π ≥ k− x + h X, Π ik since A ∗ ( π ⊕ Π) = − xπ + h X, Π i . The sametechnique as in the first proof can now be applied.4. Multi-period Model
The multi-period model is specified by an increasing sequence of times ( t j ) ≤ j ≤ n at which transactions can occur, a sequence of algebras ( A j ) ≤ j ≤ n on the set ofpossible outcomes Ω where A j represents the information available at time t j , asequence of bounded R m valued functions ( X j ) ≤ j ≤ n with X j being A j measurablethat represent the prices of m instruments, and a sequence of bounded R m valuedfunctions ( C j ) ≤ j ≤ n with C j being A j measurable that represent the cash flowsassociated with holding one share of each instrument over the preceding time period.We further assume the cardinality of A is finite, and the A j are increasing. SIMPLE PROOF OF THE FUNDAMENTAL THEOREM OF ASSET PRICING 9 A trading strategy is sequence of bounded R m valued functions (Γ j ) ≤ j ≤ n withΓ j being A j measurable that represent the amount in each security purchased attime t j . Your position is Ξ j = Γ + · · · + Γ j , the accumulation of trades over time.A trading strategy is called closed out at time t j if Ξ j = 0. Note in the one periodcase closed out trading strategies have the form Γ = γ , Γ = − γ .The amount your account makes at time t j is A j = Ξ j − · C j − Γ j · X j , 0 ≤ j ≤ n ,where we use the convention C = 0. The financial interpretation is that at time t j you receive cash flows based on the position held from t j − to t j and are chargedfor trading Γ j shares at prices X j . Definition 4.1.
Arbitrage exists if there is trading strategy that makes a strictlypositive amount on the initial trade and non-negative amounts until it is closed out.
We now develop the mathematical machinery required to state and prove theFundamental Theorem of Asset Pricing.Let B (Ω , A , R m ) denote the Banach algebra of bounded A measurable functionson Ω taking values in R m . We write this as B (Ω , A ) when m = 1.Recall that if B is a Banach algebra we can define the product yy ∗ ∈ B ∗ for y ∈ B and y ∗ ∈ B ∗ by h x, yy ∗ i = h xy, y ∗ i for x ∈ B , a fact we will use below.The standard statement of the FTAP uses conditional expectation. This versionuses restriction of measures, a much simpler concept. The conditional expectationof a random variable is defined by Y = E [ X |A ] if and only Y is A measurable and R A Y dP = R A X dP for all A ∈ A . Using the dual pairing this says h A Y, P i = h A X, P i for all A ∈ A . Using the product just defined we can write this as h A , Y P i = h A , XP i so Y P ( A ) = XP ( A ) for all A ∈ A . If P has domain A thissays Y P = XP | A .We need a slight generalization. If Y is A measurable, P has domain A , and h A Y, P i = h A X, Q i for all A ∈ A , then Y P = XQ | A . There is no requirementthat P and Q be probability measures.Let P ⊂ L nj =0 B (Ω , A j ) be the cone of all ⊕ j P j such that P > P j ≥ ≤ j ≤ n . The dual cone, P + is defined to be the set of all ⊕ j Π j in L nj =0 ba (Ω , A j )such that h P, Π i = h⊕ j P j , ⊕ j Π j i = P j h P j , Π j i > Lemma 4.1.
The dual cone P + consists of ⊕ j Π j such that Π > , and Π j ≥ for ≤ j ≤ n .Proof. Since 0 < h P , Π i for P > ( A ) > A soΠ >
0. For every ǫ > j > < ǫ Π (Ω) + h P j , Π j i for every P j ≥
0. This implies Π j ≥ (cid:3) Theorem 4.2. (Multi-period Fundamental Theorem of Asset Pricing) There is noarbitrage if and only if there exists ⊕ i Π i ∈ P + such that X i Π i = ( C i +1 + X i +1 )Π i +1 | A i , ≤ i < n. Note each side of the equation is a vector-valued measure and recall Π | A denotesthe measure Π restricted to the algebra A . Proof.
