Are American options European after all?
AAre American options European after all?
Sören Christensen ∗ Jan Kallsen † Matthias Lenga ‡ Abstract
We call a given American option representable if there exists a European claim whichdominates the American payoff at any time and such that the values of the two optionscoincide in the continuation region of the American option. This concept has interestingimplications from a probabilistic, analytic, financial, and numeric point of view. Relyingon methods from [7, 8, 3] and convex duality, we make a first step towards verifyingrepresentability of American options.
Keywords: optimal stopping, representable American option, embedded Americanoption, cheapest dominating European option, free boundary problem, duality
MSC (2010) classification:
This paper is concerned with reducing the valuation of American options to the simpler problemof computing prices of European options whose payoff is not path dependent. For ease ofexposition we consider the standard risk-neutral Black-Scholes setting of a deterministic bondand a stock whose price processes B resp. S = e X evolve according to dB t = r B t dt , B = , dX t = (cid:18) r − σ (cid:19) dt + σ dW t , (1.1)with parameters r ≥ σ > W . Relative to the probability measure P x ,the return process X is assumed to start in X = x almost surely. We denote the fair value of aEuropean option with payoff f ( X T ) for a payoff function f : R → R + , time to maturity T ∈ R + and initial logarithmic stock price x as v eu , f ( T , x ) , i.e. v eu , f ( T , x ) : = E x (cid:0) e − rT f ( X T ) (cid:1) . (1.2)Similarly, for an upper semi-continuous payoff function g : R → R + satisfying the integrabilitycondition E x (cid:18) sup t ∈[ , T ] g ( X t ) (cid:19) < ∞ , (1.3) ∗ Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Kiel, Germany, email:[email protected] † Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Kiel, Germany, email: [email protected] ‡ Philips Research Europe, Hamburg, Germany, email: [email protected] a r X i v : . [ q -f i n . M F ] F e b he fair value of an American claim with payoff process Z = g ( X ) , time to maturity T ∈ R + ,and initial stock price x is written as v am , g ( T , x ) , i.e. v am , g ( T , x ) : = sup τ ∈ T [ , T ] E x ( e − r τ g ( X τ )) , (1.4)where T [ , T ] denotes the set of [ , T ] -valued stopping times. We write C T : = (cid:8) ( ϑ, x ) ∈ [ , T ] × R : v am , g ( ϑ, x ) > g ( x ) (cid:9) (1.5)and C CT : = ([ , T ] × R ) \ C T = (cid:8) ( ϑ, x ) ∈ R + × R : v am , g ( ϑ, x ) = g ( x ) (cid:9) for the continuation region and the stopping region of the American claim, respectively.Fix a time horizon T and an initial log price X = x such that ( T , x ) is contained in C T . For this introductory section let us assume that C T is a connected set. We say that aEuropean payoff function f : R → R + represents the American payoff function g : R → R + ifthe value of f dominates the value of g everywhere and the two coincide in the continuationregion of the American claim, i.e. v eu , f ( ϑ, x ) ≥ v am , g ( ϑ, x ) for all ( ϑ, x ) ∈ [ , T ] × R and v am , g ( ϑ, x ) = v eu , f ( ϑ, x ) holds for all ( ϑ, x ) ∈ C .The main question in this paper is the following: given an American payoff function g , isthere a European payoff function f representing g ? In this case we call g representable . Ifrepresentability holds, this has several interesting consequences.• The American value function can be computed efficiently by means of linear program-ming, as is explained below.• The early exercise boundary can be obtained numerically at low computational costs, see[13, Section 3.4].• A buy-and-hold position in the European option with time- T payoff f ( X T ) hedges theAmerican claim perfectly. Put differently, the American option can be hedged staticallywith a portfolio of calls/puts that does not cost more than the American claim itself. Here, portfolio is to be understood in the limiting sense of e.g. [16].• In the continuation region, the difference v am , g − v eu , g is the fair value of a Europeanpayoff with time- T payoff f ( X T ) − g ( X T ) . Put differently, the early exercise premium ofthe American option can be interpreted as the price of a European claim with a specificpayoff profile.• The Snell envelope corresponding to the American option allows for a Markovian-styledecomposition, cf. (1.12) below.• Some analytical properties of the early exercise curve can be obtained easily. Indeed, itcoincides with the boundary of the set {( ϑ, x ) ∈ ( , T ] × R : v eu , f ( ϑ, x ) = g ( x )} . Thisallows to derive smoothness of the early exercise curve from the analyticity of v eu , f andthe implicit function theorem. In the same vein, certain analyticity properties of theEuropean payoff function v eu , f transfer to the American payoff function v am , g .2 The solution of the free boundary problem associated to the American option can beextended to a solution of the Black-Scholes partial differential equation beyond the freeboundary.On top of representability of a given option one may ask how to obtain the representing Eu-ropean payoff, at least numerically. Moreover, are possibly all American options representable?Or, if this is not the case, do representable options exist at all – except for the obvious casewhere early exercise is suboptimal and hence g itself represents g ?The concept of representability is not studied here for the first time. It was considered intwo seminal papers by Jourdain and Martini, which have not yet received the attention theydeserve. In [7] it is shown that many European payoffs represent some American payoff, whichis obtained in a natural way. Indeed, given some European payoff function f , they define anAmerican payoff function am T ( f ) : R → R + asam T ( f )( x ) : = inf ϑ ∈[ , T ] v eu , f ( ϑ, x ) , (1.6)from now on called the embedded American option (EAO) associated with f . If the infimum in(1.6) is attained in a connected curve, f represents its embedded American option am T ( f ) , cf.[7, Theorem 5]. Jourdain and Martini provide an explicit example where this is the case. Onthe other hand, they show that embedded American payoff functions satisfy certain analyticityproperties, cf. [7, Proposition 16]. From their results we conclude that representable optionsexist but that not all American payoff functions are representable.In their follow-up article [8] they study the American put option in detail. They show that itcannot be represented by any of a seemingly general and reasonable candidate family of Euro-pean claims. This suggests that this particular option may not be representable. Summing up,Jourdain and Martini provide a way to obtain an American payoff function g that is representedby a given European claim f . Our question here is rather the converse: given g , is there arepresenting European claim f , and how can it be obtained?In order to tackle these problems, we make use of the approach in [3]. Fix an Americanpayoff function g : R → R + . The key contribution of [3] is the linear optimisation problemminimise v eu , f ( T , x ) subject to f : R → R + measurable and v eu , f ( ϑ, x ) ≥ g ( x ) for all ( ϑ, x ) ∈ [ , T ] × R . (1.7)We call the minimiser f of (1.7) cheapest dominating European option (CDEO) of g relativeto ( T , x ) . The linear problem (1.7) can be solved efficiently by numerical methods, cf. [3] fordetails. It is easy to see that the fair price of a CDEO f provides an upper bound to the valueof the given American claim g .However, in [3] it remains open how large the gap between the two actually is. While thereis a priori no reason why the two should coincide, numerical studies in [3] indicate that thedifference seems to be small. In the present paper, we use the CDEO as a candidate whichmay generate the desired American payoff g . Indeed, if g is representable at all, it must berepresented by its CDEO. This also answers the question how to obtain a representing Europeanpayoff function numerically if it exists at all.It is important to distinguish the minimisation problem (1.7) and more generally the presentstudy from the well-known duality approaches put forward by [19, 5, 6]. Consider again3n American payoff function g : R → R + leading to the discounted exercise process (cid:98) Z t : = e − rt g ( X t ) . From [19] we know that v am , g ( T , x ) = inf (cid:40) E x (cid:18) sup t ∈[ , T ] ( (cid:98) Z t − M t ) (cid:19) : M martingale with M = (cid:41) . (1.8)Indeed, the inequality ≤ is obvious because E x ( (cid:98) Z τ ) = E x ( (cid:98) Z τ − M τ ) ≤ E x (cid:18) sup t ∈[ , T ] ( (cid:98) Z t − M t ) (cid:19) for any [ , T ] -valued stopping time τ and any martingale M with M =
0. For the converseinequality consider the Doob-Meyer decomposition V = V + M V − A V (1.9)of the Snell envelope V of the discounted exercise process (cid:98) Z , i.e. M V is a martingale and A V anincreasing process with M V = = A V . Since (cid:98) Z t − M Vt ≤ V t − M Vt = V − A Vt ≤ V = v am , g ( T , x ) for any t ∈ [ , T ] , we conclude that the inequality ≥ holds in (1.8) as well.Similarly, observe that v am , g ( T , x ) = inf (cid:8) E x ( Y ) : Y ≥ (cid:98) Z t ≤ E x ( Y | F t ) , t ∈ [ , T ] (cid:9) . (1.10)Again the inequality ≤ is obvious because any martingale dominating (cid:98) Z is an upper bound of thediscounted American option price process. The converse inequality ≥ follows from choosing Y = V + M VT , where V and M V are defined as above.The linear problem (1.7) can be rephrased asinf (cid:8) E x ( e − rT f ( X T )) : f : R → R + with (cid:98) Z t ≤ E x ( e − rT f ( X T )| F t ) , t ∈ [ , T ] (cid:9) , (1.11)which seems almost identical to the right-hand side of (1.10). However, the dominatingEuropean payoff Y in (1.10) may well be path dependent, which is not the case in (1.11). Andindeed, it is easy to see that the terminal value V + M VT cannot typically written as a functionof X T , e.g. in the case of an American put. Therefore, the identities (1.8) and (1.10) do not helpin deciding whether the value of the CDEO in the sense of (1.7) coincides with the price of thegiven American option g .From a different perspective, one may note that the martingale in the Doob-Meyer decom-position (1.9) is not the only one that leads to optimal choices in (1.8) and (1.10). In fact, wecould replace M V by (cid:101) M in any decomposition of the form V = V + (cid:101) M − (cid:101) A (1.12)with some martingale (cid:101) M and some nonnegative process (cid:101) A satisfying (cid:101) M = = (cid:101) A . Contraryto the unique decomposition (1.9) we do not require (cid:101) A to be increasing. As noted above, (1.11)coincides with the American option price (1.10) if we can choose (cid:101) M such that V + (cid:101) M T = − rT f ( X T ) for some deterministic function f . In this case, the decomposition (1.12) is of Markovian style in the sense that both (cid:101) M t and (cid:101) A t are functions of t and X t at any time t . Hencethe issue of representability is linked to the existence of Markovian-style decompositions (1.12)of the Snell envelope corresponding to the optimal stopping problem.The present study serves different purposes. In Section 2 we establish the link betweenembedded American options from [7], cheapest dominating European options from [3], andrepresentability. By providing an example, we show that representability may depend on thetime horizon T , cf. Section 2.3. The main contribution of this paper is contained in Section3. Firstly, we establish the existence of CDEOs in a distributional sense for sufficiently regularAmerican payoff functions g . Secondly and more importantly, we provide a sufficient criterionfor representability of a given American claim. The assumptions of this result depend onqualitative properties of the corresponding CDEO. Numerical computations suggest that theyare satisfied for the American put, cf. Section 4.Let us fix some notation that is used in the paper. (cid:107) µ (cid:107) stands for the total variation of asigned measure µ . The set of signed measures of finite variation on a measurable space ( S , S ) iswritten as M ( S ) . The vector spaces of real-valued continuous functions and continuous functionsvanishing at infinity on S are denoted by C ( S ) and C ( S ) , respectively. They are Banach spaceswith respect to the norm (cid:107) · (cid:107) ∞ which generates the topology of uniform convergence T uc . By M + ( S ) , C + ( S ) , C + ( S ) we denote the cones of nonnegative elements in the respective spaces. Theclosure and the interior of a set M in some topological space are denoted by cl M and int M .We write ∂ M : = cl M \ int M for the boundary of the set. B V ( x , r ) : = { v ∈ V : (cid:107) v − x (cid:107) ≤ r } denotes the ball with radius r around x in a normed space V . If the space is obvious, we simplywrite B ( x , r ) . The Dirac measure in x is denoted as δ x . Moreover we write ϕ ( µ, σ , ·) forthe probability density function of the normal distribution N ( µ, σ ) with mean µ and variance σ . The cumulative distribution function of N ( , ) is denoted as Φ . The gradient of a real-or complex-valued function f is denoted as D f and its partial derivatives with respect to itsfirst, second, d th argument are written as D f , D f , D d f etc. The convex conjugate and thebiconjugate in the sense of [18] of a function v are denoted by v ∗ and v ∗∗ , respectively. In this section we derive some general results about embedded, cheapest dominating, andrepresentable options. For ease of exposition, we focus on the univariate Black-Scholes market(1.1). Moreover, we use the notation (1.2, 1.4) from Section 1 for the fair values of Europeanand American options.
Fix T ∈ [ , ∞] . Let f : R → R + denote a measurable European payoff function with v eu , f ( ϑ, x ) < ∞ for all ( ϑ, x ) ∈ [ , T ] × R with ϑ < ∞ and g : R → R + an upper semi-continuous American payoff function satisfying (1.3). Let us recall the following key notionsfrom the introduction. Definition 2.1.
1. The embedded American option (EAO) of f up to T is defined as thepayoff function am T ( f ) : R → R + given byam T ( f )( x ) : = inf (cid:8) v eu , f ( ϑ, x ) : ϑ ∈ [ , T ] , ϑ < ∞ (cid:9) , x ∈ R . (2.1)5. We say that f superreplicates or dominates g up to T if v eu , f ( ϑ, x ) ≥ g ( x ) holds for allfinite ϑ ∈ [ , T ] and x ∈ R .3. If T < ∞ and an initial logarithmic stock price X = x is given, we call a Europeanpayoff function f (cid:63) cheapest dominating European option (CDEO) of g relative to ( T , x ) if f (cid:63) superreplicates g up to T and v eu , f (cid:63) ( T , x ) ≤ v eu , f ( T , x ) holds for all Europeanpayoff functions f dominating g up to time T . The set of all such CDEOs is denoted aseu T , x ( g ) . We write eu T , x ( g ) = f (cid:63) if there is a unique CDEO f (cid:63) , i.e. if eu T , x ( g ) = { f (cid:63) } .Here we identify functions which differ only on a set of zero Lebesgue measure.We state some first results. Proposition 2.2.
1. The set eu T , x ( g ) is convex.2. If f superreplicates g up to time T , we have g ( x ) ≤ v am , g ( ϑ, x ) ≤ v eu , f ( ϑ, x ) (2.2) for all finite ϑ ∈ [ , T ] and all x ∈ R and in particular g ≤ am T ( f ) ≤ f . (2.3) g ≤ am T ( eu T , x ( g )) in the sense that g is dominated by any element of the right-hand side.4. am T ( f )( x ) is decreasing in T .5. am T ( f )( x ) is increasing in f .6. If f is upper semi-continuous, so is x (cid:55)→ am T ( f )( x ) .Proof.
1. Choose f , f ∈ eu T , x ( g ) and note that for any λ ∈ ( , ) the convex combination f λ : = λ f + ( − λ ) f superreplicates g up to T . Moreover, we have v eu , f λ ( T , x ) = λ v eu , f ( T , x ) + ( − λ ) v eu , f ( T , x ) = v eu , f ( T , x ) , which implies that the payoff f λ isindeed contained in eu T , x ( g ) .2. Recall that the discounted value process V ( ϑ ) = ( e − rt v eu , f ( ϑ − t , X t )) t ∈[ ,ϑ ] of the Europeanoption with time- ϑ payoff f ( X ϑ ) is a martingale. Indeed, applying the Markov propertyyields e − rt v eu , f ( ϑ − t , X t ) = e − r ϑ E X t ( f ( X ϑ − t )) = e − r ϑ E x ( f ( X ϑ )| F t ) for any t ∈ [ , ϑ ] .Owing to the superreplication property and the optional sampling theorem, we have v am , g ( ϑ, x ) = sup τ ∈ T [ ,ϑ ] E x ( e − r τ g ( X τ ))≤ sup τ ∈ T [ ,ϑ ] E x (cid:0) e − r τ v eu , f ( ϑ − τ, X τ ) (cid:1) = v eu , f ( ϑ, x ) , which proves (2.2). Minimising both sides of this inequality with respect to ϑ yields(2.3). 6. This follows from the fact that any payoff in eu T , x ( g )) superreplicates g up to time T .4. This is obvious.5. This is obvious as well.6. By dominated convergence, x (cid:55)→ v eu , f ( ϑ, x ) = e − r ϑ ∫ ϕ (cid:0) x + ( r − σ / ) ϑ, σ ϑ, y (cid:1) f ( y ) d y is upper semi-continuous in x for finite ϑ ≤ T . Since the pointwise infimum of a familyof upper semi-continuous functions is upper semi-continuous, the assertion follows. (cid:3) Now we turn to the representability of an American claim as explained in Section 1. To thisend, we fix T ∈ ( , ∞) and assume that the continuation region C T in (1.5) is nonempty. Givenany ( T , x ) ∈ C T we denote by C T , x the connected component of C T = C T ∩ ([ , T ] × R ) which contains ( T , x ) . Definition 2.3.
