Electronic copy available at: http://ssrn.com/abstract=1335302

The Hedging Costs of Discrete Monitoring of FX

Barrier Options

Antonio Castagna∗

January 30, 2009

1 Introduction

Discretely monitored barrier options are becoming increasingly popular for FX pairs againstthe Euro, since they can be referred to an official fixing (the so called ECB37, pub-lished by the European Central Bank) and hence they offer more transparency to thenon-professional buyers (or sellers) as for the touching of the barrier level and the conse-quent cancelling (knock-out) or activation (knock-in) of the optionality in the contract.

On one hand the discrete monitoring feature is much appreciated by customers (typ-ically exporting or importing firms), on the other hand it may cause some nightmares tothe hedgers, when the FX spot rate is around the barrier level and between two fixings’dates. In these circumstances it is not possible to take the unequivocal decision whether tocompletely unwind the Delta-hedge, if the option is a knock-out (if the option is a knock-in,the decision is whether to consider the option a plain vanilla).

In what follow we propose a possible approach to include the costs produced by theabove mentioned uncertainty. To this end we first have to come up with a method to copewith a somehow related risk referred to continuously monitored barriers: the so calledslippage. We also inspect the relationship of our approach to other related works. Finallywe produce numerical results showing the impact of the inclusion of the Delta-hedgingextra-costs into the pricing of discretely monitored barrier options.

∗Antonio Castagna is a consultant at Iason ltd. Email address: antonio.castagna@iasonltd.com. I wouldlike to thank Fabio Mercurio for helpful discussions and for pointing out some errors in a preliminary versionof this work. Any remaining error is clearly only mine.

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Electronic copy available at: http://ssrn.com/abstract=1335302

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2 Stylized Pricing of Continuously Monitored Barrier

Options and the Slippage Cost

The pricing of a barrier option can be performed in the standard fashion applied in theOption Pricing Theory. The price at time t is the expected value of the discounted terminalpay-off under the risk-neutral measure. We will provide the formula only in a general termsof the variables and the parameters which it depends on, so as to include all the types ofthe first generation exotic barrier options.

We start by setting the following notation:

+1 -1ω call putθ down upζ out in

The value of a barrier option at time t expiring in T and contingent to the barrier level Hcan be thus written:

B(St, t, T, K, H, ω, θ, ζ) =

EQ

[e−

∫ T

trdsdsω(ST − K)+

[1 − ζ

2− ζI{θSt′>θH,∀t′∈[t,T ]}

]] (1)

The expectation in (1) can be calculating exactly in a Black&Scholes (hereon, BS) economywith constant domestic and foreign interest rates and implied volatility (see, amongst manyothers, Zhang [12] for a collection of pricing formulae). Approximations can be derivedfor time-dependent parameters (see Rapisarda [9] and references therein), whereas onegenerally has to resort to Montecarlo simulations or other numerical schemes in a stochasticvolatility framework.1 In what follows we assume we are working in a BS setting and thata market maker prices FX barrier options by a set of rules of thumb to include the effectof the volatility smile and of other relevant risks not taken into account by the standardmodels (not only the BS). We assume also that the BS model is used to manage the risksrelated to these options, but we will focus in this work only to the risks related to theDelta-hedging activity.2

In the managing of risks, the exposure to the underlying FX spot rate is the first onethat is hedged, by zeroing the Delta. Operating this activity is extremely difficult whenthe FX spot rate is near the barrier level and the time to the expiry of the option is short.In these situations, when a market maker is short a reverse knock-out option, they will

1One exception is the Uncertain Volatility model described in Mercurio [8]; An extensive analysis ofthe application of this model to the barrier options’ hedging is presented in Castagna [5].

2Castagna [5] examines in details this set of rules in what is therein defined as the “market approach”to the pricing of FX barrier options. Also there he analyses how to manage in an advanced fashion therisks of a wide range of exotic options within the BS framework.