Define A : L ni =0 B (Ω , A i , R m ) → L ni =0 B (Ω , A i ) by A = L ≤ i ≤ n A i . De-fine C to be the subspace of strategies that are closed out by time t n .With P as above, no arbitrage is equivalent to A C ∩ P = ∅ . Again, the normtopology ensures that P has an interior point so the Hahn-Banach theorem implies there exists a hyperplane H = { X ∈ L ni =0 B (Ω , A i ) : h X, Π i = 0 } for some Π = ⊕ n Π i containing A C that does not meet P . It is not possible that hP , Π i takeson different signs. Otherwise the convexity of P would imply 0 = h P, Π i for some P ∈ P so we may assume Π ∈ P + . Note 0 = h A ( ⊕ i Γ i ) , ⊕ i Π i i = P ni =0 h Ξ i − · C i − Γ i · X i , Π i i for all ⊕ i Γ i ∈ C . Taking closed out strategies of the form Γ i = Γ, Γ i +1 = − Γhaving all other terms zero yields, where Γ is A i measurable, gives 0 = h Ξ i − · C i − Γ i · X i , Π i i + h Ξ i · C i +1 − Γ i +1 · X i +1 , Π i +1 i = h− Γ · X i , Π i i + h Γ · C i +1 +Γ · X i +1 , Π i +1 i ,hence h Γ , X i Π i i = h Γ , ( C i +1 + X i +1 )Π i +1 i for all A i measurable Γ. Taking Γ to bea characteristic function proves X i Π i = ( C i +1 + X i +1 )Π i +1 | A i for 0 ≤ i < n . (cid:3) A simple induction shows
Corollary 4.3.
With notation as above, (1) X j Π j = X j
0, then R j = Π j / Π j +1 . In thiscase the short rate process determines the price deflators Π j = Π / ( R · · · R j − ), j > Example 9. (Zero Coupon Bonds) A zero coupon bond has a single cash flow C k = 1 at maturity t k . Since X j Π j = Π k | A j for a bond maturing at t k we have its price at time t j ≤ t k is X j ≡ D j ( k ) = Π k / Π j | A j = Π k | A j / Π j . The price at and after maturity is 0. Note D j ( j + 1) = 1 /R j . The function j D ( j ) is called the discount or zero curve. Example 10. (Forward Rate Agreement) A forward rate agreement starting at t i has price X i = 0 and two non-zero cash flows, C j = − at t j and C k = 1 + F i ( j, k ) δ ( j, k ) at t k where δ ( j, k ) is the daycount fraction for the interval [ t j , t k ] . The day count basis (Actual/360, 30/360, etc.) is a market convention thatdetermines the day count fraction and is approximately equal to the time in yearsof the corresponding interval.We have 0 = − j | A i + (1 + F i ( j, k ) δ ( j, k )Π k | A i so F i ( j, k ) = 1 δ ( j, k ) (cid:18) Π j Π k − (cid:19) | A i = 1 δ ( j, k ) (cid:18) D i ( j ) D i ( k ) − (cid:19) . Forward rates are determined by zero coupon bond prices since they are a portfolioof such.Note that if a zero coupon bond with maturity t k is available at time t j then F j ( j, k ) = (1 /D j ( k ) − /δ ( j, k ) is the forward rate over the interval. Example 11. (Bonds) A bond is specified by calculation dates t < t < · · · < t n ,cash flows C j = cδ j , < j < n , and C n = 1 + cδ n where δ j = δ ( j − , j ) . The price at time t satisfies X Π = c P nj =1 δ j Π j |A +Π n |A so X = c P j δ j D ( j )+ D ( n ). A bond is priced at par if X = 1 in which case c = (1 − D ( n )) / P j δ j D ( j )is the par coupon . Example 12. (Swaps) A swap is specified by calculation dates t < t < · · · < t n and cash flows C j = ( c − F j − ( j − , j )) δ j , < j ≤ n There are many types of swaps. This one is more accurately described as payingfixed and receiveing float without exchange of principal. It is also common for theday count basis of the fixed and floating legs to be different.A fundamental fact about the floating cash flow stream is n X j =1 F j − ( j − , j ) δ j Π j | A = n X j =1 (Π j − / Π j − | A j Π j | A = n X j =1 (Π j − − Π j ) | A j | A = Π − Π n | A . This shows the value of the floating leg is the same as receiving a cash flow of 1 at t and paying a cash flow of 1 at t n . The intuition is that the initial cash flow canbe invested at the prevailing forward rate over each interval and rolled over whileharvesting the floating