valuation bounds on barrier options under model uncertainty

Valuation Bounds on

Barrier Options Under

Model Uncertainty

YI HONG∗

This article investigates valuation bounds on barrier options under model uncer-tainty. This investigation enriches the literature on the model-free valuation ofthese exotic options. It is found that with weak assumptions on underlying priceprocesses, tight valuation bounds on barrier options can be sought from a setof European options. As a result, the numerical routine developed in this articlecan be reviewed as a new method for the evaluation of barrier options, which isindependent of model assumptions. C© 2012 Wiley Periodicals, Inc. Jrl Fut Mark

1. INTRODUCTION

This study investigates valuation bounds on barrier options under model un-certainty. Pricing these exotic derivatives poses a challenge to practitioners.The standard approach is to use parametric pricing models under whichthe prices of exotics are expressed as the discounted risk-neutral expecta-tions of their payoffs. A wide spectrum of alternative models is available to

This study is based on Chapter 2 of my dissertation. The author gratefully acknowledges the support from andinvaluable discussions with Anthony Neuberger, Nick Webber, Xing Jin, and Juan Tao as well as the fruitfulcomments of one anonymous referee. The editorial support from Editor Robert Webb is sincerely appreciatedwithout any reservation. The author also acknowledges the financial support from Warwick PostgraduateResearch Scholarship (WPRS).

∗Correspondence author, Department of Business, Economics and Management, Xi’an Jiaotong-LiverpoolUniversity, No. 111 Ren’ai Road, Suzhou, Jiangsu 215123, China. Tel: +86 512 8816 1729, Fax: +86 5128816 1730, e-mail: [email protected]

Received July 2011; Accepted November 2011

� Yi Hong is an Assistant Professor in the Department of Business, Economics andManagement, Xi’an Jiaotong-Liverpool University, Jiangsu, China.

The Journal of Futures Markets, Vol. 00, No. 0, 1–36 (2012)C© 2012 Wiley Periodicals, Inc.Published online xxxx 00, 2011 in Wiley Online Library (wileyonlinelibrary.com).DOI: 10.1002/fut.21545

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practitioners. However, this way in turn introduces an uncertainty in the choiceof model. Hirsa, Courtadon, and Madan (2002) illustrate that the prices of up-and-out call options differ noticeably when different stochastic processes arecalibrated to the market prices of vanilla options, regardless of the closeness ofcalibrations. Schoutens, Simons, and Tisteart (2004) further find that althoughall candidate price processes incorporating stochastic volatility are calibratedvery nicely to a realistic option surface, the resulting exotics prices can varysignificantly.

Also, these pricing models are subject to specification errors in estimat-ing relevant parameters. On February 28, 1997, for example, NatWest CapitalMarket revealed a £50 million loss that escalated to £90.5 million after fur-ther investigation, due to a mispriced portfolio of Deutschemark (DEM) andU.K. (GBP) interest rate options and swaptions by a single-derivative trader inLondon.1 This huge derivative loss is attributable to model risk, partly becausethe implied volatilities of underlying assets in pricing models are misspecified.

As a result, using pricing models introduces model risk. A traditional sta-tistical view about model risk is that there is a true model but we do not knowit (Uppal & Wang, 2003). Another view, associated with Knight (1921) andEllsberg (1961), is that there is an uncertainty so that no probabilistic modelcan be said to be true. In either way, I assume that there is no model thatcaptures the dynamics of prices. I therefore am concerned with what one cansay with confidence about the fair price of a barrier option in the absence of aspecific model.

A growing body of literature has used alternative ways to mitigate modelrisk.2Avellaneda, Levy, and Paras (1995) and Lyons (1995) address the issueof pricing and hedging derivatives in an uncertain volatility environment.3 In-stead of choosing a stochastic pricing model with parametric statistics, the waythat they deal with uncertainty is to impose a band on future instantaneousvolatility paths. Tight arbitrage price bounds are inferred from extreme volatil-ity paths. The “good-deal” approach proposed by Cochrane and Saa-Requejo(2000) and Bernardo and Ledoit (2000) involves a class of parametric pricing

1See http://www.independent.co.uk/news/business/natwest-staff-face-sack-over-pounds-50m-loss-1272151.html.2Jacquier and Jarrow (2000) incorporate parameter uncertainty and model error into the implementation ofcontingent claim models in the Black–Scholes framework. Hull and Suo (2002) present a method to assessmodel risk in valuing exotic options, similar to Green and Figlewski (1999). Their method involves comparingthe pricing and hedging performance of the true model and a test model. They find that model risk caused bylocal volatility models is not important for compound options, but significant for barrier options. Detlefsenand Hardle (2007) demonstrate that the choice of calibration methods may have a substantial impact onpricing exotic options and on measuring model risk. Also see Schmeidler (1989) for a preference-basedapproach, Kandel and Stambaugh (1996) for a Bayesian approach, Hansen and Sargent (2001) for the robustcontrol theory under uncertainty and Cont (2006) for a convex model risk measure.3See Avellaneda and Paras (1996) and Avellaneda, Friedman, Holmes, and Samperi (1997) for more details.

Journal of Futures Markets DOI: 10.1002/fut

Valuation Bounds on Barrier Options 3

models by excluding not only arbitrage opportunities but also favorable oppor-tunities.4 Valuation bounds on trading assets are sought within a set of pricingmodels. The measure of a “good deal” plays an important role in determiningvaluation bounds on asset prices. Restrictions on these measures are subjective.Even when these restrictions are imposed, valuation bounds still rely on modelassumptions about the dynamics of underlying assets.

This study investigates the valuation of barrier options from a differentperspective. Unlike good-deal bounds, we seek arbitrage bounds on barrieroptions that are robust to model misspecification. Seeking these bounds involvesidentifying a set of European call options as hedging instruments. It is found thatwith weak assumptions on underlying price processes, tight valuation boundson barrier options can be sought from the market prices of these instruments.More specifically, the valuation problem is expressed as an linear programming(LP) such that the relation between pricing and hedging can be identified asa duality. Rather than precise prices, valuation bounds on barrier options aretherefore determined.

Several researchers also consider optimization approaches to barrier optionpricing and hedging.5 Siven and Poulsen (2009) investigate risk-minimizinghedges of barrier options based on four risk measures (i.e., the quadratic, thepositive part, value-at-risk, and expected shortfall). Maruhn, Nalholm, and Fen-gler (2011) study model-dependent super-replicating hedges for reverse barrieroptions. Both of these two studies are set within a static context where allEuropean options at the outset are hedging instruments. This study further in-corporates dynamic trading in European options so that valuation bounds canbe sought within an augmented space of trading strategies.

In particular, both of these two studies have addressed the issue of modelrisk. Throughout two numerical examples, Siven and Poulsen (2009) investigatethe uncertainty in the choice of an option pricing model and its effects onhedging performance. Maruhn et al. (2011) develop a hedge robust to significantchanges in model parameters (parameter uncertainty) in order to reduce modeldependence. However, it is not clear that how model risk is measured, providedthat these studies mainly utilize several specific pricing models. Britten-Jonesand Neuberger (2000) point out that there exist a number of stochastic volatilityprocesses that are consistent with initial European option prices. This study thusmeasures model risk by the width of valuation bounds. In this way, the relationbetween pricing and hedging is naturally exploited by virtue of LP techniques.

4See Cerny and Hodges (2001), Staum (2004), and Bjork and Slinko (2006) for more details.5Hedging barrier options has been widely investigated. Derman, Ergener, and Kani (1995) propose thecalendar-spread approach and Carr, Ellis, and Gupta (1998) suggest the strike-spread approach. Andersen,Andreasen, and Eliezer (2002), Fink (2003), and Nalholm and Poulsen (2006) discuss the hedging of barrieroptions in specific pricing models.


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This study is relevant to the approach of robust pricing and hedging. Thelatter requires no assumptions on asset price dynamics, apart from the stan-dard assumptions of frictionless markets and absence of arbitrage. This robustapproach aims to seek model-free arbitrage bounds on derivatives that are en-forced by the prices of available trading assets in the spirit of Merton (1973)who argues that the prices of derivatives are in effect enforced by the prices ofrelevant assets so that these derivatives are neither dominating nor dominatedby these traded assets.

