No-Arbitrage Bounds on Two One-Touch Options

No-Arbitrage Bounds on Two One-Touch Options Yukihiro Tsuzuki March 30, 04 Abstract This paper investigates the pricing bounds of two one-touch options with the same maturity but different barrier levels, where the pricing bound is a range within which a one-touch option can take a price when a price of another one-touch option is given. The upper or lower bounds are the cost of a super-replicating portfolio a sub-replicating portfolio respectively. These consist of call options, put options, digital options a one-touch option. We assume that the underlying process is a continuous martingale, but do not postulate a model. Keywords: barrier option, one-touch option, model-independent, super-replication, pricing bounds JE Classification: G3 Mathematics Subject Classification (00: 9G0 Introduction This paper investigates pricing bounds within which a one-touch option can take a price when the price of another one-touch option with the same maturity but a different barrier level is given. Financial markets trade many barrier option types such as single/double barrier knock-in/-out options. Of these, one-touch no-touch options are the simplest barrier options widely are traded. A one-touch option is a barrier option that pays a unit of currency at the maturity if the barrier is hit is worthless if the barrier has not been hit. In contrast, a no-touch option is worthless if the barrier is hit. These are important instruments for traders of barrier options, because they reflect a market view of the probability of the barrier being hit. There has been considerable research on pricing hedging barrier options. In particular, researchers have proposed several methods that semi-statically hedge barrier options (see e.g. Carr Chou (997, Carr et al. (998 Derman et al. (995. Here, semi-static hedging means the replication of barrier options by trading European puts calls no more than once after inception. Hedging strategies require options, thus models that price barrier options must be calibrated to these. However, even if the model is perfectly calibrated to a volatility surface there are risks attached to the valuation of barrier options. For instance, Hirsa et al. (003, ipton McGhee (00 Schoutens et al. (005 all state that although models may produce similar European put call option prices, they give markedly different barrier option All the contents expressed in this research are solely those of the authors do not represent the view of any institutions. The authors are not responsible or liable in any manner for any losses /or damages caused by the use of any contents in this research. Graduate School of Economics, University of Tokyo, 7-3-, Hongo, Bunkyo-ku, Tokyo, 3-0033, Japan, Phone: +8-3-584-068, E-mail: yukihirotsuzuki@gmail.com The author is grateful to Professor Akihiko Takahashi (Graduate School of Economics, University of Tokyo for his guidance. Electronic copy available at: http://ssrn.com/abstract667

prices. Touch options are recognized as important products because they are used as an instrument to which a model is calibrated (see e.g. Carr Crosby (00. The model-independent approach has also been considered for exotic derivatives including barrier options (see e.g. Hobson (998, Hobson Neuberger (0, abordére et al. (0 Hobson Klimmek (0. In particular, Brown et al. (00 propose robust super-replicating sub-replicating barrier option strategies including touch-options without assuming any specific models. Cox Oblój (0a Cox Oblój (0b focus on touch options with two barrier levels in the same manner as Brown et al. (00. They use call options put options as well as digital options with the same maturity as replicating instruments trade forward contracts at the first barrier(s hitting time(s. Generally, pricing bounds derived from model-independent replications tend to be rather wide, which is also the case for touch-options. Hence, it is worth investigating how much these pricing bounds are refined if other instruments are traded. This paper investigates pricing bounds within which a one-touch option can take a price when a price of another one-touch option with the same maturity but a different barrier level is given those European options (including call, put digital options with the same maturity. Suppose there is a pricing operator on European options with a certain maturity a touch option with the same maturity a certain barrier level. The question is how to extend this pricing operator to a space spanned by a touch option with the same maturity but a different barrier level as well as these derivatives. To address this, we propose pricing operators that provide upper lower bounds for the touch option based on a super-replication a sub-replication. Our approach is in line with Brown et al. (00, Cox Oblój (0a Cox Oblój (0b, in that we assume the underlying asset process is a continuous martingale our replications consist of static portfolios transactions of a forward contract in the first instances of hitting the barrier levels. We differentiate by using a touch option as well as European options for the static portfolios. Moreover, we provide pricing bounds on a touch/no-touch option that pays one unit of currency if only if the first barrier is hit but the second is not. In Section 4, we consider the pricing bounds on this touch/no-touch option using the one-touch option with the second barrier as well as European options. If we use, instead of the one-touch option, the upper- or lower bounds the super- or sub-replications on this, we obtain the pricing bounds as well as super- sub-replications of the touch/no-touch option using only European options. The next section of this paper describes the settings notations. The third section reviews the research of Brown et al. (00. The super-replications sub-replications for a one-touch option using another with a different barrier level are derived in the fourth section. The fifth section provides numerical examples. Settings Notations The settings notations used in this paper are stated here. First, we introduce some notations. et us denote the spot price of the underlying asset at time t 0, T ] by S t, where T is some arbitrary time horizon the time-t price of a call option a put option with strike K maturity T 0, T by C t (K P t (K respectively. The one-touch option is assumed to be a single knock-in option with maturity T barrier level B (S 0, +. This option is worthless if B has not been hit by the expiration date. If the barrier is hit at any time during the option s life, the terminal payoff is. Then, the payoff of the barrier option is {τb T }, where τ B is the first time of hitting B: τ B : τ B (S : inf{t < T S t B}. (. A time-t price of this option is denoted as O t (B. The subscript t may be omitted in case of t 0 such as C(K, P (K O(B for simplicity. Second, we make some assumptions. The first assumption is that the underlying price process S is a non-negative martingale. The interest rates are also assumed to be zero. This is merely for simplicity, since Electronic copy available at: http://ssrn.com/abstract667

