Consistent upper price bounds for exotic options given a finite number of call prices and their convergence
CCONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS GIVEN AFINITE NUMBER OF CALL PRICES AND THEIR CONVERGENCE
NICOLE B ¨AUERLE ∗ AND DANIEL SCHMITHALS ∗ Abstract.
We consider the problem of finding a consistent upper price bound for exotic optionswhose payoff depends on the stock price at two different predetermined time points (e.g. Asianoption), given a finite number of observed call prices for these maturities. A model-free approachis used, only taking into account that the (discounted) stock price process is a martingale underthe no-arbitrage condition. In case the payoff is directionally convex we obtain the worst casemarginal pricing measures. The speed of convergence of the upper price bound is determinedwhen the number of observed stock prices increases. We illustrate our findings with somenumerical computations.
Key words : Martingale Optimal Transport; Directional Convexity; Convex Order;Asian Option Introduction
Given a finite number of observable call prices on the same stock for two different maturities0 < t < t and different strikes, what is an arbitrage-free upper price bound for an arbitraryoption whose payoff is a function of the stock price at time t and t ? A typical example forsuch an option would be an Asian option. We study here a model-free setting, only relying onthe assumption that the discounted stock price process is a martingale. In case the option’spayoff function is directionally convex we show that the upper bound for the price is given bythe optimal martingale transport between the marginal distributions which are obtained fromlinearly interpolating the observable call prices. Moreover, it seems intuitively clear that whenthe number of observed call prices for different strikes increases that the so constructed upperbound converges against the true upper bound. We show this and also determine the bestpossible speed of convergence.In case the marginal risk neutral distributions of the stock are completely known, the problemof finding an upper bound for the price of another derivative which is only a function of the twostock prices is know as martingale optimal transport problem (see e.g. [4]). More precisely, the martingale optimal transport problem is the problem to maximize (minimize) (cid:90) R c ( x, y ) Q ( dx, dy ) (1.1)under the constraints that the margins of Q are predefined distributions µ, ν , e.g. Q ( dx, R ) = µ ( dx ) and Q ( R , dy ) = ν ( dy ) and µ and ν are the distributions of a martingale, i.e. (cid:90) R y Q ( x, dy ) = x, for µ − a.e. x ∈ R . (1.2)According to the Lemma of Breeden and Litzenberger (see [7]) the risk neutral marginal distri-bution of a stock price at time t i can be obtained when call prices of all strikes for the maturity t i are observable. Assuming that the market is free of arbitrage this then leads to the martingaleoptimal transport problem (for more details, see [4]). For special options, these kind of problemshave already been discussed in [16] using Skorokhod embedding techniques. In case the payofffunction c satisfies certain properties (more precisely the Martingale Spence Mirrlees condition: ∗ Department of Mathematics, Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany. c (cid:13) a r X i v : . [ q -f i n . M F ] J u l N. B ¨AUERLE AND D. SCHMITHALS c xyy ≥ < t < t observable which seems to be the realistic case. But then a whole familyof risk neutral marginal distributions for both times points are available. In combination withthe martingale condition which is then the best upper price bound? I.e. we want to maximize(1.1) over all Q which satisfy the martingale condition and are consistent with observed callprices. Unlike in classical transport problems we cannot separate the margins from the copulasince they are connected via the martingale condition. However, we are able to answer thequestion when the payoff function is directionally convex, i.e. convex in the components andsupermodular on R . In this case the worst margins are the ones which are obtained from thelinearly interpolated call price function.In an older stream of literature, see e.g. [21, 13, 12] researchers have calibrated discretemodels to given option prices by constructing implied trees. The calibration is to a volatilitysurface obtained from interpolated market data. But the question of no arbitrage has not beendiscussed at that point in time directly. Moreover, in [24] it has been shown that calibratinggiven models to data might lead to fairly different prices for exotic options.On the other hand a lot of studies are concerned with price bounds for specific options. Forexample in [17] lower bounds for prices of basket options (with two stocks) are derived, given afinite number of observations of call prices of all stocks in the basket. The paper [19] considersupper bounds for spread options which are basket options where the weights may have arbitrarysign. The authors use call price information of all strikes and special properties of the payofffunction to derive the inequality. Since only one time point is important in the payoff of basketoptions, the martingale property does not give further information in this case.There are also a number of papers which consider bounds and semi-static hedging strategiesfor Asian options. Here the option payoff depends on the performance of one stock at differenttime points and the martingale property as a further information is highly relevant. Moreover,the payoff of an Asian option like (cid:16)
12 ( S t + S t ) − k (cid:17) + (1.3)as the average of the stock price at two time points, satisfies the directional convexity neededfor our main result. In [8] options written on weighted sums of asset prices are considered.The study includes basket and Asian options. Upper bounds and super-replication strategiesfor these kind of options are derived in the case that all relevant call prices on the options areobserved and in the case of a finite number of call prices are given. The convergence issue is alsotreated. The authors use comonotonicity arguments to construct the upper bound and generalizeresults in [23]. This however is (in genral) in contradiction to the martingale property (see alsoRemark 2.3 in [1]). I.e. respecting the martingale property should lead to tighter bounds. In [1]lower bounds on Asian options are derived under some further assumptions on the expectationof the stock price process which are shown to be satisfied in L´evy markets. The case of a finitenumber of observable call prices is also considered. A completely different approach is pursuedin [9] where an Asian option with continuous stock price average is considered and bounds arederived using dynamic programming techniques.Our paper is organized as follows: In the next section we summarize some facts about therelation between call prices and pricing measures and about the convex order. In Section 3 weconstruct a special distribution from a finite number of observable call prices, show that is amaximal element with respect to the convex order in the set of all consistent pricing measures anddetermine its Wasserstein distance to other consistent pricing measures. Section 4 identifies the ONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS 3 worst case margins, given the payoff function is directionally convex. The next section determinesthe best possible convergence rate of the upper bound, given the number of observable call pricesincreases. Section 6 provides some numerical results for the convergence and one example for thecalculation of an upper price bound for an Asian option using real data. The appendix containssome longer proofs and an example showing that the speed of convergence is the optimal one ingeneral. 2.
Preliminary Results
Consider a financial market with one risk-free asset and one risky asset. We consider only twofuture time points which we denote by 0 < t < t . The risk-free asset has no interest and isnormalized by 1 and for the risky asset with price process ( S t ) we write ( S t , S t ) = ( X, Y ) andassume S = 1. The random variables X, Y are non-negative and defined on a probability space(Ω , F , P ). No model for the stock price is assumed, but throughout - and this is important - weassume that the market is free of arbitrage.2.1. Call Options and Pricing Measures.
Using the assumption of no-arbitrage we mayderive some properties of the price function k (cid:55)→ C i ( k ), where C i ( k ) shall represent the priceat time t = 0 of a call option with strike price k ∈ R + and maturity t i , i = 1 , C i should have the following properties: Lemma 2.1.
The mapping C i : R + → R + , k (cid:55)→ C i ( k ) , i = 1 , is in an arbitrage free market (a) decreasing. (b) convex. (c) lim k →∞ C i ( k ) = 0 . (d) C (cid:48) i (0+) ≥ − . (e) C i (0) = S = 1 . A proof can e.g. be found in [11] or [22] Lemma 3.2 and Lemma 3.4 or in [11], Theorem 9.1.
Definition 2.2.
A function C : R + → R + is called a candidate function for call option prices ,if it satisfies conditions (a)-(e) from Lemma 2.1.The no-arbitrage condition implies by virtue of the first fundamental theorem of asset pricingin a general market the existence of a pricing measure. More precisely, let us denote by P ( R + )the set of all probability measures on R + . Then there exists a measure µ ∈ P ( R + ) such that C ( k ) = (cid:90) ( x − k ) + µ ( dx ) , k ≥ . (2.1)In this case we say that µ and C are consistent. The lemma of Breeden and Litzenberger [7]states that in this situation µ (( −∞ , x ]) = 1 + C (cid:48) ( x +) , x ∈ R . (2.2)That means when we know call prices for all strikes k > µ by (2.2). Note that contrary if µ ∈ P ( R + ) isgiven, the function C defined in (2.1) is automatically a candidate function if (cid:82) xµ ( dx ) = S = 1 . Convex Order.
