3 Hedging with a Compensated Process: Plain-Vanilla and Binary Options
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CHAPTER 6 PRICING OPTIONS IN THE PRESENCE OF SMILES
0
6.500%
7.000%
7.500%
8.000%
8.500%
−200
−400
−600
Delta(log)
Delta(Norm)
Delta(Comp)
−800
−1000
−1200
−1400
Figure 6.1 The normal, log-normal and compensated deltas as a function of the option strike (see
the text for a discussion).
The approach is tempting. Unfortunately, in the presence of smiles the implied volatility
quote does not directly tell us anything about the process of the underlying. In particular,
it does not follow that the Black delta evaluated with the higher volatility should provide
a ‘better’ hedge for the out-of-the-money option than the hedge carried out on the basis
of the at-the-money volatility.
This point is illustrated in Figures 6.1 and 6.2, and in Table 6.1. In obtaining these
values I have assumed that the true distribution for the forward rate, f , was normal
rather than log-normal, and, from its absolute volatility, σabs , obtained the percentage (lognormal) volatility, σlog , that produces the same at-the-money price. I discuss in Chapter 16
that this percentage volatility is accurately, although not exactly, given by σabs = σlog f .
The caplet prices were then calculated for several strikes. For an at-the-money strike the
normal and log-normal prices virtually coincided, but, away from the at-the-money strike,
discrepancies were obviously found between the prices obtained using the two different
distributional assumptions.
Since in this example I have assumed that the true underlying process is known, I
also know what the correct hedge ratio should be. I can therefore ask the question: If
I used the implied smiley volatility with the log-normal Black formula, would I obtain
better hedge ratios than if I used the at-the-money volatility? Or, to phrase the question
6.3 HEDGING WITH A COMPENSATED PROCESS
171
70 000
65 000
60 000
55 000
50 000
vega(log)
vega(norm)
vega(comp)
45 000
40 000
35 000
30 000
25 000
6.500%
7.000%
7.500%
8.000%
8.500%
Figure 6.2 The normal, log-normal and compensated vegas as a function of the option strike (see
the text for a discussion).
Table 6.1 The delta (in futures contracts) and the vega (in $) for a caplet with the strikes in
the left-hand column (Strike). The columns Delta(Norm) and Vega(Norm) indicate the delta and
vega under the assumption that the forward rate was normally distributed. The columns Delta(log)
and Vega(log) report the same quantities under the assumption of log-normality. The columns
Delta(Comp) and Vega(Comp) give the delta and vega obtained using the ‘compensated’ log-normal
volatility. See the text for a discussion.
Strike
6.846%
7.096%
7.346%
7.596%
7.846%
8.096%
8.346%
Delta(log)
Delta(Norm)
Delta(Comp)
Vega(log)
Vega(Norm)
Vega(Comp)
−1,285
−1,127
−950
−755
−407
−283
−202
−1,349
−1,181
−990
−776
−410
−269
−176
−1,285
−1,122
−936
−728
−358
−228
−149
67,511
68,299
66,715
62,013
53,640
44,193
35,712
66,511
65,786
62,500
56,018
45,873
34,932
25,524
67,263
67,721
65,625
59,886
49,721
38,379
28,805
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CHAPTER 6 PRICING OPTIONS IN THE PRESENCE OF SMILES
differently: Can the information about the true process contained in the smiley implied
volatilities be proﬁtably used to estimate the delta simply by using this implied volatility
in the Black formula?
In order to answer these questions I proceed as follows. First, the ‘true’ (i.e. given our
assumptions, the normal) deltas and vegas were calculated for a variety of strikes. These
true risk statistics are shown in the columns Delta(Norm) and Vega(Norm), respectively.
The same quantities were then calculated using the Black formula for the delta and vega
with the same percentage volatility as determined by the relationship σabs = σlog f as
input. (We know that this volatility will reproduce the correct at-the-money prices.) The
hedge ratios so obtained are shown in the columns Delta(log) and Vega(log). Finally, the
delta and vega statistics were calculated using the Black formula and a ‘compensated’
volatility equal to the implied smiley volatility that produces the away-from-the-money
prices. These are shown in columns Delta(Comp) and Vega(Comp).