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typically be long (short) a possibly huge amount of the underlying asset, for Delta-hedgingpurposes, when the FX spot is around the down (up) barrier. As the barrier level isbreached, they will have to unwind the Delta-hedging position by selling (buying) backthe underlying FX rate amount, with a declining (raising) spot rate. In such an event, theorder to place in the market is a stop-loss (i.e.: close the position for an adverse movementof the FX spot rate) and this will typically produce an additional cost known as slippage:The stop-loss will be executed only after the specified level is traded in the market, sothat in most of cases the order is executed at a worse price than the stop-loss level. Ifthe hedger places the stop-loss order at the barrier (i.e.: when they need to unwind theDelta-hedge position after the triggering of the barrier), the difference between this leveland the actual price at which the closing of the position is executed generates the slippagecost.

The market maker has to include the slippage into the pricing of the barrier option anda possible approach is the following. Assume we want to price a knock-out option: We mayimpose that the value of the option at the barrier level is not nil, but it has some residualvalue so as to make up for the slippage cost born by the hedger. In practice, the standardboundary condition at the barrier level B(H, t, T,K,H, ω, 1, θ) = 0 is replaced by:

B∗(H, t, T,K,H, ω, 1, θ) = −x%H∂B∗

∂St

(2)

where x% is the expected slippage cost, expressed as a percentage of the FX spot level atthe barrier H. If x% is, for example, set at 1% and the barrier level is 200, this means thatthe order is expected to be actually executed at an FX spot rate distant 200 × 1% = 2numeraire currencies units from the stop-loss level 200 (equal to the barrier). The minussign on the right-hand side of (2) is due to the fact that the cost is referred to a positionin the underlying FX that is the opposite of the Delta.

It is manifest that the slippage cost can be included into the pricing of the barrier optionby valuing a contract with a slight different condition when the FX spot rate breaches thetrigger level. To price this contract set ∆H = x%H and B∗ as the value of the contractwith the new feature we want to determine. We can re-write the equation above as:

B∗(H, t, T,K,H, ω, 1, θ) + ∆H∂B∗

∂St

= 0 (3)

Since for an ordinary barrier, with knock-out trigger at H∗ = H + ∆H, when the FX spotrate St = H we have:

B(H, t, T,K,H∗, ω, 1, θ) + ∆H∂B∂St

≈ B(H∗, t, T,K,H∗, ω, 1, θ) = 0 (4)

by inspection (assuming that3 ∂B∗

∂St= ∂B

∂St) we immediately get that B∗(H, t, T,K,H, ω, 1, θ) ≈

B(H, t, T,K,H∗, ω, 1, θ), so that the expected value (given the current FX spot rate) of the

3The assumption is not strong since we x% is usually very small and well below 1%. It should bestressed anyway that the arguments of the two dervatives are different.

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slippage cost can be included into the pricing by valuing an otherwise equal barrier option,by setting the trigger at a new level H∗ = (1 + x%)H. The slippage cost charging is thendefined as:4

slg = B(St, t, T,K,H∗, ω, 1, θ) − B(St, t, T,K,H, ω, 1, θ) (5)

We can see that the inclusion of the slippage cost increases the value of the option, sincethe barrier level is shifted farther from the starting FX spot rate. This can be consistentwith the fact that the slippage cost is borne by the hedger who is short the barrier option,so that they sell the contract at a higher price thus compensating for it. For a buyer ofthe option the cost does not have to be included since the hedger with place take profitorders at the barrier to unwind the Delta equivalent position in the underlying FX spotrate: These orders are executed exactly at the level so that no costs arise.5

The option we considered above is a knock-out barrier. For knock-in options the slip-page cost should be valued by means of the knock-out/knock-in parity (knock-in are worthless when the slippage is taken into account). The slippage cost can be included into thepricing of the bets (one-touch’s and double-no-touch’s) also. Similarly to the case of barrieroptions, the slippage cost can be calculated by pricing the one-touch at two different levelsof the barrier.6 It should be stressed that for one-touch options, moving the barrier fartherfrom the starting FX spot rate will imply a lower price, so that the slippage cost will benegative, thus decreasing the fair value when included in it. This is consistent with thefact that the hedger with a long position in the one-touch needs to place a stop-loss inthe market to unwind their Delta-hedge when the barrier level is breached, so that, rightlyenough, the buyer of a one-touch should be compensated for this additional borne cost byrequiring a discount on the fair price.

Finally, we just notice that the inclusion of the slippage has a much greater impacton reverse barriers than on standard barrier, and for short termed contract than for longterm.