our results are valid by reading all prices of all options portfolios as forward T prices in case of a non-zero interest rate. Examples to which our results are applied are that the underlying process is a forward price or that the underlying asset pays continuous dividends equal to the interest rate. We assume that forward transactions are costless all instruments such as underlying asset, forward, are traded without transaction costs. Importantly, we assume that the underlying price process is continuous. This allows us to exchange a call option with strike K, with (B K amounts of cash a put option with the same strike by trading a forward contract with zero cost at the first time of hitting B, since the following parity holds: C τb (K P τb (K B K. (. This type of trade is used throughout this paper. Moreover, we add an assumption in Section 4. 4. that the distribution of the underlying asset at maturity T under a risk-neutral measure is given. This distribution is centered at S 0. We consider the case where only a finite number of call options are known in Section 4.3. Knowledge of the distribution is equivalent to the knowledge of European call option prices without arbitrage opportunities for the continuum of strikes by Breeden itzenberger (978. The conditions for no arbitrage are well-documented in Davis Hobson (007. We assume C(B > 0 to avoid a trivial case. We denote by ν the risk-neutral distribution of the spot price at maturity T determined by prices of these options. It is also assumed that call options, put options digital call options can be used as replication, where the digital call option with strike K is an option whose payoff is {K ST } in this paper. Third, we state the aim of this paper: to extend a pricing operator φ that is a linear operator defined on X : (0, +, ν, a set of ebesgue integrable functions on 0, + with respect to ν, which associates a payoff of an European option with its initial price such as φ(k K, φ((s T K + C(K, φ((k S T + P (K. If a price of a one-touch option whose payoff is {τb T } is known, we can extend the operator φ to X Y, where Y is a linear space spanned by {τb T } means a direct sum. This paper examines how to extend the operator φ to X Y Ỹ, where Ỹ is a linear space spanned by {τ B T } with another barrier level B. To address this, we propose sharp pricing bounds on one-touch options the corresponding replicating strategies, where sharpness means that the pricing bounds can not be improved without adding any other assumption. The lower upper bounds on the option are defined as follows under our settings: ] W : inf E {τ P B(S T } (.3 ] W G : sup E {τ B(S T }, (.4 P where P is a set of all risk-neutral probability spaces (Ω, F, Q a continuous martingale process {S t } t 0,T ] on it that satisfies φ( E ] on X Y E is an expectation operator corresponding to the probability space. Prices of super-replicating sub-replicating portfolios are superior inferior, but not necessarily equal, to W G W respectively. To prove the] sharpness, we find super-replicating sub-replicating portfolios whose prices are equal to E {τ B(S T } with respect to a certain element of P. Finally, we introduce some further technical notations. Every function f considered in this paper is a combination of the call price function C. We exp the domain of the function f from 0, + to R by C(K : C(0 K for K < 0 (recall that we assume that the underlying process is non-negative. The function has left- right-sided directional derivatives as does the function C. In this paper, we denote K as the left-sided derivative operator. Moreover, the derivatives have finite total variations the derivative KK can be defined except for a countable set. The subdifferential of a function f at K can be defined is denoted by Kf(K : {k R f(κ f(k + k(κ K, κ R}. (.5 We introduce the following notation for simplicity: N ( Kf : {K R 0 Kf(K}. (.6 3

