1 A Worked-Out Example: Pricing Continuous Double Barriers
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17.1 A WORKED-OUT EXAMPLE: CONTINUOUS DOUBLE BARRIERS
565
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
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CHAPTER 17 THE DYNAMICS OF SMILE SURFACES
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
Ratio).
Vol
Mat
Black
PerfCost
Ratio
20.00%
20.00%
20.00%
20.00%
20.00%
5.00%
40.00%
40.00%
1
0.75
0.5
0.25
0.125
0.25
0.25
0.25
7.9656
6.9013
5.6372
3.9878
2.8204
0.9973
7.9656
7.9656
6.2534
5.0576
4.0183
3.0562
2.1746
0.7382
5.4182
5.5688
1.2738
1.3645
1.4029
1.3048
1.2969
1.3511
1.4702
1.4304
17.1 A WORKED-OUT EXAMPLE: CONTINUOUS DOUBLE BARRIERS
567
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
1
99
100
110
111
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.
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CHAPTER 17 THE DYNAMICS OF SMILE SURFACES
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 −
t
= 100; ) = 0
(17.1)
1 (T
−
t, T |ST −
t
= 110; ) = 0
(17.2)
and
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 −
t
= 100; ) = a = 0
(17.3)
1 (T
−
t, T |ST −
t
= 110; ) = b = 0
(17.4)
and
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 −
t
t
= 100, K = 110, T −
= 100, K = 100, T −
t, T )
t, T ) = −a
(17.5)
17.1 A WORKED-OUT EXAMPLE: CONTINUOUS DOUBLE BARRIERS
569
and
αCall(ST −
+βP ut (ST −
t
= 110, K = 110, T −
= 110, K = 100, T −
t
t, T )
t, T ) = −b
(17.6)
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 −
t
= 100; ) = 0
(17.7)
2 (T
−
t, T |ST −
t
= 110; ) = 0
(17.8)
and
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
t
= 100; ) = c = 0
(17.9)
2 (T
− 2 t, T |ST −2
t
= 110; ) = d = 0
(17.10)
and
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
t
t
= 100, K = 110, T − 2 t, T −
= 100, K = 100, T − 2 t, T −
t)
t) = −c
(17.11)
and
γ Call(ST −2
+δP ut (ST −2
t
t
= 110, K = 110, T − 2 t, T −
= 110, K = 100, T − 2 t, T −
t)
t) = −d
(17.12)
whose solutions will give the required amounts of out-of-the-money call and put, γ and δ.
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CHAPTER 17 THE DYNAMICS OF SMILE SURFACES
0.1000
Portfolio
Portfolio cost
0.0500
0.0000
92.00
97.00
102.00
107.00
112.00
117.00
−0.0500
−0.1000
−0.1500
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.
0.6000
AmtPut
AmtCall
0.4000
Call/put principal
0.2000
0.0000
0
5
10
15
20
25
30
−0.2000
−0.4000
−0.6000
−0.8000
−1.0000
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.
17.2 ANALYSIS OF THE COST OF UNWINDING
571
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) =
lim
→0,n→∞
( t, )
(17.13)
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.
17.2
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
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CHAPTER 17 THE DYNAMICS OF SMILE SURFACES
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
procedure);
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
17.2 ANALYSIS OF THE COST OF UNWINDING
573
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
prices.
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
desirable).
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CHAPTER 17 THE DYNAMICS OF SMILE SURFACES
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
volatility.
Time-dependent
volatility
Flat volatility
ATMVol
Const
Fixed
Float
10%
9%
8%
7%
6%
5%
4%
0.0061
0.0167
0.0411
0.0914
0.1834
0.3312
0.5374
0.0068
0.0173
0.0414
0.0911
0.1820
0.3283
0.5329
0.0074
0.0178
0.0416
0.0908
0.1812
0.3269
0.5315
Fixed
Float
0.0065
0.0173
0.0420
0.0926
0.1849
0.3327
0.5388
0.0069
0.0178
0.0424
0.0929
0.1848
0.3323
0.5382
17.3 THE TRADER’S DREAM
575
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
option).
17.3
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
(17.14)
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).