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3 Hedging with a Compensated Process: Plain-Vanilla and Binary Options

170

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

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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

## Volatility correlation, rebonato

## 6 Switching Between the PDE, the Expectation and the Binomial Replication Approaches

## 4 Hedging Options: Volatility of Spot and Forward Processes

## 9 Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again

## 1 Correlation, Co-Integration and Multi-Factor Models

## 6 Conclusions (or, Limitations of Quadratic Variation)

## 4 Green’s Functions (Arrow–Debreu Prices) in the DK Construction

## 2 The Financial Model: Smile Tale 2 Revisited

## 10 Jump–Diffusion Processes and Market Completeness Revisited

## 1 A Worked-Out Example: Pricing Continuous Double Barriers

## 5 The Unconditional Variance of the Short Rate in BDT – the Discrete Case

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3 Hedging with a Compensated Process: Plain-Vanilla and Binary Options