We can now ask: When used in conjunction with the (incorrect) Black formula, do
these strike-dependent ‘compensated’ volatilities provide a better approximation of the
true (normal) deltas and vegas than the single value given by the at-the-money log-normal
volatility? As one can see from Figures 6.1 and 6.2 and from Table 6.1, the answer is in
general ‘no’: a small improvement in the vega has, in fact, to be counterbalanced by a
similarly small deterioration of the approximation in the case of the delta. The fact that
the delta turned out to be slightly worse and the vega slightly better was, of course, just
due to the speciﬁc choice of the strike and of the smile-generating process (a normal,
rather than log-normal diffusion). In general, however, the trader will simply observe a
smile, and will not be able to tell by which process it has been generated. Therefore, the
implied smiley volatilities can only be reliably used for the purpose of obtaining prices.
6.3.2 Pricing a European Digital Option
A second example can make the same point even more clearly. Let us consider a European
cash-or-nothing binary option. This option pays $1 if the underlying ends above the strike
of $100 at expiry, and zero otherwise. Since there is only one ‘look time’, this option
is also called a European digital. Let us also assume again that the true process for the
underlying is a normal diffusion.
The payoff of this cash-or-nothing option can be approximated, in the absence of
transaction costs and bid–offer spreads, with arbitrary precision by means of a simple
call spread: in the limit as goes to zero, the strategy of buying a call at $100 − and
selling a call at $100 (both with a notional of 1 ) would approximate the required payoff
as closely as we may wish. See Chapter 17 for a more detailed discussion. Therefore the
analytic price of the European digital, PV(ED), is obtained as the limit as goes to zero
of the price of the two call spreads:
PV(ED) = N (h2 ) = lim
→0
1
[Call(K − ) − Call(K)]
(6.1)
In a Black-and-Scholes world and in the absence of market frictions, the cost of setting
up this strategy converges to the well-known term N (h2 ), with
h2 =
ln
S
1
− σ 2t
K √2
σ t
(6.2)
6.4 HEDGE RATIOS FOR PLAIN-VANILLA OPTIONS
173
and
S = stock price
K = strike
σ = volatility
t = time to expiry.
If there are no smiles, the formula is totally unambiguous, since there is one and only
one volatility, σ , for a given maturity. In the presence of smiles, however, the problem
changes. It is tempting to use the analytic digital formula, i.e. N (h2 ), with the quoted
implied volatility for the strike equal to the barrier, K. Alternatively, we could use the
limit of a tight call spread, with each call price obtained using the appropriate implied
volatility in the Black formula. This requires some care, because, when calculating the
value of the two calls, which have different strikes, their implied volatility is different.
This difference does tend to zero with , but the notionals, 1 , diverge towards inﬁnity,
amplifying the effect of any small difference. Fortunately, the combined result of the two
effects is to provide in the limit a ﬁnite value for the replicating strategy.3 However, in
the presence of smiles, this limit of the inﬁnitely tight call spread does not converge to the
value given by the N (h2 ) formula, with the term h2 calculated using the smiley implied
volatility corresponding to the strike level.
We therefore seem to be faced with two plausible ways of pricing the cash-or-nothing
option: either by using the formula N (h2 ), with the smiley implied volatility corresponding to the barrier; or by using the limit of the inﬁnitely tight call spread, with each component call valued using the appropriate implied volatility. The two methodologies, however,
in general give different answers. Which one gives the theoretically correct price?
In order to answer this question, recall that we deﬁned an implied volatility as ‘the
wrong number to put in the wrong formula to get the right price of plain-vanilla options’.
The question ‘Which of these two prices should one believe?’ therefore has an unequivocal
answer: the price is given by limit of the call spread that is constructed on the basis of the
prices of plain-vanilla options.4 Once again, the implied volatility, when used outside its
correct domain of applicability, does not convey any information about an ‘equivalent’
or ‘compensated’ process.
6.4
Hedge Ratios for Plain-Vanilla Options in the Presence
of Smiles
Section 6.2 showed that special care must be used when employing the Black implied
volatility function in the presence of smiles. In particular, I stressed that it can be safely
used only for the purpose of obtaining a price – as its deﬁnition implies – but that other
uses, such as the calculation of a delta or a vega via the use of the Black formula, are in
general unwarranted.