payments until maturity.Swaps are typically issued at t with price X = 0. Using the above fact showsthe swap par coupon is determined by the same formula as for a bond. Moregenerally, if X t = 0 for t ≤ t and X t = 0 we write F δt ( t , . . . , t n ) = D t ( t ) − D t ( t n ) P nj =1 δ ( j − , j ) D t ( t j )for the par coupon at time t corresponding to the underlying (forward starting)swap. Note we are using the actual times instead of the index as arguments. Alsonote that a one period swap is simply a forward rate agreement. Example 13. (Futures) The price of a futures is always zero. Given an underly-ing index S k at expiration t k , they are quoted as having ‘price’ Φ j at t j with theconstraint Φ k = S k at t k . Their cash flows are C j = Φ j − Φ j − , j ≤ k . No arbitrage implies 0 = (Φ j +1 − Φ j )Π j +1 | A j . If the deflators are predictablethen Φ j = Φ j +1 | A j = S k | A j . The standard way of making this statement is to sayfutures quotes are a martingale.If we assume there is a probability measure P on Ω such that Π t = D t P for some D t , the stochastic discount to time t , that are bounded A t measurable functionsthen we can write h X, Π t i = EXD t .If F is a forward and D is the stochastic discount to expiration we have 0 = E ( F − f ) D = EF ED + Cov(
F, D ) − f ED so the convexity is φ − f = − Cov(
F, D ) /ED ,where φ = EF is the futures rate. In general F and D have negative correlation sofutures quotes are higher than forward rates.In the equity world it is often assumed the price deflators are not stochastic andΠ t = D (0 , t ) ≡ D ( t ) is given. The (instantaneous) spot rate , r ( t ), is defined by D ( t ) = e − tr ( t ) and the (instantaneous) forward rate , f ( t ), by D ( t ) = e − R t f ( s ) ds .We also write D s ( t ) = D ( t ) /D ( s ) for the discount from time s to t . Stock volatilitiesswamp out any dainty assumptions of stochastic rates. Example 14. (Generalized Ho-Lee Model [16] ) The short rate process is R t = φ ( t )+ σ ( t ) B t . The original Ho-Lee model specifies a constant volatility. It allows the discountcurve to be fitted to market data. As in the Bachelier model, it allows inter-est rates to be negative, but it has a simple closed form solution using the fact
SIMPLE PROOF OF THE FUNDAMENTAL THEOREM OF ASSET PRICING 13 exp( − R t Θ( s ) / ds + R t Θ( s ) dB s ) is a martingale plus d (Σ( t ) B t ) = Σ( t ) dB t +Σ ′ ( t ) B t so R ut σ ( s ) B s ds = Σ( u ) B u − Σ( t ) B t − R ut σ ( s ) dB s where Σ ′ = σE t e − R ut σ ( s ) B s ds = E t e − (Σ( u ) B u − Σ( t ) B t )+ R ut Σ( s ) dB s = e − (Σ( u ) B t − Σ( t ) B t ) E t e − (Σ( u ) B u − Σ( u ) B t )+ R ut Σ( s ) dB s = e − (Σ( u ) B t − Σ( t ) B t ) E t e R ut (Σ( s ) − Σ( u )) dB s = e − (Σ( u ) B t − Σ( t ) B t ) e R ut (Σ( s ) − Σ( u )) ds and E t denotes conditional expectation with respect to time t . The generalizedHo-Lee model has discount prices D t ( u ) = e − R ut φ ( s ) − (Σ( s ) − Σ( u )) ds +(Σ( u ) − Σ( t )) B t where we reparameterize by replacing σ ( t ) with − σ ( t ). In case of constant volatilitywe have D t ( u ) = e − R ut φ ( s ) − σ ( s − u ) ds + σ ( u − t ) B t . This shows the convexity in the Ho-Lee model is φ ( t ) − f ( t ) = σ t which isquadratic in t . Example 15. (Forwards) A forward is a contract issued at time s and maturingat time t having price X s = 0 and one nonzero cash flow C t = S t − F s ( t ) at time t , where S t is the price at t of the underlying and F s ( t ) is the forward rate that isspecified at time s . Assuming no dividends S s D ( s ) = S t D ( t ) | A s so 0 = ( X s D ( s ) = ( S t − F s ( t )) D ( t ) | A s = S s D ( s ) − F s ( t )) D ( t ) and we have F s ( t ) = S s /D s ( t ). This is just the cost-of-carry formula. In the presence of dividends ( d j ) at ( t j ) this formula becomes F s ( t ) = P s 2, and Var( W