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1 A Worked-Out Example: Pricing Continuous Double Barriers

1 A Worked-Out Example: Pricing Continuous Double Barriers

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how small), and then engage in a costless self-financing trading strategy that will enable

the seller to meet her terminal obligations no matter what the final realization of the

underlying turns out to be, and keep the premium. More precisely, the idea is as follows.

Without loss of generality, assume that the stock price today is $100. Borrow $100 today.

Sell a European call option with strike $100 and maturity T . Accept any premium the

buyer is prepared to pay, no matter how small, as long as strictly positive. Then engage

in the following hedging strategy.

• If the stock price at any point in time is above or at $100, hold one stock. When

the stock trades for $100, the option seller – party A in what follows – can always

convert her $100 in cash into $100 worth of stock at no cost. In particular she can

do so today.

• If the stock price goes below $100 at any point in time party A sells the stock

exactly at $100, i.e. A converts her wealth from stock to cash (her wealth is therefore

still $100).

• If the stock price comes back above $100 party A will buy back the stock exactly at

$100, thereby converting her cash wealth into a stock holding that is worth exactly

the same (therefore incurring no profit and no loss).

• Party A should continue with the same (self-financing) strategy until maturity.

• At option maturity, if the stock price ends below zero, the option is out-of-themoney, party A must be holding $100 in cash, with which she can pay back her

debt. Party A keeps the premium.

• If the option ends in-the-money, party A delivers the stock and receives the strike

($100). With the cash from the strike, A repays the debt. Party A keeps the premium.

(Incidentally, note that this strategy corresponds to carrying out a Black delta-hedging

strategy by using zero volatility in the formula: the delta amount of stock is therefore

equal to 0 or 1.) Have we discovered a money machine?

After a moment’s thought, it is easy to see that the flaw in the argument lies in the

assumption that party A will be able to decide what to do when the stock price is exactly

at $100. Suppose in fact that, as the price rises towards $100, the trader decided to go

long the stock at $100. The price process might however fail to cross the strike and

retrace its steps. In this case the trader would have to sell the too-hastily-bought stock for

a price lower than $100. To avoid this, party A could try to trade infinitesimally close to

$100, but before the strike is crossed. But, if the trader ‘jumped the gun’ she would have

no guarantee that the path would not reverse, forcing her again to a costly unwinding.

Similarly, if party A, to be on the safe side, waited until the barrier is crossed, she will

have to pay more than $100. I will call this the ‘perforation’ effect. Exactly at $100,

the price could go either way, and the trader therefore does not know what to do. With

hindsight, even the initial description of the set-up, with the stock price exactly at $100,

was a bit of a sleight-of-hand, and the hedging strategy ambiguous: should the trader hold

the stock, or be fully invested in cash?

Owing to the finite ‘perforation’ effect, party A will always make a loss: she will

only be able to sell ‘a bit’ below $100 and to buy ‘a bit’ above $100. The magnitude of



these losses will depend on the volatility of the underlying (which the strategy decided

to ignore).

The total amount of these losses will depend on the particular realization, but the

strategy (even in the absence of transaction costs) will no longer be self-financing, and

will therefore give rise to a finite variance of portfolio returns by time T . In addition, the

cumulative losses will never be smaller than 0 (the latter value occurring only for those

paths that start at $100 and never go below their starting value).

Party A might be tempted to try to rescue her strategy by proposing to trade more

and more frequently,

√ thereby reducing each perforation cost, which one can expect to be

proportional to σ

t. However, also this attempt to rescue the money-spinning scheme

would not work, because each individual loss might well be lower, but there will be

proportionally more potential crossings of the re-hedging boundary, and one can show

that the net effect will be the same. This observation will be important in the following

discussion of the static replication of the double-barrier problem.

Table 17.1 displays the Black value for an option with maturities and volatilities as

displayed in the two left-hand columns, and the average of the costs incurred over the life

of the option in the column labelled PerfCost. The corresponding exact Black price, and

the ratio between the two values are given in the columns Black and Ratio, respectively.