If, however, we forget for a moment how the Black formula was originally obtained,
and what the links between this formula and the volatility of the underlying are, we could
can be proven by expanding the term N (h2 ) in the vicinity of K.
reality, of course, one would not be able to deal at the quote prices for a ‘market-size’ transaction in
the large (let alone inﬁnite) notionals required. None the less, the call–spread procedure is still valid in order
to obtain the theoretical ‘mid’ price of the European digital.
3 This
4 In
174
CHAPTER 6 PRICING OPTIONS IN THE PRESENCE OF SMILES
regard the market practice of quoting a plain-vanilla option price in the presence of smiles
as made up of three components:
1. an a priori agreement of an ‘arbitrary formula’, which is a function of several inputs;
2. an agreement on the procedure needed in order to obtain from the market or from
the product speciﬁcation all the inputs but one (i.e. today’s stock price, the funding
rate to maturity, the strike, the time to maturity plus an extra input);
3. the price quote, expressed in terms of the remaining unobservable input.
If we look at price-making in this light, no speciﬁc interpretation in terms of more
fundamental quantities is in general warranted for this last input, despite its being called
‘volatility’. This interpretation of the price-making process is obviously somewhat strained:
even in the presence of smiles, the Black formula, for all its imperfections, is clearly not
an arbitrary formula; and the implied volatility, despite the fact that it is not ‘truly’ a timedependent volatility, is in some imprecise way ‘similar’ to its well-deﬁned cousin that
applies when the process for the underlying is a pure diffusion driven by a time-dependent
volatility. However, this way of looking at implied volatilities and at the Black formula
provides a useful way of looking at option pricing. This can be seen as follows.5
From the deﬁnitions given, we can always write in a purely formal way the price, Opt,
of an option as
Opt = Black(S, K, r, T ; σimpl )
(6.3)
where Opt is the price of a plain-vanilla call or put, S, K, T and r are the stock price,
the strike, the time to expiry and the short rate, respectively; σimpl is the price-generating
number chosen by the trader to make a price; and Black is the agreed-upon function to
obtain a quoted price from this last input.
6.4.1 The Relationship Between the True Call Price Functional
and the Black Formula
When dealing with a non-ﬂat implied-volatility surface it is essential to draw a clear
distinction between the true price functional that gives the value of a call as a function
of the state variables, and the market-reference Black pricing formula. In the presence
of smiles, in this section I regard the latter simply as a convenient agreed-upon way
of quoting a price. To make the discussion concrete, I will assume that the underlying is a stock price, and the associated option is a call. Nothing material turns on this
choice.
In what follows I will analyse the problem of a trader who observes with inﬁnite
accuracy6 a smiley implied volatility surface. The trader is agnostic as to what the true
process for the underlying might be, and simply expects that the true call pricing functional
5 The
remainder of this section has been adapted from Section 11.2 of Rebonato (2002).
I say that the trader knows the smile surface with ‘inﬁnite’ accuracy I mean that she has carried out
an interpolation/extrapolation of the smile surface in such a way that she can associate a unique and certain
implied volatility to a continuum of strikes, and to all the maturities of interest. How to obtain these smooth
input volatility surfaces is described in Chapter 9.
6 When
6.4 HEDGE RATIOS FOR PLAIN-VANILLA OPTIONS
175
will depend on
• today’s value of the stock price,
• the residual time to maturity,
• the strike of the option,
• a discount factor (numeraire),
• an unknown set of parameters describing the ‘true’ dynamics (such as diffusion
coefﬁcients, jump amplitudes, etc.), and, possibly,
• the past history of the relevant stochastic quantities.
The parameters describing the process of the underlying (volatility, jump frequency,
jump amplitude, etc.) can, in turn, themselves be stochastic. However, they are all, obviously, strike-independent. The unknown ‘true’ parameters and the full history up to time
t will be symbolically denoted by {αt } and {Ft }, respectively.
The Black formula, on the other hand, used to translate contemporaneous observed
market prices for different strikes and maturities into a set of implied volatility numbers,
depends in the presence of smiles on
• today’s value of the stock price,
• the residual time to maturity,
• a discount factor (numeraire),
• the strike of the option and
• a single strike-dependent parameter (the implied volatility).