Another important tool that we need is the so-called convex order.
Definition 2.3.
Two measures µ, ν on R are said to be in convex order , denoted by µ ≤ c ν , iffor any convex function f : R → R such that the integrals exist, (cid:90) R f ( x ) µ ( dx ) ≤ (cid:90) R f ( x ) ν ( dx ) . Since both f ( x ) = x and f ( x ) = − x are convex as well as f ( x ) = 1 and f ( x ) = −
1, theproperty µ ≤ c ν implies that (cid:82) xµ ( dx ) = (cid:82) xν ( dx ) and µ ( R ) = ν ( R ). The next result followsfrom [25]: N. B ¨AUERLE AND D. SCHMITHALS
Lemma 2.4.
Suppose µ, ν ∈ P ( R + ) . Then µ ≤ c ν is equivalent to the existence of a probabilityspace (Ω , F , P ) and non-negative random variables X, Y on it such that X has distribution µ and Y has distribution ν and X = E [ Y | X ] . The next lemma follows from Theorem 1.5.3 and Theorem 1.5.7 in [20].
Lemma 2.5.
Let µ, ν ∈ P ( R + ) and denote by C µ and C ν the respective consistent pricingfunctions. Suppose that (cid:82) xµ ( dx ) = (cid:82) xν ( dx ) = 1 . Then µ ≤ c ν is equivalent to C µ ≤ C ν . Specially Designed Marginals
The lemma of Breeden and Litzenberger (see [7]) implies that when we can observe call pricesfor a fixed maturity for all strikes k >
0, we are able to determine the pricing measure which inturn defines prices of other European options with the same maturity. In reality however, thereare for a fixed maturity t i only a finite number of call prices c i > . . . > c in i > ≤ k i < . . . < k in i , n i ∈ N , i = 1 , Definition 3.1.
Let for i = 1 , P i := (cid:110) µ ∈ P ( R + ) : c ij = (cid:90) ( x − k ij ) + µ ( dx ) , j = 0 , . . . , n i , (cid:90) xµ ( dx ) = 1 (cid:111) be the set of all pricing measures which are consistent with the observable call prices havingmaturity t i .In what follows we assume that there is a strike price K > K , i.e. the pricing measures are concentrated on the compactinterval [0 , K ]. Moreover we assume that call prices c i > . . . > c in i = 0 are available for strikes0 = k i < . . . < k in i where by our assumption c i = 1. We denote by S i := { k i < . . . < k in i } , i =1 , C ∗ µ , C ∗ ν to be exactly the functions that result from interpolating between the observed call option prices( c j ) and ( c j ) respectively (see Figure 1). We concentrate the discussion in this section on C ∗ µ and in order to ease notation write k j instead of k j .That is, for C ∗ µ ( k j ) = c j , j = 0 , . . . , n and for k ∈ [ k j , k j +1 ) , j = 0 , . . . , n, we define C ∗ µ ( k ) := k j +1 − kk j +1 − k j C ∗ µ ( k j ) + k − k j k j +1 − k j C ∗ µ ( k j +1 ) .k i k i k i k i k i k i k i k i c i c i c i c i ... Figure 1. C ∗ µ , C ∗ ν interpolate the observed call prices.We assume that the observed prices are arbitrage-free and thus C ∗ µ is a candidate functionaccording to Lemma 2.1 (see Theorem 3.1 in [11]). ONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS 5
Lemma 3.2.
The measure µ ∗ consistent with C ∗ µ is a discrete measure of the form µ ∗ := n (cid:88) j =0 ω j δ k j := n (cid:88) j =0 (cid:20) C ∗ µ ( k j +1 ) − C ∗ µ ( k j ) k j +1 − k j − C ∗ µ ( k j ) − C ∗ µ ( k j − ) k j − k j − (cid:21) δ k j where we set C ∗ µ ( k n +1 ) − C ∗ µ ( k n ) k n +1 − k n = 0 and C ∗ µ ( k ) − C ∗ µ ( k − ) k − k − = − and δ x is the Dirac measure on point x .Proof. It is a routine calculation to see that C ∗ µ ( k ) = (cid:82) Kk ( x − k ) + µ ∗ ( dx ). A detailed calculationcan be found in [22] Appendix A.4. Note that this construction has also been used in [8]. (cid:3) By construction the measure µ ∗ has a special property. Lemma 3.3.
Suppose that µ ∈ P , i.e. µ is another probability measure consistent with theobservable call prices in P . Then µ ≤ c µ ∗ . Proof.
Let C µ be consistent with µ , i.e. it is by definition convex and passes through the points( k j , c j ) . Moreover, (cid:82) xµ ( dx ) = (cid:82) xµ ∗ ( dx ) = 1. But the convexity implies that C µ ≤ C µ ∗ on R + .This in turn implies by Lemma 2.5 that µ ≤ c µ ∗ . (cid:3) Hence with respect to convex order µ ∗ is the maximal element of the set P . It is alsopossible to determine the Wasserstein distance of µ ∗ to any other element of P explicitly. TheWasserstein distance between two probability measures is defined as follows. LetΠ( µ, µ ∗ ) := (cid:8) π ∈ P ( R ) : µ ( B ) = π ( B × R ) , µ ∗ ( C ) = π ( R × C ) , B, C ∈ B ( R ) (cid:9) Then we define
Definition 3.4.
The
Wasserstein distance of two probability measures µ, µ ∗ ∈ P ( R ) is givenby W ( µ, µ ∗ ) := inf π ∈ Π( µ,µ ∗ ) (cid:90) | x − y | π ( dx, dy ) . (3.1) Remark 3.5. (a) If F µ and F µ ∗ are the cumulative distribution functions of µ and µ ∗ , italso holds that (see [10], Sec. 1) W ( µ, µ ∗ ) = (cid:90) ∞−∞ | F µ ( t ) − F µ ∗ ( t ) | dt. (b) There is also a dual representation of the Wasserstein distance: W ( µ, µ ∗ ) = sup f ∈ C ( R ) (cid:90) f ( x )( µ − µ ∗ )( dx ) , where C ( R ) := { f : R → R : f is Lipschitz-continuous with constant 1 } , see e.g. [26]Theorem 6.9.Now we can show that Theorem 3.6.
Let µ ∈ P ( R + ) with supp ( µ ) ⊂ [0 , K ] . Moreover choose k j = jK n , j = 0 , . . . , n , n ∈ N . Then we have W ( µ, µ ∗ ) = 2 · n − (cid:88) j =0 sup k ∈ [ k j ,k j +1 ) | C µ ∗ ( k ) − C µ ( k ) | ≤ K n , (3.2) If we additionally assume that C µ ∈ C ( R + ) , then, for any n ∈ N , we have W ( µ, µ ∗ ) ≤ T µ · K n +1 , (3.3) where T µ := sup κ ∈ [0 ,K ] | C (cid:48)(cid:48) µ ( κ ) | . A proof can be found in the appendix.
