9 Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again
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CHAPTER 4 VARIANCE AND MEAN REVERSION
Factor
p (up)
p (down)
d
u
Vol
1.10517
0.47502
0.52498
0.90484
1.10517
10%
1.2214
0.4502
0.5498
0.8187
1.2214
20%
134.9859
110.5171
110.5171
90.48374
90.48374
Time 1
74.08182
Time 2
100
Time 0
Figure 4.14 The stock price in universe a, where the volatility is equal to 10% over the ﬁrst
period, and 20% over the√second. The stock moves up to Su and down to Sd , with u = Factor,
t), p(up) = (1 − d)/(u − d), p(down) = 1 − p(up).
d = 1/u, Factor = exp(σ
Factor
p (up)
p (down)
d
u
Vol
1.22140
0.45017
0.54983
0.81873
1.22140
20%
1.10517
0.47502
0.5250
0.9048
1.1052
10%
134.9859
122.1403
110.5171
100
90.48374
81.87308
74.08182
Time 0
Time 1
Time 2
Figure 4.15 The stock price in universe b, where the volatility is equal to 20% over the ﬁrst
period, and 10% over the second. All the symbols have the same meaning as in Figure 4.14.
4.9 FINITE RE-HEDGING INTERVALS AGAIN
10%
137
20%
34.98588
15.74945
0.786162
10.51709
4.734437
0.288651
0
9.57535
0.549834
Time 0
Time 1
0
Time 2
Figure 4.16 The call option value for strike 100 in universe a and the delta amount
shown in bold italics.
20%
22.14028
1.00000
Optup −Optdown
Sup −Sdown
10%
34.98588
10.51709
9.966799
0.549834
0
0
0.00000
0
Time 0
Time 1
Time 2
Figure 4.17 The call option value for strike 100 in universe b and the delta amount
shown in bold italics.
Optup −Optdown
Sup −Sdown
Note that the delta amounts do not depend on the probabilities for the up and down jumps
(see the discussion in Chapter 2), and that they are vastly different in the two universes.
For the particular value of the strike chosen, the delta amount at the origin happens to
be exactly the same, but, as Crouhy and Galai (CG in what follows) point out, for any
other value of the strike it is in general different. (The actual numerical values I obtain
are somewhat different from CG’s paper because I have chosen zero interest rates, and a
different discretization scheme.)
A few comments are in order. To begin with, when one lets the time-step approach zero,
as in most applications one would certainly want to do, the actual discretization scheme
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CHAPTER 4 VARIANCE AND MEAN REVERSION
becomes conceptually irrelevant, and only affects the speed of convergence. Therefore,
as
√
t goes to√
zero, the up state can be√
equivalently modelled as Sup = Sold (1+σ
t), Sup =
t), Sup = Sold (1 + σ
t + 12 σ 2 t), or in a variety of other asymptotically
Sold exp(σ
equivalent ways. In CG’s approach, however, the time-step cannot be allowed to be
reduced at will. Different values will therefore be obtained for the state variable, the
option price, and its delta, depending on which of the (only asymptotically equivalent)
discretization schemes is chosen. Given that CG impose a ﬁxed re-hedging interval, these
differences cannot be ignored.
Note that the ﬁxed re-hedging assumption is crucial and is justiﬁed by CG (1995) as
follows:
. . . because of transaction costs and other execution problems, hedges are
re-adjusted discretely, often once a day, or even once a week. The issue of the
appropriate volatility measure becomes important in such a trading environment.
With this observation in mind, we plot in Figure 4.18 the delta in the two universes
for a variety of strikes between the values where the deltas are exactly equal to one or
zero for both volatility regimes. Notice that, in this ﬁgure, there are three distinct linear
segments for the delta amount, with different slopes and three intersection points: the two
‘degenerate’ levels at which all the nodes in the tree are in or out of the money, giving a
delta of one or zero, and the at-the-money level. If we had subdivided the same trading
interval into more and more steps the two different delta curves would have crossed at
correspondingly more and more points, and would have progressively begun to merge into
each other. Note again, however, that, given the constant-re-hedging-interval assumption,
this limiting process cannot be undertaken. Therefore, as a ﬁrst observation, one can say
that, for the CG effect to be signiﬁcant, the length of each time-step must be of the same
order of magnitude as the residual time to expiry.