Linking back to the literature, Hobson (1998) finds arbitrage bounds onlookback options from a set of European options. Upper bounds are enforcedby hedge portfolios that involve purchasing call options, and gradually sellingthem over time depending on the behavior of the underlying asset. The proceedsfrom combined sales are sufficient to cover liabilities at maturity. Brown et al.(2001) investigate model-free valuation bounds on barrier options. They firstshow that price bounds on American digital options are enforced by simpledominating strategies, and extend the results about digital options to otherEuropean single barrier options. Neuberger and Hodges (2002) employ LPtechniques to exploit the robust pricing and hedging of exotic derivatives. Theycontend that the valuation problem of a large class of exotics can be solvedwithin an LP framework.

Although arbitrage bounds are robust to model misspecification, they aretoo wide to be useful in practice. This study further shows that under somecircumstances relatively weak restrictions on future prices of European optionscan lead to tight valuation bounds on barrier options. This result hence enrichesthe literature on the model-free valuation of derivatives. Note that a similarmethodology has been proposed by Avellaneda et al. (1995) who simply putbounds on the instantaneous volatility of underlying asset prices. However,instantaneous volatility is unobservable in real markets, which poses a problemof choosing a reliable volatility range. So I instead impose restrictions on theprices of options that will trade in future, mainly because their spot pricesare observable in markets. Because the numerical routine for searching forvaluation bounds is proposed within the LP framework, such restrictions thenhave natural implications for hedge. Also, this numerical routine can be viewedas a new method for the evaluation of barrier options that is independent ofmodel assumptions.

The remainder of this study proceeds as follows. Section 2 discusses boththe market structure and underlying price processes. Section 3 presents a char-acterization of martingale price processes that are consistent with the pricesof traded options. Section 4 proposes a numerical approach for pricing barrieroptions. I make further discussion about relaxing assumptions in Section 5.Section 6 concludes.



2. THE SETUP

In this section, I introduce the market structure, and then discuss the stockprice processes that support the prices of traded instruments in the market.

2.1. Market Structure

The market is set in a continuous-time and discrete-space framework. Discreteprices are convenient for numerical tractability. The continuous-time settingensures that the market structure itself does not impose restrictions on thevolatility of the underlying stock price S when the price space is discrete.6 Themarket is frictionless; there are no short sale restrictions, transaction costs,taxes, and other frictions.

The dividend yield of the stock is zero. During the period [0, T∗], the stockprice St takes values from a set K with 2M + 1 elements,

K = {Ki : Ki = eiuS0, u ∈ R+, i = 0, ±1, ±2, . . . , ±M},where S0 is the spot price and M is an integer number. The geometric space ischosen to keep stock prices positive and easily relates to conventional models. 7

There is a set of traded European call options written on the stock. LetC(K , t; s ) be the time-s price of a call option with strike K ∈ K and maturity t(henceforth s ∈ [0, t) ⊆ [0, T∗] unless otherwise specified). These option pricesare increasing in maturity and decreasing in strike. Let C(K , t) (=C(K , t; 0))stand for the prices of call options at inception.

Pure discount bonds that pay 1 at any maturity t ∈ (0, T∗] exist in themarket. In terms of pricing exotic derivatives, I make the following assumptions:

[A1] There is no arbitrage among all the traded assets (the stock, call options andbonds).

[A2] A complete set of European call options, C(K , t; s ) with all strikes(K ∈ K)and a continuum of maturities t ∈ (0, T∗] exists and trades.8

[A3] The call option prices are continuous with respect to maturity.[A4] The risk-free rate is zero.

It follows from assumption [A1] that there is at least one stock price pro-cess that supports the market prices of hedging assets. Assumption [A2] will

6Given this discrete-space and continuous-time market, limτ→t( Sτ −St

St)2

τ−t → ∞.7Alternatively, an arithmetic space, for example, could be employed. The approach presented here is stillappropriate.8Although European options traded in markets are finite, the method proposed by Andreasen and Huge(2011) makes it possible to construct a complete set of option prices. Their approach involves interpolationand extrapolation of a discrete set of European option quotes into an arbitrage-free, full double continuumin expiry and strike of option prices.


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be relaxed in Section 5. Assumption [A3] is a technical requirement in thecontinuous-time setting. For simplicity and without loss of generality, the zerointerest rate (assumption [A4]) is assumed. The critical requirement I need isthat interest rates are deterministic. With the zero interest rate, both the optionprices and the underlying stock price are viewed as discounted prices.

2.2. Stock Price Processes

Assumption [A1] ensures the existence of at least one risk-neutral probabilitymeasure (Harrison & Kreps, 1979). Then the prices of traded assets are theexpectations of their discounted payoffs under such a probability measure:

E[ST∗ |Ft] = St ;

E[(St − K )+ |F0] = C(K , t), St ∈ K, t ∈ [0, T∗] , (1)

where Ft (t ∈ [0, T∗]) is the available information collection at time t, andE[·|Ft] is the expectation operator conditional on the information filtration Ft .The revelation of the information embedded in the prices of all traded assets ischaracterized by the filtration {Ft}t∈[0,T∗].

Let M be the set of all underlying price processes that satisfy the equationsin (1). This set is not empty. So those processes that fulfill the first equationare martingales, whereas the second one ensures that all the options are valuedproperly. Within this discrete-space and continuous-time setup, I first defineMarkov and continuous processes.

Definition 1: A stock price process is a Markov process if the conditional proba-bility density of this price process depends only on the current stock price and noton other information,

prob{Sτ |Ft} = prob{Sτ |St}, (2)

for 0 ≤ t < τ ≤ T∗. �

Note that I use the term “Markov” to mean specifically Markov with respectto price level.

Definition 2: A stock price process is continuous if the stock price can only jumpup or down at most one level at any moment,

limτ→t

prob(Sτ |Ft)τ − t

= 0, (3)

if |ln Sτ − ln St| > u for any t < τ ∈ [0, T∗]. �



This definition equivalently states prob(|ln Sτ − ln St| > u) = o(τ − t). The“continuity” of a price process requires that stock prices jump (move) no morethan one node each time. I will show that there always exists a continuousmartingale process in M. Moreover, note that the limit on the ratio in (3) canbe viewed as the jump rate of price movements over time in the continuous-timesetting.

3. CHARACTERIZATION OF MARTINGALEPROCESSES

In this section, the information extracted from the initial option prices C(K , t)is used to infer the properties of martingale processes in M. I first presenta characterization for these martingale processes. In order to construct a setof bounded martingale processes, the restrictions on the prices of Europeanoptions C(K , t; s ) (s > 0) that will trade in future are then introduced. I finallyaddress the issue of pricing barrier options.

3.1. Probability Densities

No-arbitrage arguments ensure the existence of at least one risk-neutral proba-bility measure under which all initial options are valued properly. Breeden andLitzenberger (1978) show that the marginal probability density of the under-lying asset can be inferred from initial option prices. They establish that thismarginal probability density equals the second derivative (∂2C(K , t)/∂K 2) ofoption prices with regard to strike in a continuous-time and continuous-spaceworld. Britten-Jones and Neuberger (2000) present the discrete counterpart ofthis probability density in a discrete-time and discrete-space world.

Lemma 1: For any martingale price process in M, the marginal probabilitydensity of the stock price being K ∈ K at time t ∈ (0, T∗] is equal to

π(K , t) = C(K eu, t) − (1 + eu)C(K , t) + euC(K e−u, t)K eu − K

, for St = K . (4)

�Proof. See Appendix A. �

Lemma 1 shows that the probability of the stock price reaching any levelat any time is determined by initial option prices. This result is independent ofasset price dynamics. In terms of portfolios, this marginal density is exactly theprice of a butterfly spread, as shown in (4).