3 Review of Brown et al. (00 In this section, we review the replications for a one-touch option with only European options, as proposed by Brown et al. (00, because we use these results in Section 4. The one-touch option is assumed to have a barrier level B, where S 0 < B. First, we prepare the following lemma: emma. Suppose that there is a measurable set Ω 0 F such that S T B, + on Ω 0 ES T : Ω 0 ] BQ Ω 0 ]. Then, there exists a continuous martingale {S t } t 0,T ] such that S T S T Q τ B (S T ] Q Ω 0]. Proof. et X 0, X X be rom variables defined as X 0 : S 0, X : S T X : B Ω0 + β Ω c 0, (3. where β : B B S 0 Q Ω c 0 ]. (3. Note that β < B E S T : Ω c 0] βq Ω c 0]. Then, {X n } n0,, is a discrete martingale with respect to a filtration generated by X. By Dudley s theorem (see, for instance, p.88 of Karatzas Shreve (988, the rom variables X X 0, (X X Ω0 (X X Ω c 0 can be expressed with stochastic integrals with respect to the Winner processes. A continuous martingale process S t such that Q τ B(S T ] Q Ω 0] can be constructed by these stochastic integrals. 3. Super-Replication Consider the following self-financing strategy G(K; B for K 0, B:. At the initial outset Buy B K. At the first time of hitting B Sell B K units of a call option with strike K. units of the forward contract. The strategy G(K; B super-replicates the one-touch option with any K 0, B. optimal strategies properties. Definition. The initial value of strategy G(K; B is defined as We provide some G(K; B : C(K B K, (3.3 G (B as the infimum value of G(K; B with respect to K, K G (B as a strike price by which the infimum is attained: G (B as the corresponding strategy. G (B : inf G(K; B K (,B G(K G(B; B (3.4 4

Proposition. The infimum of Eq.(3.4 is attained by any element of N ( K G(B, an interval of 0, B. For all K G N ( K G(B, the following holds: ] ST K G Q K + < S T ] G (B E : K G S T Q K S T ], (3.5 B K G where K : inf N ( KG(B K + : sup N ( KG(B. In addition, there is a continuous martingale process { } St G such that t 0,T ] ( G (B Q τ B S G ] T. (3.6 Proof. By differentiating G with respect to K, we obtain ( KG(K B K KC(K + C(K B K ( B K KC(K + G(K (3.7 KKG(K B K KKC(K + (B K KC(K + C(K (B K 3 B K KKC(K + B K KG(K. (3.8 Since K G(0 ( + S < 0, lim B B K B K G(K + because KK G > 0 if KG > 0, the set N ( K G(B is an interval of 0, B we have Eq.(3.5. Apply emma with Ω 0 Ω such that {K G < S T } Ω 0 {K G S T } Q Ω 0] G, then we have a continuous martingale { } St G 3. Sub-Replication Consider the following self-financing strategy (K; B for K 0, B:. At the initial outset B K Buy units of a call option with strike B. Buy unit of a digital call option with strike B. Sell units of a put option with strike K. B K. At the first time of hitting B Sell B K units of the forward contract. The strategy (K; B super-replicates the one-touch option with any K 0, B. optimal strategies properties. Definition. The initial value of the strategy (K; B is defined as t 0,T ]. We provide some (K; B : C(B B K P (K B K KC(B, (3.9 (B as the supremum value of (K; B with respect to K, K (B as a strike price by which the supremum is attained: (B as the corresponding strategy. (B : sup (K; B K (,B (K ; B, (3.0 5

Proposition. The supremum of Eq.(3.0 is attained by any element of N ( K (B, an interval of 0, B. For all K N ( K (B, the following holds: ] ST K Q S T < K, B S T ] (B E : S T K, B S T Q S T K +, B S T ], (3. B K where K : inf N ( K (B K + : sup N ( K (B. In addition, there is a continuous martingale process { } St such that t 0,T ] ( (B Q τ B S ] T. (3. Proof. By differentiating with respect to K, we obtain ( K(K C(B B K B K KP (K P (K B K ( (K + B K KC(B KP (K (3.3 KK(K C(B (B K 3 B K KKP (K (B K KP (K P (K (B K 3 B K K(K B K KKP (K. (3.4 Since K (0 S > 0, lim B K B K (K because KK < 0 if K < 0, the set N ( K(B is an interval of 0, B we have Eq.(3.. Apply emma with Ω 0 Ω such that {K < S T, B S T } Ω 0 {K S T, B S T } Q Ω 0], then we have a continuous martingale { } St 4 Replication using another One-Touch Option t 0,T ]. Here, we consider super-replication sub-replication for a one-touch option with a barrier level B using European options a one-touch option with a barrier level B, where S 0 < B < B. Rather than considering the barrier option, we consider a touch/no-touch option whose payoff is {τ T <τ } where τ τ are the first times of hitting B B respectively, because of {τ T <τ } {τ T } {τ T }. For easing expression, we introduce the notation π : 0, ] F: where π(p is an element of F for p 0, ] such that Qπ(p] p, S T (ω S T (ω c for ω π(p ω c / π(p. We also define π(p, q : π(p c π(q for p, q 0, ] π(i : N n π(i n for I : N n I n, where I n are disjoint intervals. The ebesgue measure on 0, ] is denoted as µ. Then, we have µ(i Q π(i] for any interval I 0, ]. 4. Super-Replication Consider the following self-financing strategy G B (K; B, B for K 0, B :. At the initial outset Buy B K units of a call option with strike K. Sell units of a call option with strike B B K. Buy B B B units of the one-touch option with a barrier level B. K Sell B K units of a digital call option with strike B B K. 6