The second necessary condition for the CG effect to be ‘strong’ is that the volatility
should be signiﬁcantly non-constant in both universes over the (short) residual life. In the
examples above, it either doubled or halved, depending on the universe, in going from the
1
0.9
0.8
0.7
Delta
0.6
Delta(a)
Delta(b)
0.5
0.4
0.3
0.2
0.1
0
74
84
94
104
Strike
114
124
134
Figure 4.18 The delta amount of stock to hold at the origin (today) in the two universes.
4.9 FINITE RE-HEDGING INTERVALS AGAIN
139
ﬁrst to the second and last step. This, however, is hardly realistic if the residual maturity is
short. If coupled with the ﬁrst necessary condition (‘few’ steps), the second requires for the
effect to be important that the trading environment should be one of strongly non-constant
volatility, the option-maturity-to-time-step ratio small, and the re-hedging periods few.
This observation gives a ﬁrst indication that the effect presented by CG, while conceptually interesting, is perhaps not very relevant for practical hedging purposes. To use CG’s
numbers, if one considers the case of only two subdivision periods the deltas for the two
universes are indeed signiﬁcantly different: 0.90 and 0.68 for a K/S ratio of 0.90, and 0.43
and 0.67 for a K/S ratio of 1.1. Simply adding two more steps, however, already makes
the deltas very similar (0.85 vs 0.84 for K/S = 0.90, and 0.65 vs 0.63 for K/S = 1.1).
To look at the problem in a different light, one could ask two related questions: Would
a trader really keep her re-hedging interval constant if she were within a day or two of
option expiry, and the spot level were roughly at the money? and Would the same trader
really expect the volatility to be not only time-dependent, but predictably so, over such a
short trading period?
There is, however, a second, and, in my opinion, fundamental rather than practical
objection to the CG reasoning. One of the apparent strengths of the CG approach is that
the construction of their deltas does not rely on the discretization of a limiting process,
or on an asymptotically correct matching of moments. The delta amounts they obtain in
the two universes are exactly the amounts needed to replicate in discrete time the ﬁnal
and intermediate option payoffs at the various nodes, given the knowledge of the possible
realizations of the underlying in the different ‘up’ and ‘down’ states. In Chapter 2 I gave
a thorough discussion of this construction, and, in particular, of its limiting properties.
CG’s observations are certainly correct. The real interest of the two-state, discrete-time
dynamics, however, does not lie in the fact that we truly believe that over each time-step
only two realizations will be possible for the stock price. After all, if three, instead of two,
states were reachable, a possibility that makes as much, or as little, ﬁnancial sense as the
binomial-branching assumption, one would obtain the ﬁnancially very ‘ugly’ result that
another asset would have to be added in order to replicate the payoff with certainty. The
real appeal of the binomial construction lies therefore not in its descriptive realism, but
in the fact that, in the limit as t goes to zero, two suitably chosen states are all that is
needed in order to discretize a continuous Gaussian process. This justiﬁcation, however,
cannot be invoked in CG’s setting, since it is crucial to their argument that the trading
interval should remain of ﬁxed length as the option maturity approaches. This being the
case, no special meaning can be attached to the requirement that each move in the bushy
tree should only lead to an ‘up’ or a ‘down’ state. Needless to say, if three, or more,
states had been allowed, one would have had to introduce correspondingly more securities
depending on the same underlying (presumably other options) to replicate exactly all the
payoffs, and the resulting delta amounts of stock would have been different.
Given the arbitrariness, for a ﬁxed time-step size, of the choice of two as the number
of states to which the stock price can migrate, and the considerations about the actual
trading frequency close to expiry put forth in the previous paragraph, it seems fair to say
that the effect presented by CG, while interesting, should not be of serious concern to
traders in their hedging practice. More generally, this example shows that discretizations
must be handled with care, and that results that are not robust to letting the time-step
go to zero, or that rely on the computational details of the construction (e.g. number of
branches, recombination, etc.), should in general be regarded with suspicion.