3 Discretely Monitored Barrier Options

In previous section we have examined barrier contracts whose pay-off is contingent to thebreaching of a given level by the FX spot rate, monitored continuously. This means that

4In a much more formal and thorough framework Shreve et al. [10] and [11] derive a more preciseresult, although our less exact formula (5) works well for practical purposes.

5We do not discuss here the case of a partial fulfillment of the take profit order: This can produceextra-costs but their estimation is extremely difficult. Luckily enough, this event occurs rarely.

6For a double-no-touch option the shifting is operated on both barrier levels. This will produce a higherprice and then a positive slippage cost to be added to the fair value. Also in this case the procedure isconsistent with the fact that the seller of the DNT will use stop-loss orders to unwind the Delta-hedge,thus requiring a higher price to make up for this additional costs.

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it is enough that the FX spot rate traded in a good size7 when one of the main marketplaces is open (basically week-end’s are not considered in the monitoring).

Barriers can be also discretely monitored, by some predefined frequency: The mostcommon alternative to the continuous monitoring, especially for pairs against the euro, isthe daily frequency referred to the ECB fixing. Less usually the monitoring may be set at aweekly or monthly fixing. The monitoring only at the expiry deserves a special treatmentand we defer its examination to the next chapter, since what we will present here doesnot apply to this feature. Unfortunately to our knowledge exact closed form formulae toevaluate such contracts do not exist, so that one has to resort to some numerical scheme toproperly take into account the discrete monitoring: Montecarlo, trees or finite differencesare the most widely adopted. An alternative is the analytical approximation provided byBroadie, Glasserman, and Kou in [2] and [3], and by Kou [7].8 It works well for practicalpurposes for the daily frequency, then the accuracy of the approximation deteriorates andit is worth to employ one of the numerical procedures we mentioned above.

Assume we have an exotic contract contingent to the barrier level H and monitored mtimes until the expiry. The following propositions holds:

Proposition 3.1. Let E(H) be the price of a an exotic option contingent to the barrierlevel H, and Em(H) be the price of an otherwise identical exotic option with annual periodfrequency 1/m (i.e.: monitored in m equally space points until the expiry). Then for thediscrete monitored exotic options, we have the approximation:

Em(H) = E(He±βσ√

τ/m) + o(1/√

τ) (6)

with + for a barrier level above the starting FX spot rate at the evaluation time, and − fora barrier level above the starting FX spot rate. The constant

β = −ζ(1/2)√2π

≈ 0.5826

with ζ being the Riemann zeta function

For the proof, see Kou [7]. The approach can be used for any kind of contract depen-dent on a single barrier, but it can be safely employed also to calculate the adjustment forcontracts dependent on two barriers (see Horfelt [6] for a discussion on the extension ofthe method). We assume that the hedger adopts this method to price discretely monitoredbarrier options, and this can be justified for a couple of reasons: Firstly, the most commonmonitoring period is daily, so that the approximation works satisfactorily; Secondly, al-ternative methods involve numerical procedures with slow degree for convergence, so that

7It is meant by “good size” an amount of base currency that is can be considered twice or three times theminimum amount traded by professionals market-makers. For example, for the EURUSD it is 3 millionseuros.

8We refer to those article for the proofs of the results we will show here.

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an approximation error is unavoidable anyway, to keep computation times within feasiblebounds.

One of the difficulties faced by the hedger with discrete monitored barriers, is theunwinding of the Delta-hedge after the breaching of the barrier levels (we are consideringthe case on a knock-out option). This is somehow related to the slippage cost we haveexamined in the previous section, although it is much more dangerous, especially for reverseknock-out options with a short time to maturity. In fact, if the FX spot rate touches thebarrier between two observation times defined by the frequency period (e.g.: one ECBfixing and that one of the following day), then they have to decide whether to close theDelta-hedge spot position in the underlying FX pair, thus assuming that the spot will notrevert back to the levels where the option is still alive, or to leave the position unchanged.

The problem is partially solved by using the pricing formulae with adjustment (6),which implicitly suggests to keep on Delta-hedging even for levels above (or below, for adown-and-out) the contract barrier level H, although the problem is simply shifted to the

adjusted level He±βσ√

τ/m. Once this level is reached, again, one has to choose whether tounwind or not the Delta position.