. At the first time of hitting B Sell B K 3. At the first time of hitting B Buy B K units of the forward contract units of the forward contract. Fig. shows that the G B (K; B, B strategy super-replicates the touch/no-touch option with K 0, B. We investigate the optimal strategies properties. First, we define the following. Definition 3. The initial value of the G B (K; B, B strategy is defined as G B (K; B, B : C(K C(B B K + B B B K O(B + B K B K KC(B, (4. G B (B, B as the infimum value of G B (K; B, B with respect to K, K B G (B, B as a strike price by which the infimum is attained: G B (B, B as the corresponding strategy. G B (B, B : inf G B (K; B, B K (,B G B (K B G (B, B ; B, B (4. There is another super-replication: the G (B strategy combined with a short position of the one-touch option with barrier B. The following theorem states that the better of the two strategies is the sharp upper bound, because the bound is attained by an expectation of the payoff with respect to a certain martingale. Theorem. If the set N ( K G B (B, B is not empty, the infimum of Eq.(4. is attained by any element of a set N ( K G B (B, B, an interval of (, B. For all K B G N ( K G B (B, B, the following holds: Q K + < S T < B ] ] G B ST KG B (B, B E : K B B KG B G < S T < B + B B Q τ B KG B T ] Q K S T < B ], (4.3 where K : inf N ( KG B (B, B K + : sup N ( KG B (B, B. If the set N ( KG B (B, B is empty, the infimum of Eq.(4. is not attained G B (B, B Q S T < B ]. If G B (B, B < G (B O(B, then N ( K G B (B, B is a non-empty interval of (sup N ( K G(B, B. In addition, there is a continuous martingale process { } St G such that t 0,T ] { } min G B (B, B, G (B O(B Q τ (S G ( T < τ S G]. (4.4 Proof. First, by differentiating G B with respect to K, we obtain KG B (K KKG B (K B K KC(K + B K C(K C(B (B K + ( G B (K + KC(K KC(B B B ( O(B + (B K KC(B (4.5 B K KKC(K + (B K KC(K + C(K C(B (B K 3 B B ( + O(B + (B K KC(B 3 B K KKC(K + B K KG B (K. (4.6 7

Note that K GB takes at least one positive value, because lim (B K KG B (K C(B C(B + (B B ( O(B + KC(B > 0. (4.7 K B Since KK GB 0 if K GB 0, N ( K G B (B, B is empty or a non-empty interval. If N ( K G B (B, B is empty, we have G (B, B lim G(K; B, B Q ST < B]. (4.8 K If N ( K G B (B, B is not empty, we have Eq.(4.3. Moreover, if the following holds: G B (B, B < G (B O(B, (4.9 then N ( K G B (B, B is a non-empty interval of (sup N ( K G(B, B, because ] Q K G B < S T < B G B < Q K G(B S T < B ] (O(B Q B S T ], (4.0 where any KG B N ( K G B (B, B K G (B N ( K G(B. Note that if N ( K G B (B, B is not empty, we have ] ] E S T B : π (kg, B b E S T KG B : π (kg, B b + (KG B B µ (kg, B b C(KG B C(B (B KG B µ (b, ] + (KG B B µ (kg, B b ( ] C(KG B C(B (B B µ (b, ] (B KG B µ kg, B (B B Q τ T ], (4. where b QS T < B ] kg B b G B, if N ( K G B (B, B is empty, we have ] E S T B : π (kg, B b lim (B K KG B (K + (B B Q τ T ] K (B B Q τ T ]. (4. Next, suppose that Eq.(4.9 holds. We show that there an interval x, y] 0, k B G] exists such that µ (x, y b, ] Q τ T ] (4.3 E S T B : π(x, y b, ]] 0. (4.4 et x 0 y be a real number satisfied with Eq.(4.3 with x 0. Then, since y k ( ( b, we have ] E S T B : π(0, y b, ]] E S T B : π(0, k ( b, ] : (B 0. (4.5 Conversely, let y kg B x be a real number satisfied with Eq.(4.3 with y kg. B By Eq.(4.9, we have ( µ (kg, B b < G (B µ x, kg B b, ] ( ( µ k ( G, µ x, kg B b, ], (4.6 8