j ) = j/ − j / 2. Ifwe let Z j = 2 W j − j then EZ j = 0 and Var( Z j ) = j .Define [ ω ] j = [ ω ] j +1 ∩ { ω j +1 = 0 } and similarly for [ ω ] j so [ ω ] j is the disjointunion of [ ω ] j and [ ω ] j . It is easy to see Z j +1 P | A j = Z j P . More generally f ( Z j +1 ( ω )) P ([ ω ] j ) = f ( Z j +1 ( ω )) P ([ ω ] j ) + f ( Z j +1 ( ω )) P ([ ω ] j )= f ( Z j ( ω ) − P ([ ω ] j ) / f ( Z j ( ω ) + 1) P ([ ω ] j ) / f ( Z j +1 ) P | A j = ( f ( Z j − 1) + f ( Z j + 1)) P . Example 16. (Multi-period Binomial Model) Fix the annualized realized return R > , the initial stock price s , the drift µ , and the volatility σ . Define X j =( R j , S j ) = ( R j , se µj + σZ j ) . Many price deflators exist but we will look for one having the form Π j = R − j P .Clearly R j +1 Π j +1 | A j = R j Π j . Since S j +1 Π j +1 | A j = ( e µ /R ) ( e − σ + e σ ) S j Π j , themodel is arbitrage free if e µ = R/ cosh σ . Example 17. (Geometric Brownian Motion) Fix the spot rate r , the initial stockprice s the drift µ , and the volatility σ . Let B t be standard Brownian motion anddefine X t = ( e rt , se µt + σB t ) . Let P be Brownian measure and recall M λt = e − λ t/ λB t is a martingale. Look-ing for deflators of the form e − rt P ensures e − rt Π t | A s = e − rs Π s . Since S t Π t = se ( µ − r ) t + σB t , the model is arbitrage free if µ = r − σ / { k − S, } at the end of the periodis E max { k − S, } = kP ( S ≤ k ) − ES S ≤ k ) = kP ( S ≤ k ) − ESP ( Se σ t ≤ k )where we use Ee N f ( N ) = Ee N Ef ( N + Var( N )). (More generally, Ee N f ( N , ... ) = Ee N Ef ( N + Cov( N, N ) , ... ) if N , N , ... are jointly normal.) This can be written E max { k − S, } = kP ( Z ≤ z ) − f P ( Z ≤ z − σt ) where z = σt/ /σ ) log k/f ,and f = Rs is the forward price of the stock.For a European option with payoff p at time t , the value of the option is v = e − rt Ep ( S t ). The delta is ∂v/∂s = e − rt Ep ′ ( S t ) e ( r − σ / t + σB t = Ep ′ ( e σ t S t )and the gamma is ∂ v/∂s = Ep ′′ ( e σ t S t ) e σ t e ( r − σ / t + σB t = e ( r + σ ) t Ep ′′ ( e σ t S t ) . Infinitely Divisible Distrbutions. Brownian motion is characterized as astochastic process having increments that are independent, stationary, and nor-mally distributed. Dropping the last requirement characterizes L´evy processes[3].Knowing the distribution at time 1 determines the distribution at all times and thedistribution at any time is infinitely divisible.Prior to L´evy and Khintchine, Kolmogorov [20] derived a parameterization forthe characteristic function of infinitely divisible distributions having finite variance.There exists a number γ and a non-decreasing function G ( x ) such thatlog Ee iuX = iγu + Z ∞−∞ K u ( x ) dG ( x ) , where K u ( x ) = ( e iux − − iux ) /x . Note φ ′ ( u ) = iγ + i R ∞−∞ ( e iux − /x dG ( x )and φ ′′ ( u ) = − R ∞−∞ e iux dG ( x ) so EX = − iφ ′ (0) = γ and Var X = − φ ′′ (0) = R ∞−∞ dG ( x ) = G ( ∞ ) − G ( −∞ ). Lemma 4.5. If X is infinitely divisible with Kolmogorov parameters γ and G ,then Ee isX e iuX = Ee isX Ee iuX ∗ where X ∗ has Kolmogorov parameters γ ∗ = γ + R ∞−∞ ( e isx − /x dG ( x ) = − iφ ′ ( s ) and dG ∗ ( x ) = e isx dG ( x ) . SIMPLE PROOF OF THE FUNDAMENTAL THEOREM OF ASSET PRICING 15 Proof. We have Ee isX e iuX = Ee iγ ( s + u )+ R ∞−∞ K s + u ( x ) dG ( x ) = Ee isX e iγu + R ∞−∞ ( K s + u ( x ) − K s ( x )) dG ( x ) A simple calculation shows K s + u ( x ) − K s ( x ) = iu ( e isx − /x + e isx K u ( x ) so Ee isX e iuX = Ee isX Ee iuX ∗ where X ∗ is infinitely divisible with Kolmogorov pa-rameters γ ∗ = − iφ ′ ( s ) and dG ∗ ( x ) = e isx dG ( x ). (cid:3) We call X ∗ the K -transform of X .If X is standard normal, then γ = 0, G = 1 [0 , ∞ ) and φ ( u ) = − u / γ ∗ = is and dG ∗ = dG . We have e − s / Ee iu ( is + X ) = e − s / e − su − u / = e − ( s + u ) / = Ee isX e iuX . Corollary 4.6. If f and its Fourier transform are integrable, then Ee isX f ( X ) = Ee isX Ef ( X ∗ ) where X ∗ is the K-transform of X .Proof. If f and its Fourier transform are integrable, then f ( x ) = R ∞−∞ e iux ˆ f ( u ) du/ π ,where ˆ f ( u ) = R ∞−∞ e − iux f ( x ) dx is the Fourier transform of f . Ee isX f ( X ) = Z ∞−∞ Ee iuX e iuX ˆ f ( u ) du/ π = Ee isX Z ∞−∞ Ee iuX ∗ ˆ f ( u ) du/ π = Ee isX Ef ( X ∗ ) (cid:3) Example 18. (L´evy Processes) Fix the spot rate r , the initial stock price s , the drift µ , and the volatility σ . Let L t be a L´evy process and define X t = ( e rt , se µt + σL t ) . Again we look for deflators of the form e − rt P . If we define the cumulant κ t ( s ) =log Ee sL t then κ t ( s ) = tκ ( s ) and e − tκ ( σ )+ σL t is a martingale. Since S t Π t = se ( µ − r ) t + σL t , the model is arbitrage free if µ = r − κ ( σ ).The formula for the forward value of put is E ( k − S t ) + = E ( k − S t )1( S t ≤ k ) = kP ( S t ≤ k ) − se rt P ( S ∗ t ≤ k ) where S t = se ( r − κ ( σ )) t + σL ∗ t and L ∗ t is the K-transformwith is = σ . 5. Remarks • Not only do traders want to know exactly how much they make upfrontbased on the size of the position they put on, they and their risk managersalso want to hedge the subsequent gains they might make under favorablemarket conditions. • Different counterparties have different short rate processes. A large finan-cial institution can fund trading strategies at a more favorable rate than aday trader using a credit card. • As previously noted, ∂v/∂s = n in Example 2, however ∂ ( Rv ) /∂R = ns for both Example 2 and 3. In words, the derivative of the future value ofthe option with respect to realized return is the dollar delta. • It is not necessay to assume algebras for the prices and cash flows areare increasing. If they are adapted to the algebras ( B j ) and B j ⊆ A j forall j then X j Π j will be well defined. This is useful in order to model arecombining tree. In the standard binomial model the atoms of B j are { W j = j − i } , 0 ≤ i ≤ j . This can be used to give a rigorous foundationto path bundling algorithms, e.g., Tilley [35]. • This theory only allows bounded functions as models of prices and positions.This corresponds to reality, but not to the classical Black-Scholes/Mertontheory. The fact that prices are bounded has no material consequenceswhen it comes to model implementation. An unbounded price process canbe replaced by one stopped at an arbitrarily large value. Since we canmake the probability of stopping vanishingly small, calculation of optionprices can be made arbitrarily close to those computed using the unboundedmodel. Every model I have implemented had prices bounded by 1 . × . • Likewise, discrete time is not material problem since one could model yoctosecond time steps. In fact, continuous time introduces serious technicalproblems such as doubling strategies[13]. Zeno wasn’t the only one to dis-tract people’s attention with this sort of casuistry. • Measures being finitely additive is also not an issue. Countably additivemeasures are also finitely additive and so all such models fit into this frame-work. Interchanging limits and the Radon-Nikodym theorem for finitelyadditive measures are more complicated than for countably additive mea-sures, but these are not needed here. • The examples show this theory has the same expressive power as the stan-dard theory and illustrates the usefulness of distinguishing prices from cashflows to uniformly handle all types of instruments. There is no need tocook up a “real world” measure. Not only does it ultimately get replaced,it adds technical complications to the theory.6. Appendix: Origins While preparing this paper I had difficulty understanding who figured out whatwhen in the early theory. Cutting edge research is always messy. This appendixis