I will denote the stock price in the problem as S and the volatility of the diffusion part
of its unknown true process as σ . I do not assume, however, that the true process is just a
complex diffusion, and the symbol {αt } collectively denotes all the other parameters (e.g.
jump frequencies, jump amplitude ratios) that might characterize the true process. The
symbol K, as elsewhere, has been reserved for the strike of the call, and I will denote the
true functional by C(St , t, T , K, {αt }, {Ft }, σ ), and the Black formula by Black(St , T −
t, K, σimpl (t, T , S, K)).
6.4.2 Calculating the Delta Using the Black Formula
and the Implied Volatility
The trader is aware that, given the existence of a non-ﬂat implied volatility curve, she does
not inhabit a Black world, and that, in particular, the Black implied volatility, σimpl , is not
linked in any simple way to the volatility, σ , of the true process – more speciﬁcally, she
knows that is not equal to its root mean square: the implied volatility is just ‘the wrong
number to put in the wrong (Black) formula to get the right call price’. Therefore, as of
‘today’ (time 0), by the very deﬁnition of implied volatility, the trader can only write
C(S0 , 0, T , K, {α0 }, {F0 }, σ ) = Black(S, T − 0, K, σimpl (0, T , S, K))
(6.4)
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CHAPTER 6 PRICING OPTIONS IN THE PRESENCE OF SMILES
(Since, for simplicity, I will always deal in the following example with a single expiry
time, T , as seen from today, I will lighten the notation by writing σimpl (0, T , S, K) =
σimpl (S, K).)
The trader would then like to be able to calculate the delta, i.e. to compute
=
∂C(S, 0, T , K, {α0 }, {F0 }, σ )
∂S
(6.5)
The task appears difﬁcult because the trader does not know the true functional. Today,
however, thanks to (6.4), she can always write,
=
∂Black(S, T , K, σimpl (S, K))
∂S
(6.6)
which, because of the dependence of the implied volatility on S, she knows to be given by
= N (h1 ) +
∂Black(S, T , K, σimpl (S, K)) ∂σimpl (S, K)
∂σimpl (S, K)
∂S
= N (h1 ) + BlackVega(S, T , K, σimpl (S, K))
∂σimpl (S, K)
∂S
(6.7)
In Equation (6.7) the quantity BlackVega is the derivative of the Black function with
respect to the implied volatility, N (.) denotes the cumulative normal distribution, and h1
is the usual argument of N (.) in the Black formula calculated with the implied volatility.
From Equation (6.7) it is clear that the only difﬁculty in calculating the delta is associated
∂σ
(S,K)
. How can the trader estimate this quantity?
with the term impl∂S
6.4.3 Dependence of Implied Volatilities on the Strike
and the Underlying
From today’s plain-vanilla option market the trader can, at least in principle, observe a
continuous series of implied volatilities as a function of the call and put strikes, or, more
precisely, she can observe a series of contemporaneous call prices for different strikes
given today’s stock price, S0 . The trader, by inversion of the Black’s formula, then knows
how to convert these prices into implied volatilities. In other words, for a given S0 , she
observes the variation of the function σimpl (S, K) along the K dimension, σimpl (S0 , K).
However, she cannot say anything, just from this market information, about how this
function varies, if at all, when S changes. For the sake of concreteness, let us assume
that the trader observes that, as a function of K, the quantity σimpl (S0 , K) increases as
K decreases (for a ﬁxed S0 ). In order to make some progress, the trader decides to make
a few assumptions about the ‘true’ process, namely that it is a diffusion, that the true
process volatility – which bears no direct relationship to the implied volatility – is of the
form σ (S), and that this true volatility function decreases when S (not K!) increases, i.e.