Knowing the marginal probability density of the underlying asset is insuffi-cient to identify a stock price process. In the presence of path dependence, there


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exist many martingale processes that could lead to the same marginal density. Ifthe price process is continuous, however, the joint probability density of pricemovements can be inferred from option prices. I introduce a function,

�(K ∗, τ ; K , t) ≡ prob(St = K ∧ Sτ = K ∗),

to represent the joint probability of the stock price being K at time t and beingK ∗ at time τ (> t), starting from the spot price S0.

Lemma 2: Given the initial set of option prices, the joint probability density ofany continuous martingale process in M satisfies the following conditions,

limτ→t

�(K ∗, τ ; K , t)τ − t

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1K eu − K

∂C(K , t)∂t

, if K∗ = K eu;

1K − K e−u

∂C(K , t)∂t

, if K∗ = K e−u ;

0 if K∗ ∈ K \ {K eu, K , K e−u} ,

(5)and

∑K ∗∈K

�(K ∗, τ ; K , t) = π(K , t), (6)

where π(K , t) is given in Lemma 1. �

Proof. See Appendix B. �

Lemma 2 asserts that under a continuous price process initial option pricescan determine the probability of a price movement.9 Then all continuous mar-tingale processes consistent with option prices have the same joint probabilitiesabove. This lemma together with Lemma 1 also determines the jump rates of acontinuous Markov martingale process. These jump rates, associated with themarginal probabilities, show the construction of this process, and thus thereexists a continuous martingale process in M.

3.2. Bounded Martingale Processes

A large and fruitful body of financial literature has sought to identify underly-ing processes from market prices of relevant traded assets (see Bates (1996),Bakshi, Cao, and Chen (1997) and Carr and Wu (2007) for references). The

9Britten-Jones and Neuberger (2000) present a discrete version of Lemma 2.



specification of volatility processes is the central issue. Apart from the prob-ability densities in Lemmas 1 and 2, initial option prices have put no furtherrestrictions on stock price processes.

As a result, the price range of barrier options could be very wide. The strate-gies that enforce this price range usually involve buying options at inceptionand holding them to maturity. Dynamic trading in options is useless for nar-rowing this range. Because there are no restrictions on processes (apart frommartingale), options that one might want to sell could be traded at intrinsicvalue, making it pointless selling them, while options that one wants to buymay be so expensive (trading at their arbitrage upper bounds) that they are notworth buying.

To narrow valuation bounds on barrier options, a natural and tractable wayis to put restrictions on the future prices of options. Market data suggest thatin terms of implied volatilities option prices tend to lie within a range. Forinstance, in the top panel in Figure 1, I report the volatility index (VIX) on theStandard Poor’s 500 index (henceforth S&P500 index). All data are observed atthe first trading day of each month, and the time period is over 20 years fromJanuary 1990 to June 2010. Because the VIX is based on the implied volatilitiesof S&P500 index options, it can be viewed as an indicator of implied volatility.As shown in the top panel, the series appears to be stationary, and all valuestend to lie within the range of [10%, 70%] over 20 years.

In order to seek the sort of bounds that might be reasonable in practice, Iam particularly interested in VIX(t + j) relative to VIX(t), because the boundson prices at time t + j will be fixed given information at time t. The bottompanel in Figure 1 reports the ratios of V I X (t + j) against V I X (t) for j = 6(12) months. The ratios seem to be bounded within a band of [0.25, 3.50].This observation suggests that conditional on information at the beginning of ahedge, it may be very possible to impose bounds on implied volatilities duringthe life of the hedge.

A convenient way to impose restrictions on implied volatilities is to putbounds on the future prices of at-the-money (ATM) European options. Thisway may avoid the diverse choice of strikes. Within our setup, restrictionson implied volatilities further imply that the jump rate of the stock would bebounded.10 Therefore, it is required that the range of the jump rate has to beconsistent with initial option prices. Note that for a continuous stock priceprocess, the price level can only have at most one jump at any time. Then if

10To be consistent with the quoted implied volatility at time t, σt,imp , the expected squared returns, conditionalon Ft , can be expressed as

limτ→t

E

[(Sτ − St

St

)2∣∣∣∣∣Ft

]

τ − t= σ 2

t,imp .


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FIGURE 1VIX ratios. For the monthly data on S&P500 index options over 20 years (January 1990 to June

2010), a VIX ratio is calculated by ratiot, j = VIX(t+ j)V I X (t) , where VIX(t) is the index at time t, and

VIX(t + j) is the index after j months. The solid (dashed) line presents all ratios for j = 6 (12)months.



this process supports initial option prices, as implied by Lemma 1 and 2, thejump-up rate of the stock from St to Sτ (=euSt) is equal to

1St(eu − 1)π(St, t)

∂C(St, t)∂t

, for τ → t.

I now introduce the following assumption to impose bands on the future pricesof ATM European calls C(St, τ ; t) (St ∈ K and τ ∈ (t, T∗]) that trades at time tand matures at τ .

[A5] Given a pair of (λmin, λmax) at node (S, t), the future prices of ATM Europeancall options C(St, τ ; t) are bounded as follows:

limτ→t

C(St, τ ; t)St(eu − 1)(τ − t)

∈ [λmin(St, t), λmax(St, t)], (7)

where

0 ≤ λmin(St, t) ≤∂C(St ,t)

∂t

St(eu − 1)π(St, t)≤ λmax(St, t). (8)

Price bounds in (7) are expressed as the restrictions on the jump rate of thestock. λmin and λmax, which depend on price level and time, can be viewed as thelower and upper bound on jump rate so that implied volatilities are bounded aswell. To be consistent with initial option prices, the condition in (8) is required.In this way, those processes that generate extremely high (or low) future pricesfor ATM options are excluded. Then we can concentrate on a subset of M. Thissubset consists of bounded martingale processes that satisfy both the conditionsin (1) and the constraint in (7), and thus it is not empty.

3.3. Implications for Pricing Barrier Options

The valuation of barrier options is path-dependent. A barrier option’s payoffdepends on two components. The first component is the stock price at maturity,and the second one is whether or not the barrier has been breached during theoption’s lifetime. Define It (t ∈ [0, T∗]) to be an auxiliary variable that captures

By referring to (3), let λt (K ) be the jump rate of the stock price being K from St , e.g. λt (K ) =limτ→t

pr ob(Sτ =K |Ft )τ−t . Then the conditional expected squared returns can be re-expressed as

∑K ∈K

λt (K )(

K − St

St

)2

= σ 2t,imp .


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barrier-hitting events. Let It take the value 1 if the barrier has been breachedby time t and zero otherwise. Let b(ST∗, IT∗) be the terminal payoff of a barrieroption at maturity. I am looking for the valuation bounds on this option.

In the spirit of Definition 1, I call a process that is Markovian with re-spect to the stock price S and the variable I an extended-Markov process. Themain feature of an extended-Markov process is that each path of this processcarries information that indicates the occurrence of barrier-hitting events. Thefollowing proposition presents the properties of extended-Markov processes andshows that valuation bounds on barrier options can be sought within a set ofextended-Markov martingale processes.

Proposition 1: For any process P ∈ M, there exists a function that maps processesin M into processes in a set MEM ⊆ M:

f : M → MEM

P → f (P )

such that

(i) the process f (P ) is martingale in S and is Markovian in S and I ;(ii) the process f (P ) is consistent with initial option prices;(iii) if the process P is continuous, so is the process f (P );(iv) if the process P is bounded in (7), so is the process f (P );(v) the following relations hold:

maxP∈M

EP [b(ST∗, IT∗)|F0] = maxf (P )∈MEM

E f (P )[b(ST∗, IT∗)|F0];

minP∈M

EP [b(ST∗, IT∗)|F0] = minf (P )∈MEM

E f (P )[b(ST∗, IT∗)|F0]. (9)

�Proof. See Appendix C. �

In general, valuation bounds on barrier options are attained by seeking mar-tingale processes in M. Proposition 1 establishes that these valuation boundscan literally be sought within a set of extended-Markov martingale processesMEM ⊆ M. The set of Markov processes is much smaller than the set of allmartingale processes, so this result reduces the size of the valuation problem.These extended-Markov martingale processes then play a central role in pricingbarrier options. If assumption [A5] is imposed, the results in Proposition 1 arestill valid for a family of bounded martingale processes. A numerical procedureis proposed to investigate the tightness of valuation bounds on barrier optionsin the next section.