We say that the European payoff function f represents g relative to ( T , x ) ∈ C T if f superreplicates g up to time T and v am , g ( ϑ, x ) = v eu , f ( ϑ, x ) holds for all ( ϑ, x ) ∈ C T , x . Inthis case we write f T , x −→ g . We call g representable relative to ( T , x ) if there exists some f representing it.If an American payoff is representable, it is in fact represented by its CDEO: Proposition 2.4.
Suppose that the American payoff function g is continuous and satisfies thegrowth condition g ( x ) ≤ C ( + | x | k ) , x ∈ R for some constants C , k < ∞ . Let ( T , x ) ∈ C T . If f T , x −→ g , the following holds.1. For any ( (cid:101) T , (cid:101) x ) ∈ C T , x we have f (cid:101) T , (cid:101) x −→ g .2. The representing function is unique up to a Lebesgue-null set, i.e. (cid:101) f T , x −→ g implies (cid:101) f = f Lebesgue-almost everywhere.3. We have f = eu T , x ( g ) Lebesgue-almost everywhere.4. We have g ( x ) = am T ( f )( x ) and hence g ( x ) = am T ( eu T , x ( g ))( x ) for all x ∈ cl π ( C T , x ) , where π ( C T , x ) : = { x ∈ R : ( ϑ, x ) ∈ C T , x for some ϑ ∈ R + } denotes the projection of the set on its second coordinate.5. The set C T , x is a connected component of the continuation region C (cid:48) T associated to theAmerican value function v am , am T ( f ) . We have f T , x −→ am T ( f ) and therefore f = eu T , x ( am T ( f )) . . Suppose that (cid:101) g ≤ g is an upper semi-continuous American payoff function with (cid:101) g ( x ) = g ( x ) for all x ∈ cl π ( C T , x ) . Then f T , x −→ (cid:101) g and C T , x is a connected component of thecontinuation region (cid:101) C T associated to the American value function v am , (cid:101) g .Proof.
1. This is obvious because C ( (cid:101) T , (cid:101) x ) is a subset of C T , x .2. Assume that f and (cid:101) f represent g relative to T , x . Clearly, we have v eu , f ( ϑ, x ) = v eu , (cid:101) f ( ϑ, x ) = v am , g ( ϑ, x ) < ∞ for any ( ϑ, x ) ∈ C T , x . Lemma A.5 implies that the valuefunctions v eu , f and v eu , (cid:101) f have an analytic extension on some C -domain containing the set ( , T ) × R . The set C T , x contains an open ball B . First, we apply the identity theoremto the variable ϑ which shows that the mappings v eu , f and v eu , (cid:101) f coincide on the openstrip ( , T ) × π ( B ) . Then we apply the identity theorem to the variable x which yields v eu , f ( ϑ, x ) = v eu , (cid:101) f ( ϑ, x ) < ∞ for any ( ϑ, x ) ∈ ( , T ) × R . Consequently, it is easy to seethat the functions u ( y ) : = ϕ (cid:0) x + (cid:98) r ϑ , σ ϑ , y (cid:1) f ( y ) , (cid:101) u ( y ) : = ϕ (cid:0) x + (cid:98) r ϑ , σ ϑ , y (cid:1) (cid:101) f ( y ) , are both integrable on R , where we set ϑ : = T / (cid:98) r : = r − σ /
2. Lemma A.4(2)yields v eu , f ( ϑ / , x / ) = ∫ ϕ (cid:0) x / + (cid:98) r ϑ / , σ ϑ / , y (cid:1) ϕ (cid:0) x + (cid:98) r ϑ , σ ϑ , y (cid:1) u ( y ) d y = √ (cid:18) ( x − x / + (cid:98) r ϑ / ) σ ϑ (cid:19) ∫ exp (cid:18) − ( y − x + x ) σ ϑ (cid:19) u ( y ) d y for any x ∈ R . This equation remains valid after replacing f and u by (cid:101) f and (cid:101) u , respectively.The mappings v eu , f and v eu , (cid:101) f coincide on ( , T ) × R and consequently ∫ ϕ (cid:0) x , σ ϑ , x − y (cid:1) u ( y ) d y = ∫ ϕ (cid:0) x , σ ϑ , x − y (cid:1)(cid:101) u ( y ) d y holds for any x ∈ R . We multiply both sides of this equation by e izx , z ∈ R and integratethe x variable over the real line. After a few simplifications we obtain ∫ e iz y u ( y ) d y = ∫ e iz y (cid:101) u ( y ) d y for any z ∈ R . The injectivity of the Fourier transform on L ( R ) yields that u , (cid:101) u andtherefore f , (cid:101) f coincide up to a Lebesgue-null set.3. The growth condition on g implies (1.3). Moreover, v am , g is continuous on [ , T ] × R by[11, Theorem 4.1.1]. Observe that C CT is closed and τ T : = inf { t ≥ ( T − t , X t ) (cid:60) C T } ∧ T = inf (cid:8) t ≥ v am , g ( T − t , X t ) = g ( X t ) (cid:9) ∧ T (2.4)8s an optimal stopping time for the stopping problem in (1.4), cf. [17, Corollary 2.9].By (2.2) the function f is contained in eu T , x ( g ) . It remains to be shown that this setis a singleton. To this end choose a function h ∈ eu T , x ( g ) and note that v eu , h ( T , x ) = v eu , f ( T , x ) = v am , g ( T , x ) < ∞ . By Lemma A.5 the mappings v eu , h and v eu , f are analyticon a C -domain containing the set ( , T ) × R . Define N : = (cid:8) ( ϑ, x ) ∈ C T , x ∩ (( , T ) × R ) : v eu , h ( ϑ, x ) (cid:44) v eu , f ( ϑ, x ) (cid:9) , which is open because both v eu , h and v eu , f are continuous on C T , x . Moreover, let τ T bethe corresponding optimal stopping time as in (2.4).Assume by contradiction that there is an interior point ( ϑ , ξ ) of N ⊂ C T , x , we have { ϑ } × [ ξ − ε, ξ + ε ] ⊂ N ⊂ C T , x (2.5)and P x ( τ T > t , X t ∈ [ ξ − ε, ξ + ε ]) > ε > t < T − ϑ because C T , x is connected and open in [ , T ]× R . (2.5) implies [ ϑ , T ]×[ ξ − ε, ξ + ε ] ⊂ C T , x . From the properties of Brownian motion it also follows that the probability of X staying in the interval [ ξ − ε, ξ + ε ] from time t till T − ϑ is strictly positive. Hence P x (( T − τ T ) ∨ ϑ , X ( T − ϑ )∧ τ T ) ∈ N ) > . (2.6)On the other hand, we have E x (cid:16) e − r (( T − τ T )∨ ϑ ) (cid:0) v eu , h − v eu , f (cid:1) (cid:0) ( T − τ T ) ∨ ϑ , X ( T − ϑ )∧ τ T (cid:1) (cid:17) = E x (cid:16) e − r (( T − τ T )∨ ϑ ) (cid:0) v eu , h − v am , g (cid:1) (cid:0) ( T − τ T ) ∨ ϑ , X ( T − ϑ )∧ τ T (cid:1) (cid:17) = v eu , h ( T , x ) − v am , g ( T , x ) = . The second equality follows from the fact that the discounted European value process aswell as the optimally stopped Snell envelope of the discounted exercise price process aremartingales, see [17, Theorem 2.4 and Remark 2.6]. Since it is nonnegative, we concludethat (cid:0) v eu , h − v eu , f (cid:1) (cid:0) ( T − τ T ) ∨ ϑ , X ( T − ϑ )∧ τ T (cid:1) = P x -almost surelyin contradiction to (2.6). Hence N is empty.By the proof of the second assertion we conclude that h and f coincide up to a Lebesgue-null set.4. Choose any x ∈ π ( C T , x ) and a ϑ C ∈ ( , T ] such that ( ϑ C , x ) ∈ C T , x . Due to compactness,there is a largest ϑ S ∈ [ , ϑ C ) such that ( ϑ S , x ) is contained in the stopping region. In viewof [11, Theorem 4.1.1], v am , g is continuous and therefore g ( x ) ≤ am T ( f )( x ) ≤ lim inf ϑ ↓ ϑ S v eu , f ( ϑ, x ) = lim inf ϑ ↓ ϑ S v am , g ( ϑ, x ) = v am , g ( ϑ S , x ) = g ( x ) . This proves the assertion for x ∈ π ( C T , x ) . For any x b ∈ ∂π ( C T , x ) there is some ϑ b ∈ ( , T ] such that ( ϑ b , x b ) is in the boundary of the set C T , x . For an approximatingsequence C T , x (cid:51) ( ϑ n , x n ) → ( ϑ b , x b ) as n → ∞ we have g ( x b ) = v am , g ( ϑ b , x b ) = lim inf n →∞ v am , g ( ϑ n , x n ) = lim inf n →∞ v eu , f ( ϑ n , x n ) . n →∞ v eu , f ( ϑ n , x n ) ≥ v eu , f ( ϑ b , x b ) ≥ am T ( f )( x b ) ≥ g ( x b ) , which yields g ( x b ) = am T ( f )( x b ) .5. The European payoff f superreplicates am T ( f ) up to time T . Owing to Proposition2.2(2), we have g ( x ) ≤ am T ( f )( x ) and hence v am , g ( ϑ, x ) ≤ v am , am T ( f ) ( ϑ, x ) ≤ v eu , f ( x ) (2.7)for any ( ϑ, x ) ∈ [ , T ] × R . Moreover, equality in (2.7) holds on the set C T , x becausethe payoff f represents g relative to ( T , x ) . For any ( ϑ, x ) ∈ C T , x the fourth assertionwarrants that g ( x ) = am T ( f )( x ) and therefoream T ( f )( x ) = g ( x ) < v am , g ( ϑ, x ) = v am , am T ( f ) ( ϑ, x ) . This shows that C T , x is a connected subset of C (cid:48) T . For any boundary point ( ϑ, x ) ∈ ∂ C T , x with ϑ > g ( x ) = v am , g ( ϑ, x ) = v eu , f ( ϑ, x ) . In view of (2.7), we obtain v am , am T ( f ) ( ϑ, x ) ≤ v eu , f ( x ) = g ( x ) ≤ am T ( f )( x ) , which shows that ( ϑ, x ) is located in the stopping region of the American payoff am T ( f ) .Summing up, the set C T , x is indeed a connected component of C (cid:48) T and am T ( f ) isrepresented by f relative to ( T , x ) .6. Choose any ( ϑ, x ) ∈ C T , x and let τ ϑ be the optimal stopping time as in (2.4). Due to X τ ϑ ∈ cl π ( C T , x ) we have v am , g ( ϑ, x ) = E x ( e − r τ ϑ g ( X τ ϑ )) = E x ( e − r τ ϑ (cid:101) g ( X τ ϑ )) ≤ v am , (cid:101) g ( ϑ, x ) . The reverse inequality follows immediately from the assumption (cid:101) g ≤ g . Therefore (cid:101) g ( x ) = g ( x ) < v am , g ( ϑ, x ) = v am , (cid:101) g ( ϑ, x ) . This shows that C T , x is a connected subset of (cid:101) C T .Now choose any boundary point ( ϑ, x ) ∈ ∂ C T , x and an approximating sequence ( ϑ n , x n ) n ∈ N in C T , x , i.e. ( ϑ n , x n ) → ( ϑ, x ) as n → ∞ . We have g ( x ) = (cid:101) g ( x ) . Since v am , (cid:101) g ( ϑ n , x n ) = v am , g ( ϑ n , x n ) for any n ∈ N , we conclude v am , (cid:101) g ( ϑ, x ) = lim n →∞ v am , (cid:101) g ( ϑ n , x n ) = lim n →∞ v am , g ( ϑ n , x n ) = g ( x ) = (cid:101) g ( x ) . Consequently ( ϑ, x ) is located in the stopping region of the American payoff (cid:101) g . Summingup, C T , x is a connected component of the set (cid:101) C T and (cid:101) g is represented by f relative to ( T , x ) . (cid:3) A European payoff often – but not always – generates its embedded American option:
Proposition 2.5.
Suppose that f is continuous. Let T ∈ ( , T ] and assume that there exists acontinuous function ˘ ϑ : R → [ , T ] such that the the infimum in the definition of am T ( f ) , cf.(2.1), is reached in ˘ ϑ ( x ) for any x ∈ R . Then we have: . am T ( f ) is continuous.2. f T , x −→ am T ( f ) and hence f = eu T , x ( am T ( f )) for any x ∈ R with ( T , x ) ∈ C T .3. ˘ ϑ ( x ) corresponds to the early exercise curve of am T ( f ) in the sense that τ : = inf (cid:8) t ≥ T − t = ˘ ϑ ( X t ) (cid:9) ∧ T (2.8) is an optimal stopping time for the stopping problem in the definition of v am , am T ( f ) ( T , x ) ,cf. (1.4).4. g : = am T ( f )( x ) satisfies the concavity condition σ g (cid:48)(cid:48) ( x ) + (cid:18) r − σ (cid:19) g (cid:48) ( x ) − r g ( x ) ≤ on the set G : = { x ∈ R : 0 < ˘ ϑ ( x ) < T and g is twice differentiable in x } . Proof. v eu , f is continuous on ( , T ) × R + by Lemma A.5. The integrability condition v eu , f ( T , x ) < ∞ , x ∈ R and dominated convergence yield that continuity actually holds on ( , T ]× R + . Since f is uniformly integrable relative to P X ϑ x for ( ϑ, x ) ∈ [ , T ]×[ x − ε, x + ε ] and since P X ϑ x → δ x weakly for ( ϑ, x ) → ( , x ) , the function v eu , f is in fact continuouson [ , T ] × R . Since ˘ ϑ is continuous, we have that x (cid:55)→ am T ( f )( x ) = v eu , f ( ˘ ϑ ( x ) , x ) iscontinuous as well.2. f superreplicates the payoff am T ( f ) up to T by definition. Since ϑ (cid:55)→ v am , g ( ϑ, x ) is increasing, ( ϑ, x ) ∈ C T , x implies ( (cid:101) ϑ, x ) ∈ C T , x for any (cid:101) ϑ ≥ ϑ . Now am T ( f )( x ) = v eu , am T ( f ) ( ˘ ϑ ( x ) , x ) ≥ v am , am T ( f ) ( ˘ ϑ ( x ) , x ) implies ( ˘ ϑ ( x ) , x ) (cid:60) C T , x and therefore ϑ > ˘ ϑ ( x ) for any ( ϑ, x ) ∈ C T , x . Set M : = (cid:8) ( ϑ, x ) ∈ [ , T ] × R : am T ( f )( x ) = v eu , f ( ϑ, x ) (cid:9) . Since˘ ϑ is continuous, this implies that P x (( ϑ − τ ϑ, M , X τ ϑ, M ) ∈ M ) = ( ϑ, x ) ∈ C T , x andthe stopping time τ ϑ, M : = inf { t ∈ R + : ( ϑ − t , X t ) ∈ M } . Since the discounted Europeanvalue process is a martingale, we obtain v am , am T ( f ) ( ϑ, x ) ≥ E x (cid:0) e − r τ ϑ, M g ( X τ ϑ, M ) (cid:1) = E x (cid:0) e − r τ ϑ, M v eu , f ( ϑ − τ ϑ, M , X τ ϑ, M ) (cid:1) = v eu , f ( ϑ, x ) by optional sampling. The reverse inequality is (2.2) from Proposition 2.2.3. This follows now from v am , am T ( f ) ( T , x ) ≥ E x (cid:0) e − r τ am T ( f )( X τ ) (cid:1) = E x (cid:0) e − r τ v eu , f ( T − τ, X τ ) (cid:1) = v eu , f ( T , x )≥ v am , am T ( f ) ( T , x ) .
