Tải bản đầy đủ - 0 (trang)
9 Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again

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

Tải bản đầy đủ - 0trang

136



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



138



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)



141



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



Tài liệu bạn tìm kiếm đã sẵn sàng tải về

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

Tải bản đầy đủ ngay(0 tr)

×