17.1.2 Static Replication of a Continuous Double Barrier

Having established these results, we can now tackle the problem of the static replication,

by means of plain-vanilla options, of the continuous double barrier. For the sake of

concreteness we will consider the case of a continuous double barrier with upper and

lower boundaries at $100 and $110.

If neither of these values is touched by the reference price throughout the life of the

option, the buyer of the knock-out receives $1 at the end of year 1. For this she pays

today a premium, strictly less than $1. For the sake of simplicity we ignore interest rates.

For the problem to make sense, today’s value of the reference price must be strictly more

Table 17.1 The Black values (column Black ) of options

with maturities and volatilities as displayed in the lefthand columns (column Vol and Mat), the average of the

perforation costs incurred over the life of the option (column PerfCost), and the ratio between the two (column

















































than $100 and less than $110. We assume no transaction costs, no bid–offer spreads, and

the ability to deal in arbitrarily large sizes.

As mentioned earlier, we will attempt to construct a replicating portfolio that satisfies

the boundary and initial conditions to within an arbitrary degree of accuracy. We will then

invoke the uniqueness theorem to claim that, if we have found a solution that satisfies the

aforementioned conditions, then it must be the solution. In addition we will show that the

replicating portfolio built using the procedure described below will need no re-balancing

until expiry of the option, or breach of either barrier. For this reason, the construction

is sometimes referred to as the ‘static-replicating-portfolio’ strategy. However, I will not

make use of this term, which I reserve for those instruments for which a replicating

portfolio can always be left in place until expiry of the option, irrespective of the path of

the underlying. Let us look at the procedure in detail.

The first step is the matching of the terminal1 conditions at the final expiry (time

T ). The true profile (see Figure 17.1) displays two discontinuities, at $100 and at $110,

where the payoff profile jumps from $0 to $1 and vice versa. This terminal payoff can

be approximated by purchasing today a T -maturity call struck at $100 − , selling a

T -maturity call struck at $100, selling another T -maturity call struck at $110, and finally

purchasing a T -maturity call struck at $110 + . See Figure 17.1.

The notional of each call option is simply equal to 1/ . As goes to zero the profile of

the call spread approximates as closely as we may wish the discontinuous terminal payoff

of the knock-out option, and the notionals become larger and larger. At time t the set-up

cost of this portfolio, 1 , will depend on the prevailing spot value of the underlying

reference price, and on the maturity of the options. We will denote this dependence using

the notation 1 (t, T |St = St ; ), where the parametric dependence on the call spread






Figure 17.1 The true initial (i.e. expiry) condition of a unit double-barrier option (shaded area),

and the approximating payoff with a width of 1. The approximating payoff is obtained by

purchasing a call struck at $99, selling a call at $100, selling a call at $110 and purchasing a call

at $111. The notional for all these calls is equal to 1/ . The two axes in the figure are not drawn

to scale.

1 In PDE language this condition is typically referred to as the ‘initial’ condition. Pace PDE enthusiasts, I

will use the more financially transparent term ‘terminal’ instead.



width, , has been explicitly included to emphasize the fact that the value of the portfolio

does depend on .

Let us now consider an arbitrarily small time interval t, and let us place ourselves

at the boundary levels, $100 and $110, t years before expiry (i.e. at time T − t).

We can evaluate the values of the four T -maturity options contingent on our being at

$100 and $110 at time T − t, i.e. we can evaluate 1 (T − t, T |ST − t = 100; )

and 1 (T − t, T |ST − t = 110; ). The only assumption required for the evaluation of

these quantities is that the future volatility of the underlying should be known today. This

is certainly the case if the volatility is assumed to be a deterministic function of time.

But this is also the case if we assume, for instance, that the future volatility is a known

function of the absolute future level of the underlying price, as it is for local-volatility

models. We will discuss this point at greater length later in the chapter.

For the moment I should point out that the value at time T − t of the four-option

portfolio set up to satisfy the terminal condition is not equal to zero either at $100 or at

$110, i.e.