11. The mapping Ψ : v eu , f − g is twice differentiable on the set ( , T ) × G . If we define theoperator A : = (cid:16) r − σ (cid:17) D + σ D − r , It ¯o’s formula and the martingale property of ( e − rt v eu , f ( T − t , X t )) t ∈[ , T ] yield that ( A − D ) v eu , f = ( , T ) × R and hence A g = A v eu , f − A Ψ = D v eu , f − (cid:18) r − σ (cid:19) D Ψ − σ D Ψ − r Ψ = c (cid:62) D Ψ − σ D Ψ − r Ψ (2.10)where c : = ( , σ − r ) and g is interpreted as a mapping on ( , T ) × R via g ( t , x ) : = g ( x ) .Now choose any x ∈ G . By definition we have Ψ ( ˘ ϑ ( x ) , x ) =
0. Due to the fact that Ψ onlyassumes nonnegative values, the first order condition D Ψ ( ˘ ϑ ( x ) , x ) = D Ψ ( ˘ ϑ ( x ) , x ) ≥ ( A g )( x ) = − σ D Ψ ( ˘ ϑ ( x ) , x ) ≤ , which concludes the proof. (cid:3) We start with a simple explicit example of a representable American option.
Example 2.6.
Consider the market of Section 1 with interest rate r = σ = √ f ( x ) = e x / + e x / . Since E x ( S α t ) = exp (cid:0) α x + ( α − α ) t (cid:1) for α ∈ R , x >
0, its value function equals v eu , f ( ϑ, x ) = (cid:18) x − ϑ (cid:19) + exp (cid:18) x + ϑ (cid:19) . We conclude that the embedded American option and the associated early exercise curve aregiven by am ∞ ( f )( x ) = e x / (−∞ , ) ( x ) + f ( x ) R + ( x ) , ˘ ϑ ( x ) = argmin ϑ ∈ R + v eu , f ( ϑ, x ) = − x (−∞ , ) ( x ) . More specifically, τ in (2.8) is optimal for the stopping problem (1.4) for g = am ∞ ( f ) and timehorizon T . Indeed, τ is optimal foram T ( f )( x ) = (cid:40) e x / − T / + e x / − T / if x ≤ − T , am ∞ ( f )( x ) otherwise.by Proposition 2.5. Since am ∞ ( f ) ≤ am T ( f ) , it follows easily that it is optimal for am ∞ ( f ) aswell. 12he following example shows that the embedded American option am T ( f ) of f may berepresentable without necessarily being represented by f itself. On top, we observe that theearly exercise curve can have jumps. Example 2.7.
Consider the Black-Scholes market of Section 1 with interest rate r = σ = √
2. The European value function associated to the payoff f : = [ , ] is given by v eu , f ( ϑ, x ) = e − ϑ (cid:18) Φ (cid:18) − x √ ϑ (cid:19) − Φ (cid:18) − x √ ϑ (cid:19) (cid:19) . An elementary calculation yields D v eu , f ( ϑ, x ) = − e − ϑ ( ϑ ) − / (cid:18) ϕ (cid:18) − x √ ϑ (cid:19) ( − x ) + ϕ (cid:18) x √ ϑ (cid:19) x (cid:19) − v eu , f ( ϑ, x ) < ( ϑ, x ) ∈ ( , ∞) × [ , ] . Moreover, we havelim ϑ ↓ v eu , f ( ϑ, x ) = (cid:40) < f ( x ) if x ∈ { , } , f ( x ) otherwise. (2.12)Fix some time horizon T ∈ ( , ∞) . In view of (2.11, 2.12), the embedded American optionis given by g ( x ) : = am T ( f )( x ) = v eu , f ( T , x ) [ , ] ( x ) . The infimum in (2.1) is attained at the unique point˘ ϑ ( x ) = T [ , ] ( x ) , x ∈ R . This shows that neither the embedded American option nor the associated curve x (cid:55)→ ˘ ϑ ( x ) ofunique minima need to be continuous if the underlying European payoff is discontinuous. Thereader may compare this result to the statements of Proposition 2.5 and [7, Remark 4].Let C T denote the continuation region as (1.5). Since g ( x ) = < v am , g ( ϑ, x ) for ( ϑ, x ) ∈( , T ] × ( R \ [ , ]) , it is evident that ( , T ] × ( R \ [ , ]) ⊂ C T . For ( ϑ, x ) ∈ [ , T ] × [ , ] we have g ( x ) ≤ v am , g ( ϑ, x ) ≤ v am , g ( T , x ) ≤ v eu , f ( T , x ) = g ( x ) because the American value function v am , g ( ϑ, x ) is increasing in ϑ . Consequently, the continu-ation region is of the form C T = ( , T ] × ( R \ [ , ]) . At the end of this example we prove thatthe stopping time τ ϑ : = inf { t ≥ X t ∈ [ , ]} ∧ ϑ is optimal for the stopping problem (1.4)with time horizon ϑ ∈ [ , T ] .Beforehand we show that the embedded American payoff g is not represented by its generat-ing European claim f . To this end choose any ( ϑ, x ) ∈ [ , T ] × R + from the continuation region C T . Since D v eu , f ( ϑ, x ) < ( , ∞) × [ , ] and by the optional sampling theoremapplied to the discounted European option price process, we obtain v am , g ( ϑ, x ) = E x (cid:0) g ( X τ ϑ ) e − r τ ϑ (cid:1) = E x (cid:0) [ , ] ( X τ ϑ ) v eu , f ( T , X τ ϑ ) e − r τ ϑ (cid:1) < E x (cid:0) v eu , f ( ϑ − τ ϑ , X τ ϑ ) e − r τ ϑ (cid:1) = v eu , f ( ϑ, x ) ( ϑ, x ) ∈ C with ϑ ≤ T . Therefore the payoff g is indeed not represented by f .Nonetheless, there exist European payoff functions which represent g on the connectedcomponents C T , − = ( , T ] × (−∞ , ) and C T , = ( , T ] × ( , ∞) of the continuation region. Weverify that h ( x ) : = g ( ) cosh ( x ) R + ( x ) represents g on the left connected component C T , − . By symmetry, one can show that the sameholds for (cid:101) h ( x ) : = g ( ) cosh ( x − ) (−∞ , ] ( x ) on the right connected component C T , .The European value function associated to h is given by v eu , h ( ϑ, x ) = g ( ) e − ϑ H ( ϑ, x ) , where H ( ϑ, x ) : = E x ( cosh ( X ϑ ) R + ( X ϑ )) . Since P X ϑ x = N ( x , ϑ ) , a straightforward calculationyields H ( ϑ, x ) = (cid:18) e ϑ − x Φ (cid:18) x √ ϑ − √ ϑ (cid:19) + e ϑ + x Φ (cid:18) x √ ϑ + √ ϑ (cid:19) (cid:19) for ϑ ∈ ( , T ] . In particular, we have v eu , h ( ϑ, ) = g ( ) . (2.13)Let us verify that h superreplicates the American payoff g up to time T . Since g ( x ) = e − T (cid:18) Φ (cid:18) − x √ T (cid:19) − + Φ (cid:18) x √ T (cid:19) (cid:19) ≤ e − T (cid:18) Φ (cid:18) √ T (cid:19) − + Φ (cid:18) √ T (cid:19) (cid:19) = g ( )≤ h ( x ) = v eu , h ( , x ) for any x ∈ [ , ] , it suffices to verify v eu , h ≥ g on the set ( , T ] × [ , ] . In view of D v eu , h ( ϑ, x ) = − g ( ) xe − ϑ ( ϑ ) / ϕ (cid:18) x √ ϑ (cid:19) < ( ϑ, x ) ∈ ( , T ] × [ , ] , we only need to show that v eu , h ( T , x ) − g ( x ) = v eu , h − f ( T , x ) isnonnegative for any x ∈ [ , ] . We have v eu , h − f ( T , x ) = E x (( h − f )( X T )| X T ≥ ) P x ( X T ≥ ) because h − f vanishes on (−∞ , ) . Since h − f is increasing, Lemma A.1 yields v eu , h − f ( T , ) P ( X T ≥ ) ≤ v eu , h − f ( T , x ) P x ( X T ≥ ) for any x ∈ [ , ] . Using (2.13) we conclude0 = v eu , h ( T , ) − v eu , f ( T , ) = v eu , h − f ( T , ) , x ∈ [ , ] and hence 0 ≤ v eu , h − f ( T , x ) as desired. 14igure 1: The EAO (blue) associated to the European put with strike K =
100 (red) in theBlack-Scholes market with T = r = . σ = .
4, and stock price s = e x We already observed that the functions v eu , h and g coincide on the stopping boundaryassociated to C T , − , i.e. v eu , h ( , x ) = = g ( x ) for any x < v eu , h ( ϑ, ) = g ( ) for any ϑ ∈ [ , T ] . Consequently, optional sampling yields v am , g ( ϑ, x ) = E x (cid:0) g ( X τ ϑ ) e − r τ ϑ (cid:1) = E x (cid:0) v eu , h ( ϑ − τ ϑ , X τ ϑ ) e − r τ ϑ (cid:1) = v eu , h ( ϑ, x ) for any ( ϑ, x ) ∈ C T , − . In particular, we observe that τ ϑ is an optimal stopping time for thestopping problem (1.4) with time horizon ϑ . Altogether, this shows that the American payoff g is represented by h on the left connected component C T , − . The embedded American option of the European put has some interesting properties. It isrepresentable, but only for sufficiently small time horizons.
Lemma 2.8. By f ( x ) : = ( K − e x ) + we denote the European put option with some strike price K > . There are positive finite time horizons T < T such that1. am T ( f ) is represented by f relative to ( T , x ) for all x ∈ [ log K , ∞) ,2. If r > , there are no T ∈ R + , x ∈ [ log K , ∞) such that am T ( f ) is representable relativeto ( T , x ) . In particular, neither am T ( f ) nor am T ( f ) are represented by f relative to ( T , log K ) . ϑ ( x ) associated to the European put with strike K =
100 in the Black-Scholes market with T = r = . σ = .
4, and stock price s = e x Proof.
1. Owing to the Black-Scholes formula, the value function of the European put isgiven by v eu , f ( ϑ, x ) = e − r ϑ K Φ (− d ( ϑ, e x )) − s Φ (− d ( ϑ, e x )) (2.14)with d ( ϑ, s ) : = log ( s / K ) + ( r + σ / ) ϑσ √ ϑ , d ( ϑ, s ) : = log ( s / K ) + ( r − σ / ) ϑσ √ ϑ , where s = e x denotes the spot price of the underlying and ϑ ∈ R + the maturity of theoption. We show that for sufficiently small terminal time T , there exists a continuousfunction ˘ ϑ ( x ) : R → [ , T ] with am T ( f )( x ) = v eu , f ( ˘ ϑ ( x ) , x ) for any x ∈ R . Proposition2.5 then warrants that the American payoff am T ( f ) is represented by f relative to any x ∈ R with ( T , x ) ∈ C T . We recall the following well-known partial derivatives of v eu , f : D v eu , f ( ϑ, x ) = e x σ √ ϑ ϕ ( d ( ϑ, e x )) − r K e − r ϑ Φ (− d ( ϑ, e x )) , e − x D v eu , f ( ϑ, x ) = − Φ (− d ( ϑ, e x )) , e − x D v eu , f ( ϑ, x ) = (cid:0) r + σ / (cid:1) ϑ − log ( e x / K ) ϑ / σ ϕ ( d ( ϑ, e x )) . Consequently, for any ( ϑ, x ) ∈ R + × R we have D v eu , f ( ϑ, x ) > e x < K exp (cid:0) ( r + σ / ) ϑ (cid:1) . Moreover, one easily verifies that the followingproperties are satisfied for any T > x ↓−∞ sup ϑ ∈[ , T ] D v eu , f ( ϑ, x ) < , (2.16)lim ϑ ↓ D v eu , f ( ϑ, log K ) = ∞ , (2.17)lim ϑ ↓ D v eu , f ( ϑ, x ) = − r K , x ∈ (−∞ , log K ) . (2.18)By (2.17) there is some constant T max > D v eu , f ( ϑ, log K ) > ϑ ∈( , T max ) . Let T ∈ ( , T max ) . Property (2.16) warrants that lim inf x ↓−∞ D v eu , f ( T , x ) < K T ∈ ( , K ) suchthat D v eu , f ( T , log K T ) = D v eu , f ( T , x ) < x ∈ ( , log K T ) , and D v eu , f ( T , x ) > x ∈ ( log K T , log K ] . Taking (2.18) into account, we conclude that m ( x ) : = min ϑ ∈[ , T ] v eu , f ( ϑ, x ) < v eu , f ( , x ) ∧ v eu , f ( T , x ) , x ∈ ( log K T , log K ) . Put differently, the nonempty compact set M x : = { ϑ ∈ [ , T ] : v eu , f ( ϑ, x ) = m ( x )} is contained in the open interval ( , T ) for any x ∈ ( log K T , log K ) . We write˘ ϑ ( x ) : = max M x for the largest value of the set M x . For any x ∈ ( log K T , log K ) we have D v eu , f ( ˘ ϑ ( x ) , x ) =
0. By decreasing the bound T max we can always achieve that D v eu , f ( ˘ ϑ ( x ) , x ) > x ∈ ( log K T , log K ) . This can be verified by analysing the asymptotic behaviour of thederivative D v eu , f as ϑ →
0. The calculation is elementary but somewhat lengthy andtherefore omitted. Theorem A.3 yields that the mapping x (cid:55)→ ˘ ϑ ( x ) is analytic on someopen complex domain containing the interval ( log K T , log K ) . Moreover, owing to (2.15)we have ˘ ϑ (cid:48) ( x ) = − D v eu , f ( ˘ ϑ ( x ) , x ) D v eu , f ( ˘ ϑ ( x ) , x ) < x ∈ ( log K T , log K ) , which implies that the limits lim x ↓ log K T ˘ ϑ ( x ) and lim x ↑ log K ˘ ϑ ( x ) exist. Note that the mapping [ , T ] (cid:51) ϑ (cid:55)→ v eu , f ( ϑ, x ) attains its unique minimum at ϑ = x ≥ log K .A simple calculation shows that ˘ ϑ ( log K ) : = ϑ continuously to x = log K . Indeed, assuming v : = lim x ↑ log K ˘ ϑ ( x ) > ϑ ( x ) ∈ ( v , T ] for any x ∈ ( log K T , log K ) . The mapping D v eu , f is continuous on ( , ∞) × R . Thus we obtainthe contradiction 0 < D v eu , f ( v , log K ) = lim x ↑ log K D v eu , f ( ˘ ϑ ( x ) , x ) = T max further, we can achieve that ˘ ϑ ( log K T ) : = T extends the curvecontinuously into x = log K T .A similar argument as above shows that for any x ∈ ( , log K T ) the minimum of themapping [ , T ] (cid:51) ϑ (cid:55)→ v eu , f ( ϑ, x ) is attained at ϑ = T . Indeed, by (2.18) no minimum canbe located at ϑ =
0. Now assume that for some x ∈ ( , log K T ) a minimum is attained at17ome maturity (cid:37) ∈ ( , T ) . Denoting by ˘ ϑ − the inverse function of ˘ ϑ | ( log K T , log K ) , property(2.15) yields 0 = D v eu , f ( (cid:37), x ) < D v eu , f ( (cid:37), ˘ ϑ − ( (cid:37) )) = ϑ : R → [ , T ] .2. With regard to the Black-Scholes formula (2.14), it is apparent that lim ϑ →∞ v eu , f ( ϑ, x ) = x ∈ R . For fixed x < log K choose T large enough with v eu , f ( T , x ) < am T ( f )( x ) . Let T ∈ R + be arbitrary and T ≤ T as in the first assertion.Assume by contradiction that am T ( f ) is represented by some (cid:101) f relative to ( T , x ) withsome x ≥ log K .For sufficiently large x < log K we have am T ( f ) = am T ( f ) on [ x , ∞) . Indeed,am T ( f )( x ) → x → x (cid:55)→ inf ϑ ∈[ T , T ] v eu , f ( ϑ, x ) has a positive lower boundon the compact interval [ log K − , log K ] because v eu , f is continuous and strictly positiveon [ T , T ] × [ log K − , log K ] .After possibly decreasing T we can apply Proposition 2.4(6) and obtain that f representsam T ( f ) relative to ( T , x ) . Proposition 2.4(1,2) yields that the mappings f and (cid:101) f coincideup to a Lebesgue-null set. Hence we obtain the contradictionam T ( f )( x ) ≤ v eu , (cid:101) f ( T , x ) = v eu , f ( T , x ) < am T ( f )( x ) . (cid:3) The embedded American option of the European put and the curve ˘ ϑ in the proof of theprevious lemma are illustrated in Figures 1, 2. The aim of this section is to establish the existence of cheapest dominating European optionsand, more importantly, to verify that that a given American option is represented by its CDEO.For ease of exposition we focus on payoffs of a particular form.