Chapter 5
Instantaneous and Terminal
Correlation
5.1
Correlation, Co-Integration and Multi-Factor Models
The importance of correlation has often been emphasized, both in the academic literature
and by practitioners, in the context of the pricing of derivatives instruments whose payoffs
depend on the joint realizations of several prices or rates. Examples of derivative products
for which correlation is important are:
1. basket options: often calls or puts on some linear combinations of equity indices or
of individual stocks;
2. swaptions: calls or puts on a swap rate, where the latter is seen as a linear combination of imperfectly-correlated forward rates;
3. spread options: calls or puts on the difference between two reference assets (e.g.
equity indices) or rates (typical examples could be the spread between, say, the 2and 10-year swap rates in currency A, or the spread between the 10-year swap rate
in currency A and the 10-year swap rates in currency B);
4. tranched credit derivatives: their price will depend on the correlation among the
credit spreads of (or, in certain models, on the default correlation among) a certain
number of reference assets.
The observation that ‘correlation is important’ in the pricing of these types of option is
not controversial. It is important, however, to understand precisely what correlation can
and cannot achieve. To this effect, consider the SDEs of two diffusive variables, say x1
and x2 :
dx1 = µ1 dt + σ11 dz1
(5.1)
dx2 = µ2 dt + σ21 dz1 + σ22 dz2
(5.2)
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142
CHAPTER 5 INSTANTANEOUS AND TERMINAL CORRELATION
with orthogonal increments dz1 and dz2 . The coefﬁcient of linear correlation between x1
and x2 , ρ12 , is
ρ12 =
E[dx1 dx2 ]
E[dx12 ]E[dx22 ]
σ11 σ21
=
(σ11 σ21 )2 + (σ11 σ22 )2
For simplicity set the drift terms to zero, and consider the new variable y = x1 + λx2 :
dy = σ11 dz1 + λ (σ21 dz1 + σ22 dz2 )
= (σ11 + λσ21 ) dz1 + λσ22 dz2
(5.3)
In general the variable y will be normally distributed, with zero mean and instantaneous
variance var(y) = (σ11 + λσ21 )2 + (λσ22 )2 dt. After a ﬁnite time t, its distribution will
be
y ∼ N (0, [(σ11 + λσ21 )2 + (λσ22 )2 ]t)
(5.4)
Now choose λ such that the coefﬁcient in dz1 for the increment dy is zero:
λ=−
σ11
σ21
(5.5)
It is easy to see that this value of λ gives the lowest possible variance (dispersion) to the
variable y:
2
y ∼ N (0, λ2 σ22
t),
λ=−
σ11
σ21
(5.6)
If λ were equal to −1 the variable y would simply give the spread between x1 and x2 ,
and in this case
y ∼ N (0, [(σ11 − σ21 )2 + (σ22 )2 ]t),
λ = −1
(5.7)
Equation (5.7) tells us that, no matter how strong the correlation between the two variables
might be, as long as it is not 1, the variance of their spread, or, for that matter, of any
linear combination between them, will grow indeﬁnitely over time. Therefore, a linear
correlation coefﬁcient does not provide a mechanism capable of producing long-term
‘cohesion’ between diffusive state variables.
Sometimes this is perfectly appropriate. At other times, however, we might believe
that, as two stochastic variables move away from each other, there should be ‘physical’
(ﬁnancial) mechanisms capable of pulling them together. This might be true, for instance,
for yields or forward rates. Conditional on our knowing that, say, the 9.5-year yield in 10
years’ time is at 5.00%, we would expect the 10-year yield to be ‘not too far apart’, say,
somewhere between 4.50% and 5.50%. In order to achieve this long-term effect by means
of a correlation coefﬁcient we might be forced to impose too strong a correlation between
the two yields for the short-term dynamics between the two variables to be correct. Or,
conversely, a correlation coefﬁcient calibrated to the short-term changes in the two yields