where k ( G : G (B. Then, we have x > k ( G. In addition, by Eq.(4., we have ( ] ( ] E S T B : π x, kg B b, ] E S T B : π x, kg B b, ] + (B B Q τ T ] ( ] E S T B : π k ( G, kb G b, ] + (B B Q τ T ] ] E S T B : π (k G, B b + (B B Q τ T ] 0. (4.7 Therefore, we can find an interval x, y have ( ]] ] E S T : π x, y kg, B B µ (x, y b, ] + E S T : π (kg, B b ( B µ x, y kg, B, (4.8 using Eq.(4. again. Then, we construct a martingale { } St G. et X t 0,T ] X be rom variables defined as { B, π ( x, y k B X : G, ], (4.9 β, otherwise B, π (x, y b, ] X : β, π ( k B G, b β, otherwise, (4.0 where β 0, B, β 0, B are taken as in emma S t is a stochastic process defined as S t : S 0 {t< 3 T } + X { 3 T t< 3 T } + X { 3 T t<t } + S T {tt }. (4. Then, applying the same argument from emma to {St } t 0,T ], we obtain a continuous martingale with respect to a certain filtration. We obtain Q τ T < τ ] G B. ( ]] Finally, suppose that Eq.(4.9 does not hold. If O(B G (B, we have E S T B : π k ( G, 0, where k ( G E S T B : G (B. If Eq.(4.9 holds with equality, we have ( ] ] ( : π k ( G, kb G b, ] E S T B : π k ( E S T B : π G, kb G ] ] b, ] (k B G, b ]] + (B B µ ( + (B B µ ( k ( G, kb G k ( G, kb G ] ] b, ] b, ] 0. (4. Then, we can take an interval x, y k ( G, b which is satisfied with Eq.(4.3 Eq.(4.4, because of k ( G holds. < k( G. Similar to the previous case, a continuous martingale can be constructed such that Eq.(4.4 4. Sub-Replication Consider the following self-financing strategy B (K; B, B for K 0, B :. At the initial outset Sell B K units of a put option with strike K. 9

Buy B B B K units of the one-touch option with a barrier level B.. At the first time of hitting B Sell B K units of a forward contract. 3. At the first time of hitting B Buy B K units of the forward contract. Fig. shows that the B (K; B, B strategy sub-replicates the touch/no-touch option with K 0, B. We investigate the optimal strategy properties. First, we define the following. Definition 4. The initial value of the strategy B (K; B, B is defined as B (K; B, B : P (K B K + B B B K O(B, (4.3 B (B, B as the supremum value of B (K; B, B with respect to K, K B (B, B as a strike price by which the supremum is attained: B (B, B as the corresponding strategy. B (B, B : sup B (K; B, B K (,B B (K B (B, B ; B, B (4.4 There is another sub-replication: the strategy (B combined with a short position of the one-touch option with barrier B. The following theorem states that the better of the two strategies is the sharp lower bound, because the bound is attained by an expectation of the payoff with respect to a certain martingale. Theorem. The supremum of Eq.(4.4 is attained by any element of N ( K B (B, B, an interval of (0, sup N ( K (B ]. For all K B N ( K B (B, B, the following holds: Q S T < K ] B ST K B (B, B E B K B : S T K B where K : inf N ( K B (B, B K + : sup N ( K B (B, B. In addition, there is a martingale process { } St such that { } max B (B, B, (B O(B t 0,T ] Q Proof. First, by differentiating B with respect to K, we obtain ] + B B Q τ B K B T ] Q S T K + ], (4.5 τ (S T < τ ( S ]. (4.6 KK B (K K B (K B K KP (K P (K (B K + B B (B K O(B ( B K KP (K + B (K (4.7 P (K B K KKP (K (B K KP (K (B K + B B 3 (B K O(B 3 B K K B (K B K KKP (K. (4.8 0