N. B ¨AUERLE AND D. SCHMITHALS Bounds for Exotic Options
As explained in the beginning we consider a risky asset at two time points ( S t , S t ) = ( X, Y ) . We are interested in finding upper bounds for prices of contingent claims of the form c ( X, Y ) , given a finite number of call prices on the same stock for both maturities t and t . A typicalexample for such a contingent claim would be an Asian option with payoff c ( X, Y ) = ( ( X + Y ) − K ) + . In what follows we denote by ( C µ , µ ) a consistent pair of price function and measurefor X and ( C ν , ν ) for Y . Since the general pricing theory implies that ( X, Y ) is a martingale, i.e. E [ Y | X ] = X we must have by Lemma 2.4 that µ ≤ c ν . We will restrict our consideration nowto finite discrete measures µ and ν since every measure µ can be approximated arbitrary well bya sequence of finite measures. Moreover we assume that S ⊃ S i.e. if a call price is observablefor strike k j for maturity t then a call price for the same strike is also observable for maturity t . We assume here that the observable call prices are such that an arbitrage-free pricing modelexists. Conditions for this are e.g. given in [11], Theorem 3.1. In case S = S these boil downto C ∗ µ and C ∗ ν being candidate functions and C ∗ µ ≤ C ∗ ν (see Corollary 4.1 in [11]).In what follows suppose that P is the space of all finite, discrete probability measures on R .We will denote by Q ∈ P the joint distribution of ( X, Y ). Then we define for two (discrete)probability measures µ ∈ P , ν ∈ P with µ ≤ c ν M ( µ, ν ) := { Q ∈ P : Q ( · , R + ) = µ, Q ( R + , · ) = ν, (cid:90) y Q ( x, dy ) = x, Q − a.s. } , (4.1) M ( µ, · ) := { Q ∈ M ( µ, ν ) : ν ∈ P , µ ≤ c ν } , µ ∈ P (4.2) M ( · , ν ) := { Q ∈ M ( µ, ν ) : µ ∈ P , µ ≤ c ν } , ν ∈ P (4.3) M := { Q ∈ M ( µ, ν ) : µ ∈ P , ν ∈ P , µ ≤ c ν } . (4.4)Note that by Lemma 2.4 the set M ( µ, ν ) is not empty if and only if µ ≤ c ν. The set M ( µ, ν ) isthe set of all so-called martingale transports given the knowledge of the marginal distributions µ and ν , i.e. the set of all potential pricing measures when we know the marginal distributions.However this is not the case in reality. Thus we consider the set M which consists of all potentialpricing measures which are consistent with the finitely many observable call prices. The nextresult can easily be checked: Lemma 4.1.
Let Q ∈ P be a probability measure and µ ≤ c ν . Then Q ∈ M ( µ, ν ) if and onlyif (a) (cid:80) x Q ( x, y ) = ν ( y ) , for all y ∈ supp ( ν ) , (b) (cid:80) y Q ( x, y ) = µ ( x ) for all x ∈ supp ( µ ) , (c) (cid:80) y Q ( x, y ) y = xµ ( x ) for all x ∈ supp ( µ ) , where supp ( µ ) and supp ( ν ) are the supports of measures µ and ν respectively. The first two equations guarantee the correct marginals, equation (c) expresses the martingaleproperty.We are now interested in finding sup Q ∈M E Q [ c ( X, Y )] (4.5)an upper price bound for the exotic option with payoff c , consistent with the given finite numberof call price observations. Unlike in classical transportation problems we cannot separate theoptimization problem into finding the best marginals and then the best copula, because inoptimal martingale transport the optimal transportation plan depends crucially on the margins.However, under some assumptions we can nevertheless solve the problem. Theorem 4.2. If µ ∈ P is an arbitrary, fixed (discrete) probability measure and y (cid:55)→ c ( · , y ) isconvex, then sup Q ∈M ( µ, · ) E Q [ c ( X, Y )] = sup Q ∈M ( µ,ν ∗ ) E Q [ c ( X, Y )] (4.6)
ONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS 7
Proof.
Let ν ∈ P be an arbitrary, consistent, discrete probability measure with µ ≤ c ν . ByLemma 3.3 we know that ν ≤ c ν ∗ . Choose ¯ Q ∈ M ( µ, ν ) and ˜ Q ∈ M ( ν, ν ∗ ) arbitrary. Then wedefine a new probability measure Q ∈ P as follows: Q ( x, z ) := (cid:88) y ¯ Q ( x, y ) ν ( y ) ˜ Q ( y, z ) , x ∈ supp ( µ ) , z ∈ supp ( ν ∗ ) . We show that Q ∈ M ( µ, ν ∗ ). First we check that Q has the correct marginal distributions. Forthis we use that ˜ Q and ¯ Q satisfy the condition of Lemma 4.1: (cid:88) z Q ( x, z ) = (cid:88) z (cid:88) y ¯ Q ( x, y ) ν ( y ) ˜ Q ( y, z ) = (cid:88) y ¯ Q ( x, y ) ν ( y ) (cid:88) z ˜ Q ( y, z ) = (cid:88) y ¯ Q ( x, y ) = µ ( x ) , (cid:88) x Q ( x, z ) = (cid:88) x (cid:88) y ¯ Q ( x, y ) ν ( y ) ˜ Q ( y, z ) = (cid:88) y ˜ Q ( y, z ) = ν ∗ ( z ) . Second the martingale property is also satisfied: (cid:88) z Q ( x, z ) z = (cid:88) z (cid:88) y ¯ Q ( x, y ) ν ( y ) ˜ Q ( y, z ) z = (cid:88) y ¯ Q ( x, y ) ν ( y ) (cid:88) z ˜ Q ( y, z ) z = (cid:88) y ¯ Q ( x, y ) y = xµ ( x ) . Finally the expected value of the payoff c ( X, Y ) under Q is larger than under ¯ Q , because (cid:88) x (cid:88) z Q ( x, z ) c ( x, z ) = (cid:88) x (cid:88) z (cid:88) y ¯ Q ( x, y ) ν ( y ) ˜ Q ( y, z ) c ( x, z )= (cid:88) x (cid:88) y ¯ Q ( x, y ) (cid:88) z ˜ Q ( y, z ) ν ( y ) c ( x, z ) ≥ (cid:88) x (cid:88) y ¯ Q ( x, y ) c (cid:16) x, (cid:88) z ˜ Q ( y, z ) ν ( y ) z (cid:17) = (cid:88) x (cid:88) y ¯ Q ( x, y ) c ( x, y ) . Thus E Q [ c ( X, Y )] ≥ E ¯ Q [ c ( X, Y )] for all ν ∈ P with µ ≤ c ν . This proves the statement. (cid:3) Theorem 4.2 implies that if the payoff function c is convex in the second component, then wecan fix ν ∗ as worst margin for the second component of the pricing measure, irrespective of thefirst component. Corollary 4.3. If y (cid:55)→ c ( · , y ) is convex, then sup Q ∈M E Q [ c ( X, Y )] = sup Q ∈M ( · ,ν ∗ ) E Q [ c ( X, Y )] (4.7)Discussing the first component is more involved. We need the following definition.
Definition 4.4.
Let Q be a measure on R .a) Q is called direct transport if Q = ωδ ( x,x ) for some ω ∈ (0 ,
1) and x ∈ R .b) Q is called two-way transport if Q = ϑ δ ( x,y ) + ϑ δ ( x,y ) for some ϑ , ϑ ∈ (0 ,
1) and x, y , y ∈ R with y < x < y and with the property( ϑ + ϑ ) x = ϑ y + ϑ y . Note that if Q ∈ P in part b), then Q ∈ M ( δ x , ϑ δ y + ϑ δ y ). The next lemma shows thatevery Q ∈ M ( µ, ν ) can be decomposed into a finite number of direct and two-way transports. N. B ¨AUERLE AND D. SCHMITHALS
Lemma 4.5.
Suppose µ, ν are two discrete probability measures. Then Q ∈ M ( µ, ν ) if and onlyif Q has margins µ and ν and can be written as a finite sum of direct and two-way transports.Proof. Suppose Q ∈ M ( µ, ν ). Consider the measure ˜ Q which we obtain when we fix x ∈ supp ( µ ):˜ Q = (cid:80) y ∈ supp ( ν ) δ ( x,y ) Q ( x, y ). The margins ˜ µ, ˜ ν of ˜ Q are by Lemma 4.1 in convex order and ˜ Q is the unique element in M (˜ µ, ˜ ν ). It can e.g. be constructed using the primal algorithm in Sec.5 of [2]. From this algorithm we see that ˜ Q can be decomposed into a finite number of directand two-way transports. But one could also start constructing the two-way transports from theextreme points in ˜ ν .This can be done with all x ∈ supp ( µ ) and the decomposition of Q follows.Contrary suppose Q = (cid:80) nj =1 Q j where Q j are direct or two-way transports and Q has theright margins. It is left to show that Q ∈ M ( µ, ν ), i.e. we have to show the martingale property.But this is true if it separately holds for all Q j which is the case by definition. (cid:3) The next lemma will be crucial for our main result:
Lemma 4.6.