We consider the basic model of (1.1) with initial logarithmic stock price X = x and fixedtime horizon T . We are primarily interested in the American put but for the theorems belowit satisfies to assume a certain more general structure. Specifically, we consider payoffs of theform g ( x ) = (−∞ , K ] ( x ) ϕ ( x ) , x ∈ R (3.1)with K ∈ R and an analytic function ϕ : U → C on some domain U ⊂ C such that1. ϕ ( x ) ∈ ( , ∞) for x ∈ (−∞ , K ) ,2. ϕ ( K ) = R (cid:51) x →−∞ e ( r / σ ) x ϕ ( x ) = c ( x ) : = g (cid:48)(cid:48) ( x ) − r σ ( g ( x ) − g (cid:48) ( x )) − g (cid:48) ( x ) ≤ , x ∈ (−∞ , K ) (3.3)from (2.9) in Proposition 2.5 holds.These assumptions are satisfied and in fact motivated by the payoff g ( x ) = (cid:0) e K − e x (cid:1) + , whichcorresponds to the American put.Our first goal is to show that the cheapest dominating European option of g relative to ( T , x ) exists in a suitably generalised sense. If f : R → R + denotes a European payoff function, wehave v eu , f ( ϑ, x ) = e − r ϑ ∫ ϕ (cid:0) x + (cid:98) r ϑ, σ ϑ, y (cid:1) f ( y ) d y , (3.4)where (cid:98) r : = r − σ . Put differently, we obtain v eu , f ( ϑ, x ) = e − r ϑ ∫ ϕ (cid:0) x + (cid:98) r ϑ, σ ϑ, y (cid:1) ϕ (cid:0) x + (cid:98) rT , σ T , y (cid:1) µ ( d y ) (3.5)for the measure µ on R with density f relative to N ( x + (cid:98) rT , σ T ) .In the European valuation problem, the payoff function f is only needed for defining thepricing function v eu , f . In view of (3.5) we can and do therefore extend the notion of a payoff“function” to include all µ ∈ M + ( R ) , where M + ( R ) denotes the set of measures on R . In linewith (3.5), we define the pricing operator v eu ,µ : R + × R → [ , ∞] by v eu ,µ ( ϑ, x ) : = e − r ϑ ∫ ϕ (cid:0) x + (cid:98) r ϑ, σ ϑ, y (cid:1) ϕ (cid:0) x + (cid:98) rT , σ T , y (cid:1) µ ( d y ) , ( ϑ, x ) ∈ ( , ∞) × R (3.6)and v eu ,µ ( , x ) : = lim inf ( ϑ, y )→( , x ) v eu ,µ ( ϑ, y ) , x ∈ R . (3.7)In terms of our generalised domain, the linear problem (1.7) now reads asminimise v eu ,µ ( T , x ) subject to µ ∈ M + ( R ) , v eu ,µ ( ϑ, x ) ≥ g ( x ) for any ( ϑ, x ) ∈ [ , T ] × R . (3.8)In line with Definition 2.1(3), a minimiser µ ∗ is called cheapest dominating European option(CDEO) of g relative to ( T , x ) . Our first main result establishes its existence. Theorem 3.1.
The optimal value of programme (3.8) is obtained by some µ ∗ ∈ M + (−∞ , K ] ,i.e. some measure on R which is concentrated on (−∞ , K ] . In particular, a CDEO of g relativeto ( T , x ) exists in the present generalised sense. The proof is to be found in Section 3.2. We now turn to the question whether the CDEOactually generates the American claim g under consideration. Theorem 3.2.
Let µ ∗ denote an optimal measure from Theorem 3.1. Suppose that the followingassumptions are satisfied for some constant δ > : . There exists some x ∈ R such that v eu ,µ ∗ ( T + δ, x ) < ∞ .2. For any ϑ ∈ ( , T + δ ) the function x (cid:55)→ v eu ,µ ∗ ( ϑ, x ) − g ( x ) assumes its unique mini-mum within the interval (−∞ , K ] at some point ˘ x ( ϑ ) ∈ (−∞ , K ) . Moreover, we have lim inf ϑ → ˘ x ( ϑ ) = K = : x ( ) .3. The well-defined quantity H ( ϑ, x ) : = σ D v eu ,µ ∗ ( ϑ, x ) + r σ ( v eu ,µ ∗ ( ϑ, x ) − g ( x )) − c ( x ) (3.9) is strictly positive on the set {( ϑ, ˘ x ( ϑ )) : ϑ ∈ ( , T ]} .4. We have lim inf ϑ → v eu ,µ ∗ ( ϑ, K ) < ∞ .Define C T , x as in Section 2.1 and set (cid:101) C T , x : = {( ϑ, x ) ∈ ( , T ] × R : ˘ x ( ϑ ) < x } . (3.10) Then the following statements hold.1. The function ϑ (cid:55)→ ˘ x ( ϑ ) is strictly increasing and it can be extended to some analyticfunction on a complex domain containing ( , T ] .2. We have v eu ,µ ∗ ( ϑ, ˘ x ( ϑ )) = g ( ˘ x ( ϑ )) for any ϑ ∈ [ , T ] .3. The CDEO µ ∗ is the unique measure that represents g relative to ( T , x ) in the sense that v am , g ( ϑ, x ) ≤ v eu ,µ ∗ ( ϑ, x ) for any ( ϑ, x ) ∈ [ , T ] × R and equality holds on C T , x .4. The payoff g coincides on cl π ( (cid:101) C T , x ) = [ min ϑ ∈( , T ] ˘ x ( ϑ ) , ∞) with the embedded Americanoption of µ ∗ up to T in the sense that g ( x ) = inf ϑ ∈[ , T ] v eu ,µ ∗ ( ϑ, x ) = : am T ( µ ∗ )( x ) , x ∈ π ( (cid:101) C T , x ) . (cid:101) C T , x = C T , x , which can be interpreted in the sense that ˘ x parametrises the optimalstopping boundary.6. The stopping time τ ϑ : = inf { t ∈ [ , ϑ ] : X t ≤ ˘ x ( ϑ − t )} ∧ ϑ (3.11) is optimal in (1.4) , i.e. v am , g ( ϑ, x ) = E x ( e − r τ ϑ g ( X τ ϑ )) holds for any ( ϑ, x ) ∈ [ , T ] × R . The proof of this theorem is to be found in Section 3.3.What are the strengths and weaknesses of this result? The assumptions above concern certainqualitative properties of the cheapest dominating European option. On the negative side, thismeans that Theorem 3.2 does not warrant representability unless one can prove that theseproperties hold for the CDEO of the specific claim under consideration. This is complicated bythe fact that this CDEO is typically not known explicitly. However, numerical approximationsare obtained quite easily as it is explained in [13, Chapter 3]. While such approximations cannottell whether the CDEO represents the American claim or just provides a relatively close upperbound, they provide evidence whether the qualitative properties needed for Theorem 3.2 holdtrue. As an illustration, we study the prime example of the American put in Section 4.20 .2 Proof of Theorem 3.1
First we verify that in programme (3.8) it suffices to consider measures µ ∈ M + (−∞ , K ] . Tothis end we define by M + ( R ) (cid:51) µ (cid:55)→ s ( µ ) : = ν + ν d ν : = (−∞ , K ] d µ d ν : = µ (( K , ∞)) d δ K the mapping which relocates any mass in ( K , ∞) to K . Here δ K denotes the Dirac measure at K . One easily verifies that s maps onto the cone M + (−∞ , K ] and preserves the total variation,i.e. (cid:107) s ( µ )(cid:107) = (cid:107) µ (cid:107) . Let µ ∈ M + ( R ) be admissible in programme (3.8). We have v eu ,µ ( T , x ) = e − rT (cid:107) µ (cid:107) = e − rT (cid:107) s ( µ )(cid:107) = v eu , s ( µ ) ( T , x ) . By Lemma A.4(2) there is some c ( ϑ, x ) > ϕ (cid:0) x + (cid:98) r ϑ, σ ϑ, y (cid:1) ϕ (cid:0) x + (cid:98) rT , σ T , y (cid:1) = c ( ϑ, x ) exp (cid:18) − ( y − A ( ϑ, x )) B ( ϑ ) (cid:19) for any ( ϑ, x ) ∈ ( , T ) × (−∞ , K ) , where A ( ϑ, x ) : = x + ( x − x ) T /( T − ϑ ) and B ( ϑ ) : = σ T ϑ /( T − ϑ ) . Recalling that x < K < x , we obtain A ( ϑ, x ) = x + ( x − x ) T /( T − ϑ ) < K and therefore v eu , s ( µ ) ( ϑ, x ) = v eu ,ν ( ϑ, x ) + µ (( K , ∞)) v eu ,δ K ( ϑ, x ) = v eu ,ν ( ϑ, x ) + e − r ϑ ∫ ( K , ∞) c ( ϑ, x ) exp (cid:18) − ( K − A ( ϑ, x )) B ( ϑ ) (cid:19) µ ( d y )≥ v eu ,ν ( ϑ, x ) + e − r ϑ ∫ ( K , ∞) c ( ϑ, x ) exp (cid:18) − ( y − A ( ϑ, x )) B ( ϑ ) (cid:19) µ ( d y ) = v eu ,ν ( ϑ, x ) + v eu ,µ − ν ( ϑ, x ) = v eu ,µ ( ϑ, x ) ≥ g ( x ) for any ( ϑ, x ) ∈ ( , T ) × (−∞ , K ) . Along the same lines we can apply Lemma A.4(3) in order toobtain v eu , s ( µ ) ( T , x ) = v eu ,ν ( T , x ) + µ (( K , ∞)) v eu ,δ K ( T , x )≥ v eu ,ν ( T , x ) + v eu ,µ − ν ( T , x ) = v eu ,µ ( T , x ) ≥ g ( x ) for any x < K . Summing up, the calculations above imply that the inequality v eu , s ( µ ) ≥ g holds on the set ( , T ] × (−∞ , K ) . Since g is assumed to vanish on [ K , ∞) , the measure s ( µ ) is admissible in programme (3.8). Hence, it suffices to consider measures µ ∈ M + (−∞ , K ] in(3.8). 21 .2.1 Transformation to r = (cid:101) B t = , d (cid:101) X t = (cid:101) r dt + σ dW t (3.12)with (cid:101) r : = − r − σ / < (cid:101) g ( x ) : = e ( r / σ ) x g ( x ) , (3.13)where W denotes a Wiener process and (cid:101) X = x holds P x -almost surely. The growth condition(3.2) warrants that (cid:101) g is a continuous function vanishing at infinity. Invoking a measure changewith density process ( exp (−( r / σ )( X t − X ) − rt )) t ∈[ , T ] , it is easy to see that E x ( e − r τ g ( X τ )) = e −( r / σ ) x E x (cid:0) e ( r / σ ) (cid:101) X τ g ( (cid:101) X τ ) (cid:1) = e −( r / σ ) x E x (cid:0)(cid:101) g ( (cid:101) X τ ) (cid:1) for any stopping time τ ≤ T . Likewise, we have E x ( e − r ϑ f ( X ϑ )) = e −( r / σ ) x E x ( (cid:101) f ( (cid:101) X ϑ )) for anyEuropean payoff function f : R → R + and any ϑ ∈ [ , T ] where (cid:101) f ( x ) : = e ( r / σ ) x f ( x ) . Somesimple algebraic manipulations yield that v eu ,µ ( ϑ, x ) = e − r ϑ ∫ ϕ (cid:0) x + (cid:98) r ϑ, σ ϑ, y (cid:1) ϕ (cid:0) x + (cid:98) rT , σ T , y (cid:1) µ ( d y ) (3.14) = e (− r / σ ) x ∫ ϕ (cid:0) x + (cid:101) r ϑ, σ ϑ, y (cid:1) ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) e ( r / σ ) x − rT µ ( d y ) . Consequently, the linear programme (3.8) is, up to renormalising the target functional, equivalentto minimise (cid:107) µ (cid:107) subject to µ ∈ M + (−∞ , K ] , (cid:101) v eu ,µ ( ϑ, x ) ≥ (cid:101) g ( x ) for any ( ϑ, x ) ∈ [ , T ] × (−∞ , K ) , (3.15)where (cid:101) v eu ,µ ( ϑ, x ) : = ∫ ϕ (cid:0) x + (cid:101) r ϑ, σ ϑ, y (cid:1) ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) µ ( d y ) , ( ϑ, x ) ∈ ( , ∞) × R , (cid:101) v eu ,µ ( , x ) : = lim inf ( ϑ, y )→( , x )( ϑ, y )∈( , ∞)× R (cid:101) v eu ,µ ( ϑ, x ) , x ∈ R . More specifically, any admissible measure µ in (3.8) corresponds to the admissible measure e ( r / σ ) x − rT µ for the programme (3.15). Note that (cid:101) v eu ,µ ( T , x ) = (cid:107) µ (cid:107) < ∞ for any µ ∈ M + ( R ) . We define the set Ω : = ( , T ) × (−∞ , K ] and linear operators T : M (−∞ , K ] → C ( Ω ) T µ ( t , x ) : = ∫ κ ( t , x , y ) µ ( d y ) , T (cid:48) : M ( Ω ) → B ((−∞ , K ] , R ) T (cid:48) λ ( y ) : = ∫ κ ( t , x , y ) λ ( d ( t , x )) κ ( t , x , y ) : = ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y (cid:1) ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) , = (cid:114) TT − t exp (cid:18) − ( y − A ( x , t )) B ( t ) (cid:19) exp (cid:18) ( x − x − (cid:101) rt ) σ t (cid:19) = Tt ϕ ( A ( x , t ) , B ( t ) , y ) ϕ ( x + (cid:101) rt , σ t , x ) , (3.16)where B ((−∞ , K ] , R ) denotes the set of measurable functions from (−∞ , K ] to R . and A ( t , x ) : = x + ( x − x ) T / t , B ( t ) : = σ T ( T − t )/ t , cf. Lemma A.4(2). Taking the specific structure of theintegral kernel κ into account, we can show that for any measure µ ∈ M (−∞ , K ] the mapping Ω (cid:51) ( t , x ) (cid:55)→ T µ ( t , x ) is analytic on the open C -domain G : = (cid:110) ϑ ∈ C : (cid:112) ( Re ϑ − T / ) + ( Im ϑ ) < T / (cid:111) × C . This is a special case of step 1 from Section 3.3 below, where a proof can be found. In particular,the range of the operator T is indeed contained in C ( Ω ) . Lemma 3.3. If T µ = on some open subset of Ω then µ = . In particular, the operator T isinjective.Proof. Let µ ∈ M (−∞ , K ] be a measure such that T µ vanishes on some open subset of Ω .Denote by µ = µ + − µ − the Hahn-Jordan decomposition of µ . By analyticity and the identitytheorem we conclude that T µ = ( , T )× R . Since A ( x + x / , T / ) = x and B ( T / ) = σ T ,we obtain from (3.16) that ϕ (cid:18) x + (cid:101) rT , σ T , x (cid:19) T µ ± (cid:18) T , x + x (cid:19) = ∫ ϕ (cid:0) x , σ T , y (cid:1) d µ ± ( y ) . T µ + = T µ − implies ∫ ϕ (cid:0) y , σ T , x (cid:1) d µ + ( y ) = ∫ ϕ (cid:0) y , σ T , x (cid:1) d µ − ( y ) , x ∈ R . Multiplying both sides with e izx and integrating in x yields ∫ exp (cid:18) i y z − σ T z (cid:19) d µ + ( y ) = ∫ exp (cid:18) i y z − σ T z (cid:19) d µ − ( y ) for all z ∈ R . Since the Fourier transform is injective, we conclude that the orthogonal measures µ − and µ + coincide. This implies µ = (cid:3) After these preliminary remarks we return to our optimisation problem. The convex pro-gramme (3.15) can be rephrased in functional analytic terms asminimise (cid:107) µ (cid:107) subject to T µ − (cid:101) g ∈ C + ( Ω ) ,µ ∈ M + (−∞ , K ] . ( P )23he requirement that the European value function dominates the payoff is expressed by theconic constraint. To this primal minimisation problem we associate the Lagrange dualmaximise (cid:104) (cid:101) g , λ (cid:105) subject to T (cid:48) λ ( y ) ≤ ∀ y ∈ (−∞ , K ] ,λ ∈ M + ( Ω ) , ( D )where (cid:104) (cid:101) g , λ (cid:105) : = ∫ Ω (cid:101) g ( x ) λ ( d ( t , x )) . This dual problem allows for a probabilistic or physical interpretation. To this end, supposethat particles move in space-time Ω ⊂ R + × R , where the first coordinate of ( t , x ) stands fortime and the second for the location at this time. In the space coordinate x the particles areassumed to follow a Brownian motion with drift rate (cid:101) r and diffusion coefficient σ . Let us injectparticles of total mass λ ( Ω ) into Ω , distributed according to λ , i.e. mass λ ( A ) is assigned to anymeasurable subset A of Ω . Where in R are the particles to be found at the final time T ? Sincethey follow Brownian motion, they are distributed according to the Lebesgue density y (cid:55)→ ∫ ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y (cid:1) λ ( d ( t , x )) . On the other hand, the constraint ∫ κ ( t , x , y ) λ ( d ( t , x )) ≤ ∫ ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y (cid:1) λ ( d ( t , x )) ≤ ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) . (3.17)The right-hand side is the probability density function at time T of a Brownian motion started in x at time 0. Put differently, the constraint (3.17) means that we consider only laws λ on space-time Ω such that the resulting final distribution on R is dominated by the Gaussian law stemmingfrom a Brownian motion started in x at time 0. If equality holds in (3.17), the distribution ofparticles at time T is the same as for a Brownian motion with drift rate (cid:101) r , diffusion coefficient σ , and starting in x at time 0.Regarding the primal problem P and its formal dual D , we may wonder whether weakor even strong duality holds, if optimisers exist and if they are linked by some complementaryslackness condition. The following first main result shows that this is indeed the case, at leastif the CDEO payoff strictly dominates the American payoff function (cid:101) g at all x < K . Lemma 3.4.