1 (T

t, T |ST −


= 100; ) = 0


1 (T

t, T |ST −


= 110; ) = 0



The boundary conditions at time T − t, which would require the value of the replicating

portfolio to be equal to 0, are therefore not met. How can we fix this? Let us denote the

values of the four-option portfolio at $100 and $110 by a and b:

1 (T

t, T |ST −


= 100; ) = a = 0


1 (T

t, T |ST −


= 110; ) = b = 0



To satisfy the boundary conditions at time T − t we need to add to the portfolio of four

options built thus far a suitable position in two plain-vanilla options whose discounted

values at $100 and $110 should be exactly equal to −a and −b, respectively. At the same

time these two options should be chosen in such a way as not to ‘spoil’ the terminal

condition at time T .

It is easy to see that it is possible to fulfil both conditions by choosing an out-of-themoney call expiring at time T and struck at $110, and an out-of-the-money put, also

maturing at time T , and struck at $100. The payoffs of these two options are exactly

zero inside the range [$100 $110] and at the boundary values; the initial conditions are

therefore still satisfied. If we denote by α and β the unknown holdings of call and put,

with (hopefully) obvious notation, we will then impose that

αCall(ST −

+βPut(ST −



= 100, K = 110, T −

= 100, K = 100, T −

t, T )

t, T ) = −a





αCall(ST −

+βP ut (ST −


= 110, K = 110, T −

= 110, K = 100, T −


t, T )

t, T ) = −b


This set of two linear equations in two unknowns, α and β, constitutes a 2 × 2 linear

system that will yield as a solution the amounts of out-of-the-money call and put to sell.

We will denote the value at time T − t of the portfolio of six options, now satisfying

both the terminal conditions and the boundary conditions, at time T − t and when the

stock price is equal to S, by 2 (T − t, T |ST − t = S; ). In particular, note that, by

construction, we now have

2 (T

t, T |ST −


= 100; ) = 0


2 (T

t, T |ST −


= 110; ) = 0



The procedure should now be clear: we move one further step t away from the terminal

maturity and we evaluate the values of the six-option portfolio at time T −2 t conditional

on the underlying being at both barriers: 2 (T − 2 t, T |ST −2 t = 100; ) and 2 (T −

2 t, T |ST −2 t = 110; ). Neither of these two values will be equal to zero, but we will

find that

2 (T

− 2 t, T |ST −2


= 100; ) = c = 0


2 (T

− 2 t, T |ST −2


= 110; ) = d = 0



From the previous reasoning, we should now look for two additional out-of-the-money

options (a call struck at $110 and a put struck at $100) so as to match the boundary

conditions at time T − 2 t. We must make sure, however, that these new options will not

spoil the construction up to this point. A moment’s thought suggests that their expiration

will have to be at time T − t, and their strikes at $100 (for the put) and at $110 (for

the call), so that the boundary conditions at time T − t, which were already taken care

of, are not affected. Once again, we will find a 2 × 2 system, given by

γ Call(ST −2

+δP ut (ST −2



= 100, K = 110, T − 2 t, T −

= 100, K = 100, T − 2 t, T −


t) = −c



γ Call(ST −2

+δP ut (ST −2



= 110, K = 110, T − 2 t, T −

= 110, K = 100, T − 2 t, T −


t) = −d


whose solutions will give the required amounts of out-of-the-money call and put, γ and δ.





Portfolio cost












Spot price

Figure 17.2 The cost of the replicating portfolio as a function of today’s spot price. Note that the

solution to the double-barrier problem only makes sense (and can therefore be meaningfully read)

within the 100/110 band. The value of the replicating portfolio, however, is given by the dotted

line for any value of the underlying, inside, at, or outside the barrier.





Call/put principal















Time-step index

Figure 17.3 The amount of calls (AmtCall ) and puts (AmtPut) necessary to replicate the payoff

of the $100/$110 double-barrier problem with 30 steps between today and option expiry. The step

number is on the x-axis. Step number 1 corresponds to the time slice t years before expiry.



We can repeat this procedure all the way to the origin. Altogether, 4 + 2(n − 1) plainvanilla options will be required to carry out the portfolio strategy (the integer n is the

number of time-steps, i.e. T / t). Let us indicate the final portfolio by ( t, ), where

only the parametric dependence on the call spread width, , and on the chosen time-step,

t, has been retained.