4. VALUATION BOUNDS ON BARRIEROPTIONS

I focus on single barrier options. The problem of pricing barrier options isformulated as an LP. Following the results in Proposition 1, I search for val-uation bounds on barrier options within a set of extended-Markov martingaleprocesses.

4.1. The Problem Formulation

Consider a European option with barrier B (>S0) that pays b(ST∗, IT∗) at ma-turity. Pricing this barrier option is equivalent to searching for an extended-Markov martingale process that maximizes its price. To seek the least upperprice bound on this option, the valuation problem may be expressed as follows:

maxQ∈MEM

EQ[b(ST∗, IT∗)], (10)

subject to the constraints in (7). The price bound is finite as the payoff to thebarrier option is also finite (bounded from above). Furthermore, there existsa feasible price for this valuation problem, because there exists at least oneextended-Markov martingale process that supports initial option prices and isconsistent with the price band in (7).

By discretizing time into finite intervals, I am able to express the valua-tion problem in (10) as a finite LP. The convergence of valuation bounds isinvestigated through the numerical examples in Section 4.4. I now employ LPtechniques to present a numerical procedure that is used to solve this valuationproblem within a set of extended-Markov martingale processes.

4.2. LP Construction

Suppose that the time period [0, T∗] is discretized with a time interval so thatthe T time points are equally spaced and the length of each time interval isequal to δ = T∗/T . With initial value of 0, the time index t takes values fromthe set

t ∈ T = {0, 1, 2, . . . , T}.

Price levels are distributed geometrically with K j+1 = eu K j for some positiveu where the stock index j takes values from the set

j ∈ N = {1, 2, . . . , N}.


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A stock price set K is constructed. In this way, a discrete-space and continuous-time setup is converted into a discrete-space and discrete-time one.

Within this setup, the marginal probability density in (4) is represented byan N × (T + 1) matrix P = {p j,t}. Let an N × (T + 1) × 2 matrix {b j,t,i } be thepayoff of a barrier option b. Hence, its terminal payoff is b j,T,i = b(ST∗, IT∗) forST∗ = K j and IT∗ = i (0/1), and otherwise b j,t,i ≡ 0 for t < T .

Assumption [A5] may be expressed as the restriction on the future pricesof one-period call options:

C(St, t + 1; t) ∈ [Cl, Cu], (11)

where Cl ≡ St(eu − 1)δλmin(St, t) and Cu ≡ St(eu − 1)δλmax(St, t).I introduce three matrices F = { f j,t,i }, G = {g j,m,t} and H = {h j,m,t} where

each element of these matrices is defined as follows:

f j,t,i ≡ prob{St = K j ∧ It = i};g j,m,t ≡ prob{St = K j ∧ St+1 = Km ∧ It = 1}; (12)

h j,m,t ≡ prob{St = K j ∧ St+1 = Km ∧ It = 0};

for j, m ∈ N, t ∈ T and g j,m,T = h j,m,T ≡ 0. These matrices are used to char-acterize extended-Markov martingale processes in MEM under which initialoptions are priced properly and the future prices of one-period call options arebounded in the band in (11).

To seek the least upper bound on this barrier option, I now express thevaluation problem in (10) as an LP:

(LP1) find the N × (T + 1) × 2 matrix F , and N × N × (T + 1)

matrices G, H to maximize the expected payoff

maxF,G,H

∑j,t,i

b j,t,i f j,t,i ,

s.t.

(1)∑

i f j,t,i = p j,t , for j and t;(2)

∑m(Km − K j )g j,m,t = 0 and

∑m(Km − K j )h j,m,t = 0 for all j, m and t <

T ;(3)

∑m g j,m,t = f j,t,1, for all j, m;

(4)∑

m h j,m,t = f j,t,0, for all K j < B and h j,m,t ≡ 0, for K j ≥ B;(5)

∑m gm, j,t−1 + hm, j,t−1 = f j,t,1, for all K j ≥ B;

(6)∑

m gm, j,t−1 = f j,t,1, for all K j < B;(7)

∑m hm, j,t−1 = f j,t,0, for all K j , Km < B and hm, j,t−1 ≡ 0 for Km ≥ B;



(8) Cl∑

m g j,m,t ≤ ∑m[Km − K j ]+g j,m,t ≤ Cu

∑m g j,m,t , for all j, m;

(9) Cl∑

m h j,m,t ≤ ∑m[Km − K j ]+h j,m,t ≤ Cu

∑m h j,m,t , for all j, m;

(10) f j,t,i ≥ 0, g j,m,t ≥ 0 and h j,m,t ≥ 0, for all j, m, t.

The no-arbitrage argument (assumption [A1]) implies that within this dis-crete setup there exists at least one extended-Markov martingale processthat is characterized by the matrices F, G, and H. The solution set forthis program is not empty. The objective function is bounded below bymin j,i b j,T,i and above by max j,i b j,T,i . Therefore, this program must have asolution.

I now explain these constraints in detail. The first constraint specifies thatinitial option prices are priced properly.

(a) prob{St = K j } = prob{St = K j ∧ It = 0} + prob{St = K j ∧ It = 1} , for allj, t.

The second constraint ensures that the underlying process is a martingale.

(b) E[St+1 − St |St] = 0, for all St and t < T .

The constraints from (3) to (7) specify the relations between two adjacentnodes in terms of probabilities. The constraints (3) and (4) state the relation ina forward way.

(c) prob{St = K j ∧ It = i} = ∑m prob{St = K j ∧ St+1 = Km ∧ It = i}, for all

j, m and t < T ,

while the constraints from (5) to (7) state the relation in a backward way.

(d) prob{St = K j ∧ It = 1} = ∑m,i prob{St−1 = Km ∧ St = K j ∧ It−1 = i}, for all

K j ≥ B and t ≤ T ;(e) prob{St = K j ∧ It = i} = ∑

m prob{St−1 = Km ∧ St = K j ∧ It−1 = i}, for allK j < B and t ≤ T .

Finally, assumption [A5] leads to the constraints (8) and (9):

(f) Cl f j,t,1 ≤ ∑m[Km − K j ]+g j,m,t ≤ Cu f j,t,1, for St = K j and It = 1;

(g) Cl f j,t,0 ≤ ∑m[Km − K j ]+h j,m,t ≤ Cu f j,t,0, for St = K j and It = 0.

If all extended-Markov processes are restricted to be continuous, the stockprice can only jump at most one level or stay over δ. Then the program LP1 canbe expressed in a compact form. In the light of Lemma 2, I simplify LP1 to anew LP as follows:

(LP2) To find the N × 3 × T matrices G and H to maximize∑j

∑m∈{ j−1, j, j+1}

(gm, j,T−1b j,T,1 + hm, j,T−1b j,T,0)


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subject to

(1′) g j, j+1,t + h j, j+1,t = C(K ,t+1)−C(K ,t)K eu−K , for all j, t;

(2′) g j, j,t + h j, j,t = p j,t − (1 + eu)C(K ,t+1)−C(K ,t)K eu−K , for all j, t;

(3′)∑

m(Km − K j )g j,m,t = 0;∑

m(Km − K j )h j,m,t = 0, for all j, t;(4′)

∑m(g j,m,t − gm, j,t−1) = 0;

∑m(h j,m,t − hm, j,t−1) = 0, for K j < B or Km ≥

B;(5′)

∑m(g j,m,t − (gm, j,t−1 + hm, j,t−1)) = 0, for K j ≥ B and K j−1 < B;

(6′) λ−g j, j,t ≤ g j, j+1,t ≤ λ+g j, j,t ; λ−h j, j,t ≤ h j, j+1,t ≤ λ+h j, j,t , for all j, t,

where λ− = λmin1−(1+eu)λmin

and λ+ = λmax1−(1+eu)λmax

.The final constraints interpret the restrictions on one-period option prices

under continuous extended-Markov martingale processes. Two parameters λmin

and λmax are given in assumption [A5]. The solution set for LP2 is not empty,because at least one continuous extended-Markov martingale process can beconstructed in a discrete setup by following the way in Proposition 1. Also, LP2is bounded from above as both G and H are bounded (0 ≤ g j,m,t (or h j,m,t) ≤p j,t). Therefore, this program has a solution.