dS
= σ (S) dz
S
(6.8)
∂σ (S)
<0
∂S
(6.9)
6.4 HEDGE RATIOS FOR PLAIN-VANILLA OPTIONS
177
(The drift has been ignored for simplicity.) The question faced by the trader is: How
0 },{F0 },σ )
can she compute the delta, ∂C(S,0,T ,K,{α
? Naturally enough, given her imperfect
∂S
knowledge of the process, she would like to make as much use as possible of the information embedded in the quoted market price of options. In particular, she would like to
use the expression
= N (h1 ) + BlackVega(S, T , K, σimpl (S, K))
∂σimpl (S, K)
∂S
This approach might be promising even if the true process (6.8) were known, because it
might be difﬁcult to calculate its derivative with respect to S analytically. The marketgiven plot of σimpl (S0 , K), however, only gives the trader information about how the
implied volatility changes as a function of the strike, K, not of the underlying, S. In order
to calculate the delta the trader must therefore make some much stronger assumptions
about the true process. If the trader assumed, for instance, that the implied volatility
∂
were of the form σimpl (S, K) = σimpl (S − K), then we she could easily switch from ∂S
∂
to ∂K
, and could read the information she needed directly off today’s market function
σimpl (S0 , K). Alternatively, if the trader knew that the implied volatility function was of
the form σimpl = σimpl (ln S/K), the change in implied volatility as a function of S could
be read off today’s chart for the change of σimpl as a function of K. For a process of
the general form (6.8), however, she cannot know a priori whether the implied volatility
function will indeed turn out to be of the form, say, σimpl (S − K), i.e. whether the prices
implied by the true process (6.8) would actually give rise to an implied volatility that
only depends on the difference (S − K). In general, therefore, the trader will have to add
to her information set some views about the behaviour of the implied volatility function
along the S dimension.
In order to make some progress, the trader decides to test the hypothesis that the true
process volatility should be of the form σ = g(S), for some particular function g that gives
∂σ (S)
∂S < 0. The dependence of the volatility on the underlying might be very complex, and
perhaps only available following an algorithmic route (see, for instance, the local-volatility
models described in Chapters 11 and 12). If this is the case, how can the trader go about
ﬁnding the dependence of the implied volatility on the underlying (i.e. ultimately, the
delta)? She could conceptually attempt to answer this question by proceeding as follows.
She could begin by running a Monte Carlo simulation of the process (6.8) starting with
S0 + δS and σ (S0 + δS) using for the initial process volatility. By averaging over discounted expectations, she could then calculate the prices of several calls for different
strikes, and convert these prices into an implied volatility function, σimpl (S0 + δS, K).
She could then repeat the exercise with today’s price for the underlying shifted down by
the same amount, i.e. she could re-run the Monte Carlo simulation with a starting point
equal to S0 − δS (and volatility σ (S0 + δS)), calculate the prices for the same set of strikes
and convert them into a new function, σimpl (S0 − δS, K). Now, for each strike K, she
∂σ
(S)
would ﬁnally be in a position to compute the term impl
by approximating it as
∂S
∂σimpl (S, K)
∂S
σimpl (S0 + δS, K) − σimpl (S0 − δS, K)
2δS
(6.10)
As we shall see in Chapters 23 and 24, if the true process volatility does display a
decreasing behaviour in S (see Equation (6.8) and the assumption above) the implied
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CHAPTER 6 PRICING OPTIONS IN THE PRESENCE OF SMILES
volatility curve will indeed display a monotonically decreasing shape in K. The important
point, however, is that the true process volatility was assumed to be a decreasing function
of the underlying, S, and that the implied volatility was found to be a decreasing function
of the strike, K. A priori, i.e. before doing the price calculations outlined above, one could
not have immediately concluded that a decreasing true volatility with S would give rise
to a decreasing implied volatility with K (although some good ﬁnancial intuition could
have suggested that this might have been the case).
From this thought experiment one can draw a ﬁrst conclusion: in general, even if the
trader knew perfectly today’s implied volatility curve as a function of the strike K for the
current stock price, S0 , she could not in general and in a model-neutral way determine
the delta. There is only one alternative to specifying the process for the underlying, as
the trader had to do in order to run the Gedanken Monte Carlo simulation described
above: the trader could try to make some assumptions directly about the dependence of
the smile surface on the underlying, such as, for instance, σimpl (S, K) = σimpl (S − K) or
σimpl (S, K) = σimpl (ln(S)−ln(K)), or, more generally, σimpl (S, K) = σimpl (h(S)−g(S)),
where h(.) is some function.