In order to cope with this problem and to include the related risks into the pricing ofthe discrete barrier options, we propose the following Delta-hedging strategy for a discretebarrier options (we assume we are working with knock-out options):

Strategy 3.1. The market maker completely unwinds the Delta-hedge positions whenever

the barrier level He±βσ√

τ/m is breached as they were hedging a continuously monitoredbarrier (the pricing formulae in this case evaluated the contracts as zero, and the Delta isnil); If the FX spot rate reverts so that the contract is back in the “live” region, then theDelta-hedge suggested by the pricing formula is set up again.

This strategy has been proposed also for plain vanilla options, with reference to thestrike price, and it is called stop-loss-start-gain (SLSG).9 Clearly this rule implies somecosts that should be borne, since the Delta is traded (for example, in case of a call up-and-

out) at some level of the FX rate St∗ = He±βσ√

τ/m + ε, and it is set up again at a level

St∗ = He±βσ√

τ/m − ε. Letting ε → 0 we have that the expected cost of the rebalancingof this strategy is equal to the so called expected local time of the FX spot rate’s processduring a given period τ .10 As far as discrete barriers are concerned, the period is given bythe monitoring frequency.

To make things concrete, let us consider a reverse knock-out call option with barrier H,daily monitored. It is rather clear that we will perform the (SLSG) Delta-hedging strategy

9To be precise, for a plain vanilla option the SLSG strategy is operated by keeping, at any time, acounterbalancing underlying amount equal to 0 when the contract is out-of-the-money, and equal to thenotional amount when the contract is in-the-money. See Carr and Jarrow [4].

10See Carr and Jarrow [4]. The local time is the amount of time spent around a given level by the FXspot rate, in an interval of time.

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only if the FX spot is around the barrier level and for the period between two observations,i.e.: one day in this case. Let us set for a moment ∆ = 1, i.e.: Around the barrier we eitherkeep 0, or the (reverse signed) option’s notional amount. At time t0 = 0 we calculate theexpected costs associated to the SLSG strategy over a given period τ = T − t0 and arounda level X, given that the underlying spot rate reaches the level X, as the local time:

P d(0, T )ET [Λ0(X)] = P d(0, T )ET

[ ∫ T

0

1{Fu>X}dFu

](7)

where we operated under a forward-risk adjusted measure and Ft is the forward price ofthe FX rate at time t with expiry in T . Carr and Jarrow [4] show the general resultthat the current expected value local time for a SLSG strategy performed with one unitof underlying asset, during a given period and around a given level X, is equal to thetime value of an option struck at X and expiring in the time interval we are considering.Applying this result, we can calculate the quantity in (7) as the time value of a plain vanillaoption struck at X, when S0 = X and expiring in T .

For a daily monitored barrier, we are interested in determining the costs associatedwith the proposed SLSG strategy over a period of one day when the underlying FX spot

rate reaches the adjusted barrier level St = Heβσ√

τ/m. We calculated then the discountedlocal time as the time value of a (either a call or a put) plain vanilla struck at a level equalto the barrier and maturing in one day:

Λ =P d(0, T )ET [Λ0(Heβσ√

τ/m)] =

Bl(Heβσ√

τ/m, 0, 1/365, Heβσ√

τ/m, P d(0, 1/365), P f (0, 1/365), σ, ω)(8)

where ω = 1 if the barrier level is above the FX spot rate at time t = 0, or ω = −1otherwise. Equation (8) valuates the expected local time assuming that we are in a BSenvironment:

Bl(St, t, T,K, P d(t, T ), P f (t, T ), σT , ω) = P d(t, T ) [ωF (t, T )Φ(ωd1) − ωKΦ(ωd2)] (9)

where

d1 =ln

F (t,T )K

+σ2

2(T−t)

σ√

T−t, d2 = d1 − σ

√T − t

P d(t, T ) = e−∫ T

trdsds, P f (t, T ) = e−

∫ T

trfs ds

σ is the implied volatility, and rdt and rf

t are respectively the domestic and the foreigninstantaneous interest rates.