1. The optimal value of P is obtained by some µ ∈ M + (−∞ , K ] and itcoincides with the optimal value of D . The measure µ puts mass on every open subsetof (−∞ , K ) .2. If (cid:101) v eu ,µ ( , x ) > (cid:101) g ( x ) for any x ∈ (−∞ , K ) , the optimal value of D is obtained by somemeasure λ ∈ M + ( Ω ) . In this case the following complementary slackness conditions aresatisfied: T µ ( ϑ, x ) = (cid:101) g ( x ) λ -a.e. on Ω , (3.18) T (cid:48) λ ( x ) = µ -a.e. on (−∞ , K ] . (3.19)In view of the discussion from Subsection 3.2.1 this theorem can be easily restated in termsof the quantities v eu ,µ and g associated to the programme (3.8). We immediately obtain Theorem3.1 from the first assertion of Lemma 3.4. 24 .2.3 Proof of Lemma 3.4 For any ε ∈ ( , T ) we define the set Ω ε : = [ ε, T − ε ] × [− / ε, K ] and the following linear operator: T ∗ : M ( Ω ε ) → C (−∞ , K ] T ∗ λ ( y ) : = ∫ κ ( t , x , y ) λ ( d ( t , x )) The range of the operator T ∗ is contained in C (−∞ , K ] due to Lebesgue’s dominated con-vergence theorem, by (3.16) and the compactness of the set Ω ε . On the Cartesian products C ( Ω ε ) × M ( Ω ε ) and C (−∞ , K ] × M (−∞ , K ] we consider the algebraic pairing (cid:104) f , ν (cid:105) (cid:55)→ ∫ f d ν. (3.20)This mapping is finitely valued, bilinear and separates points. We endow C ( Ω ε ) , M ( Ω ε ) and M (−∞ , K ] with the weak topologies σ ( C , M ) , σ ( M , C ) and σ ( M , C ) induced by (3.20). Thefunction space C (−∞ , K ] is endowed with the topology of uniform convergence T uc . Thisturns all four spaces into locally convex Hausdorff spaces. Moreover, each space of measuresis the continuous dual of the associated function space and vice versa, cf. [20, Theorem 6.19].Fubini’s theorem yields that for all measures µ ∈ M (−∞ , K ] and λ ∈ M ( Ω ε ) we have (cid:104) T µ, λ (cid:105) = (cid:104) µ, T ∗ λ (cid:105) . (3.21)By [13, Lemma 5.17] we find that the operator T is σ ( M , C ) - σ ( C , M ) continuous and T ∗ is σ ( M , C ) - σ ( C , M ) continuous.We want to find a measure µ ∈ M + (−∞ , K ] which solves the linear programme P fromSubsection 3.2.2. Our strategy is to approximate this optimisation problem by the followingsequence of linear programmes with milder constraintsminimise (cid:107) µ (cid:107) subject to ( T µ − (cid:101) g ) | Ω ε ∈ C + ( Ω ε ) ,µ ∈ M + (−∞ , K ] . ( P ε )The solution to P will be obtained by compactness from the family of P ε -extremal elements.For each ε ∈ ( , T / ) , the Lagrange dual problem of P ε is given bymaximise (cid:104) (cid:101) g , λ (cid:105) subject to 1 − T ∗ λ ∈ C + (−∞ , K ] ,λ ∈ M + ( Ω ε ) . ( D ε )The optimal values of P ε and D ε are denoted by p ε and d ε , respectively. By construction wehave that weak duality ≤ d ε ≤ p ε holds. Indeed, in view of the adjointness relation (3.21) weobtain 0 ≤ (cid:104) (cid:101) g , λ (cid:105) ≤ (cid:104) T µ, λ (cid:105) = (cid:104) µ, T ∗ λ (cid:105) ≤ (cid:104) µ, (cid:105) = (cid:107) µ (cid:107) (3.22)25or any primal admissible µ ∈ M + (−∞ , K ] and any dual admissible λ ∈ M + ( Ω ε ) . Next, weverify primal and dual attainment . The nonnegative measure (cid:101) µ with Lebesgue density y (cid:55)→ (cid:107) (cid:101) g (cid:107) ∞ ϕ (cid:16) x + (cid:101) rT , σ T , y (cid:17) (−∞ , K ) ( y ) is P ε -admissible because for any ( t , x ) ∈ Ω ε we have T (cid:101) µ ( t , x ) = (cid:107) (cid:101) g (cid:107) ∞ ∫ K −∞ ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y (cid:1) d y = (cid:107) (cid:101) g (cid:107) ∞ Φ (cid:32) K − x σ √ T − t − (cid:101) r √ T − t σ (cid:33) ≥ (cid:107) (cid:101) g (cid:107) ∞ Φ ( ) = (cid:107) (cid:101) g (cid:107) ∞ . (3.23)The total mass of the measure (cid:101) µ is bounded by the constant 2 (cid:107) (cid:101) g (cid:107) ∞ . Therefore solving theminimisation problem P ε is equivalent to minimising the total variation norm over the σ ( M , C ) -compact set C ε p : = T − (cid:0)(cid:101) g + C + ( Ω ε ) (cid:1) ∩ M + (−∞ , K ] ∩ B M ( R ) ( , (cid:107) (cid:101) g (cid:107) ∞ ) . (3.24)The σ ( M , C ) -compactness of C ε p is established as follows. First we note that the set (cid:101) g + C + ( Ω ε ) is homeomorphic to the σ ( C , M ) -closed cone C + ( Ω ε ) = (cid:92) λ ∈ M + ( Ω ε ) { f ∈ C ( Ω ε ) : (cid:104) f , λ (cid:105) ≥ } and that the continuity properties of the operator T warrant the σ ( M , C ) -closedness of thepreimage T − ( (cid:101) g + C + ( Ω ε )) . Secondly, we observe that the cone M + (−∞ , K ] = (cid:92) f ∈ C + (−∞ , K ] { µ ∈ M (−∞ , K ] : (cid:104) f , µ (cid:105) ≥ } is σ ( M , C ) -closed as well and that B M ( R ) ( , (cid:107) (cid:101) g (cid:107) ∞ ) is a σ ( M , C ) -compact set due to Alaoglu’stheorem, cf. [15, Theorem 23.5]. The target functional µ (cid:55)→ (cid:107) µ (cid:107) is lower semi-continuouswith respect to the topology σ ( M , C ) and therefore its minimal value p ε is attained by somemeasure µ ε ∈ C ε p .Next, we prove the attainment of the D ε -optimal value. For any measure λ ∈ M ( Ω ε ) and y ∈ (−∞ , K ] we define U λ ( y ) : = ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) T ∗ λ ( y ) . Obviously U is a σ ( M , C ) - σ ( C , M ) -continuous, linear operator from M ( Ω ε ) into the space C (−∞ , K ] . The inequalityconstraint of the programme D ε is equivalent to U λ ( y ) ≤ ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) for all y ∈ (−∞ , K ] .Integrating this inequality over the interval (−∞ , K ] yields ∫ ∫ K −∞ ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y (cid:1) d y λ ( d ( t , x )) ≤ ∫ K −∞ ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) d y ≤ . A calculation similar to (3.23) yields ∫ K −∞ ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y (cid:1) d y ≥ Φ ( ) = ( t , x ) ∈ Ω ε and consequently any D ε -admissible measure λ satisfies (cid:107) λ (cid:107) ≤
2. Solvingthe maximisation problem D ε is therefore equivalent to maximising the σ ( M , C ) -continuousmapping λ (cid:55)→ (cid:104) (cid:101) g , λ (cid:105) over the set C ε d : = U − (cid:16) ϕ (cid:0) x + (cid:101) rT , σ T , · (cid:1) − C + (−∞ , K ] (cid:17) ∩ M + ( Ω ε ) ∩ B M ( Ω ε ) ( , ) . One easily modifies the arguments following (3.24) in order to verify that C ε d is a σ ( M , C ) -compact subset of M ( Ω ε ) . Hence the target functional of the Lagrange dual D ε attains itsmaximal value d ε at some measure λ ε ∈ C ε d .In order to prove strong duality d ε = p ε , we use some well-established techniques fromconvex optimisation. We refer the reader to [18] for a well-written introduction to conjugateduality and optimisation on paired spaces. A short summary for our needs can be found in [13,Section 5.4]. The Lagrange function K : M (−∞ , K ] × M ( Ω ε ) → [−∞ , ∞] associated to the P ε - D ε -duality is defined by K ( µ, λ ) : = (cid:107) µ (cid:107) + (cid:104) (cid:101) g , λ (cid:105) − (cid:104) T µ, λ (cid:105) + I M + (−∞ , K ] ( µ ) − I M + ( Ω ε ) ( λ ) , (3.25)where I M ( x ) : = (cid:40) x ∈ M , ∞ if x (cid:60) M for any set M . For later reference we provide the following explicit calculations:sup λ ∈ M ( Ω ε ) inf µ ∈ M (−∞ , K ] K ( µ, λ ) = sup λ ∈ M + ( Ω ε ) inf µ ∈ M + (−∞ , K ] ((cid:107) µ (cid:107) + (cid:104) (cid:101) g − T µ, λ (cid:105)) (3.26) = sup λ ∈ M + ( Ω ε ) (cid:18) (cid:104) (cid:101) g , λ (cid:105) + inf µ ∈ M + (−∞ , K ] (cid:104) − T ∗ λ, µ (cid:105) (cid:19) = sup λ ∈ M + ( Ω ε ) T ∗ λ ≤ (cid:104) (cid:101) g , λ (cid:105) = d ε , (3.27)inf µ ∈ M (−∞ , K ] sup λ ∈ M ( Ω ε ) K ( µ, λ ) = inf µ ∈ M + (−∞ , K ] sup λ ∈ M + ( Ω ε ) ((cid:107) µ (cid:107) + (cid:104) (cid:101) g − T µ, λ (cid:105)) (3.28) = inf µ ∈ M + (−∞ , K ] (cid:32) (cid:107) µ (cid:107) + sup λ ∈ M + ( Ω ε ) (cid:104) (cid:101) g − T µ, λ (cid:105) (cid:33) = inf µ ∈ M + (−∞ , K ] T µ ≥ (cid:101) g (cid:107) µ (cid:107) = p ε . One easily verifies that the mapping M (−∞ , K ] (cid:51) µ (cid:55)→ K λ ( µ ) : = K ( µ, λ ) is closed in the senseof [18, Section 3] and convex for any λ ∈ M ( Ω ε ) . Lemma 3.5.
The dual value function v : C (−∞ , K ] (cid:55)→ (−∞ , ∞] , v ( f ) : = inf λ ∈ M ( Ω ε ) K ∗ λ ( f ) is convex and we have v ( ) = − d ε ≥ v ∗∗ ( ) = − p ε . Here K ∗ λ denotes the convex conjugate ofthe mapping K λ . roof. By Fenchel’s inequality and (3.26) we have v ∗∗ ( ) ≤ v ( ) = inf λ ∈ M ( Ω ε ) K ∗ λ ( ) = − sup λ ∈ M ( Ω ε ) inf µ ∈ M (−∞ , K ] K ( µ, λ ) = − d ε . The conjugate v ∗ : M (−∞ , K ] (cid:55)→ [−∞ , ∞] of the function v is given by v ∗ ( µ ) : = sup f ∈ C (−∞ , K ] ((cid:104) f , µ (cid:105) − v ( f )) = sup λ ∈ M ( Ω ε ) sup f ∈ C (−∞ , K ] (cid:0) (cid:104) f , µ (cid:105) − K ∗ λ ( f ) (cid:1) = sup λ ∈ M ( Ω ε ) K ∗∗ λ ( µ ) = sup λ ∈ M ( Ω ε ) K ( µ, λ ) . The last equality follows from the Fenchel-Moreau theorem because the mapping K λ is closedand convex, cf. [18, Theorem 5]. Hence the biconjugate of the dual value function is given by v ∗∗ ( f ) : = sup µ ∈ M (−∞ , K ] ((cid:104) f , µ (cid:105) − v ∗ ( µ )) = sup µ ∈ M (−∞ , K ] inf λ ∈ M ( Ω ε ) ((cid:104) f , µ (cid:105) − K ( µ, λ )) . (3.29)Owing to (3.28) we obtain v ∗∗ ( ) = − p ε .Next, we show that the mapping v does not assume the value −∞ . Suppose to the contrarythat there exists some f ∈ C (−∞ , K ] with v ( f ) = −∞ . Fenchel’s inequality yields v ∗∗ ≤ v andhence v ∗∗ ( f ) = −∞ . Equation (3.29) now implies that sup λ ∈ M ( Ω ε ) K ( µ, λ ) = ∞ for any measure µ ∈ M (−∞ , K ] and therefore p ε = ∞ . This is impossible because the set of P ε -admissiblemeasures has already been shown to be nonempty.In order to verify that v is convex, suppose that α ∈ ( , ) and f , f ∈ C (−∞ , K ] . From(3.25) it is apparent that the Lagrange function K is concave in the second component and thisyields v ( α f + ( − α ) f ) = inf λ ∈ M ( Ω ε ) sup µ ∈ M (−∞ , K ] (cid:0) (cid:104) α f + ( − α ) f , µ (cid:105) − K ( µ, λ ) (cid:1) ≤ sup µ ∈ M (−∞ , K ] (cid:0) (cid:104) α f + ( − α ) f , µ (cid:105) − K ( µ, αλ + ( − α ) λ ) (cid:1) ≤ α sup µ ∈ M (−∞ , K ] (cid:0) (cid:104) f , µ (cid:105) − K ( µ, λ ) (cid:1) + ( − α ) sup µ ∈ M (−∞ , K ] (cid:0) (cid:104) f , µ (cid:105) − K ( µ, λ ) (cid:1) (3.30)for any choice of λ , λ ∈ M ( Ω ε ) . Minimising with respect to λ , λ proves that v is indeed aconvex function. (cid:3) Hence strong duality holds if we can show that v ∗∗ ( ) = v ( ) is true. By virtue of Lemma3.5 and the Fenchel-Moreau biconjugate theorem, cf. [18, Theorem 5], we obtain v ∗∗ ( ) = lsc ( v )( ) = sup O ∈ U ( ) inf f ∈ O \{ } v ( f ) , where lsc ( v ) denotes the semi-continuous hull of the mapping v , cf. [18, Equation 3.7] and U ( ) the set containing all T uc -open neighbourhoods of 0. Put differently, in order to verify28trong duality it suffices to show that the mapping v is continuous at the origin with respect tothe topology of uniform convergence. We use the following adaptation of [1, Theorem 5.42] tolocally convex spaces: Lemma 3.6.