Are these assumptions reasonable? The question can be rephrased as: How do call
prices across strikes (i.e. the implied volatility curve) change in the real world when S
changes? Clearly, the question is difﬁcult because, when the trader observes a market
change in S, and she simultaneously observes a change in the call price, she cannot be
sure that the true (and possibly stochastic) volatility function might not have changed
as well at the same time. However, if it is reasonable to invoke a sort of ‘adiabatic
approximation’, i.e. to say that the volatility should change more slowly than the price,
the trader can attempt to answer the question empirically. Indeed, this is the type of
analysis undertaken in the interest-rate case in Chapter 23.
Useful as this information might be, one must be very careful in assigning a priori the
behaviour of the implied volatility as a function of the underlying. Chapter 17 discusses
in detail if, when and to what extent this exercise can be carried out.
6.4.4 Floating and Sticky Smiles and What They Imply about
Changes in Option Prices
We have reached the conclusion that the evaluation of the delta is equivalent to a statement
as to how the implied volatility changes as a function of the stock price. If, for instance, the
assumption σimpl (S, K) = σimpl (S − K) were true, then, as S moves to S + δS, the trader
would observe that the price of the call with strike K + δS would now be recoverable by
inputting the same implied volatility in the Black formula. If the assumption σimpl (S, K) =
σimpl (ln S − ln K) = σimpl (ln S/K) were true, then the trader would observe that, as S
moves to (1 + δ)S, the call price with strike K(1 + δ) would be recoverable by inputting
in the Black formula the same implied volatility and strike K(1 + δ); etc. In Chapter 17, I
call this behaviour the (absolute or relative, respectively) ﬂoating smile. Recall, however,
that the Black formula is homogeneous of degree one in S and K, i.e. if both S and K
are multiplied by the same constant (1 + δ), the call price is simply multiplied by (1 + δ).
Therefore, if the smile were, for instance, relatively ﬂoating, when both the underlying
and the strike move from S and K to S(1 + δ) and K(1 + δ), respectively, the ratio of
the stock price to the strike would not change, the (relatively ﬂoating) implied volatility
would be the same, and the price of a call would simply be multiplied by (1 + δ). Once
6.4 HEDGE RATIOS FOR PLAIN-VANILLA OPTIONS
179
again, it is only by observing how real call prices change when the underlying changes
that one can deduce the dependence on S of the implied volatility, and, therefore, calculate
the correct delta.
This scaling (‘ﬂoating’) behaviour of the option price as a function of the underlying appears very ‘natural’. The most different type of behaviour is probably the following. Let us suppose that, when S = S0 , the true process gives rise to the price
Black(S0 , K, σimpl (S0 , K)) for a K-strike call. Let us also suppose that, when the underlying instantaneously moves to S0 + δS, the price of the same-(K)-strike call is given by
Black(S0 + δS, K, σimpl (S0 , K)). When prices behave this way the underlying process is
said to give rise to a sticky smile. Note carefully that for a ﬁxed strike K the implied
volatility σimpl (S0 , K) used to calculate the price when S moved from S0 to S0 + δS
has not changed to σimpl (S0 + δS, K) (although the call price is now obviously different). If this is the case, then the implied volatility is independent of S, one can write
σimpl (S, K) = σimpl (K),
∂σimpl (S, K)
=0
(6.11)
∂S
and in evaluating the delta
= N (h1 ) + BlackVega(S, K, σimpl (S, K))
∂σimpl (S, K)
∂S
(6.12)
the last term disappears and the trader recovers the simple Black expression (with the
implied volatility as input):
(6.13)
Black = N (h1 )
Summarizing: if a smile were truly sticky, a K-strike option would always have the
same implied volatility irrespective of where the underlying moved, i.e. irrespective of
the degree of in-the-moneyness of the option itself. If a smile were perfectly ﬂoating,
the implied volatility would always remain the same for the same degree of (log-) inthe-moneyness, and the price of same-delta options would be exactly proportional to
the underlying. Perfectly ﬂoating and perfectly sticky constitute two extreme cases of
smiles. A given process will in general produce a behaviour intermediate between the
two. It is impossible to deduce just by inspection of the smile curve as a function of
K for a given S0 whether the smile is sticky, ﬂoating or intermediate in nature. Indeed,
one can prove that an inﬁnity of processes is compatible with any (admissible) set of
market call prices today (no matter how ﬁnely spaced in expiry and/or strike). See, for
example, Britten-Jones and Neuberger (1998). More fundamentally, smiles can be of an
altogether more complex nature, i.e. they can be stochastic. This means that knowledge
of the future realization of the underlying and of the sticky or ﬂoating character of the
smile does not tell us what the future smile will be like. Stochastic smiles are dealt with
in Chapter 17. Today’s smile just gives us a snapshot but what we would actually need
is a (short) movie.