Suppose µ ∈ P and µ has a positive mass κ > on a point ˜ x / ∈ S . Denote by x := max { x ∈ S ∪ supp ( µ ) : x < ˜ x } and by x := min { x ∈ S ∪ supp ( µ ) : x > ˜ x } . Then ˜ µ := µ − κδ ˜ x + κ x − ˜ xx − x δ x + κ ˜ x − x x − x δ x satisfies ˜ µ ∈ P , µ ≤ c ˜ µ and ˜ µ can be obtained from µ by a two-way transport.Proof. The situation is best illustrated with Figure 2 which shows a part of the functions C µ and C ˜ µ . x ˜ x x f f f Figure 2. C µ (solid line) and C ˜ µ (dashed line) on the interval [ x , x ] in thesituation of Lemma 4.6.Recall that C µ ( k ) := (cid:80) x ∈ supp ( µ ) ( x − k ) + µ ( x ), i.e. the function is piecewise linear and haskinks at points which have positive mass. The (black) solid line belongs to C µ . We will showthat the dashed (red) line belongs to C ˜ µ . In order to do so let us write down the three linearfunctions f , f , f in Figure 2.For x ∈ [ x , ˜ x ] we have f ( x ) = sx + c where s < c ∈ R .For x ∈ [˜ x, x ] we have f ( x ) = ( s + κ ) x + c . Both functions coincide at ˜ x , thus c = κ ˜ x + c . The functional form of the dashed (red) line is f ( x ) = f ( x ) − f ( x ) x − x x + ˜ c. Since f ( x ) = sx + κ ˜ x + c and f ( x ) = ( s + κ ) x + c we obtain f ( x ) − f ( x ) x − x = 1 x − x (cid:0) s ( x − x ) + κx − κ ˜ x (cid:1) = s + κ x − ˜ xx − x . ONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS 9
Thus we get f ( x ) = (cid:16) s + κ x − ˜ xx − x (cid:17) x + ˜ c. Since f ( x ) = f ( x ) and f ( x ) = f ( x ) we obtain ˜ c = c + κx x − x x − x . The additional mass ˜ µ places on x is the difference of the slopes of f and f at x + which is θ := κ x − ˜ xx − x ∈ (0 , κ )and the additional mass ˜ µ places on x is the difference of the slope of f and f at x − whichis given by κ − θ = κ ˜ x − x x − x . ˜ µ has no mass at ˜ x any more since there is no kink here. Hence f really corresponds to ˜ µ asdefined in the statement. That ˜ µ ≥ c µ is clear from the construction and Lemma 2.5. Moreover, C ˜ µ is still the same as C µ on ( −∞ , x ] and on [ x , ∞ ) thus it satisfies ˜ µ ∈ P . Finally note that δ (˜ x,x ) κ ˜ x − x x − x + δ (˜ x,x ) κ x − ˜ xx − x is a two-way transport. (cid:3) In order to deal with the second worst case margin for price bounds we need a special propertyof the payoff c . Definition 4.7.
A function c : R → R is called directionally convex if c is convex in bothcomponents and c is supermodular , i.e. for all x, y ∈ R f ( x ) + f ( y ) ≤ f ( x ∧ y ) + f ( x ∨ y ) , where x ∧ y = (min { x , y } , min { x , y } ) and x ∨ y = (max { x , y } , max { x , y } ). Remark 4.8.
For equivalent definitions of the directionally convex property, see [20] Theorem3.12.2. In particular if f ∈ C , then f is directionally convex if and only if ∂ ∂x i ∂x j f ( x ) ≥ , for all x ∈ R , i, j = 1 , . Next we can show
Theorem 4.9. If c is directionally convex, then sup Q ∈M E Q [ c ( X, Y )] = sup Q ∈M ( µ ∗ ,ν ∗ ) E Q [ c ( X, Y )] (4.8)
Proof.
We know already that we can fix the upper margin to be ν ∗ . Now let µ ∈ P be arbitraryand µ ∗ the convex upper bound construction from Lemma 3.3. Let Q ∈ M ( µ, ν ∗ ) be an arbitrarymartingale transport. We will show that as long as µ possess mass on atoms ˜ x which are nostrike prices (like in Lemma 4.6), the expectation E Q [ c ( X, Y )] can be increased.Suppose µ has a positive mass on a point ˜ x / ∈ S . By Lemma 4.6 there is a two-way transportfrom ˜ x which shifts mass to neighbouring points x and x which are either strike prices or otheratoms of µ . By Lemma 4.5 there is also a two-way transport from ˜ x to y , y ∈ supp ( ν ∗ ) in Q .In both transports a certain mass from ˜ x is transported and we consider the smaller of the twomasses. I.e. in the other transport we consider the respective part of the mass. Suppose thismass is κ >
0. More precisely we consider now the following two transports. The first between µ and ν ∗ : αδ (˜ x,y ) + ( κ − α ) δ (˜ x,y ) with 0 ≤ α ≤ κ ≤ µ and˜ µ (see Figure 3): θδ (˜ x,x ) + ( κ − θ ) δ (˜ x,x ) with 0 ≤ θ ≤ κ ≤ . Note that by definition and ourassumption on the strike prices we have that y ≤ x < x ≤ y .In particular by the definition of a two-way transport it holds that(i) αy + ( κ − α ) y = κ ˜ x .(ii) θx + ( κ − θ ) x = κ ˜ x . ˜ x y y ακ − α ˜ x x x θκ − θ Figure 3.
Left: Two-way transport from µ to ν ∗ . Right: Two-way transportfrom µ to ˜ µ Now we will replace the transport Q from µ to ν ∗ by one which yields a higher expectationand where the first margin is ˜ µ . We consider the following transport (see Figure 4): Q ∗ := Q − αδ (˜ x,y ) − ( κ − α ) δ (˜ x,y ) + ϑ δ ( x ,y ) + ( θ − ϑ ) δ ( x ,y ) + ϑ δ ( x ,y ) + ( κ − θ − ϑ ) δ ( x ,y ) where it holds that(iii) ϑ + ϑ = α .(iv) θx = ϑ y + ( θ − ϑ ) y .(v) θ − ϑ + κ − θ − ϑ = κ − α .(vi) ( κ − θ ) x = ϑ y + ( κ − θ − ϑ ) y .It is not difficult to see that (v) follows from (iii) and that (vi) follows from (ii), (iii) and (iv). x x y y ϑ κ − θ − ϑ ϑ θ − ϑ Figure 4.