4.3. Solving the LPs

Both LP1 and LP2 are large-scale programs when the number of time step islarge. In the standard form Ax ≤ b, the structure of the coefficient matrix A hasa crucial impact on computational efforts required to solve the LP. Because thecoefficient matrices in both LP1 and LP2 are sparse, more efficient algorithmsare needed to solve these large-scale LPs.

To solve a large-scale LP, Zhang (1996) suggests that Cholesky factorizationshould be used to decompose a sparse matrix as the product of a lower triangularand its transpose (A = LLT ). This decomposition involves three steps. First, therows and columns of the matrix A are reordered to minimize all new nonzeroelements that are generated by a Cholesky factorization. Second, a symbolicfactorization is used to explore the sparse pattern of the triangular matrix L.Finally, one performs a real factorization. To a large extent, the efficiency ofan approach to solving large-scale LPs depends on the numerical stability ofCholesky factorization. Associated with this factorization, one can use somemethods to determine the step length of next search at each iteration, that is,Newton’s method and its variants. I therefore employ a primal-dual infeasible-interior-point algorithm (LIPSOL) developed by Zhang (1996). This algorithmhas been integrated into the MATLAB environment by taking advantage of



MATLAB’s sparse-matrix functions and of the Fortran code package for sparseCholesky factorization.11

At this stage, I solve the program LP2 by focusing on a set of continuousextended-Markov martingale processes. This program is solved in a 201 × 100grid. Because all underlying price processes are continuous, the stock priceevolves along a trinomial tree where total time steps are T = 100 and totalprice states are N = 201, given a time interval of δ = T∗/100. Usually, this isthe case that after constructing the coefficient matrix, LP2 is solved within 4min of computing time on a 3 GHz desktop computer. Solving LP under generalextended-Markov martingale processes (LP1) will be numerically investigatedin Section 5.

4.4. Numerical Analysis

I first present a model that is used to generate call option prices, and demon-strate the tightness and robustness of valuation bounds on barrier options.

4.4.1. Model implementation

The discrete setup has been specified in Section 4.2. I first have to set optionprices at time zero, and then define the bounds on the future prices of one-period ATM options.

All initial options are determined by setting the implied volatility equal toa constant σ . More specifically, the option prices C(K , t) (K ∈ K, t ∈ T) aregenerated by a continuous price process (starting from S0) with the annualimplied volatility of σ :

St+δ =

⎧⎪⎨⎪⎩

Steu with probability λδ;

Ste−u with probability euλδ;

St otherwise,

(13)

with a jump rate λ = σ 2

(eu−1)2(1+e−u) . The rate λ is calculated by matching thesecond moment of the underlying process. The process in (13) is dependent onthe jump size u and time interval δ. As both of them go to zero, jumps becomesmaller and more frequent.

11The version of the LIPSOL algorithm I use is 1.1.6.2 in the MATLAB environment (v7.5).


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Given a set of European options, C(K , t), the marginal density of the stockprice is known from Lemma 1:

p j,t = C(K j+1, t) − (1 + eu)C(K j , t) + euC(K j−1, t)K j+1 − K j

, (14)

which is represented by an N × (T + 1) matrix P = {p j,t}.The restriction on the future prices of one-period call options in (11) is

expressed as

C(St, t + 1; t) ∈ [Cl, Cu], (15)

where Cl = e−h[eu − 1]Stλδ and Cu = e+h[eu − 1]Stλδ; the positive real numberh is used to control the width of price band. In terms of the condition in(7), I set λmin = e−hλ and λmax = ehλ. I can also express the price band interms of implied volatilities in numerical examples. Then the restriction in (15)equivalently states that future implied volatilities are bounded within a range of[σ e−h/2, σ eh/2].

In the limit as h goes to zero, the model converges to the local volatilitymodel of Derman and Kani (1994), and so barrier options can be priced exactlyusing the continuous process specified in (13). In the limit as h goes to infinity,I have arbitrage bounds derived by Neuberger and Hodges (2002).

4.4.2. Rational valuation bounds (h = ∞)

I first examine valuation bounds on barrier options when the price band in(15) becomes very wide (h = ∞). There is no restriction on the prices of ATMcall options, apart from the initial option prices, C(K , t) (K ∈ K, t ∈ T). Givena complete set of European call options C(K , t), both Brown et al. (2001)and Neuberger and Hodges (2002) derive valuation bounds on barrier optionsfrom static portfolios of these vanilla options. I call these valuation boundsrational bounds. Brown et al. (2001) derive these rational bounds under generalmartingale processes, denoted as Rg . Neuberger and Hodges (2002) search forrational bounds if martingale processes are continuous, denoted as Rc. Ourbounds with h = ∞ are the same as the bounds derived by Neuberger andHodges (2002).

The bounds Rg and Rc are dependent only on the prices of European calloptions with the same maturity as the barrier option. Following the analyticalresults in Brown et al. (2001) that show the construction of hedging strategies,I calculate rational bounds Rg in this discrete model. Following the approachby Neuberger and Hodges (2002), rational bounds Rc are sought through aone-dimensional search.



4.4.3. Convergence of valuation bounds (h < ∞)

For h < ∞ in (15), valuation bounds that are yielded under continuous Markovmartingales are denoted as Bc(h). These valuation bounds must stay withinrational bounds Rc (h = ∞). The upper and lower bound span a price of thebarrier option, denoted as V L, which is calculated by assuming that the futureprices of the stock follow the continuous process specified in (13).

I now introduce a measure ρ(h) to gauge the tightness of valuation bounds:

ρ(h) ≡ 1 − Bc(h) − Bc(h)

Rc(∞) − Rc(∞)

, (16)

where Bc(h) and Bc(h) are the upper and lower bounds of Bc for a fixed h;

Rc(∞) and Rc(∞) are the upper and lower bounds of rational bounds (Rc),

respectively. ρ(h) takes values between 0 and 1. In particular, a unique price isthen determined if the upper and lower bounds are identical such that valuationbounds can be substantially tightened (ρ(h) = 1). However, valuation boundson barrier options cannot be improved at all if the price bounds Bc are nottighter than rational bounds Rc (so ρ(h) = 0).

Before looking at the bounds, I investigate the sensitivity of the numericalanalysis to the choice of time and space grid. To implement the LP, I need fixon a time step and choose an appropriate jump size. The impact of coarsenessof the grid on valuation bounds is examined through two numerical examples.In both examples, the theoretical prices are calculated using the formula inRubinstein and Reiner (1991) in a Black–Scholes model.

Note that in all the numerical examples I examine, the restrictions on theprices of options are expressed as the volatility band of [σ e−h/2, σ eh/2], and theinitial parameters are set as (S0, σ ) = (100, 40%). Now consider an Americandigital option with barrier B = 108 and maturity T∗ = 1 month. The figures inTable I show that ρ(1) is apparently converging to a range of [57%, 58%] as thetime step T increases from 60 to 250. This is consistent with the results forother exotic options’ payoffs that are not reported here. To support a volatilityrange determined by h = 1, the minimum time step is 60 for the parameters Iuse. Our objective is to measure ρ(h), the relative tightness of valuation boundsdue to restrictions on the future prices of call options. For h = 1 (equal to avolatility band of [24%, 65%]), in Table I, it is suggested that I can attain reliablenumerical results even with T as low as 60. Therefore, I set T = 100 throughthe following numerical examples.