my attempt to clear that up and point out the repercussions. Priority is thecurrency of academics, legacy is the other side of that coin.Currency is both sides of the coin for practitioners and I make my living tryingto provide them with tools they find useful. They usually don’t understand thesubtleties of mathematical models but they know if the software implementationprovides numbers that make sense.As George Box said “all models are wrong, but some are useful.” MathematicalFinance is still in its infancy, but it has notched up some significant victories. Dollardenominated fixed income derivatives having maturity less than 4 years trade atbasis point spreads. Every bank has a different implementation, but they all getthe same answer. “Practitioners” in that market can no longer rely on cunning andmakeshift.As Haug and Taleb[15] carefully delineate, the Black-Scholes and even moresophisticated formulas were used well before Black, Scholes, and Merton showed upon the scene. They underscore the importance of the no arbitrage condition andare entirely correct that traders still use ad hoc devices to produce numbers they SIMPLE PROOF OF THE FUNDAMENTAL THEOREM OF ASSET PRICING 17 find useful. Options are used to determine model parameters and now play the roleof primary securities in hedging more complex derivatives. Such is financial marketprogress.However, they don’t seem to appreciate the power of the mathematical underpin-nings. Ed Thorpe came up with a formula for calls and puts, but didn’t know howto extend that to price bonds with embedded options . Academics have time toreflect on the paths blazed by practicioners. Exotic option pricing formulas requirenontrivial mathematics unobtainable through seat-of-the-pants methods.It is beyond the scope of this appendix to review the tenor of the time laid downby Markowitz[23], Tobin[36], Sharpe[32], Lintner[22] and other pioneers in the fieldof quantitative finance, but they developed an economic theory to quantify howdiversification reduced risk. The Capital Asset Pricing Model showed how to createportfolios that could minimize systemic market risk.Many of the fundamental results in the FTAP can be traced back to Merton’sunpublished, but widely circulated, technical report[24] that ultimately becamechapter 11 in his book on continuous time finance[27]. It uses a general equilibriumpricing model (intertemporal CAPM) to derive the Black-Scholes option model. Hisproof did not require normally distributed returns or a quadratic utility function,as CAPM did, foreshadowing Ross’s Arbitrage Pricing Theory[30].Merton also derived what is now called the Black-Scholes partial differentialequation and showed how individual sample paths could be used to model pricesdirectly instead of only considering expected values. Black and Scholes introducedthe idea of dynamic trading when people were thinking in terms of portfolio selec-tion. They showed continuous time trading with prices modeled by an It¯o diffusionallows perfect replication and that the problem of estimating mean stock returnswas irrelevant to pricing options.This had some deleterious knock on effects in the theory of mathematical finance.Merton was so far ahead of his time with the mathematical tools he introduced thatgenerations of people in his field overestimated the power of mathematics when itcame to modeling the complicated world we live in. People that did not have hisability to understand the math latched on to binomial models. Brownian motionis a binomial model in wolves clothing.Haug and Taleb are on the right track when it comes to pointing out the conse-quences of a theory that no practitioner would find plausible. I embarrassed myselfin my early career when a trader asked me how to price a barrier option that wastriggered on the second touch. For some reason he