What we have constructed is a portfolio of options that matches at the discrete

times i t, i = 0, 1, 2, . . . , n, the boundary conditions of the double barrier, and which

super-replicates the terminal (expiry) condition. It is plausible to surmise (and it can

be proven that this is indeed the case) that the present value of the double barrier,

P V (Double-Barrier), will be given by

P V (DoubleBarrier) =



( t, )


With this method for finding the present value of a continuous double knock-out option

the solution is expressed in terms of an infinite series of closed-form expressions (the

Black prices of calls and puts). The series does not converge particularly rapidly, nor are

the individual computations (i.e. the evaluations of the Black formulae) extremely fast.

The main strength of the static-portfolio replication technique lies in its flexibility, since

it can be applied not only to the case of deterministic time-dependent volatilities, but

whenever the user can assume that the future implied volatility of an option of a given

maturity can be known (or ‘guessed’) today. In particular, if the trader believed the smile

to be floating or sticky, she could be tempted to embed this belief in the construction of

the replicating portfolio.

In the next section I show that this way of looking at the portfolio-replication problem

has taken us right to the heart of the problem of how a future smile can be assigned,

and why it is unavoidable to do so. In particular, I will analyse in detail what costs will

be incurred if the portfolio has to be unwound. When we tackle this topic, the previous

discussion of the paradox about the trading strategy consisting of selling an option and

keeping the premium at no cost will prove useful. In addition, we will have to reconcile

our strategy with the current market prices of the full series of plain-vanilla options with

maturities at all the various time-steps. In other words, we will have to ask the fundamental

question: what beliefs about the values of future option prices (i.e. the future smile) are

compatible with today’s plain-vanilla option prices? These topics are dealt with in the rest

of this chapter. In the meantime Figures 17.2 and 17.3 show the price of the replicating

portfolio (and therefore the option value), and the necessary amounts of calls and puts

for the $100/$110 double-barrier problem.


Analysis of the Cost of Unwinding

The portfolio built using the procedure described above will certainly produce the required

payoff if the underlying stock price ends up anywhere between $100 and $110 without

having ever touched either boundary throughout the life of the option. But what happens

if the stock price breaches either barrier before expiry? The trader who put in place the

static strategy made up of a very large number of calls and puts will have to unwind all

the unexpired options as soon as the barrier is touched. By looking at Figure 17.2 one



can immediately see, in fact, that, outside and away from either barrier, the value of the

plain-vanilla portfolio and the value of the double barrier rapidly diverge.

In this context, the discussion of the finite perforation cost becomes very useful: once

again, the trader in reality will not be able to ‘catch’ the stock price exactly at $100 (or

$110), but will incur a cost due to the finite difference between the actual transaction

price and the theoretical unwinding level. As in the money-for-nothing paradox explored

above, by making the trading interval smaller and smaller, the perforation

cost (which,

t) can be made

as we argued in Chapter 2, should depend for a diffusive process on

arbitrarily small. Unlike the previous example, however, increasing the trading frequency

does not expose the trader to more and more potential crossings of the barrier level (and,

therefore, to more and more small losses). With a continuous-double-barrier case, once

either trigger level is touched, the option is dead, the finite perforation cost is incurred

only once and can therefore be truly reduced (in our frictionless world) without having

to pay any price.

There still remains, however, one problem. When determining the amounts of calls and

puts (displayed in Figure 17.3) necessary to meet the boundary conditions at any of the

intermediate times, we had to calculate the discounted value of the later payoffs contingent

on the future stock price being at either barrier. The volatility that entered this calculation

was therefore the future implied volatility for the stock price at the chosen intermediate

time and at the appropriate barrier. If the volatility were truly perfectly deterministic (and

perfectly known to the trader), the market in plain-vanilla options would not only give us

information about the implied future volatility (see Chapter 3), but would also provide the

trader with the instruments to ‘lock-in’ this future volatility. In a deterministic-volatility

world, therefore, giving a series of spot option prices for different maturities is equivalent

to providing a series of future option prices starting on all the intermediate expiries.