We can use the current market value for the interest rates and set σ equal to the currentover-night level. We are interested, though, in finding the value of the local time for aSLSG strategy operated with an amount ∆t (the Delta of the exotic option), which is timedependent. To this end we make the simplifying assumption that the local time for each day

is constant, and express it as a percentage of the adjusted barrier Λ = Λ/(He±βσ√

τ/m): At

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any time t we then have that the expected local time (or, alternatively said, the expected

rebalancing cost around the barrier level) is ∆tΛHe±βσ√

τ/m.By a reasoning similar to that made for the slippage cost of a continuous barrier we

want that when the barrier is breached, the option is worth an amount equal to zero plusa value compensating for the loss due to the local time at the barrier during one day (theDelta amount held to hedge has the opposite sign of the mathematical first derivativeswith respect to the spot):

B∗(Heβσ√

τ/m, t, T,K,Heβσ√

τ/m, ω, 1, θ) = −∆ΛHeβσ√

τ/m (10)

and by repeating the same argument as above for the slippage cost, we have:

mtg =B(St, t, T,K, H∗, ω, 1, θ)

− B(St, t, T,K, H, ω, 1, θ)(11)

where H = He±βσ√

τ/m, H∗ = (1 + Λ)H and mtg is the extra-cost for the risk of Deltahedging for a discritely monitored barrier.

Equation (11) can be employed also for lower frequencies, as weekly or monthly ob-servation periods, by changing the expiry of the option in (8), although in these cases itis more appropriate to resort to numerical procedures as mentioned above. On the otherhand, equation (11) can be used also for standard knock-out’s, double knock-out’s and betcontracts.

4 Relationship with a Previous Work

Becker and Wystup [1] conducted an analysis on the costs related to discretely monitoredbarriers, with the ECB fixing. Apparently their work seems strongly related to ours, butactually it deals with a different problem and namely: The delay in the announcementof the fixing. More specifically, the ECB fixing is usually published on the Reuters page“ECB37” around 14.20/14.25 CET, and it refers to an average of FX spot rates’ quotesagainst EUR, asked to a pool of market makers at 14.15 CET. The lag between the surveyand the publishing of the fixing entails a risk for the hedger, since the FX spot ratesmove and they do not know, for example, whether a barrier has been breached so that thecorresponding Delta-hedge has to be dismantled.

The risk exists and it is real, but we think it is not so dangerous for a few reasons:Firstly, it is possible to infer, within a couple of pips of error, which the ECB fixing willbe just by looking at the screens’ quotes (e.g.: EBS platform) at 14.15 CET.11 Secondly,it is extremely unlikely that the spot rate at 14.15 CET will be exactly in a couple of basispoints distance from the barrier level: The probability, calculated at the inception of the

11On a light note, but not so irrelevant as it may appear, the risk can be mitigated by having on thetrading desk one of those clocks linked via radio to some time observatory.

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Eur Int. Rate 3.85%Usd Int. Rate 2.25%Spot EURUSD 1.3000

Strike 1.3000Implied Volatility 10.00%

Table 1: Values of relevant variables and parameters used in examples.

contract is in practice nil, so that one cannot reasonably charge an extra-cost for this risk.In any case, when this plight occurs, the hedger decides by assuming that the FX spot rateat 14.15 CET will be the fixing and in the end they will simply have to revise the decisionif the ECB fixing actually turns out to be different form the expected one. We may safelyaffirm that also in these circumstances the expected cost is small, due to the short lag ofmaximum 10 minutes between the survey and the publishing.

For these reasons, the expected losses to charge over the discretely monitored barrierfor the delayed announcement is definitely small, if any. In fact, Becker and Wystup [1]have to calculate a sort of maximum unexpected loss, via a simulation, within a 99.9%degree of confidence, to get some appreciable adjustments to the option’s fair value.

The risk that we have examined in this work is much more dangerous, since it in-volves the decisions that the hedger has to take regarding the Delta-hedge of the discretelymonitored option, when the underlying FX spot rate is hovering around the barrier. Theresolution of the uncertainty cannot occur before the next fixing, in a period of at least oneday, which is a significantly longer time than the 10 minutes announcement’s delay. In thiscase the spot can move a lot and a knock-out options, for example, can die and resurrectmany times: Quite a disturbing situation for the trader. The expected cost related to thisrisk is not negligible and has a material impact when charged over the option’s fair value.