Exercise 1 Assume that the true process is given by a normal diffusion, i.e. in Equation (6.8),
assume σ (S) = σS0 . Calculate, using the appropriate closed-form expression for normal
diffusions, a set of call prices for different strikes starting from a given S0 . Repeat the exercise
for S0 + δS and S0 − δS. Obtain σimpl (S0 , K), σimpl (S0 + δS, K) and σimpl (S0 − δS, K).
∂σ
(S,K)
∂σ
(S,K)
Calculate the derivatives impl∂S
and impl∂K
. Comment on their relative magnitude,
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CHAPTER 6 PRICING OPTIONS IN THE PRESENCE OF SMILES
and compare with the relative magnitude of the same derivatives in the log-normal (Black)
case. What can you conclude about the degree of ‘stickiness’ of the normal-diffusion smile?
We have reached the conclusion that a good starting point towards choosing an appropriate process and evaluating the delta (and, by extension of the reasoning, the other
derivatives) of a call option can be proﬁtably split into two components:
1. An empirical part, which addresses the question: What is the functional dependence
of the implied volatility on S and K, σimpl (S, K)? The answer to this question can
only be obtained by observing price changes (not just prices). See, for instance, the
analysis in Chapter 23. The observation of these price changes should, in theory,
be separated by small time intervals (to avoid the possibility that the true process
volatility might change, at the same time polluting the price picture).
2. A modelling part: once the function σimpl (S, K) has been determined, one can
either determine the delta directly using formula (6.7); or one can ﬁt a model that
simultaneously produces today’s prices with S0 (a relatively easy task) and the prices
that give rise to the function σimpl (S, K) (a much more difﬁcult task).
In theory, if we ‘just’ want to price and hedge plain-vanilla options, the two approaches
are equivalent. The second route in point 2 above is however more appealing because,
if one can think in terms of a model, one’s intuition is signiﬁcantly enhanced, and we
can understand why, say, the delta displays a certain dependence on the underlying. If
the model allows for closed-form solutions, then the deltas, gammas, vegas, etc. can be
obtained easily and quickly, without having to contend with numerical problems which,
especially for gammas, can often be burdensome. More generally, once a model is speciﬁed one can do a lot more than just calculating the risk statistics, and the evaluation of
complex derivatives becomes possible.
On the other hand, the desirability of a model is often assessed on the basis of its
ability to reproduce an evolution of the smile surface (as a function of time and of the
underlying) congruous with the trader’s intuition. It is therefore also tempting to try to
assign directly how the smile surface is deformed when the underlying moves, and as
time goes by. Appealing as this approach might be, it is prone to risk of arbitrage. Why
this is the case is explained in Chapter 17.
In the discussion above I have made reference to ‘sticky’ and ‘ﬂoating’ smiles, and I
have pointed out that the associated behaviours of the smile surface are at the opposite
ends of the spectrum of the possible evolution patterns of the smile as a function of the
underlying. In order to gain some ﬁnancial intuition about these important, albeit stylized,
modes of deformation, I present a couple of ‘smile tales’ in the following sections.
6.5
Smile Tale 1: ‘Sticky’ Smiles
An option trader considers entering a strategy consisting of a long and a short position in
plain-vanilla options on forward rates spanning different portions of the same steep yield
curve. The early-expiring forward rates trade at around 3%, and the long end of the curve
has forward rates in the 6% area. The trader wants to delta-hedge dynamically her option
positions through time, and ‘trade the gamma’, much as discussed in Chapter 4. In other
words, she hopes that the realized volatility will be greater than the implied volatility. If