Let V be a locally convex space, f : V → (−∞ , ∞] a convex function and x ∈ V .If there exists an open neighbourhood O of x such that sup x ∈ O f ( x ) < ∞ , then f is continuousat x . The set O : = {(cid:107) f (cid:107) ∞ < } is a T uc -open neighbourhood of 0. For any f ∈ O we have v ( f ) = inf λ ∈ M + ( Ω ε ) sup µ ∈ M + (−∞ , K ] (cid:0) (cid:104) f , µ (cid:105) − (cid:107) µ (cid:107) − (cid:104) (cid:101) g , λ (cid:105) + (cid:104) T µ, λ (cid:105) (cid:1) ≤ sup µ ∈ M + (−∞ , K ] (cid:0) (cid:107) µ (cid:107) (cid:107) f (cid:107) ∞ − (cid:107) µ (cid:107) (cid:1) = . Lemma 3.6 warrants that the mapping v is indeed continuous at 0 and therefore p ε = − v ∗∗ ( ) = − v ( ) = d ε . Next, we verify that the optimisers λ ε and µ ε satisfy the complementary slackness property.Using the strong duality we obtain0 ≤ (cid:104) T µ ε − (cid:101) g , λ ε (cid:105) = (cid:104) T µ ε , λ ε (cid:105) − p ε = (cid:104) µ ε , T ∗ λ ε (cid:105) − d ε = (cid:104) µ ε , T ∗ λ ε − , (cid:105) ≤ . (3.31)In other words, T µ ε = (cid:101) g holds λ ε -a.e. on Ω ε and T ∗ λ ε = µ ε -a.e. on (−∞ , K ] . Moreover,the structure of the dual problem D ε implies that we can always choose a D ε -optimal elementwhich assigns no mass to the zeros of the function (cid:101) g , i.e. λ ε ({( t , x ) ∈ Ω ε : (cid:101) g ( x ) = }) = Lemma 3.7.
For any ε ∈ ( , T / ) the linear programmes P ε , D ε have solutions µ ε , λ ε andtheir optimal values p ε , d ε coincide. The total mass of both optimisers is bounded by a constant (cid:37) ∈ ( , ∞) that does not depend on ε . Moreover, no mass of the measure λ ε is located on thezero set of the function (cid:101) g . The equation T µ ε = (cid:101) g holds λ ε -a.e. on Ω ε and T ∗ λ ε = holds µ ε -a.e. on (−∞ , K ] . We now turn our attention to programme P and the associated dual D from Subsection3.2.2. Lemma 3.4 is proved in two steps. First, we show that the primal optimisers ( µ ε ) ε> cluster at some P -optimal measure µ ε and that the family ( λ ε ) ε> contains a D -admissibleaccumulation point λ . Subsequently we show that the measure λ is D -optimal. The otherassertions of Lemma 3.4 are verified on the way. Step 1
Let p and d denote the optimal values of P and D . The weak duality 0 ≤ d ≤ p follows literally from the same calculation as in (3.22). Recall that for any ε > µ ε ∈ M + (−∞ , K ] and λ ε ∈ M + ( Ω ) is bounded by some constant (cid:37) >
0, whichdoes not depend on ε . General theory tells us that the vague topology is metrisable on the totalvariation unit balls in both spaces. Alaoglu’s theorem warrants that they are vaguely compactsets. Hence we can find a sequence ε n ↓ µ ∈ M + (−∞ , K ] , λ ∈ M + ( Ω ) with (cid:107) µ (cid:107) ∨ (cid:107) λ (cid:107) ≤ (cid:37) such that µ ε n converges vaguely to µ and λ ε n converges vaguely to λ . For29ny ( t , x ) ∈ Ω the mapping y (cid:55)→ κ ( t , x , y ) is continuous on (−∞ , K ] and vanishes at infinity, see(3.16). By vague convergence we conclude that T µ ( t , x ) = ∫ κ ( t , x , y ) d µ ( y ) = lim n →∞ ∫ κ ( t , x , y ) d µ ε n ( y ) ≥ (cid:101) g ( x ) , ( t , x ) ∈ Ω . This ensures that µ is indeed P -admissible. Next we verify that the measure λ is D -admissible. Obviously, for any δ ∈ ( , T / ) we have ∅ (cid:44) Ω δ ⊂ Ω δ ⊂ Ω . By Urysohn’s lemma,cf. [12, Theorem 4.2], there exists a continuous function ϕ δ : Ω → [ , ] such that ϕ δ ( t , x ) = ( t , x ) ∈ Ω δ and ϕ δ ( t , x ) = ( t , x ) ∈ cl ( Ω \ Ω δ ) . For any y ∈ (−∞ , K ] thecontinuous mapping Ω (cid:51) ( x , t ) (cid:55)→ κ ( t , x , y ) ϕ δ ( t , x ) vanishes at infinity. By vague convergenceof the sequence λ ε n → λ we obtain ∫ κ ( t , x , y ) λ ( d ( t , x )) = lim δ ↓ ∫ κ ( t , x , y ) Ω δ ( t , x ) λ ( d ( t , x ))≤ lim δ ↓ ∫ κ ( t , x , y ) ϕ δ ( t , x ) λ ( d ( t , x )) = lim δ ↓ lim n →∞ ∫ κ ( t , x , y ) ϕ δ ( t , x ) λ ε n ( d ( t , x ))≤ lim sup δ ↓ lim sup n →∞ ∫ κ ( t , x , y ) λ ε n ( d ( t , x )) ≤ . In other words, the measure λ is dual admissible.Next, we establish the strong duality p = d by putting together several of the previousresults. The vague convergence of the measures µ ε n to µ implies that (cid:107) µ (cid:107) ≤ lim inf n →∞ (cid:107) µ ε n (cid:107) is true. Recalling that strong duality holds in the P ε - D ε -setting yields d ≤ p ≤ (cid:107) µ (cid:107) ≤ lim inf n →∞ (cid:107) µ ε n (cid:107) = lim inf n →∞ p ε n = lim inf n →∞ d ε n ≤ d . (3.32)The last inequality follows from the fact that all D ε -admissible elements are D -admissible.Along the way we have shown that the P -optimal value is attained by the measure µ .We prove by contradiction that any P -admissible element assigns mass to any open subsetof (−∞ , K ) . Otherwise there is some P -admissible measure µ and a bounded, open interval I : = ( c − ν, c + ν ) ⊂ (−∞ , K ) such that µ ( I ) =
0. Obviously we have 0 < δ : = inf x ∈ I (cid:101) g ( x ) . Thisyields δ < (cid:101) g ( c ) ≤ T µ ( t , c ) = ∫ {| y − c |≥ ν } κ ( t , c , y ) µ ( d y ) (3.33)for all t ∈ ( , T ) . In view of (3.16), the right-hand side of (3.33) converges to 0 as t ↑ T . Thiscontradiction proves the claim. Step 2
We show that the D -optimal value is attained by λ if some additional requirement ismet. Recall that the measure λ is D -admissible and that the sequence λ ε n converges to λ withrespect to the vague topology on M ( Ω ) . Due to the lack of compactness, we cannot directlyconclude that λ ε n converges weakly to λ . Observe that the functional M ( Ω ) (cid:51) λ (cid:55)→ (cid:104) (cid:101) g , λ (cid:105) isweakly but not vaguely continuous.First we prove that the sequence λ ε n converges weakly in M ( cl Ω ) , where cl Ω = [ , T ] ×(−∞ , K ] . It is sufficient to show that the family { λ ε n : n ∈ N } is tight. For any ε > K ε : = [ , T ] × [− / ε, K ] a compact subset of cl Ω . The mass of λ ε n is concentratedon Ω ε n ⊂ K ε n . Let us assume by contradiction that the family of measures is not tight. Thenthere exists a constant δ > n ∈ N there is some integer M n ≥ n with λ ε Mn ( Ω \ K ε n ) > δ . Pick a sufficiently small constant C ∈ (−∞ , K ) with ∫ C −∞ ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) d y ≤ δ . Due to the fact that all measures λ ε n are D -admissible, we have ∫ ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y (cid:1) λ ε Mn ( d ( t , x )) ≤ ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) for any y ∈ (−∞ , K ] . Integrating this inequality over the set (−∞ , C ) yields ∫ ∫ C −∞ ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y (cid:1) d y λ ε Mn ( d ( t , x )) ≤ δ . Due to the positivity of measure and integrand, we conclude that δ ≥ ∫ Ω \ K ε n ∫ C −∞ ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y (cid:1) d y λ ε Mn ( d ( t , x ))≥ λ ε Mn (cid:0) Ω \ K ε n (cid:1) inf ( t , x )∈ Ω \ K ε n ∫ C −∞ ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y (cid:1) d y ≥ δ inf x < − / ε n inf t ∈[ , T ] ∫ C −∞ ϕ (cid:0) x + (cid:101) rt , σ t , y (cid:1) d y , n ∈ N . Taking the limit n → ∞ yields δ ≥ δ lim n →∞ inf x < − / ε n inf t ∈[ , T ] ∫ C −∞ ϕ (cid:0) x + (cid:101) rt , σ t , y (cid:1) d y = δ as ε n →
0. This is impossible and consequently the family { λ ε n : n ∈ N } must be tight. Hencethe sequence λ ε n converges weakly in M ( cl Ω ) to some measure λ with λ | Ω = λ .In order to assure that the measure λ is D -optimal, it suffices to show that λ assigns nomass to the borders M : = { } × (−∞ , K ) and M : = { T } × (−∞ , K ) . Indeed, in this case wehave ∫ Ω (cid:101) g ( x ) λ ( d ( t , x )) = ∫ cl Ω (cid:101) g ( x ) d λ ( t , x ) = lim n →∞ ∫ (cid:101) g ( x ) λ ε n ( d ( t , x )) = lim n →∞ d ε n = d . The second equality follows from the weak convergence of the sequence λ ε n in the space M ( cl Ω ) and the boundedness of the continuous function (cid:101) g . The last equality has already beenestablished in (3.32).Assume by contradiction that λ assigns mass to the set M . In this case there is some α < K with λ ({ } × [ α, K )) >
0. Due to weak convergence in the space M ([ , T / ] × [ α, K ]) we have ∫ { }×[ α, K ) κ ( t , x , y ) λ ( d ( t , x )) ≤ ∫ [ , T / ]×[ α, K ] κ ( t , x , y ) λ ( d ( t , x )) = lim n →∞ ∫ [ , T / ]×[ α, K ] κ ( t , x , y ) λ ε n ( d ( t , x )) ≤ y ∈ (−∞ , K ] . Fatou’s lemma, Lemma A.4(3), and K < x now yield the followingcontradiction1 ≥ lim inf y →−∞ ∫ { }×[ α, K ) ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) λ ( d ( t , x ))≥ ∫ { }×[ α, K ) lim inf y →−∞ exp (cid:16) y x − x σ T (cid:17) exp (cid:32) x − x + (cid:101) rT ( x − x ) σ T (cid:33) λ ( d ( t , x )) = ∞ Hence the assumption was wrong and therefore λ ( M ) = M . For any ( t , x ) ∈ [ , T ] × R we define by V ( t , x ) : = lim inf ( t (cid:48) , x (cid:48) )→( t , x )( t (cid:48) , x (cid:48) )∈( , T )× R T µ ( t (cid:48) , x (cid:48) ) (3.34)the lower semi-continuous extension of the function T µ to the set [ , T ] × R . We show thatimposing the additional assumption V ( T , x ) > (cid:101) g ( x ) ∀ x ∈ (−∞ , K ) (3.35)warrants that the measure λ assigns no mass to the set M . The mapping V is lower semi-continuous and bounded from below and hence attains its minimum on any compact subset of [ , T ] × R . Moreover, assumption (3.35) ensures that the minimal value of the function V − (cid:101) g isstrictly positive on any set { T } × [ a , b ] ⊂ M with a < b < K . By lower semi-continuity thereis some n ∈ N and δ > V ( t , x ) − (cid:101) g ( x ) ≥ δ (3.36)for any ( t , x ) ∈ [ T − / n , T ] × [ a , b ] . Assume by contradiction that the measure λ assigns massto M . We can choose some strip { T } × ( a , b ) ⊂ M and a constant (cid:37) > λ ( Q m ) ≥ (cid:37) holds for any m ∈ N , where Q m : = ( T − / m , T ] × ( a , b ) . The measures λ ε n converge weakly in M ( cl Ω ) to λ . Owing to [9, Theorem 13.16], we can pass to a subsequence (again denoted by ε n ) such that λ ε n ( int Q n ) ≥ (cid:37) for all n ∈ N . The strong duality in the D ε - P ε -setting yields (cid:104) T µ − (cid:101) g , λ ε n (cid:105) = (cid:104) µ , T ∗ λ ε n (cid:105) − p ε n = (cid:104) µ , T ∗ λ ε n − (cid:105) + (cid:107) µ (cid:107) − (cid:107) µ ε n (cid:107)≤ (cid:107) µ (cid:107) − (cid:107) µ ε n (cid:107) . Moreover, (3.36) implies (cid:104) T µ − (cid:101) g , λ ε n (cid:105) ≥ ∫ int Q n V ( t , x ) − (cid:101) g ( x ) λ ε n ( d ( t , x ))≥ δλ ε n ( int Q n )≥ δ (cid:37) > , n ≥ n . However, we already know from (3.32) that (cid:107) µ (cid:107) − (cid:107) µ ε n (cid:107) → n → ∞ . This yields acontradiction, which finally shows that λ ( M ) = µ and λ in the case of primal and dual attainment. Thismeans that the equation ∫ κ ( t , x , y ) d µ ( y ) = (cid:101) g ( x ) (3.37)holds λ -a.e. on Ω and ∫ Ω κ ( t , x , y ) λ ( d ( t , x )) = µ -a.e. on (−∞ , K ] . Let us summarise our results from above. Lemma 3.8.
1. For any ε ∈ ( , T / ) the linear programmes P ε and D ε have solutions µ ε and λ ε . The optimal values p ε and d ε of the latter programmes coincide. The total massof the optimisers is bounded by some constant (cid:37) ∈ ( , ∞) which does not depend on ε .Moreover, the measure λ ε assigns no mass to the zero set of the function (cid:101) g . The equation T µ ε = (cid:101) g holds λ ε -a.e. on Ω ε and T ∗ λ ε = holds µ ε -a.e. on (−∞ , K ] .2. There exists a sequence ε n ↓ such that µ ε n converges vaguely in M (−∞ , K ] to some P -admissible measure µ and λ ε n converges vaguely in M ( Ω ) to some D -admissiblemeasure λ . The optimal value of P is obtained by µ and coincides with the optimal valueof D . The measure µ assigns mass to any open subset of (−∞ , K ) and (cid:107) µ (cid:107) ∨ (cid:107) λ (cid:107) ≤ (cid:37) .3. Let V be defined as in (3.34) . If V ( T , x ) > (cid:101) g ( x ) for any x ∈ (−∞ , K ) , the optimal value ofthe programme D is obtained by λ . In this case the complementary slackness equations (3.37) and (3.38) hold. Lemma 3.4 is nothing but a slight reformulation of statements 2 and 3.