In the presence of smiles, however, the problem is much more complex. In order to

tackle the problem in the presence of smiles, the trader could take either of two different

routes: the first one (model-based) would be

1. to start from an assumption for the dynamics for the underlying; for the sake of

concreteness, let us assume that the trader has chosen a local-volatility process, such

as the one described by Equation (8.6) of Chapter 8;

2. to determine, using the prices of traded plain-vanilla options, the local volatility

surface σ (St , t);

3. to place herself at either boundary at the appropriate future point in time;

4. to evaluate the future discounted values of the payoffs (i.e. the future option prices)

by using the chosen numerical method (e.g. a trinomial tree if the Derman-and-Kani

construction had been used to extract the local volatilities, or, perhaps, a Monte Carlo


5. to solve the 2 × 2 system on the basis of these future option values in order to

determine the required amounts of calls and puts.

Note that, given the assumption in point 1 above, recovering the future option values

obtained from a given chosen model is as essential in order to prevent arbitrage as the

recovery of today’s prices of plain-vanilla options. In other words, if the model were

correct, assuming any other value for the future values of the plain-vanilla options would



be an arbitrage violation no less severe than using a non-market price for a spot-starting

plain-vanilla option. The only difference between the present and future prices, in this

respect, is that different models will predict different (model-dependent) future values for

future option prices, but all (perfectly calibrated) models must accept the same current

set of option prices on immediate penalty of arbitrage. Therefore the arbitrage violation

incurred by assuming future option prices not identical to the ones predicted by the model

is model-contingent, whilst failure to recover spot option prices is a model-independent

arbitrage violation.

The second (theoretically more dubious, but intuitively appealing) strategy would be

the following.

1. The trader could make an assumption about the nature of the smile (e.g. sticky,

floating, forward-propagated, etc.) and about its degree of time homogeneity. This

assumption could be made on the basis of statistical information, trading views, or

a combination of the two.

2. Given this assumption, the trader could then calculate the future value of the discounted payoffs by enforcing directly the sticky or floating assumption about the

smile and using the Black formula with the appropriate future implied volatility as

an input. In particular, if the trader assumed the smile to be floating, she would

translate today’s smile surface to either barrier. Depending on the barrier, the future

options would therefore have either today’s at-the-money implied volatility, or the

volatility corresponding to today’s option with a strike out-of-the-money by an

amount of dollars equal to the width of the barrier. Alternatively, if the trader had

assumed a sticky smile, the relevant volatilities would simply correspond to today’s

volatilities for the upper and lower barrier levels. Note that assuming to know the

future smile surface greatly simplifies the task of producing the future option prices:

since a smiley volatility is ‘the wrong number to put in the wrong formula to get

the right price’, we do not need a complex pricing model, and, by virtue of the very

definition of smile, the Black formula can be directly employed to calculate call


3. From the values thus calculated, the trader would then have to solve the 2 × 2

systems described above and determine the amounts of calls and puts to match the

boundary condition.

It must be stressed again that, if one believed in a particular model description of

the dynamics of the underlying, the second procedure would, in general, be theoretically

incorrect (prone to arbitrage), because the assumed future prices would not be consistent

with the underlying financial model and with today’s market prices. This requirement,

however, should be taken with a pinch of salt. I have argued, for instance, that a process

description such as the one provided by Equation (8.6) for equity or FX products leaves a

lot to be desired. Yet, the prices it produces are, within the local-volatility model, arbitragefree. So, the local-volatility model might well protect the trader from arbitrage, but only if

the true process is indeed a local-volatility diffusion, and the trader has correctly estimated

the local-volatility surface. As a consequence, feeling overly constrained by the model

output should probably not be considered an imperative (or, sometimes, perhaps even




One could therefore be tempted to approach the pricing problem from a different

angle. Since, after all, the desirability of a given model is assessed on the basis of

its ability to recover today’s prices and to produce a plausible future evolution of the

smile surface, why not dispense with the model step entirely and directly specify the

evolution of the smile surface? Perhaps we can find some clearly identifiable statistical

features of the smile dynamics that could guide us as to how this smile dynamics could

be specified. See, in this respect, Section 17.3 below, suggestively called ‘The Trader’s

Dream’. Unfortunately, I will show that matters are not that simple. For the moment,

the example presented above makes perfectly clear the link about the future re-hedging

costs, the option value and the underlying assumptions about the smile type. To give an

idea of the impact of different assumptions about future smiles, Table 17.2 shows the

portfolio replication option prices obtained with a constant (no smile) volatility (column

Const), a sticky smile (column Fixed ), a floating smile (column Float), for a constant or

a time-dependent volatility.