Note that K B (0 B B O(B B > 0 by Eq.(3. + K B (K + B K KP + (K + P (K + + (B K + + B B (B K + O(B C(B (B K + + B B (B K + O(B 0, (4.9 where K + : sup N ( K (B + K is the right-sided derivative operator. Since KK B 0 if K B 0, N ( K B (B, B is an interval of (0, K + ] we have Eq.(4.5. Note that ( ] ( ] ( ] E B S T : π 0, k B E K B S T : π 0, k B + E B K B : π 0, k B ( P (K B + (B K B µ 0, k B where k B : B. Next, suppose that the following holds: (B B Q τ T ], (4.30 (B O(B < B (B, B. (4.3 We show that there exists an interval x, y] k B, b ], where b QS T < B ], such that µ (x, y b, ] Q τ T ] (4.3 E S T B : π (x, y b, ]] 0. (4.33 We can take an interval that satisfies Eq.(4.3 because Eq.(4.3 implies ( ( Q τ T ] > µ 0, k ( b, ] µ 0, k B µ (b, ], (4.34 where k ( : (B ( b. et y b ( x be a solution of Eq.(4.3 with y b. We have x k ( G : G (B because Q τ T ] µ k ( G., ] Then, we have ( ] E S T B : π(x, ] E S T B : π k ( G, 0. (4.35 Conversely, let ( x k B y be the solution of Eq.(4.3 with x k B. We have y k (, because Q τ T ] > µ k B, k ( b, ] by Eq.(4.3. Then, by Eq.(4.30, we have ( ] E S T B : π k B, y b, ] ( ] E S T B : π k B, y b, ] + (B B Q τ T ] ( ] E S T B : π k B, k ( b, ] + (B B Q τ T ] ( ] E B S T : π 0, k B + (B B Q τ T ] 0. (4.36

Therefore, we can find the interval x, y we have for this interval ( ] ( ] E S T : π 0, k B x, y b, ] E S T : π 0, k B + B µ(x, y b, ] ( B µ 0, k B x, y b, ], (4.37 using Eq.(4.30 again. Then, we construct a martingale { } St. et X t 0,T ] X be rom variables defined as { B, π ( 0, k B x, y b, ] X :, (4.38 β, otherwise B, π(x, y b, ] X : β, π ( 0, k B β, otherwise, (4.39 where β 0, B, β 0, B are taken as in emma S t be a stochastic process defined as S t : S 0 {t< 3 T } + X { 3 T t< 3 T } + X { 3 T t<t } + S T {tt }. (4.40 Then, applying the same argument from emma to {St } t 0,T ], we obtain a continuous martingale with respect to a certain filtration. We obtain Q τ T < τ ] B. Finally, suppose that Eq.(4.3 does ] not hold. If O(B (B let k ( : (B ( b, we have E S T B : π(0, k ( b, ] 0. If Eq.(4.3 holds with equality, we have by Eq.(4.30 E S T B ( : π k B, k ( ] b, ] E E S T B S T B ( : π : π k B, k ( ( 0, k B ] b, ] + (B B Q τ T ] ] + (B B Q τ T ] 0. (4.4 Note that k ( < k(, because (B K K(K; B is decreasing with respect to B K. Then, we can take a set D 0, k ( ] b, b which is satisfied with µ (D b, ] Q τ T ] (4.4 E S T B : π (D b, ]] 0. (4.43 Similar to the previous case, a continuous martingale can be constructed such that Eq.(4.6 holds. 4.3 The Finite Basis Situation In this section, we consider the case where only a finite number of strikes are given. Suppose that call options with strikes K 0 < K < < K N, where K 0 0 B K N, are traded with no-arbitrage prices {C n } N n0. We consider super-replication sub-replication for the touch/no-touch option with barrier levels B B using the one-touch option with a barrier level B. We assume that a no-arbitrage price of the digital call option with strike B is given as D in case of the super-replication that with strike B