We use the notation from the preceding sections. In particular, see Section 3.2.2 for the definitionof the operator T and the optimisation problem P . Let µ ∗ be a cheapest dominating Europeanoption in the sense of Theorem 3.1. In view of (3.14) we have v eu ,µ ∗ ( ϑ, x ) = e − r ϑ ∫ ϕ (cid:0) x + (cid:98) r ϑ, σ ϑ, y (cid:1) ϕ (cid:0) x + (cid:98) rT , σ T , y (cid:1) µ ∗ ( d y ) (3.39) = e (− r / σ ) x ∫ ϕ (cid:0) x + (cid:101) r ϑ, σ ϑ, y (cid:1) ϕ (cid:0) x + (cid:101) rT , σ T , y (cid:1) e ( r / σ ) x − rT µ ∗ ( d y ) = e (− r / σ ) x T µ ( T − ϑ, x ) for any ( ϑ, x ) ∈ ( , T )× R . Here we denote by µ = e ( r / σ ) x − rT µ ∗ the corresponding P -optimalmeasure from Lemma 3.4. 33 tep 1: Analyticity of the European value function First, we show that the first assumptionof Theorem 3.2 ensures the analyticity of the function v eu ,µ ∗ on the open C -domain E : = (cid:110) ϑ ∈ C : (cid:112) ( Re ϑ − ( T + δ )/ ) + ( Im ϑ ) < ( T + δ )/ (cid:111) × C . It suffices to verify that the function e r ϑ v eu ,µ ∗ ( ϑ, x ) = ∫ ϕ (cid:0) x + (cid:98) r ϑ, σ ϑ, y (cid:1) ϕ (cid:0) x + (cid:98) r ( T + δ ) , σ ( T + δ ) , y (cid:1) µ ∗∗ ( d y ) (3.40)is analytic on E , where d µ ∗∗ d µ ∗ ( y ) : = ϕ (cid:0) x + (cid:98) r ( T + δ ) , σ ( T + δ ) , y (cid:1) ϕ (cid:0) x + (cid:98) rT , σ T , y (cid:1) , y ∈ R . In view of assumption 1 we have (cid:107) µ ∗∗ (cid:107) = v eu ,µ ∗ ( T + δ, x ) e r ( T + δ ) < ∞ . Due to Hartogs’ theoremit is enough to show that the function from (3.40) is partially analytic, cf. [10, Paragraph 2.4].Lemma A.4(2) implies that (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x + (cid:98) r ϑ, σ ϑ, y ) ϕ ( x + (cid:98) r ( T + δ ) , σ ( T + δ ) , y ) (cid:12)(cid:12)(cid:12)(cid:12) = | h ( ϑ, x )| (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − ( y − A ( ϑ, x )) B ( ϑ, x ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = | h ( ϑ, x )| exp (cid:32) − Re B ( ϑ, x ) | B ( ϑ, x )| (cid:18) y − Re A ( ϑ, x ) − Im A ( ϑ, x ) Im B ( ϑ, x ) Re B ( ϑ, x ) (cid:19) (cid:33) for any ( ϑ, x ) ∈ E and y ∈ R , where h , h denote certain functions which are continuous on E .The quantities A ( ϑ, x ) and B ( ϑ, x ) are defined as in (A.2). For any ( ϑ, x ) ∈ E we haveRe B ( ϑ, x ) = Re σ ϑ ( T + δ ) T + δ − ϑ = σ ( T + δ )| T + δ − ϑ | (cid:16) ( T + δ ) Re ϑ − | ϑ | (cid:17) > y ∈ R (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x + (cid:98) r ϑ, σ ϑ, y ) ϕ ( x + (cid:98) r ( T + δ ) , σ ( T + δ ) , y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | h ( ϑ, x )| . The quantity on the right-hand side is bounded on every compact subset of E . Hence we canuse a standard argument involving the theorems of Morera and Fubini in order to prove partialanalyticity. For a detailed exposition of the technique, we refer the reader to the proof of LemmaA.5, which is to be found in [13, Section 5.1]. In view of Hartogs’ theorem we conclude thatthe mapping v eu ,µ ∗ is indeed analytic on E . Step 2: Analyticity of the curve
We show that the curve ϑ (cid:55)→ ˘ x ( ϑ ) is analytic on an opencomplex domain containing the interval ( , T ] . In view of step 1 and the assumptions imposedon the American payoff g , the function Ψ ( ϑ, x ) : = v eu ,µ ∗ ( ϑ, x ) − g ( x ) (3.41)34s analytic on the open C -domain D (cid:48) × D , where D (cid:48) : = (cid:110) ϑ ∈ C : (cid:112) ( Re ϑ − ( T + δ )/ ) + ( Im ϑ ) < ( T + δ )/ (cid:111) and D denotes the domain of analyticity of g . The set D (cid:48) is simply connected and ( , T + δ ) ×(−∞ , K ) is a subset of D (cid:48) × D . The continuity of Ψ and the uniqueness of the minima warrant thatthe curve x is continuous on the interval ( , T + δ / ) . Indeed, assume by contradiction that ˘ x isdiscontinuous at ϑ ∈ ( , T + δ / ) . Then there is a sequence ϑ n → ϑ and some x ∞ ≤ K , ε > x ( ϑ n ) → x ∞ as n → ∞ and | x ∞ − ˘ x ( ϑ )| > ε . Hence there exists a constant T max > Ψ ( ϑ , ˘ x ( ϑ )) + T max < Ψ ( ϑ , x ∞ ) . Consequently, we can choose two disjointballs B (( ϑ , x ∞ ) , r ) and B (( ϑ , ˘ x ( ϑ )) , r ) of radius r ∈ ( , ε / ) with Ψ ( ϑ, x ) + T max / < Ψ ( (cid:101) ϑ, (cid:101) x ) for any ( ϑ, x ) ∈ B r ( ϑ , ˘ x ( ϑ )) and ( (cid:101) ϑ, (cid:101) x ) ∈ B (( ϑ , x ∞ ) , r ) . This yields a contradiction because ( ϑ n , ˘ x ( ϑ n )) is contained in B (( ϑ , x ∞ ) , r ) for any sufficiently large integer n . Hence the curve ˘ x is continuous.For any ϑ ∈ ( , T + δ ) we have D Ψ ( ϑ, ˘ x ( ϑ )) = D Ψ ( ϑ, ˘ x ( ϑ )) ≥ D Ψ = D v eu ,µ ∗ − g (cid:48)(cid:48) = σ D v eu ,µ ∗ + (cid:18) − r σ (cid:19) D v eu ,µ ∗ + r σ v eu ,µ ∗ − g (cid:48)(cid:48) = σ D v eu ,µ ∗ + (cid:18) − r σ (cid:19) D Ψ + r σ (cid:0) v eu ,µ ∗ − g (cid:1) − c = H + (cid:18) − r σ (cid:19) D Ψ (3.42)on ( , T + δ ) × R , where c and H are defined as in (3.3) and (3.9), respectively. Assumption 3warrants that H and therefore D Ψ are strictly positive on the set Γ : = {( ϑ, ˘ x ( ϑ )) : ϑ ∈ ( , T ]} .Therefore Theorem A.3 is applicable to the function D Ψ at any point of Γ . We obtain thatfor any ( (cid:101) ϑ, (cid:101) x ) ∈ Γ there exist open neighbourhoods (cid:101) ϑ ∈ U (cid:101) ϑ , (cid:101) x ∈ U (cid:101) x and an analytic curve χ (cid:101) ϑ : U (cid:101) ϑ → U (cid:101) x with χ (cid:101) ϑ ( ϑ ) = ˘ x ( ϑ ) for any ϑ ∈ U (cid:101) ϑ ∩ ( , T ] . The identity theorem implies thatany two curves χ (cid:101) ϑ , χ (cid:101) ϑ coincide on U (cid:101) ϑ ∩ U (cid:101) ϑ . Since the mapping ϑ (cid:55)→ ˘ x ( ϑ ) is continuous,there exists an analytic function χ such that χ | U ϑ = χ ϑ for any ϑ ∈ ( , T ] . In particular, wehave χ ( ϑ ) = ˘ x ( ϑ ) for any ϑ ∈ ( , T ] . This proves that x is indeed analytic on some complexdomain containing the interval ( , T ] . Step 3: Proof of statement 2
We verify that v eu ,µ ∗ ( ϑ, ˘ x ( ϑ )) = g ( ˘ x ( ϑ )) for any ϑ ∈ [ , T ] .Since the measure µ ∗ assigns no mass to the set ( K , ∞) , we have v eu ,µ ∗ ( , x ) = x > K .Lower semi-continuity even implies v eu ,µ ∗ ( , K ) = v eu ,µ ∗ ( , x ( )) = v eu ,µ ∗ ( , K ) = = g ( K ) = g ( ˘ x ( )) . In view of (3.39) we have e ( r / σ ) x (cid:0) v eu ,µ ∗ ( T − t , x ) − g ( x ) (cid:1) = T µ ( t , x ) − (cid:101) g ( x ) for any ( t , x ) ∈ [ , T )× R , where (cid:101) g is defined as in (3.13). Assumption 2 implies that v eu ,µ ∗ ( , x )− g ( x ) > x < K . Consequently, Lemma 3.8 warrants strong duality, primal and dual35ttainment as well as complementary slackness. In view of (3.37), the dual maximiser λ assigns no mass to the complement of the set {( t , ˘ x ( T − t )) : 0 < t < T } . We claim that thereexists a sequence ϑ n ↑ T with ϑ n ∈ ( , T ) and v eu ,µ ∗ ( ϑ n , ˘ x ( ϑ n )) = g ( ˘ x ( ϑ n )) , n ∈ N . (3.43)Assume to the contrary that this is false. Then there is some ε ∈ ( , T ) with v eu ,µ ∗ ( ϑ, ˘ x ( ϑ )) > g ( ˘ x ( ϑ )) for all ϑ ∈ ( T − ε, T ) . Equation (3.37) tells us that the measure λ is concentrated onthe set Γ ε : = {( t , ˘ x ( T − t )) : ε < t < T } . From Lemma 3.8 we already know that the primalminimiser µ assigns mass to any open subset of (−∞ , K ) . By (3.38) we can find a sequence y n ↓ −∞ with max n ∈ N y n < min ϑ ∈[ , T ] ˘ x ( ϑ ) + (cid:101) r ( T − ε ) = : z and ϕ (cid:0) x + (cid:101) rT , σ T , y n (cid:1) = ∫ Γ ε ϕ (cid:0) x + (cid:101) r ( T − t ) , σ ( T − t ) , y n (cid:1) λ ( d ( t , x )) , n ∈ N . In view of (cid:101) r < ϕ (cid:0) x + (cid:101) rT , σ T , y n (cid:1) ≤ ∫ Γ ε ϕ (cid:0) z , σ ( T − t ) , y n (cid:1) λ ( d ( t , x ))≤ ϕ (cid:0) z , σ ( T − ε ) , y n (cid:1) λ (cid:0) Γ ε (cid:1) , n ∈ N . This yields the contradiction1 ≤ λ ( Γ ) lim n →∞ ϕ (cid:0) z , σ ( T − ε ) , y n (cid:1) ϕ (cid:0) x + (cid:101) rT , σ T , y n (cid:1) = . Consequently a sequence with the desired property (3.43) exists.In view of steps 1 and 2, the mapping ϑ (cid:55)→ v eu ,µ ∗ ( ϑ, ˘ x ( ϑ )) − g ( ˘ x ( ϑ )) is analytic on someopen complex domain containing the interval ( , T ] . Equation (3.43) and the identity theoremfinally yield that v eu ,µ ∗ ( ϑ, ˘ x ( ϑ )) = g ( ˘ x ( ϑ )) for any ϑ ∈ ( , T ] . Step 4: Proof of statement 4
We verify that µ ∗ is the unique measure representing ourAmerican payoff on the set (cid:101) C T , x as defined in (3.10). Moreover, we show that (cid:101) C T , x is aconnected subset of C T , x . For any T ∈ [ , T ] the process V ( T ) t : = e − rt v eu ,µ ∗ ( T − t , X t ) is amartingale on the interval [ , T ) . Indeed, for 0 ≤ u < t + u < T the Markov property of theprocess X yields E x (cid:0) V ( T ) t + u (cid:12)(cid:12) F u (cid:1) = e − r ( t + u ) E X u (cid:0) v eu ,µ ∗ ( T − t − u , X t ) (cid:1) = e − rT ∫ E X u (cid:0) ϕ (cid:0) X t + (cid:98) r ( T − t − u ) , σ ( T − t − u ) , y (cid:1) (cid:1) ϕ (cid:0) x + (cid:98) rT , σ T , y (cid:1) µ ∗ ( d y ) = e − rT ∫ ϕ (cid:0) X u + (cid:98) r ( T − u ) , σ ( T − u ) , y (cid:1) ϕ (cid:0) x + (cid:98) rT , σ T , y (cid:1) µ ∗ ( d y ) = e − ru v eu ,µ ∗ ( T − u , X u ) = V ( T ) u . (3.44)The third equality follows from the convolution property of the normal distribution.36he martingale condition may fail to hold up to T . Nevertheless, Fatou’s lemma yields thesupermartingale property. Indeed, for any u ∈ [ , T ] we have E x (cid:0) V ( T ) T (cid:12)(cid:12) F u (cid:1) = E x (cid:18) e − rT lim inf t ↑ T v eu ,µ ∗ ( T − t , X t ) (cid:12)(cid:12)(cid:12)(cid:12) F u (cid:19) ≤ lim inf t ↑ T E x (cid:0) e − rt v eu ,µ ∗ ( T − t , X t ) (cid:12)(cid:12) F u (cid:1) = V ( T ) u . Due to superreplication, we have e − rt g ( X t ) ≤ V ( T ) t for any t ∈ [ , T ] and consequently theoptional sampling theorem yields E x ( e − r τ g ( X τ )| F t ) ≤ E x (cid:0) V T τ (cid:12)(cid:12) F t (cid:1) ≤ V ( T ) t for any [ t , T ] -valued stopping time τ . Maximising the left-hand side over all such stopping timesshows that v am , g ( ϑ, x ) ≤ v eu ,µ ∗ ( ϑ, x ) for any ( ϑ, x ) ∈ [ , T ] × R .Now we verify that the value functions v am , g and v eu ,µ ∗ coincide on the set (cid:101) C T , x . To this endlet τ ϑ be defined as in (3.11). Assumption 4 warrants that the measure µ ∗ has no atom at K , i.e. µ ∗ ({ K }) =
0. Indeed, assuming µ ∗ ({ K }) > ϑ → v eu ,µ ∗ ( ϑ, K ) ≥ T max lim inf ϑ → e − r ϑ ϕ (cid:0)(cid:98) r ϑ, σ ϑ, (cid:1) = ∞ , where T max denotes some positive constant. Owing to the geometric properties of the curve ˘ x ,we have E x (cid:16) ϕ (cid:0) X τ ϑ + (cid:98) r ( ϑ − τ ϑ ) , σ ( ϑ − τ ϑ ) , y (cid:1) { τ ϑ = ϑ } (cid:17) ≤ E x (cid:0) { y } ( X ϑ ) { X ϑ ≥ K } (cid:1) = y < K . Hence monotone convergence yields v am , g ( ϑ, x ) ≥ E x (cid:0) e − r τ ϑ g ( X τ ϑ ) (cid:1) (3.45) ≥ E x (cid:0) e − r τ ϑ g ( X τ ϑ ) { τ ϑ <ϑ } (cid:1) = E x (cid:0) e − r τ ϑ v eu ,µ ∗ ( ϑ − τ ϑ , X τ ϑ ) { τ ϑ <ϑ } (cid:1) = lim x (cid:48) ↑ K E x (cid:32) e − r ϑ ∫ x (cid:48) −∞ ϕ (cid:0) X τ ϑ + (cid:98) r ( ϑ − τ ϑ ) , σ ( ϑ − τ ϑ ) , y (cid:1) ϕ (cid:0) x + (cid:98) rT , σ T , y (cid:1) µ ∗ ( d y ) { τ ϑ <ϑ } (cid:33) = lim x (cid:48) ↑ K e − r ϑ ∫ x (cid:48) −∞ E x (cid:0) ϕ (cid:0) X τ ϑ + (cid:98) r ( ϑ − τ ϑ ) , σ ( ϑ − τ ϑ ) , y (cid:1) (cid:1) ϕ (cid:0) x + (cid:98) rT , σ T , y (cid:1) µ ∗ ( d y ) = v eu ,µ ∗ ( ϑ, x )≥ v am , g ( ϑ, x ) (3.46)for any ( ϑ, x ) ∈ (cid:101) C T , x . Summing up, we have shown that v am , g ( ϑ, x ) = v eu ,µ ∗ ( ϑ, x ) > g ( x ) holdsfor any ( ϑ, x ) ∈ (cid:101) C T , x . Moreover, this directly implies that (cid:101) C T , x is a connected subset of C T , x .Finally we verify that the representing measure µ ∗ is unique. Assume that there is anothermeasure ν such that v eu ,µ ∗ ( ϑ, x ) = v am , g ( ϑ, x ) = v eu ,ν ( ϑ, x ) for any ( ϑ, x ) ∈ (cid:101) C T , x . Recall that thevalue functions v eu ,µ ∗ , v eu ,ν are analytic on a C -domain containing the set ( , T ) × R . The set (cid:101) C T , x contains some open ball. By applying the identity theorem in each variable, we concludethat the mappings v eu ,µ ∗ and v eu ,ν ( ϑ, x ) coincide on the set ( , T ) × R . Equation (3.39) impliesthat T µ ∗ = T ν holds on ( , T ) × R . By Lemma 3.3 the operator T is injective on the Borelmeasures and therefore µ ∗ = ν . 37 tep 5: Proof of statement 5 By statement 2 we have C T , x = (cid:0) C T , x ∩ {( ϑ, x ) ∈ ( , T ] × R : ˘ x ( ϑ ) < x } (cid:1) ∪ (cid:0) C T , x ∩ {( ϑ, x ) ∈ ( , T ] × R : ˘ x ( ϑ ) > x } (cid:1) . Since C T , x is connected and the first set in the union is nonempty, the second set must be empty.Therefore C T , x ⊂ (cid:101) C T , x and hence C T , x = (cid:101) C T , x by step 4. Step 6: Proof of statement 3
For x ≥ K we have v eu ,µ ∗ ( , x ) = g ( x ) =
0. For any x ∈ [ min ϑ ∈( , T ] ˘ x ( ϑ ) , K ) there is a maturity ϑ ( x ) ∈ ( , T ] such that ( ϑ ( x ) , x ) is located onthe curve, i.e. ˘ x ( ϑ ( x )) = x . Due to superreplication and assertion 2, we have g ( x ) ≤ inf ϑ ∈[ , T ] v eu ,µ ∗ ( ϑ, x ) ≤ v eu ,µ ∗ ( ϑ ( x ) , x ) = g ( x ) . (3.47) Step 7: Proof of statement 6
This follows from (3.45, 3.46).