Going back to the general pricing philosophy behind this example, one could object

that, if the trader were to follow the second approach, she would, in a sense, assume to

know the answer beforehand, since she would be imposing the future implied volatility

surface, rather than obtaining it from the no-arbitrage dynamics of a model. I take a

different view: the acceptance of a model ultimately depends on its ability to reproduce

qualitative features of the smile that the trader feels confident with. I have argued at

length in this book that the at-least-approximate reproduction of today’s smile surface

could be one such qualitative feature. When looked at in this light the model-based and

the replication approaches would appear to be not so different after all.

Probably a more constructive approach is to recognize that different plain-vanillaoption strategies will be needed to hedge different exotic products: if the hedging strategy

were truly static in nature, as might be the case, for instance, for a single-look European

digital option, then today’s option prices would be all that matters. If, on the other hand,

substantial future re-hedging is required, then recovering future option prices becomes

essential, and therefore the trader will want to ensure that the smile predictions of a given

Table 17.2 The double-barrier option prices calculated

using the portfolio replication technique with a constant

volatility (column Const), a sticky smile (column Fixed ),

a floating smile (column Float) and flat or time-dependent




Flat volatility



















































model are reasonable, and/or to correct and supplement these predictions with exogenous

(and, possibly, theoretically incompatible) information.

It is also useful to revisit at this point the example of an option with a forwardsetting strike, analysed in the case of purely time-dependent volatilities in Section 3.8 of

Chapter 3. I argued in that context that, in order to hedge the volatility exposure arising

from the option, the trader had to ‘lock-in’ the future portion of the volatility between the

time of the strike reset and the option maturity. I also pointed out how a plausible hedging

strategy would have to display, before the reset of the strike, an appreciable sensitivity

to volatility (to match the vega exposure), but no delta and no gamma. I proposed a long

and a short position in wide, symmetric strangles to fulfil, at least approximately, these

combined requirements. Finally, I noted that, as the strike-resetting time was approached,

the chosen strangles ceased to produce a flat profile, needed to give no delta and no

gamma, for smaller and smaller movements of the underlying from the original at-themoney position. During the life of the option the strikes would therefore have to be

adjusted; the closer to the time of the strike reset, the more frequently these readjustments

are likely to take place. All these readjustments expose the trader to the future price of

options. The trader will therefore have to take into account in her price-making the future

implied volatility of options with different degrees of in- and out-of-the-moneyness and

different maturities. As a consequence, future option prices (i.e. future implied volatility

surfaces), might well turn out to have as great an impact as the spot option prices on

today’s cost of the hedging strategy (and therefore on the value of the forward-resetting



The Trader’s Dream

One of the more interesting approaches to giving a direct specification of the future

smile dynamics is probably Samuel’s (2002) methodology. His reasoning goes as follows.

First of all the trader will need a smooth fitted implied volatility surface. Any of the

methods presented in Chapter 9 would do, although a parametric approach is probably

desirable. For a given maturity the smile is symbolically shown in the top left-hand

corner of Figure 17.4 (reproduced with thanks from Samuel (2002)). From the smile

surface (i.e. from a collection of current prices) one can distill an implied risk-neutral

probability density, φT . Ignoring discounting, this can be done using Equation (9.4) in

Chapter 92 :

φT =

∂ 2C

∂K 2


This step is represented pictorially by the top right-hand quadrant in Figure 17.4. Given

this density, one can determine the values of the strike, K, that correspond to the 25th,

50th and 75th percentiles of the density. See the bottom right-hand corner of Figure 17.4.

Let these three values of the strike be denoted by K− , K0 , and K+ . At this point one

can determine the coordinates K− , K0 , and K+ on the x-axis of the implied volatility

plot, and directly read the corresponding values of the implied volatility, σ− , σ0 and σ+ .

2 For simplicity, in this chapter I will assume zero interest rates. Therefore there is no discounting factor in

Equation (17.14).

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