didn’t buy my explanation aboutthe infinite oscillatory behavior of Brownian motion and that even using the 100thtime it touched the barrier would have the same theoretical price.The work of Boyce and Kalotay [6] was far ahead of its time. They took apractical Operations Research approach to modeling what happens at the cash flowlevel, including counterparty credit and tax considerations. Something clumsilybeing rediscovered in our post September 2008 world.The origin of the modern theory of derivative securities is based on StephenRoss’s 1977 paper “A Simple Approach to the Valuation of Risky Streams.” Hewas the first to realize that the assumption of no arbitrage and the Hahn-Banachtheorem placed a constraint on the dynamics of sample paths. It is a purely geo-metric result. The price deflator is simply a positive measure used to find a point in Andrew Kalotay, personal communication a cone. Normalizing that to a probability measure does not tell you the probabilityof anything, although the normalizing factor does tell you the price of a zero couponbond if your model has one.Ross’s approach was not as rigorous as Merton’s and the attempts to place hisresults on sound mathematical footing led to the the escalation of increasinglyabstract mathematical machinery outlined in the Review section. This paper is anendeavor to provide a statement of the fundamental theorem of asset pricing thatpracticioners can understand and a mathematically rigorous proof that is accessibleto masters level students. References [1] Louis Bachelier. Th´eorie de la sp´eculation . PhD thesis, La Sorbonne, May 1900.[2] S. Banach and S. Mazur. Zur theorie der linearen dimension. Studia Math. , 4:100–112, 1933.[3] J. Bertoin. L´evy Processes . Cambridge Tracts in Mathematics. Cambridge University Press,1998.[4] Sara Biagini and Rama Cont. Model-free representation of pricing rules as conditional ex-pectations. In Jiro Akahori, Shigeyoshi Ogawa, and Shinzo Watanabe, editors, StochasticProcesses and Applications to Mathematical Finance , pages 53–66, 2006.[5] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. The Journalof Political Economy , 81(3):637–654, 1973.[6] W. M. Boyce and A. J. Kalotay. Tax differentials and callable bonds. The Journal of Finance ,34(4):825–838, 1979.[7] John Cox and Stephen A. Ross. The valuation of options for alternative stochastic processes. J. Fincancial Econ. , 3:145–166, 1976.[8] R.C. Dalang, A. Morton, and W. Willinger. Equivalent martingale measures and no-arbitragein stochastic securities market model. Stochastics and Stochastic Reports , 29:185–201, 1990.[9] Freddy Delbaen and Walter Schachermayer. A general version of the fundamental theoremof asset pricing. Mathematische Annalen , 300:463–520, 1994.[10] Darrell Duffie. 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A dynamic general equilibrium model of the asset market and its applica-tion to the pricing of the captial structure of the firm. Massachusets Institure of TechnologyWorking Paper , 1970.[25] Robert C. Merton. Theory of rational option pricing. The Bell Journal of Economics andManagement Science , 4(1):141–183, 1973.[26] Robert C. Merton. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance , 29(2):449–470, 1974.[27] Robert C Merton. Continuous-Time Finance . 1992.[28] Giuseppe Peano. Geometric calculus : according to the Ausdehnungslehre of H. Grassmann .Birkhauser, Boston, 1999.[29] Stanley R. Pliska. Introduction to Mathematical Finance: Discrete Time Models . Wiley-Blackwell, 1997.[30] Stephen A Ross. The arbitrage theory of capital asset pricing. Journal of Economic Theory ,13(3):341 – 360, 1976.[31] Stephen A. Ross. A simple approach to the valuation of risky streams. The Journal of Busi-ness , 51(3):453–475, 1978.[32] William F. Sharpe. 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