Besides considering a more relevant, in our opinion, risk, we have also proposed a clearDelta-hedging strategy when the FX spot is near the barrier level, within a simple andcomputationally efficient framework provided by the BS formula and the adjustment dueto the discrete monitoring effect.

5 Numerical Results

We now study which is in practice the extra-cost for Delta-hedging risk of a discretemonitored barrier. As an example we assume that the relevant variables and parameterstake values shown in table 1.

We first present how, for a fixed barrier level equal to 1.4000, the extra-cost changesand which is its percent weight on the premium of the barrier option. Figure 1 shows thatin absolute terms the cost is increasing in the expiry of the option up to 3 month expiry,and then decreasing. Nonetheless, in percent terms on the premium of the option, the costis increasing and quite significant, since it is above 10% for the 1 year expiry.

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0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

1m 3m 6m 9m 1y

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

Figure 1: Cost for Delta-hedging risk of a discrete barrier. Right-hand scale and bars:Absolute cost in US pips. Left-hand scale and line: Cost as a percentage of the option’spremium.

To examine the effect of the Delta-hedging risk for different barriers, we fix the expiryat 5 months and calculate the cost for increasingly distant knock-out levels. Results arein figure 2: The percentage on the options’ premium is decreasing with the distance of thebarrier from the starting level of the spot rate, but in absolute terms it is first increasingand then decreasing but in very mild way. This can be understood since there are twocounter-balancing effects operating when the barrier is farther away from the starting spotrate: On one hand there is a lower probability that the spot rate will reach the knock-outlevel, on the other and the amount of underlying to use in the Delta-hedging, once theknock-out level is approached, is the greater the wider the gap between the strike priceand the barrier.

6 Conclusions

This work proposes a possible method to determine the costs related to the Delta-hedgingrisks of a discretely monitored barrier option. The method hinges on a Delta-hedge strategyaround the barrier level, it is simple to implement and it does not resort to any historicaldata analysis, such other authors suggest in the study of similar, though not equal, risks.We then think that the method can be beneficial to the actual risk management of theabove mentioned class of exotic options.

References

[1] C. Becker and U. Wystup. On the cost of delayed currency fixing announcements.Working paper, HfB - Business School of Finance and Management, 2005.

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0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

1.3750 1.4000 1.4250 1.4500

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

Figure 2: Cost for Delta-hedging risk of a discrete barrier. Right-hand scale and bars:Absolute cost in US pips. Left-hand scale and line: Cost as a percentage of the option’spremium.

[2] M. Broadie, P. Glasserman, and S. G. Kou. A continuity correction for discrete barrieroptions. Mathematical Finance, 7:325–349, 1997.

[3] M. Broadie, P. Glasserman, and S. G. Kou. Connecting discrete and continuous path-dependent options. Finance and Stochastics, 3:5582, 1999.

[4] P. P. Carr and R. A. Jarrow. The stop-loss start-gain paradox and option valuation:A new decomposition into intrinsic value and time value. Review of Financial Studies,3(3):469–492, 1990.

[5] A. Castagna. FX Options and Volatility Smile Risk. Wiley & Sons, forthcoming, 2009.

[6] P. Horfelt. Extension of the corrected barrier approximation by broadie, glassermanand kou. Finance and Stochastics, 7(2):231–243, 2003.

[7] S. G. Kou. On pricing of discrete barrier options. Working Paper, Department ofIEOR, Columbia Universtity, 2001.

[8] F. Mercurio. A simple uncertain volatility model. Banca IMI Internal Report, 2004.

[9] F. Rapisarda. Pricing barriers on underlyings with time-dependent parameters. BancaIMI Internal report, 2003.

[10] S.E. Shreve, U. Schmock, and U. Wystup. Dealing with dangerous digitals. ForeignExchange Risk, Risk Publications, London, 2002.

[11] S.E. Shreve, U. Schmock, and U. Wystup. Valuation of exotic options under shortselling constraints. Finance and Stochastics, 2(4), 2002.

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[12] P.G. Zhang. Exotic Options: A Guide to Second Generation Options. World Scientific,1997.