Constructed transport in Q ∗ .We claim now that the expected reward from this new transport is larger, i.e. we show that c (˜ x, y ) α + c (˜ x, y )( κ − α ) ≤ c ( x , y ) ϑ + c ( x , y )( θ − ϑ )+ c ( x , y ) ϑ + c ( x , y )( κ − θ − ϑ ) . (4.9)In order to do this we first obtain from (ii) that θκ x + κ − θκ x = ˜ x ONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS 11 and thus by convexity of c in the first component: c (˜ x, y ) α + c (˜ x, y )( κ − α ) ≤ c ( x , y ) θκ α + c ( x , y ) θκ ( κ − α )+ c ( x , y ) κ − θκ α + c ( x , y ) κ − θκ ( κ − α ) . From (iii) and (iv) we get that ϑ = θ ( x − y ) y − y , ϑ = α − θ ( x − y ) y − y . (4.10)Thus it remains to show c ( x , y ) θκ α + c ( x , y ) θκ ( κ − α ) + c ( x , y ) κ − θκ α + c ( x , y ) κ − θκ ( κ − α ) ≤ c ( x , y ) ϑ + c ( x , y )( θ − ϑ ) + c ( x , y ) ϑ + c ( x , y )( κ − θ − ϑ )= c ( x , y ) θ ( x − y ) y − y + c ( x , y ) (cid:16) θ − θ ( x − y ) y − y (cid:17) + c ( x , y ) (cid:16) α − θ ( x − y ) y − y (cid:17) + c ( x , y ) (cid:16) κ − θ − α + θ ( x − y ) y − y (cid:17) . Rearranging the inequality is equivalent to0 ≤ c ( x , y ) (cid:16) θ ( x − y ) y − y − θκ α (cid:17) + c ( x , y ) (cid:16) θ − θ ( x − y ) y − y − θκ ( κ − α ) (cid:17) + c ( x , y ) (cid:16) α − θ ( x − y ) y − y − κ − θκ α (cid:17) + c ( x , y ) (cid:16) κ − θ − α + θ ( x − y ) y − y − κ − θκ ( κ − α ) (cid:17) . Simplifying the terms yields0 ≤ c ( x , y ) (cid:16) ( x − y ) − ακ ( y − y ) (cid:17) − c ( x , y ) (cid:16) ( x − y ) − ακ ( y − y ) (cid:17) − c ( x , y ) (cid:16) ( x − y ) − ακ ( y − y ) (cid:17) + c ( x , y ) (cid:16) ( x − y ) − ακ ( y − y ) (cid:17) . Since the expression in brackets is positive if and only if x > ˜ x which is true by assumptionthis is equal to 0 ≤ c ( x , y ) − c ( x , y ) − c ( x , y ) + c ( x , y )which is exactly the supermodularity from our assumption. Thus, when we replace Q by Q ∗ wehave by definition a new martingale transport from ˜ µ := µ − κδ x + θδ x + ( κ − θ ) δ x to ν ∗ whichstill satisfies ˜ µ ≤ c µ ∗ but yields a higher expectation. This procedure can be repeated until allmass is concentrated on the strike prices. But this yields a unique measure which is µ ∗ and thestatement is shown. (cid:3) Example 4.10.
An important example for a directionally convex payoff function is the payoff ofan Asian option c ( x, y ) = ( ( x + y ) − K ) + , x, y ≥
0. It is obviously convex in both componentsand supermodularity follows from the fact that if f : R → R is increasing and supermodu-lar, then max { f, c } for all c ∈ R is supermodular (see e.g. [3], Lemma 2.1 a)). For furtherconstructions of supermodular functions, see Lemma 2.1 in [3]. Remark 4.11. (a) Suppose that (4.8) is solved by Q ∗ which has not necessarily discretemargins µ and ν . Then it is possible to approximate µ and ν by discrete measures µ d ∈ P and ν d ∈ P such that for the optimal martingale transport Q d (cid:12)(cid:12)(cid:12) E Q d [ c ( X, Y )] − E Q ∗ [ c ( X, Y )] (cid:12)(cid:12)(cid:12) ≤ ε for arbitrary ε >
0. This follows since ( µ, ν ) (cid:55)→ E Q l ( µ,ν ) [ c ( X, Y )] is continuous withrespect to weak convergence where Q l is the left-monotone martingale transport (see[18]). It immediately implies that Theorem 4.9 is also true when we allow to optimize over all measures µ, ν (not only discrete ones) which are consistent with observable callprices.(b) Note that the method we use here can only be applied to upper bounds since there is nominimal element with respect to the convex order in the sets P i .(c) Once having the optimal transport for the upper bound, a super hedging strategy canbe obtained from the dual problem (see e.g. [4] for details).5. Convergence of Price Bounds
Of course we expect that the more call prices we observe, the closer the upper bound for theprice of the exotic option is to the true upper bound which we could compute if we would knowthe true pricing measures for the asset. Indeed under the assumption that the strikes of observedcall prices are given in a certain way, it is not only possible to show the convergence of the upperprice bound to the true upper price bound but also the speed of convergence can be estimated.We restrict here to the case that both margins have a compact support [0 , K ]. The situationwith unbounded support is more complicated, see e.g. [15]. Moreover, we assume that in model n call prices for strikes k nj := j n K, j = 0 , . . . , n are observable for both margins at t and t .We suppose that µ and ν are the true marginal distributions of S t and S t respectively. In thecase of a finite number of observable call prices we take as marginal distribution the constructionin Lemma 3.3 which yields µ ∗ n and ν ∗ n irrespective of whether or not this really yields the upperbound (for c directionally convex this is the case by Theorem 4.2). The lower index n in thenotation µ ∗ n refers to the number of observable call prices. In order to establish convergence andthe speed of convergence we need some more properties of c . More precisely we obtain: Theorem 5.1.
Let µ ≤ c ν with supp ( µ ) , supp ( ν ) ⊂ [0 , K ] . Let c : R → R be a Lipschitzcontinuous payoff function such that c yy exists. We denote by ˆΛ the Lipschitz constant of c andassume max { ˆΛ , sup ( x,y ) ∈ [0 ,K ] | c yy ( x, y ) |} ≤ Λ . Then, for any n ∈ N , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup Q ∈M ( µ ∗ n ,ν ∗ n ) E Q [ c ( X, Y )] − sup Q ∈M ( µ,ν ) E Q [ c ( X, Y )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M c n , (5.1) where M c = 12 K ˜Λ with ˜Λ = Λ · max { K, } . If we additionally suppose that C µ , C ν ∈ C ( R + ) ,then, for any n ∈ N , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup Q ∈M ( µ ∗ n ,ν ∗ n ) E Q [ c ( X, Y )] − sup Q ∈M ( µ,ν ) E Q [ c ( X, Y )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M d n +1 , (5.2) where M d = (7 T µ + 5 T ν ) K ˜Λ with T µ = sup κ ∈ [0 ,K ] | C (cid:48)(cid:48) µ ( κ ) | and T ν = sup λ ∈ [0 ,K ] | C (cid:48)(cid:48) ν ( λ ) | .Proof. We may reformulate the difference in (5.1) as the difference of the values of the dualproblems. Note that c is bounded on [0 , K ] . Thus we obtain with Theorem 1.1 in [6]sup Q ∈M ( µ ∗ n ,ν ∗ n ) E Q [ c ( X, Y )] = inf ( ϕ,ψ ) ∈D (cid:26)(cid:90) R + ϕ ( x ) µ ∗ n ( dx ) + (cid:90) R + ψ ( y ) ν ∗ n ( dy ) (cid:27) , sup Q ∈M ( µ,ν ) E Q [ c ( X, Y )] = inf ( ϕ,ψ ) ∈D (cid:26)(cid:90) R + ϕ ( x ) µ ( dx ) + (cid:90) R + ψ ( y ) ν ( dy ) (cid:27) , where D := { ( ϕ, ψ ) : ϕ + ∈ L ( R , µ ) , ψ + ∈ L ( R , ν ) , and for some h ∈ L ∞ ( R ) ,ϕ ( x ) + ψ ( y ) + h ( x )( y − x ) ≥ c ( x, y ) , ( x, y ) ∈ R } Now let us apply Theorem 2.4 and Remark 2.5 in [6]. For this purpose, we have to provethat the conditions are satisfied. By assumption and by construction respectively, we have that µ ≤ c ν and µ ∗ n ≤ c ν ∗ n are compactly supported. The payoff function c is Lipschitz continuouson [0 , K ] × [0 , K ] = conv( supp ( ν )) × conv( supp ( ν )) with constant ˆΛ ≤ Λ. It remains to show
ONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS 13 that there is a Lipschitz continuous function u : [0 , K ] = conv( supp ( ν )) → R such that y (cid:55)→ c ( x, y ) + u ( y ) is concave on [0 , K ] for µ -almost every x ∈ R + . As c yy ≤ Λ on [0 , K ] , it is clearthat u ( y ) := − Λ2 y is such a function with Lipschitz constant Λ K . We define ˜Λ := Λ · max { K, } .Thus, by Theorem 2.4 in [6] there exist solutions ( ϕ ∗ , ψ ∗ ) and ( ϕ ∗ n , ψ ∗ n ) for the dual problemswith respect to ( µ, ν ) and ( µ ∗ n , ν ∗ n ) respectively. By Remark 2.5 in [6] ϕ ∗ and ϕ ∗ n are Lipschitzcontinuous with constant 7 ˜Λ, and ψ ∗ and ψ ∗ n are Lipschitz continuous with constant 5 ˜Λ. Hence,we have sup Q ∈M ( µ ∗ n ,ν ∗ n ) E Q [ c ( X, Y )] − sup Q ∈M ( µ,ν ) E Q [ c ( X, Y )]= inf ( ϕ,ψ ) ∈D (cid:26)(cid:90) R + ϕ ( x ) µ ∗ n ( dx ) + (cid:90) R + ψ ( y ) ν ∗ n ( dy ) (cid:27) − inf ( ϕ,ψ ) ∈D (cid:26)(cid:90) R + ϕ ( x ) µ ( dx ) + (cid:90) R + ψ ( y ) ν ( dy ) (cid:27) ≤ (cid:90) R + ϕ ∗ ( x ) µ ∗ n ( dx ) + (cid:90) R + ψ ∗ ( y ) ν ∗ n ( dy ) − (cid:18)(cid:90) R + ϕ ∗ ( x ) µ ( dx ) + (cid:90) R + ψ ∗ ( y ) ν ( dy ) (cid:19) = (cid:90) R + ϕ ∗ ( x ) ( µ ∗ n − µ ) ( dx ) + (cid:90) R + ψ ∗ ( y ) ( ν ∗ n − ν ) ( dy ) ≤ W ( µ, µ ∗ n ) + 5 ˜Λ W ( ν, ν ∗ n ) , where in the last inequality we scale the integrands by their Lipschitz constants and then usethe dual representation of the Wasserstein distance in Remark 3.5. Completely analogous, butusing ϕ ∗ n and ψ ∗ n in the first inequality instead of ϕ ∗ and ψ ∗ , we obtainsup Q ∈M ( µ,ν ) E Q [ c ( X, Y )] − sup Q ∈M ( µ ∗ n ,ν ∗ n ) E Q [ c ( X, Y )] ≤ W ( µ, µ ∗ n ) + 5 ˜Λ W ( ν, ν ∗ n ) . Using the estimates in (3.2) and (3.3), we have the claimed convergence speed estimates. (cid:3)
Remark 5.2.
Note that the rate of convergence in Theorem 5.1 cannot be improved. We showthis by an example in the appendix.6.
Numerical Examples
An Upper Price Bound Example with Real Data.
Table 1.
Observed call prices on May 29th, 2019 in euro.Figure 5 shows the interpolated call price function. Using Lemma 3.3 we can derive µ ∗ and ν ∗ . What we clearly see in the picture is that C µ ∗ ≤ C ν ∗ hence µ ∗ ≤ c ν ∗ follows from Lemma Figure 5.
Interpolated call prices on SAP stock with maturity 17.6.19 (solid)and 12.8.19 (dashed).We obtain atom 90 95 100 105 110 115 120 125 µ ∗ ( · ) 0.895 0.002 0.006 0.01 0.03 0.05 0.004 0.003 ν ∗ ( · ) 0.9004 0.001 0.011 0.015 0.023 0.0167 0.0148 0.0181 Table 2.
Density of marginal distributions.Note that the expectation of µ ∗ and ν ∗ is 92 . c ( x, y ) = (cid:16) ( x + y ) − K (cid:17) + where x is the stock price at 17.6.2019 and y the stock price at 12.08.2019. When we choose k =120 we obtain the price bound 0 . µ ∗ and ν ∗ we obtain for the same option the price 0 . Numerical Convergence.
Let us now discuss the convergence speed of the price boundapproximation for different compactly supported theoretical marginals and payoff functions nu-merically. Therefore, we calculate the approximating upper price bounds for several n ∈ N aswell as the real upper price bounds as far as possible. Additionally, we calculate the correspond-ing normalized price bound differences d n := 2 n (cid:16) sup Q ∈M ( µ ∗ n ,ν ∗ n ) E Q [ c ( X, Y )] − sup Q ∈M ( µ,ν ) E Q [ c ( X, Y )] (cid:17) . In the case of uniform distributed margin, it is possible to compute the real upper price boundexplicitly (see e.g. [14]) when c has certain properties (like the Martingale Spence Mirrleescondition).In what follows U [ a, b ] denotes the uniform distribution on interval [ a, b ].1. Let µ ∼ U [1 , , ν ∼ U [0 , k = 4. Here, we partition the support of ν in maximally2048 intervals, i.e. we have a difference of between two partition points. We considerdifferent exotic options. ONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS 15 (a) c ( x, y ) = xy . Note that c is directionally convex on R . The real upper pricebound can be computed explicitly here, since the upper bound is attained at theleft-monotone martingale transport (see e.g. [14]). It is given bysup Q ∈M ( µ,ν ) E Q [ c ( X, Y )] = E µ (cid:34) X (cid:32) (cid:18) X + 12 (cid:19) + 14 (cid:18) − X (cid:19) (cid:33)(cid:35) = 12 . . For all n = 3 , . . . ,
11, we calculate the approximate upper price bound P ( µ ∗ n , ν ∗ n ) =sup Q ∈M ( µ ∗ n ,ν ∗ n ) E Q [ c ( X, Y )] and d n . This yields the results of Table 3. n 3 4 5 6 7 8 9 10 11 P ( µ ∗ n , ν ∗ n ) 12.808 12.57 12.517 12.504 12.501 12.5002 12.50006 12.50002 12.500004 d n Table 3.
Approximation results in the case 1.a)(b) c ( x, y ) = exp( x ) · y . Note that c is directionally convex on R . The real upperprice bound is 61 . . The approximate upper price bounds P ( µ ∗ n , ν ∗ n ) are given inTable 4. n 3 4 5 6 7 8 9 10 11 P ( µ ∗ n , ν ∗ n ) 65.8620 62.7911 62.0990 61.9338 61.8934 61.8834 61.8810 61.8803 61.8802 d n Table 4.
Approximation results in the case 1.b)2. Let µ ∼ U [9 , , ν ∼ U [0 , k = 20. Here, we partition the support of ν inmaximally 2048 intervals, i.e. we have a difference of between two partition points.(a) c ( x, y ) = xy . The real upper price bound issup Q ∈M ( µ,ν ) E Q [ c ( X, Y )] = E µ (cid:34) X (cid:32) (cid:18) X − (cid:19) + 920 (cid:18) − X (cid:19) (cid:33)(cid:35) = 1356 . . For all n = 3 , . . . ,
11, we calculate the approximate upper price bound P ( µ ∗ n , ν ∗ n )and d n . This yields the results of Table 5. n 3 4 5 6 7 8 9 10 11 P ( µ ∗ n , ν ∗ n ) 1421 1367.2 1359.35 1357.206 1356.676 1356.543 1356.511 1356.503 1356.501 d n Table 5.
Approximation results in the case 2.a)(b) c ( x, y ) = exp( x ) · y . The real upper price bound is 4041627 . . The approximateupper price bounds ( µ ∗ n , ν ∗ n ) are given in Table 6. n 4 5 6 7 8 9 10 11 P ( µ ∗ n , ν ∗ n ) 4826637 4236165 4093466 4054268 4044652 4042391 4041818 4041675 d n Table 6.
Approximation results in the case 2.b)
Appendix
Proof of Theorem 3.6.
Proof.