Step 8: Monotonicity of the curve
We show by contradiction that ˘ x is strictly increasing.First assume that it is not increasing. Since ˘ x is continuous, there are 0 < ϑ < ϑ < ϑ < T and x < K such that ˘ x ( ϑ ) = ˘ x ( ϑ ) = x and ˘ x ( ϑ ) < x . From step 4 we know that (cid:101) C T , x is aconnected subset of C T , x . In particular, ( ϑ , x ) is located within the continuation set. In viewof assertion 2 we conclude that g ( x ) = v am , g ( ϑ , x ) < v am , g ( ϑ , x ) ≤ v am , g ( ϑ , x ) = g ( x ) .This is impossible and hence the mapping ϑ (cid:55)→ ˘ x ( ϑ ) must be increasing.Now assume that there are some 0 < ϑ < ϑ ≤ T such that x is constant on the interval ( ϑ , ϑ ) . Since the curve ˘ x is analytic, the identity theorem implies that ˘ x is constant on ( , T ] .In view of the second assumption we find that K > ˘ x ( ϑ ) = lim inf ϑ (cid:48) → ˘ x ( ϑ (cid:48) ) = K for any ϑ ∈ ( , T ] , which is impossible. Therefore ϑ (cid:55)→ ˘ x ( ϑ ) is strictly increasing. (cid:3) While Theorem 3.1 warrants that the American put allows for a cheapest dominating Europeanoption in the distributional sense, Theorem 3.2 does not fully answer the question whether itis actually representable. Numerically, the CDEO is easily obtained by semi-infinite linearprogramming, cf. [3, 13]. In this section we investigate whether the numerical approximationsatisfies the qualitative assumptions of Theorem 3.2.In our numerical experiment we consider the put payoff g ( x ) = ( e K − e x ) + with log-strikeprice K = log 100 and maturity T = .
5. The parameters of the model are chosen as r = . x = log K + .
1, and σ = .
4. Figure 3 displays the price surface of the approximate CDEOalong with the put payoff plane. The s -axis represents the stock price s = e x while the ϑ -axisindicates the time to maturity of the option.If assumption 1 in Theorem 3.2 were violated, we would observe an infinite CDEO pricefor ϑ > . s = e x ∈ ( , ∞) . This is obviously not supported by Figure 3. The minimaof the functions x (cid:55)→ v eu ,µ ∗ ( ϑ, x ) − g ( x ) for ϑ ∈ ( , T + δ ) are represented by the white curvesin Figures 3 and 4, using the variable s = e x instead of x . The graphs are in line with therequirements of assumption 2 in Theorem 3.2. The colours in Figure 4 stand for the level of thefunction ( ϑ, x ) (cid:55)→ v eu ,µ ∗ ( ϑ, x ) − g ( x ) that is to be mimimised in x for fixed ϑ .38igure 3: The price surface of the CDEO associated to the American putFigure 4: The curve ϑ (cid:55)→ exp ˘ x ( ϑ ) for the CDEO of the American put39igure 5: The mapping ϑ (cid:55)→ H ( ϑ, ˘ x ( ϑ )) in Theorem 3.2(3)Figure 6: Comparison of the CDEO minima curve and a finite difference approximation to theearly exercise boundary of the American put 40he numerical approximation of the function ϑ → H ( ϑ, ˘ x ( ϑ )) in assumption 3 is shown inFigure 5. It stays well away from 0 as required. Given that representability holds, it should infact have the constant value 2 r σ − e K =
75, which explains the particular shape in Figure 5. Ifassumption 4 in Theorem 3.2 were violated, we would observe an exploding CDEO price for ϑ → s = e x = e K = ϑ (cid:55)→ exp ( ˘ x ( ϑ )) suggested by Theorem 3.2(6),see Figure 6.How can these findings be reconciled with the negative result of [8] which states thatno sufficiently regular European payoff function can represent the American put? Usingthe language of [8], a candidate representing function ϕ should satisfy an ordinary differ-ential equation A ϕ = m with an as yet unknown generalised function m , where A f ( x ) = ( σ x / ) f (cid:48)(cid:48) ( x ) + r x f (cid:48) ( x ) − r f ( x ) . As stated in [8, equations (2.1, 2.2)], the general solution tothis ODE is of the form ϕ ( x ) = ax + bx − α − σ x − α ∫ x y α ∫ K y z − m ( z ) dzd y = ax + bx − α − σ ( α + ) x − α ∫ K ( z ∧ x ) α + z m ( z ) dz (4.1)if ∫ K z α − | m ( z )| dz < ∞ , (4.2)where α : = r / σ , the strike is denoted as K , and a , b ∈ R are constants. For ϕ to representthe American put, we need ϕ ( x ) = x ≥ K , which implies a = b >
0. However,the positivity of b ultimately yields that ϕ in (4.1) cannot represent the American put, see [8,Thereom 15].The integral in (4.1) does not make sense if (4.2) is violated. But A ϕ = m may still besolved for such m , namely by ϕ ( x ) = ax + bx − α + σ x − α ∫ Kx y α ∫ K y z − m ( z ) dzd y = ax + bx − α − σ ( α + ) x − α ∫ Kx z α + − x α + z m ( z ) dz . In this case ϕ ( x ) = x ≥ K implies a = = b , which means that the fateful b -term does notappear. Hence the positive result of our study does not contradict the findings of [8] becausethe CDEO as a candidate for the representing European claim is not subject to the rather strictintegrability condition (4.2). As noted in the introduction, the representability of an American option in terms of a Europeanpayoff has several both numerically and conceptually interesting consequences. In this paper41e have made a first step towards verifying that a given American option is representable. Theresults of Section 4 suggest in particular that representability holds for the prime example ofan American put in the Black-Scholes model, contrary to the evidence from the analysis in [8].This gives new hope that the original endeavour of Jourdain and Martini may ultimately leadto a positive answer and that their concept of embedded American options has a broader scopethan expected.As an ambitious goal for future research it remains to fully characterise representabilityof American options in the Black-Scholes model and more general markets driven by uni- ormultivariate diffusions. In particular, a rigorous proof for the American put is still wanting.
Acknowledgement
The authors thank Josef Teichmann for fruitful discussions and for bringing the papers [7, 8] totheir attention.
A Auxiliary results
Lemma A.1.
Let X ∼ N ( µ X , σ ) , Y ∼ N ( µ Y , σ ) be Gaussian random variables with µ X ≤ µ Y .Then the conditional law P ( X ∈ ·| X ≥ ) is dominated by is counterpart P ( Y ∈ ·| Y ≥ ) in theusual stochastic order, i.e. P ( X > a | X ≥ ) ≤ P ( Y > a | Y ≥ ) for any a ∈ R or, equivalently, E ( f ( X )| X ≥ ) ≤ P ( f ( Y )| Y ≥ ) for any increasing function f such that the integrals exist.Proof. It is easy to verify that P ( X > a | X ≥ ) ≤ P ( Y > a | Y ≥ ) holds if and only if a isbelow some threshold. This naturally implies the claimed stochastic dominance. (cid:3) The following factorisation theorem from multivariate complex analysis gives a sufficientcondition for the analytic dependence of zeros. It is a direct consequence of the Weierstrasspreparation theorem. More details can be found in [2, Chapter 1].
Theorem A.2.
For d ≥ let f be an analytic function on a domain G = D (cid:48) × D ⊂ C d withsimply connected D (cid:48) ⊂ C d − . Assume that the function f ( z (cid:48) , ·) has exactly m distinct zeros inthe set D for any z (cid:48) ∈ D (cid:48) . Then there exist analytic functions α , ..., α m : D (cid:48) → D , positiveintegers k , ..., k m and an analytic function Φ : G → C that does not vanish on G such that f ( z (cid:48) , z ) = m (cid:89) l = ( z − α l ( z (cid:48) )) k l Φ ( z (cid:48) , z ) , ( z (cid:48) , z ) ∈ G . The following version of the analytic implicit function theorem is well suited for our pur-poses. It can be obtained as a corollary of Theorem A.2 by applying well-known ideas from theproof of Rouché’s theorem, cf. [4, page 125].
Theorem A.3.
For d ≥ let f be an analytic function on a domain G = D (cid:48) × D ⊂ C d with simply connected D (cid:48) ⊂ C d − . Assume that f ( z (cid:48) , z ) = and D d f ( z (cid:48) , z ) (cid:44) for some ( z (cid:48) , z ) ∈ G . Then there are open neighbourhoods U ( z (cid:48) ) ⊂ D (cid:48) and V ( z ) ⊂ D of z (cid:48) and z aswell as an analytic function g : U ( z (cid:48) ) → V ( z ) such that the equivalence f ( z (cid:48) , z ) = ⇔ z = g ( z (cid:48) ) holds for all z (cid:48) ∈ U ( z (cid:48) ) and z ∈ V ( z ) . roof. Due to D d f ( z (cid:48) , z ) (cid:44) ε > B ( z , ε ) is contained in D and f ( z (cid:48) , z ) (cid:44) z ∈ B ( z , ε ) \ { z } . Moreover, there are constants c , ε > B ( z (cid:48) , ε ) is contained in D (cid:48) and | f ( z (cid:48) , z )| > c holds for any z (cid:48) ∈ B ( z (cid:48) , ε ) and any z ∈ C with | z − z | = ε . By the choice of ε and the argument principle we have12 π i ∮ | z − z | = ε D d f ( z (cid:48) , z ) f ( z (cid:48) , z ) dz = . The triangle inequality for line integrals yieldssup | z (cid:48) − z (cid:48) | <ε / n (cid:12)(cid:12)(cid:12)(cid:12) − π i ∮ | z − z | = ε D d f ( z (cid:48) , z ) f ( z (cid:48) , z ) dz (cid:12)(cid:12)(cid:12)(cid:12) = π sup | z (cid:48) − z (cid:48) | <ε / n (cid:12)(cid:12)(cid:12)(cid:12)∮ | z − z | = ε D d f ( z (cid:48) , z ) f ( z (cid:48) , z ) − D d f ( z (cid:48) , z ) f ( z (cid:48) , z ) f ( z (cid:48) , z ) f ( z (cid:48) , z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ α sup | z (cid:48) − z (cid:48) | <ε / n | z − z | = ε (cid:12)(cid:12) D d f ( z (cid:48) , z ) f ( z (cid:48) , z ) − D d f ( z (cid:48) , z ) f ( z (cid:48) , z ) (cid:12)(cid:12) , n ∈ N for some α ∈ ( , ∞) which does not depend on n . Since f and its derivatives are continuous,we conclude that the right-hand side of this inequality converges to 0 as n tends to infinity.Moreover, π i ∮ | z − z | = ε D d f ( z (cid:48) , z ) f ( z (cid:48) , z ) dz is integer-valued. Consequently there is some n ∈ N suchthat 12 π i ∮ | z − z | = ε D d f ( z (cid:48) , z ) f ( z (cid:48) , z ) dz = z (cid:48) ∈ B ( z (cid:48) , ε / n ) . Put differently, for any z (cid:48) ∈ B ( z (cid:48) , ε / n ) the mapping z (cid:55)→ f ( z (cid:48) , z ) has exactly one zero within the set B ( z , ε ) . By Theorem A.2 there exists an analytic function g : B ( z (cid:48) , ε / n ) → B ( z , ε ) with f ( z (cid:48) , g ( z (cid:48) )) = (cid:3) Proofs of the following results can be found in [13, Section 5.1].
Lemma A.4.
Set ϕ ( µ, σ , y ) : = √ πσ exp (cid:18) − ( x − µ ) σ (cid:19) . (A.1)
1. For any y ∈ R , µ, σ ∈ C with Re σ > we have | ϕ ( µ, σ , y )| = exp (cid:18) − Re σ | σ | (cid:16) y − Re µ − Im µ Im σ Re σ (cid:17) + ( Im µ ) σ (cid:19)(cid:112) π | σ | .
2. For any µ, (cid:101) µ ∈ C and σ, (cid:101) σ ∈ C \ { } with σ (cid:44) (cid:101) σ we have ϕ ( µ, σ , y ) ϕ ( (cid:101) µ, (cid:101) σ , y ) = (cid:101) σσ exp (cid:18) − ( y − A ) B (cid:19) exp (cid:18) − ( µ − (cid:101) µ ) ( σ − (cid:101) σ ) (cid:19) with A : = (cid:101) µσ − µ (cid:101) σ σ − (cid:101) σ , B : = (cid:101) σ σ (cid:101) σ − σ . (A.2)43 . For any µ, (cid:101) µ ∈ C and σ ∈ C \ { } we have ϕ ( µ, σ , y ) ϕ ( (cid:101) µ, σ , y ) = exp (cid:18) y µ − (cid:101) µσ (cid:19) exp (cid:18) (cid:101) µ − µ σ (cid:19) . Lemma A.5.
For (cid:98) r ∈ R , σ > and any measure µ ∈ M + ( R ) we define the generalised Europeanvalue function V ( ϑ, x ) : = ∫ ϕ (cid:0) x + (cid:98) r ϑ, σ ϑ, y (cid:1) µ ( d y ) , where ϕ is defined as in (A.1) . Suppose there exists some ( T , x ) ∈ ( , ∞) × R with V ( T , x ) < ∞ .Then the mapping V is analytic on the open C -domain (cid:110) ϑ ∈ C : (cid:112) ( Re ϑ − T / ) + ( Im ϑ ) < T / (cid:111) × C . Lemma A.6.
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