is given as D in case of the sub-replication. Here, a no-arbitrage price D of digital call option with strike B (K n, K n ] satisfies C n+ C n K n+ K n D C n C(B K n B, (4.44 where C(B : C n+ C n K n+ K n (B K n + C n. Even if these digital call options are not liquid, we can regard the lower bound as the digital call price with strike B in case of the super-replication the upper bound as the digital call price with strike B in case of the sub-replication. First, we consider the super-replication. We suppose that the no-touch option with a barrier level B is traded the price of this no-touch option satisfies sup (K n; B O(B inf G(K n; B. (4.45 K n<b K n <B The upper bound on the touch/no-touch option derived from the super-replication is { ( } min G B B, B ; {K n} N n0, G (B ; {K n} N n0 O(B, (4.46 ( ( where G B B, B ; {K n } N n0 : infkn<b G B (K n ; B, B G B ; {K n } N n0 : infkn<b G(K n ; B. Although the marginal distribution of S T is not uniquely specified in this case, the following corollary shows that there is a distribution consistent with the given option prices under which we can construct a martingale attaining the upper bound. Corollary. There is a distribution µ C of S T which is consistent with the given call prices, the given digital call option with a strike B the given no-touch option with a barrier level B satisfying Eq.(4.45 such that Eq.(4.46 is equal to Eq.(4.4 with distribution µ C. Proof. First, we assume B {K n } n0,,n D C n C n K n K n, where B K n, et us consider call options prices {C(K} K 0,+ : C(K is the linear interpolation of C n if K K 0, K N ] an arbitrary extrapolation excluding arbitrage opportunities if K K N, +. et µ C be a distribution implied by the call option prices C. We can apply Proposition with the distribution µ C to the no-touch option with a barrier level B obtain the optimal strikes K G(B K (B. These may not be uniquely determined, but can be taken as one of the given strikes, since the distribution µ C consists of atoms at K n on 0, B. Hence, the distribution µ C is consistent with Eq.(4.45. By the same reason, Eq.(4.4 with distribution µ C is attained by one of the given strikes. Then, Eq.(4.46 is equal to Eq.(4.4 with distribution µ C. In the general case, two call prices, C( K C(B, with strikes, K : B ε B, can be added into the given call price set as: C(K : D (K ˆK + Ĉ for K K, B, where B (K n, K n ], ( ˆK, Ĉ (Kn, Cn in case of D > C n C n K n K n, ( ˆK, Ĉ (Kn, Cn in the other case, ε is a sufficiently small positive value such that K is not the optimal strike for G (B (B. Then, the same argument from the first case can be applied. Next, we consider the sub-replication. This is more involved than the super-replication. The lower bound on the touch/no-touch option derived from the sub-replication is { ( } max B B, B ; {K n } N n0, (B ; {K n } N n0 O(B, (4.47 ( ( where B B, B ; {K n } N n0 : supkn<b B (K n ; B, B B ; {K n } N n0 : supkn<b (K n ; B. We assume B {K n } n0,,n sup (K n; B < O(B < inf G(K n; B. (4.48 K n <B K n<b Owing to these assumptions, we have the similar result to Corollary. 3

Corollary. There is a distribution µ C of S T which is consistent with the given call prices which includes that with strike B, the given digital call option with a strike B the given no-touch option with a barrier level B satisfying Eq.(4.48 such that Eq.(4.47 is equal to Eq.(4.6 with distribution µ C. Proof. et n {0,,, N} be such that B K n. The proof is the same as the first part of Corollary, if D C n C n K n K n. In the general case, let K : B ε > K n for a sufficiently small positive value ε C( K : D ( K K n + C n. We can take ε such that sup (K i; B < O(B < K,K i <B inf G(K i; B. (4.49 K,K i <B et µ C be a distribution implied by an interpolation of the given call option prices C C( K. Then, we have the conclusion for the distribution µ C by the same argument from Corollary. 5 Numerical Examples This section provides numerical examples. We regard Heston s stochastic volatility model (Heston (993 as the underlying asset process. process underlying the Heston model is as follows: The ds t S t (r qdt + σ tdw t, (5. dσ t κ(η σ t dt + θσ td W t, (5. where W W are Brownian motions with correlation ρ under a risk-neutral measure. In addition, we assume that the parameters of the Heston model are as shown in Table. r q σ 0 κ η θ ρ 0.0 0.0 0.5 3.0 0. 0.4 0.0 Table : Parameters of the Heston Model The one-touch option considered has a 3-month maturity a barrier level of.05 USD. We calculate the pricing bounds of our method, those of Brown et al. (00 exact prices by a Monte Carlo simulation with the initial spot price varied from 0.9 USD to.04 USD. We calculate pricing bounds derived from G B B strategies using another one-touch option with B.06. This is evaluated by the Heston model with the same parameter set. The results are shown in Fig.3 Table. Our lower bounds are proved to be higher than those of Brown et al. (00 across the entire range our upper bounds proved lower in the 0.9, 0.98] range. Additionally, Fig.4 shows a relationship between pricing bounds on the two one-touch options with barrier levels.05 USD.06 USD, where the market conditions are the same as for the above example the initial spot price is fixed at USD. The pricing bounds of Brown et al. (00 on the two one-touch options are 0.35, 0.609] for the barrier level.05 USD 0.63, 0.59] for barrier level.06 USD. However, we established that a condition for no-arbitrage prices of these two options does not lie within these ranges but is within the range indicated in Fig.4. References Breeden, D. itzenberger, R. (978. Prices of state contingent claims implicit in option prices. Journal of Business, 5:6 65. 4