By Remark 3.5, we have W ( µ, µ ∗ ) = (cid:90) ∞−∞ | F µ ( t ) − F µ ∗ ( t ) | dt. In order to calculate the integral, we plug in the distribution function representations (2.2) usingthe call option price function C µ and use Lemma 3.2 to obtain F µ ∗ . Then we have W ( µ, µ ∗ ) = (cid:90) K (cid:12)(cid:12) C (cid:48) µ ( t +) − F µ ∗ ( t ) (cid:12)(cid:12) dt = n − (cid:88) j =0 (cid:90) k j +1 k j (cid:12)(cid:12)(cid:12)(cid:12) C (cid:48) µ ( t +) − C µ ( k j +1 ) − C µ ( k j ) k j +1 − k j (cid:12)(cid:12)(cid:12)(cid:12) dt. Note that C ∗ µ ( k j ) = C µ ( k j ) for j = 0 , . . . n . In the following, let us abbreviate m j := C µ ( k j +1 ) − C µ ( k j ) k j +1 − k j . Since F µ ∗ is constant on [ k j , k j +1 ) for all j = 0 , . . . , n − θ j ∈ [ k j , k j +1 ]such that for all t ∈ [ k j , θ j ), we have F µ ( t ) ≤ F µ ∗ ( t ) , or equivalently C (cid:48) µ ( t +) ≤ m j , and for all t ∈ [ θ j , k j +1 ), we have F µ ( t ) ≥ F µ ∗ ( t ) , or equivalently C (cid:48) µ ( t +) ≥ m j . Thus, we have W ( µ, µ ∗ ) = n − (cid:88) j =0 (cid:34)(cid:90) θ j k j (cid:0) m j − C (cid:48) µ ( t +) (cid:1) dt + (cid:90) k j +1 θ j (cid:0) C (cid:48) µ ( t +) − m j (cid:1) dt (cid:35) . (6.1)We stress that the set of points t ∈ R + such that C (cid:48) µ ( t − ) (cid:54) = C (cid:48) µ ( t +) is a Lebesgue null set.Hence, integrating over the right derivative C (cid:48) µ ( · +), we receive C µ ( · ). Based on (6.1), we thusobtain W ( µ, µ ∗ ) = n − (cid:88) j =0 (cid:20) m j ( θ j − k j ) − ( C µ ( θ j ) − C µ ( k j )) + ( C µ ( k j +1 ) − C µ ( θ j )) − m j ( k j +1 − θ j ) (cid:21) = n − (cid:88) j =0 (cid:34) C µ ( k j +1 ) − C µ ( k j ) k j +1 − k j ( θ j − k j ) − ( C µ ( θ j ) − C µ ( k j ))+ ( C µ ( k j +1 ) − C µ ( θ j )) − C µ ( k j +1 ) − C µ ( k j ) k j +1 − k j ( k j +1 − θ j ) (cid:35) = n − (cid:88) j =0 k j +1 − k j (cid:20) ( C µ ( k j +1 ) − C µ ( k j )) ( θ j − k j ) − ( C µ ( θ j ) − C µ ( k j )) ( k j +1 − k j )+ ( C µ ( k j +1 ) − C µ ( θ j )) ( k j +1 − k j ) − ( C µ ( k j +1 ) − C µ ( k j )) ( k j +1 − θ j ) (cid:21) . If we now add a suitable zero and rearrange the terms, then we obtain W ( µ, µ ∗ ) = n − (cid:88) j =0 k j +1 − k j (cid:20) ( C µ ( k j +1 ) − C µ ( θ j )) ( θ j − k j ) − ( C µ ( θ j ) − C µ ( k j )) ( k j +1 − θ j ) (cid:21) = 2 n − (cid:88) j =0 λ j C µ ( k j +1 ) + (1 − λ j ) C µ ( k j ) − C µ ( θ j )= 2 n − (cid:88) j =0 C µ ∗ ( θ j ) − C µ ( θ j ) , where we use λ j := θ j − k j k j +1 − k j and the linearly interpolating definition of C µ ∗ . By the choice of θ j , we have that the slope of the line through C µ ( k j ) and C µ ( k j +1 ) is in [ C (cid:48) µ ( θ j − ) , C (cid:48) µ ( θ j +)], i.e. ONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS 17 it equals C (cid:48) µ ( θ j ) whenever the derivative exists. In particular, the distance of C µ ∗ and C µ on[ k j , k j +1 ) is maximal in θ j . That is, θ j = argmax t ∈ [ k j ,k j +1 ) | C µ ∗ ( t ) − C µ ( t ) | . Thus, we have the desired representation W ( µ, µ ∗ ) = 2 · n − (cid:88) j =0 sup t ∈ [ k j ,k j +1 ) | C µ ∗ n ( t ) − C µ ( t ) | . Now we turn to the estimate in (3.2). For this purpose, we consider the slope C (cid:48) µ ( t +) for t ∈ [ k j , k j +1 ). In particular, we have C (cid:48) µ ( t +) (cid:40) ≥ C (cid:48) µ ( k j +) , t ∈ [ k j , θ j ) ≤ C (cid:48) µ ( k j +1 +) , t ∈ [ θ j , k j +1 ) . Using equation (6.1), we get W ( µ, µ ∗ ) ≤ n − (cid:88) j =0 (cid:34)(cid:90) θ j k j (cid:0) m j − C (cid:48) µ ( k j +) (cid:1) dt + (cid:90) k j +1 θ j (cid:0) C (cid:48) µ ( k j +1 +) − m j (cid:1) dt (cid:35) = n − (cid:88) j =0 (cid:34) (cid:90) θ j k j (cid:18) C µ ( k j +1 ) − C µ ( k j ) k j +1 − k j − C (cid:48) µ ( k j +) (cid:19) dt + (cid:90) k j +1 θ j (cid:18) C (cid:48) µ ( k j +1 +) − C µ ( k j +1 ) − C µ ( k j ) k j +1 − k j (cid:19) dt (cid:35) = n − (cid:88) j =0 (cid:34) (cid:18) C µ ( k j +1 ) − C µ ( k j ) k j +1 − k j − C (cid:48) µ ( k j +) (cid:19) ( θ j − k j )+ (cid:18) C (cid:48) µ ( k j +1 +) − C µ ( k j +1 ) − C µ ( k j ) k j +1 − k j (cid:19) ( k j +1 − θ j ) (cid:35) . (6.2)When we apply the inequalities θ j ≤ k j +1 and − θ j ≤ − k j in (6.2) (note that the terms inbrackets are non-negative), then we obtain W ( µ, µ ∗ ) ≤ n − (cid:88) j =0 (cid:0) C (cid:48) µ ( k j +1 +) − C (cid:48) µ ( k j +) (cid:1) ( k j +1 − k j )= K n n − (cid:88) j =0 (cid:0) C (cid:48) µ ( k j +1 +) − C (cid:48) µ ( k j +) (cid:1) ≤ K n . (6.3)In order to obtain the estimate in (3.3), we use the fact that the slopes get closer and closerwhen n increases. We assume that C µ ∈ C ( R + ) and rewrite the right hand side of (6.2). Thenwe have W ( µ, µ ∗ ) ≤ n − (cid:88) j =0 (cid:34) (cid:0) C µ ( k j +1 ) − C µ ( k j ) − C (cid:48) µ ( k j )( k j +1 − k j ) (cid:1) (cid:18) θ j − k j k j +1 − k j (cid:19) + (cid:0) C (cid:48) µ ( k j +1 )( k j +1 − k j ) − C µ ( k j +1 ) + C µ ( k j ) (cid:1) (cid:18) k j +1 − θ j k j +1 − k j (cid:19) (cid:35) . Now let us use the Theorem of Taylor. In particular, for two times continuously differentiablefunctions f : R → R , we obtain f ( x ) = f ( a ) + f (cid:48) ( a )( x − a ) + (cid:90) xa ( x − t ) f (cid:48)(cid:48) ( t ) dt. If we now apply this formula in the form f ( x ) − f ( a ) − f (cid:48) ( a )( x − a ) = (cid:90) xa ( x − t ) f (cid:48)(cid:48) ( t ) dt =: R f ( x, a ) , for f ≡ C µ with x = k j +1 and a = k j , and with x = k j and a = k j +1 , then we obtain W ( µ, µ ∗ ) ≤ n − (cid:88) j =0 (cid:18)(cid:18) θ j − k j k j +1 − k j (cid:19) R C µ ( k j +1 , k j ) + (cid:18) k j +1 − θ j k j +1 − k j (cid:19) R C µ ( k j , k j +1 ) (cid:19) . The well-known general Taylor residual estimate states that we have | R f ( x, a ) | ≤ sup ξ ∈ ( a − r,a + r ) (cid:12)(cid:12)(cid:12)(cid:12) f (cid:48)(cid:48) ( ξ )2 ( x − a ) (cid:12)(cid:12)(cid:12)(cid:12) for all x ∈ ( a − r, a + r ). Choosing r = K n + ε , ε >
0, we achieve | R C µ ( k j +1 , k j ) | ≤ sup t ∈ ( k j − − ε,k j +1 + ε ) (cid:12)(cid:12)(cid:12)(cid:12) C (cid:48)(cid:48) µ ( t )2 ( k j +1 − k j ) (cid:12)(cid:12)(cid:12)(cid:12) = sup t ∈ ( k j − − ε,k j +1 + ε ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C (cid:48)(cid:48) µ ( t )2 (cid:18) K n (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ T µ · K · − (2 n +1) and analogously | R C µ ( k j , k j +1 ) | ≤ T µ · K · − (2 n +1) . Thus, we get W ( µ, µ ∗ ) ≤ n − (cid:88) j =0 (cid:18) θ j − k j k j +1 − k j + k j +1 − θ j k j +1 − k j (cid:19) T µ · K · − (2 n +1) = 2 n · T µ · K · − (2 n +1) = T µ · K n +1 , which is the desired estimate and thus ends the proof. (cid:3) Speed of Convergence in Theorem 5.1 cannot be improved.