In Table II, I showthe convergence of the valuation bounds on a digitaloption with barrier B = 112 and maturity T∗ = 1 month, as the jump size u


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TABLE I

Convergence of Valuation Bounds Against Time Step

Valuation Bounds on an American Digital

T Rc Bc V L Bc

Rc

ρ(h = 1) (%)

60 0.2864 0.4224 0.4868 0.5395 0.5607 57.30100 0.2860 0.4217 0.4859 0.5386 0.5604 57.40150 0.2858 0.4214 0.4855 0.5384 0.5603 57.37200 0.2857 0.4216 0.4853 0.5380 0.5602 57.60250 0.2856 0.4212 0.4851 0.5382 0.5602 57.39

Note. The jump size u is 4% and time interval δ is determined by T ∗/T. The valuation bounds on an American digitalwith barrier B = 108 and one-month maturity (T ∗ = 1 month) are shown, and its theoretical price is 0.4849. The price V L

is calculated using the constant implied volatility of 40%. The numbers in the final column are calculated according to 16.

TABLE II

Convergence of Valuation Bounds Against Jump Size

Valuation Bounds on an American Digital

u(%) Rc Bc V L Bc

Rc

ρ(h = 1) (%)

2.0 0.1558 0.2386 0.2822 0.3206 0.3374 54.852.4 0.1596 0.2407 0.2822 0.3230 0.3373 53.693.0 0.1653 0.2439 0.2822 0.3216 0.3375 54.884.0 0.1746 0.2492 0.2823 0.3201 0.3317 54.876.0 0.1926 0.2590 0.2829 0.3243 0.3279 51.7412.0 0.2387 0.2840 0.2923 0.3003 0.3003 73.54

Note. The time step is T = 100 and time interval δ is determined by T ∗/T. The valuation bounds on an American digitalwith barrier B = 112 and one-month maturity (T ∗ = 1 month) are shown, and its theoretical price is 0.2811. The price V L

is calculated using the constant implied volatility of 40%. The numbers in the final column are calculated according to (16).

increases. The valuation bounds on this digital option are relatively sensitiveto the changes in jump size. This table seems to suggest that the numericalresults are relatively reliable if there are serval nodes (three or more) betweenthe initial stock price and the barrier. Furthermore, ρ(1) tends to lie in a rangeof [53%, 55%] for those jump sizes less than u < 6% so that all the valuationbounds show the convergence as the jump size u becomes smaller. I hence setthe jump size u < 6% through the following numerical examples.

4.4.4. Tightness of valuation bounds

I now investigate the tightness of valuation bounds on barrier options. Consideran American digital option with barrier B = 115 and maturity T∗ = 6 months.The valuation bounds on this digital option are reported in Figure 2. Overall,the substantially tight valuation bounds are yielded by changing h.

For h = 0.5 (equal to an implied volatility band of [31%, 51%]), for example,the valuation bounds Bc are 0.54 and 0.61, and the rational bounds Rc are 0.30



FIGURE 2Valuation bounds on an American digital option. The relevant parameters are

(u, δ) = (4.66%, 0.005). The valuation bounds on an American digital with barrier B = 115 andmaturity T∗ = 6 (months) are shown in the panel. The rational bounds Rc and valuation boundsBc on this digital option are indicated by the thick and solid lines, respectively. The price V L is

calculated using the constant implied volatility of 40%.

and 0.65. The price V L, equal to 0.58, is calculated using a constant impliedvolatility of 40%. As a result, the restriction on future implied volatilities tightensthe valuation bounds on this digital by about 80%.

I further investigate the tightness of valuation bounds on call and putbarrier options. Figure 3 presents all valuation bounds on eight barrier options.These options are in-the-money (ITM) with respect to strike when they knockin or out, and so are called reverse barrier options. As shown in Figure 3, thevaluation bounds on these single barrier options suggest the similar patternas the previous example. Namely, the restrictions on future implied volatilitiesdo have a very substantial effect on valuation bounds. Taking a reverse down-and-in put option with strike 100, barrier 85 and three-month maturity as anexample, the same restriction on future implied volatilities (h = 0.5) can tightenthe bounds on this option by over 58%.

For both the up-and-out call and the down-and-out put presented inFigure 3, they lose the intrinsic value when a knock-out event occurs. Thetable in the figure shows that the rational bounds (both Rc and Rg ) on bothoptions are too loose to be useful. However, in the panels in Figure 3, I suggestthat the valuation bound Bc can be substantially tight even in these hard, butmost realistic cases of reverse barrier options. With the same restriction onfuture implied volatilities (h = 0.5), for instance, the valuation bounds on theup-and-out call with strike 100, barrier 115 and three-month maturity can betightened by about 55%.


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FIGURE 3Valuation bounds on European barrier options. (a) Call barrier options and (b) put barrier

options. The time interval is set as δ = 0.0025, and the jump size u is 4.66% (5.42%) for optionswith the up(down) barrier of B = 115 (85). All barrier options have the same strike and maturity(K = 100 and T∗ = 3 months). The bounds Rc and Bc are indicated by the thick and solid lines.The dashed line indicates the price V L that is calculated using the constant implied volatility of40%. The rational bounds Rc (presented in the two panels above) and Rg are reported as follows:

Type RcL Rc

U RgL Rg

U Type RcL Rc

U RgL Rg

U

UI-C 6.720 7.916 6.720 7.916 UO-C 0.000 1.195 0.000 1.195DI-C 0.000 1.668 0.000 1.668 DO-C 6.231 7.892 6.231 7.896UI-P 0.074 2.093 0.000 5.058 UO-P 5.826 7.845 5.826 7.916DI-P 6.321 7.896 6.321 7.896 DO-P 0.000 1.573 0.000 1.575

Note. D, down; O, out; I, in; C, call; U, up; P, put; RcL/Rg

L (RcU/Rg

U ), the lower (upper) bound.



FIGURE 4Robustness of valuation bounds on an American digital option. The relevant parameters are(u, δ) = (4.66%, 0.005). For an American digital with barrier B = 115 and maturity T∗ = 6

(months), the left panel presents the sensitivity of ρ(h) to barrier level(B = 105, 110, 115, 120, 126), and the right panel shows the sensitivity of ρ(h) to maturity

(T∗ = 1, 3, 6, 9, 12 months).

4.4.5. Robustness of valuation bounds

To explore how sensitive these bounds are to the specific parameters of thedigital, I first investigate the sensitivity of the measure ρ to different barrierlevels and option maturities. The sensitivity of the valuation bounds on theAmerican digital mentioned above is presented in Figure 4. As I vary barrierlevels and maturities across a wide range, the relation between the tightnessof the price bounds ρ and h is broadly similar. If implied volatilities of optionsin future are constrained to lie within a range of [31%, 51%] (h = 0.5), thevaluation bounds can be tightened by at least 80%, compared with rationalbounds for a wide range of barrier levels and option maturities.

I now explore whether the result holds for other barrier options. Consideran up-and-in European call. The results are shown in the first panel in Figure 5.As before, bounding the future prices of options substantially tightens the valu-ation bounds on this up-and-in call. The improvement in the tightness of all thevaluation bounds is not very sensitive to the choice of parameters (i.e., barrierlevels, strikes, and maturities). For h = 0.5 (equal to an implied volatility bandof [31%, 51%]), for example, the valuation bounds are tightened by about 60%across different parameters.


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FIGURE 5Sensitivity of valuation bounds to parameters. (a) Robustness of valuation bounds on a

European up-and-in call and (b) sensitivity of ρ(h) to types of barrier option. The time interval isset as δ = 0.0025. In part (a), the jump size is set as u = 4.66%. The top-left panel shows the

valuation bounds on an up-and-in call option with B = 115, K = 90, and T∗ = 3 (months). Inthis panel, the rational bounds Rc and valuation bounds Bc are indicated by the thick and solidlines, respectively. The price V L is calculated using the constant implied volatility of 40%. The

sensitivity of ρ(h) to strike (K = 82, 86, 90, 94, 100) is shown in the top-right panel givenB = 115 and T∗ = 3 months. The bottom-left panel exhibits the sensitivity of ρ(h) to barrier

level (B = 105, 110, 115, 120, 126) given K = 96 and T∗ = 3 months. The bottom-right panelpresents the sensitivity of ρ(h) to maturity (T∗ = 1, 3, 6, 9, 12 months) given K = 100 and

B = 115. In Part (b), the jump size u is 4.66% (5.42%) for options with the up(down) barrier ofB = 115 (85). All barrier options have the same strike and maturity (K = 100 and T∗ = 3

months).