Brown, H., Hobson, D., Rogers,. C. G. Robust hedging of barrier options. Mathematical Finance, (3:85 34. (00. Carr, P. Chou, A. Breaking barriers. Risk, 0(9:39 45. (997. Carr, P. Crosby, J. A class of levy process models with almost exact calibration of both barrier vanilla fx options. Quantitative Finance, 0(0:5 36. (00. Carr, P., Ellis, K., Gupta, V. Static hedging of exotic options. Journal of Finance, 53(3:65 90. (998. Cox, A. M. G. Oblój, J. Robust hedging of double touch barrier options. SIAM Journal on Financial Mathematics, :4 8. (0a. Cox, A. M. G. Oblój, J. Robust pricing hedging of double no-touch options. Finance Stochastics, 5(3:573 605. (0b. Davis, M. H. A. Hobson, D. G. The range of traded option prices. Mathatical Finance, 7(: 4. (007. Derman, E., Ergener, D., Kani, I. Static options replication. Journal of Derivatives, :78 95. (995. Heston, S. A closed-form solution for options with stochastic volatility with applications to bond currency options. Review of Financial Studies, 6(:37 343. (993. Hirsa, A., Courtadon, G., Madan, B. The effect of model risk on the valuation of barrier options. The Journal of Risk Finance, Winter: 8. (003. Hobson, D. Robust hedging of the lookback option. Finance Stochastics, (4:39 347. (998. Hobson, D. Klimmek, M. Model independent hedging strategies for variance swaps. Finance Stochastics, 6(4:6 649. (0. Hobson, D. Neuberger, A. Robust bounds for forward start options. Mathematical Finance, (:3 56. (0. Karatzas, I. Shreve, S.-E. (988. Brownian motion stochastic calculus. Graduate Texts in Mathematics. Springer-Verlag, 3. New York etc. abordére, P. H., Oblój, J., Spoida, P., Touzi, N. Maximum maximum of martingales given marginals. http://arxiv.org/abs/03.6877. (0. ipton, A. McGhee, W. Universal barriers. Risk, 5:8 85. (00. Schoutens, W., Simons, E., Tistaert, J. Model risk for exotic moment derivatives. In Kyprianou, A., Schoutens, W., Wilmott, P., editors, Exotic Options Advanced évy Models, pages 67 97. Wiley,New York. (005. 5

Figure : Payoff of Strategy G B (K; B, B with S 0, K 0.95, B.05 B.06.5 before hitting to B or after hitting to B after hitting to B before hitting to B payoff 0.5 0-0.5 0.9 0.95.05. spot Figure : Payoff of Strategy B (K; B, B with S 0, K 0.95, B.05 B.06.5 before hitting to B or after hitting to B after hitting to B before hitting to B payoff 0.5 0-0.5 0.9 0.95.05. spot 6

Figure 3: Pricing bounds on a one-touch option 00 Brown s upper bound our upper bound Heston price (B.05 our lower bound Brown s lower bound 80 price(% 60 40 0 0 0.9 0.9 0.94 0.96 0.98.0.04 spot Table : Pricing bounds on a one-touch option (% spot 0.90 0.9 0.94 0.96 0.98.00.0.04 Brown et al. (00 s upper bound 8.0 3.0 0.4 30.8 44.5 60.9 78.5 94.4 Our upper bound W G 7.3. 9. 9.6 43.6 6.6 8.8 05.8 Heston price (B.05 5. 8.5 3.7.4 3. 46. 63.6 83. Our lower bound W 4. 7.0.5 8. 7.6 40.5 57. 77. Brown et al. (00 s lower bound 3.3 5.5 8.9 4.0.3 3.5 45.9 68.3 Heston price (B.06 4. 6.8. 7.6 6.7 39.0 54.7 73.3 7

Figure 4: Pricing bounds on two one-touch options 00 price of one-touch option with a barrier level.05 (% 80 60 40 0 0 0 0 40 60 80 00 price of one-touch option with a barrier level.06 (% 8