In this example, weshow that the convergence speed proven in Theorem 5.1 is maximal in the sense that it can notbe improved in general. For this purpose, we consider the two discrete measures µ = 14 δ + 12 δ + 14 δ and ν = 14 δ + 12 δ + 14 δ . These measures have mass 1, mean and the call option price functions C µ ( k ) = (cid:18) − k (cid:19) { ≤ k ≤ } + (cid:18) − k (cid:19) { ≤ k ≤ } + (cid:18) − k (cid:19) { ≤ k ≤ } ,C ν ( (cid:96) ) = (cid:18) − (cid:96) (cid:19) { ≤ (cid:96) ≤ } + (cid:18) − (cid:96) (cid:19) { ≤ (cid:96) ≤ } . We easily see that C µ ≤ C ν and thus µ ≤ c ν . As payoff function we choose c ( x, y ) = xy . Theupper price bounds can be computed as328 c (cid:18) , (cid:19) + 1128 c (cid:18) , (cid:19) + 116 c (cid:18) , (cid:19) + 316 c (3 ,
4) = 91354 . In order to prove the optimality of the convergence speed, we calculate the approximatingmeasures µ ∗ n and ν ∗ n and the associated price bounds for general n ≥
3. The measures have thestructure µ ∗ n = 14 ( δ + δ ) + µ rn and ν ∗ n = 14 ( δ + δ ) + ν rn , for all n ∈ N , where µ rn and ν rn are also measures with two atoms close to each. Thisfollows from the determination technique of the approximating measures based on the associatedpiecewise linearly interpolated call option price functions C µ ∗ n and C ν ∗ n . Indeed, these deviate ONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS 19 from the functions C µ and C ν only on the interval (cid:0) k j ( n ) , k j ( n )+1 (cid:1) , where j ( n ) is such that k j ( n ) < < k j ( n )+1 . Consequently, k j ( n ) and k j ( n )+1 are the atoms of the residual measures.We determine the general structure of these values and denote j = j ( n ) and k j = k j ( n ) forthe rest of the example. We have k j < < k j +1 ⇐⇒ · j n < < · j + 12 n ⇐⇒ j < · n < j + 1 . As j ∈ N , we clearly have j = (cid:22) · n (cid:23) = (cid:40) · n − , n even , · n − , n odd , from which we immediately get k j = 4 · j n = (cid:40) − · n , n even , − · n , n odd , and k j +1 = 4 · j + 12 n = (cid:40) + · n , n even , + · n , n odd . The masses of µ ∗ n and ν ∗ n in the atoms δ k j and δ k j +1 are the differences of the slopes of C µ ∗ n and C ν ∗ n on the intervals ( k j , k j +1 ) and ( k j − , k j ), and ( k j +1 , k j +2 ) and ( k j , k j +1 ) respectively.As C µ and C ν have the same slopes in these areas, we know that the masses ω nj , ϑ nj of the atomsare equal for µ ∗ n and ν ∗ n . Hence, we have ω nj = ϑ nj = m nj − (cid:18) − (cid:19) and ω nj +1 = ϑ nj +1 = − − m nj , where m nj = C µ ( k j +1 ) − C µ ( k j ) k j +1 − k j = n ( C µ ( k j +1 ) − C µ ( k j )) = n ( C ν ( k j +1 ) − C ν ( k j )) . Using therepresentations of C µ and C ν , we deduce m nj = 2 n (cid:18) − k j +1 − (cid:18) − k j (cid:19)(cid:19) = 2 n (cid:18) − −
14 ( k j +1 − k j ) + 12 k j (cid:19) = 2 n (cid:18) k j − (cid:19) − (cid:40) n (cid:0) (cid:0) − · n − (cid:1)(cid:1) − = − , n even , n (cid:0) (cid:0) − · n − (cid:1)(cid:1) − = − , n odd.This finally implies ω nj = ϑ nj = (cid:40) , n even , , n odd , and ω nj +1 = ϑ nj +1 = (cid:40) , n even , , n odd . In total, we have the general structure µ ∗ n = 14 ( δ + δ ) + 16 (cid:0) δ k j + δ k j +1 (cid:1) + 16 (cid:0) δ k j { n even } + δ k j +1 { n odd } (cid:1) ,ν ∗ n = 14 ( δ + δ ) + 16 (cid:0) δ k j + δ k j +1 (cid:1) + 16 (cid:0) δ k j { n even } + δ k j +1 { n odd } (cid:1) . By construction we have µ ∗ n ≤ c ν ∗ n . The approximate upper price bound is then given by c (1 , k j ) · k j + c ( k j , k j ) · (cid:18) − k j (cid:19) + c ( k j , k j +1 ) · k j +1 + c ( k j +1 , k j +1 ) · (cid:18) − k j +1 (cid:19) + c ( k j +1 , ·
116 + c (3 , · k j k j − k j k j · k j +1 k j +1 − k j +1 k j +1 + 9 . Plugging in the derived representations of k j and k j +1 , we get9 + 712 − · n + 73 + 83 · n + k j k j +1 k j · k j +1 − k j − k j +1 . If we now also plug in the representations for the higher degree terms and rearrange the former,then we achieve 14312 + 73 · n + 13 (cid:32)(cid:18) (cid:19) − (cid:18) (cid:19) · n + 73 · · n − · n (cid:33) + 16 (cid:32)(cid:18) (cid:19) + (cid:18) (cid:19) · n + 73 · · n + 51227 · n (cid:33) + 14 (cid:18)
499 + 289 · n − · n − (cid:18) − · n + 169 · n (cid:19) − (cid:18)
499 + 1129 · n + 649 · n (cid:19) (cid:19) = 91354 + 8454 · n + O (cid:18) n (cid:19) . Analogously, for n ≥ · n + O (cid:18) n (cid:19) . This is indeed the convergence speed from Theorem 5.1.
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Department of Mathematics, Karlsruhe Institute of Technology (KIT), D-76128Karlsruhe, Germany
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