The second panel in Figure 5 exhibits the sensitivity of the measure ρ tothe type of barrier options whose valuation bounds are reported in Figure 3.In all examples I have examined, restricting the implied volatility to ±25% ofits forward level (say 40% in the numerical analysis) can tighten bounds by atleast 50% and even up to 80% in some cases. As suggested in this figure, thesepatterns are relatively insensitive to the type of barrier options.

5. FURTHER DISCUSSIONS

In all the numerical examples I have investigated in Section 4, there are twomajor assumptions: (1) the existence of a complete set of European call options(assumption [A2]); (2) the continuity of underlying price processes. I hereafterrelax these assumptions.

5.1. Finite Number of European Options

The assumption of a complete set of options is implausible in practice. InSection 4, I have assumed the existence of a complete set of European options,that is, C(K , t) (K ∈ K, t ∈ T). I now assume that there exists only a subset ofthese European options that trade with prices C(K , t) (K ∈ K′ ⊂ K, t ∈ T′ ⊂ T)at inception.

Fewer European options would widen valuation bounds. But the linearfeature of the valuation problem in (10) (the objective function and constraints)is unchanged. Following the notation in Section 4, the valuation problem canbe expressed as follows:

maxP

EP [b(ST , IT )],

subject to

(1) C(K , t) = EP [(St − K )+|F0], for K ∈ K′, t ∈ T′;(2) S0 = EP [St |F0], for St ∈ K, t ∈ T;(3) EP [(St+1 − St)+|Ft] ∈ [Cl, Cu] for St ∈ K, t ∈ T;(4) P is within a set of continuous extended-Markov processes.

To illustrate how to account for a finite set of trading options, the valuationbounds on an American digital option with one-year maturity are reported. Iassume that there exist a finite number of European options available C(K , t)for K ∈ K and t ∈ {3, 6, 9, 12} (months). Because the subset contains all theEuropean options with one-year maturity, the rational bounds Rc are known inthis example. In Figure 6, I report the valuation bounds on this digital option.These bounds are slightly wider than the bounds derived from a complete set ofEuropean options. For h = 1, for example, the finite set of options can widen


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FIGURE 6Valuation bounds on an American digital option with finite number of European call options.The relevant parameters are (u, δ) = (4.66%, 0.005). A complete set of European option priceswith all strikes and maturities is consistent with an annual implied volatility of 40%. A finite setof European options consists of those options with all strikes and the maturities of 3, 6, 9, and12 months. The American digital option with barrier B = 115 has one-year maturity. The solidlines indicate the valuation bounds Bc derived from a finite set of European options. The dashed

lines show the valuation bounds Bc derived from a complete set of European options. Therational bounds Rc are shown by the thick solid lines. The solid line with “x” stands for the

numerical price V L that is calculated using the constant implied volatility of 40%.

the bounds by about 6%. Because I have assumed the presence of options withall possible strikes for each maturity, the existence of a finite set of options haslimited impact on the bounds in this particular case.

5.2. Price Processes with Multiple Jumps

Throughout all the numerical examples, I have only considered continuousprice processes. This is a relatively strong restriction. In real markets, jumps instock prices are observed. As implied in Proposition 1, there exists an extended-Markov martingale process under which all initial call options are valued prop-erly. I now follow the same assumptions as before but drop the continuityassumption. The stock price can jump more than one level at any time.

The American digital option with barrier 115 and one-month maturity isrevisited. Figure 7 illustrates the valuation bounds on this option under bothgeneral and continuous martingale processes. If the stock price is restricted tojump at most one level at any time (continuity), the restrictions on option pricesdo, as I have seen, substantially tighten valuation bounds on barrier options.However, the valuation bounds directly jump to the rational bounds (Rg = Rc)when the continuity assumption is relaxed. These results suggest that the priceband on future option prices has no effect on the tightness of valuation boundsunder general martingale processes, and ρ nearly goes to zero.



FIGURE 7Valuation bounds on an American digital option under martingale price processes. The relevantparameters are (u; δ) = (4.66%, 0.0025). The risk-free interest rate is zero. The American digital

option with barrier B = 115 has one-month maturity (T∗ = 1 month). The rational bounds(Rc = Rg ) on the digital option are 0.16 and 0.24. The valuation bounds Bc under a general

martingale process are represented by the thick solid lines. The solid lines report the valuationbounds under a continuous martingale process. The dotted lines present the rational bounds Rc.The line with “x” presents the price V L that is calculated using the constant implied volatility of

40%.

It is useful to understand this in terms of processes. The way of limiting theprices of ATM European options imposes weak restrictions on the jump rate ofstock prices, as stock prices can have multiple jumps. This implies that impliedvolatilities of price processes may be bounded loosely. As a result, the valuationbounds on the digital could become wide, close (or even equal) to the rationalbounds Rc.

If the continuity assumption is imposed, the jump size is restricted so thatthe constraints on ATM option prices impose bounds on the jump rate of stockprices. Hence, the valuation bounds on the digital are tightened, as shown inFigure 7. These numerical figures suggest that with the restrictions on ATMoption prices (even on future implied volatilities), the continuity assumption isrequired in order to seek tight valuation bounds on barrier options.

In particular, the continuity assumption equivalently states that the futureprices of out-of-the-money (OTM) and ITM options are bounded as follows:

limτ→t

C(Steiu, τ ; t)τ − t

→ 0, for i �= 0, (17)

so that a costless strategy that pays nonnegative cash can be constructed ifthe price jumps more than one node. In order to seek tight bounds without


28 Hong

requiring continuity, the alternative approaches would involve either limitingthe prices of OTM (or ITM) option contracts or directly restricting the prices ofshort-dated variance contracts, e.g. a limit on the future prices of log contracts.

6. CONCLUSIONS

This study investigates the tightness of valuation bounds on single barrier op-tions under model uncertainty. This investigation further enriches the literatureon the model-free valuation of financial derivatives. It is found that with weakassumptions on underlying price processes, tight valuation bounds on barrieroptions can be sought from exchange traded European options.

As a result, this study proposes a new approach for pricing these complexoptions. Unlike the traditional model building approaches that generate preciseprices for these exotics, I derive valuation bounds on them in a frictionlessmarket. These valuation bounds are dependent on European option prices.Numerical results suggest that fairly weak bounds on future option prices canlead to the substantial tightness of valuation bounds.

To attain these valuation bounds, I have shown how to set up LPs to searchfor martingale processes that are consistent with initial option prices. The priceprocesses that support valuation bounds indicate scenarios under which barrieroptions are more or less valuable. As the implication of the LP Duality Theorem,hedging strategies that enforce these valuation bounds can be identified fromthe dual problem.

APPENDIX

A. Proof of Lemma 1

Proof. Consider a trading strategy that consists of long 1/(K eu − K ) calls withstrike K eu and 1/(K − K e−u) calls with strike K e−u, and short (1 + e−u)/(K −K e−u) calls with strike K , whereas all calls have the same maturity t. Becausethe risk-free interest rate is zero, the initial prices of these calls are

C(K eu, t) =∑

St≥K eu

(St − K eu)π(St, t);

C(K , t) =∑St≥K

(St − K )π(St, t); (A.1)

C(K e−u, t) =∑

St≥K e−u

(St − K e−u)π(St, t),



for St ∈ K. The terminal payoff to this strategy would be equal to

∑St≥K eu

St − K eu

K eu − K−

∑St≥K

(1 + e−u)(St − K )K − K e−u

+∑

St≥K e−u

St − K e−u

K − K e−u= 1St=K ,

(A.2)where 1(·) is an indicator function. The cost of this strategy is the marginalprobability of St being K ∈ K at time t ∈ [0, T∗]:

π(K , t) = E[1St=K |F0] = C(K eu, t) − (1 + eu)C(K , t) + euC(K e−u, t)K eu − K

. (A.3)

��

B. Proof of Lemma 2

Proof. Consider two European calls with the same strike K and differentmaturities,

C(K , t) =∑St≥K

(St − K )π(K , t); C(K , τ ) =∑Sτ ≥K

(Sτ − K )π(K , τ ), (B.1)

where τ > t.Conditional on the stock price being K at time t, the martingale condition

in (1) states:

E[Sτ |St = K ] = K , for Sτ ∈ K, t ∈ [0, τ ]. (B.2)

The definition of continuity in (2) equivalently states

prob(|ln Sτ − ln St| > u) = o(τ − t). (B.3)

It means that the stock price St can only jump one level or stay when τ goes to t,conditional on St = K , for example, Sτ ∈ {K eu, K , K e−u} for τ → t. Conditionalon the price level at time τ , Sτ = K , we also have St ∈ {K eu, K , K e−u} for t → τ .Therefore, taking the limit on the martingale condition (B.2) with respect totime yields:

limτ→t

�(K eu, τ ; K , t)(K eu − K ) + �(K e−u, τ ; K , t)(K e−u − K ) = 0. (B.4)

It follows from Lemma 1 that the probability of the stock price reachingany level at any time is determined by initial option prices. The definition of


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continuity implies the following relations:

π(K , τ ) = �(K , τ ; K eu, t) + �(K , τ ; K , t)

+ �(K , τ ; K e−u, t) + o(τ − t), for t → τ ; (B.5)

π(K , t) = �(K eu, τ ; K , t) + �(K , τ ; K , t)

+ �(K e−u, τ ; K , t) + o(τ − t), for τ → t. (B.6)

The first equation states that there are only three possible paths that reach node(K , τ ) when t goes to τ , and the second states that there are only three possiblepaths that come out from node (K , t) when τ goes to t.

For a European call with strike K and maturity τ , its initial price may beexpressed as

C(K , τ ) =∑Sτ ≥K

(Sτ − K )π(K , τ )

=∑Sτ ≥K

(Sτ − K )(�(K , τ ; K eu, t) + �(K , τ ; K , t) + �(K , τ ; K e−u, t)

+ o(τ − t)), for t → τ

=∑St≥K

(St − K )π(K , t) + (K eu − K )�(K eu, τ ; K , t)

+∑St≥K

(St − K )o(τ − t), for τ → t

= C(K , t) + (K eu − K )�(K eu, τ ; K , t)

+∑St≥K

(St − K )o(τ − t), for τ → t. (B.7)

This expression then yields the first result

limτ→t

�(K eu, τ ; K , t)τ − t

= 1K eu − K

∂C(K , t)∂t

. (B.8)

Similarly, we can derive the second result

limτ→t

�(K e−u, τ ; K , t)τ − t

= 1K − K e−u

∂C(K , t)∂t

. (B.9)



For any price level K ∗ ∈ K \ {K eu, K , K e−u}, the continuity of the underlyingprocess implies that

limτ→t

�(K ∗, τ ; K , t)τ − t

= 0. (B.10)

From the definition of the function �, initial option prices imply that

∑K ∗∈K

�(K ∗, τ ; K , t) = π(K , t). (B.11)

��C. Proof of Proposition 1

Proof. Consider a process P in M on a filtered probability space (,F, P) thatis defined as follows:

P � {Xt(w) : t ∈ [0, T∗], w ∈ }, (C.1)

and

F0 = {�, }, Ft = σ (Xt ; t ∈ (0, T∗]), F = {Ft}t∈[0,T∗], (C.2)

where X is a vector of state variables, including the stock S, and w is a samplepath generated by the state vector X. Under the process P , all initial optionsare priced properly.

EP[(St − K )+|F0] = C(K , t), for St, K ∈ K and t ∈ [0, T∗], (C.3)

or

P{St = K } = π(K , t), (C.4)

where π(K , t) is determined in Lemma 1.An extended-Markov process Q is constructed from the process P .

This extended-Markov process is defined on a new filtered probability space(∗,F∗, Q) as follows:

Q � {St(ω) : t ∈ [0, T∗], ω ∈ ∗}, (C.5)


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where S = (S, I ) and ω indicates a sample path of the underlying stock S,conditional on the variable I . The construction of the sample space ∗ involvesselecting all sample paths generated by the process P :(a) Pick up an outcome from ;(b) following this outcome, one runs the process P to generate a sample path

w;(c) repeating (a) and (b), one may generate all sample paths {w} = ;(d) one picks up a sample path from the set {w} without repetition to generate

a stock price path ω, and the variable I takes the value 1 if the barrier hasbeen breached and zero otherwise;

(e) all stock price paths generate a new sample space, {ω} = ∗.

Given this new sample space ∗, define a new filtration {F∗τ }τ∈[0,T∗] where

F∗τ = σ (Sτ , Iτ ; τ ∈ [0, T∗]):

(1′) F∗0 = {�, ∗};

(2′) Pτ = ⋃i∈{0,1}

⋃K ∈K

{AK ,iτ }, where AK ,i

τ � {ω : Sτ (ω) = (K , i), ω ∈ ∗} forτ > t;

(3′) F∗τ = F∗

t ∪ Pτ for t ∈ [0, τ ).

Here AK ,iτ is an event for the stock price being K at time τ , conditional on Iτ = i .

Pτ is a partition of all events at time τ . Associated with (∗,F∗), a probabilitymeasure Q is defined as follows:

Q(St(ω) = (K , i)) = P(St = K , It = i);

Q(Sτ (ω)|St(ω) = (K , i),F∗t ) = P(Sτ , Iτ |St = K , It = i), (C.6)

for K ∈ K, ω ∈ ∗ and t ≤ τ . Therefore, I define a function f that maps pro-cesses in M into processes in a set MEM

f : M → MEM

P → Q = f (P ). (C.7)

Following the construction of the measure Q, we can infer the propertiesas follows:

(i) Because the process P is a martingale in M, the process Q is also amartingale about the stock price under the measure Q. It follows from thesecond equality in (C.6) that the process Q has the Markov property withrespect to the stock price and indicator variable,

Q(Sτ |F∗t ) = Q(Sτ |St).



(ii) From the first equality in (C.6), all initial options are correctly priced underthe process Q, and thus Q ∈ MEM ⊆ M.

(iii) If the process P is continuous, the construction of the process Q impliesthat Q is also continuous.

(iv) If assumption [A5] is made, no-arbitrage arguments ensure that there existsat least one martingale process in M that supports initial option prices.From the construction specified above, an extended-Markov process canbe constructed and also satisfy assumption [A5].

The rest of the proof is to complete the part (v). For the equalities in (9), Ijust need to prove the supremum. The case of infimum can be proved similarly.Let

V Q ≡ supQ∈MEM

EQ[b(ST∗, IT∗)|F0]; V P ≡ supP∈M

EP [b(ST∗, IT∗)|F0]. (C.8)

On the one hand, it is impossible that V Q > V P due to V Q ≤ V P . On the otherhand, it is also impossible that

V Q < V P . (C.9)

Otherwise there exists at least one extended non-Markov martingale processP ∗ ∈ M so that V Q < V P ∗

holds and

V P = EP ∗[b(ST∗, IT∗)|F0] = sup

P∈MEP [b(ST∗, IT∗)|F0]. (C.10)

From the construction above, an extended-Markov process Q∗ ∈ M can beconstructed from the process P ∗. This process produces a feasible price for theterminal payoff b(ST∗, IT∗):

V Q∗ = EQ∗[b(ST∗, IT∗)|F0] ≤ V Q. (C.11)

At maturity, the marginal probability densities respect to the stock price andindictor variable are identical under both the processes P ∗ and Q∗. This resultimmediately yields

V Q∗ = EQ∗[b(ST∗, IT∗)|F0] = EP ∗

[b(ST∗, IT∗)|F0] = V P . (C.12)


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Combined with (C.11), this result contradicts with the assumption in (C.9) that

V Q∗ ≤ V Q = supQ∈MEM

EQ[b(ST∗, IT∗)|F0] < V P . (C.13)

Therefore, the only possible outcome is V Q = V P , which completes theproof. ��

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