The case of a Digital option

3.5 Numerical results

3.5.1 The case of a Digital option

Weconsider the problemof hedging and pricinga Digital call,with payo

f (s) = 1 [K,∞) (s)

of maturity

T > 0

^. ^F^rom ⁽³⁵⁾ ⁱⁿ ^[49], ^the ^payo ôf ^this ôption^can ^beêxpressed âs

f (s) = lim

c→∞

1 2πi

Z R+ic R−ic

s ^z K ^−z

z dz ,

^(5.35)

for anarbitrary

R > 0

^. ^This ^implies ^that ^the ^complex ^measure

Π

^is^given ^by

Π(dz) = 1

2πi K ^−z

z dz .

^(5.36)

However, suchmeasureisonly

σ

^-nite^so^thatapplicationofTheorem3.4.1isnotrigourously valid. Nevertheless, using improper integrals one should be able torecover the main

state-ment. In this section,this willbe assumed sothat formula(4.20)willbeused inthe case of

a Digitaloption.

The underlying process

S ^c

^is ^given ^as ^the exponential of a Normal Inverse Gaussian Lévy process (see Appendix 3.5.2) i.e. for all

t ∈ [0, T ]

S _t ^c = e ^X ^t ^c ,

^where

X ^c

^is ^a^Lévy ^process ^with

X ₁ ^c ∼ NIG(α, β, δ, µ) .

Given

N + 1

^discrete ^dates

0 = t 0 < t 1 < · · · < t N = T

^, ^we ^associate ^the ^discrete ^model

pricing

X = X ^N

^where

X k = X _t ^c _k , k ∈ {0, . . . , N}

X

^is ^a ^discrete ^time ^process ^with

indepedent increments. The related cumulant generating function

z 7→ m(z, k)

^associated

to the increment

∆X k = X k − X k−1 = X _t ^c _k − X t ^c _k−1

^for

k ∈ {1, · · · N}

^is ^dened ^on

D = [ −α − β; α − β]

^. ^W^e ^refer^for ^this ^to^chapter² ^Remark ^2.3.21^2.,^since

X ^c

^is ^a^NIG ^process.

By additivity we can show that

m(z, k) = E[exp(z∆X k )] = exp

∆t k

µz + δ p

α ² − β ² − p

α ² − (β + z) ²

(5.37)

for

z ∈ D, k ∈ {0, . . . , N}

For otherinformationson the NIG law, the readercan referto Appendix 3.5.2.

Assumption 13 1. is triviallyveried,Assumption 132. isveried assoon as

2 ≤ α − β

Thanks to Remark 3.3.15 Assumption 14 is automatically veried for the call and put

rep-resentations given by Lemma 2.4.26, and, by similararguments, even for the digitaloption.

The time unit is the year and the interest rate is zero in all our tests. The initial value of

the underlying is

s 0 = 100

Êuros. ^The ^maturity ôf ^the ôption îs

T = 0.25

^i.e. ^three ^months

from now. Four dierentsets of parameters for the NIG distribution have been considered,

going from the case of almost Gaussian returns corresponding to standard equities, to the

case of highly non Gaussian returns. The standard set of parameters is estimated on the

Month-ahead base forward prices of the French Power market in2007:

α = 38.46 , β = −3.85 , δ = 6.40 , µ = 0.64 .

^(5.38)

Those parameters imply a zero mean, a standard deviation of

41%

^, ^a ^skewness ^(measuring

the asymmetry) of

−0.02

ând ân êxcess ^kurtosis ^(measuring ^the ^fatness ôf ^the ^tails) ôf

0.01

^. ^The ôther^sets ôf ^parametersâre ôbtained ^by multiplyingparameter

α

^by ^a^coecient

C

^, ⁽

β, δ, µ

⁾ ^being ^such ^that ^the ^rst ^three ^moments ^are ^unchanged. ^Note ^that ^when

C

grows to innity the tails of the NIG distribution get closer to the tails of the Gaussian

distribution. For instance, Table 2.1 shows how the excess kurtosis (which is zero for a

Gaussiandistribution)ismodiedwiththefourvaluesof

C

^chosenⁱⁿ^our^tests. ^We^compute

the Variance Optimal(VO) hedgingerror given by (4.20), for dierent grids of rebalancing

dates. Thecorrespondinginitialcapital

V 0

^denoted^by

V ₀ ^∗ = H 0

ⁱⁿ^Theorem^3.4.1^is^computed

using Proposition3.3.19.

In particular, we consider the parametric grid introducedin [39], [40] and [42]

π ^b,N := {0 = t ^b,N 0 , t ^b,N ₁ , · · · , t ^b,N N }

Coecient

C = 0.14 C = 0.2 C = 1 C = 2

α 5.38 7.69 38.46 76.92

Excess kurtosis

0.61 0.30 0.01 4. 10 ⁻³

Figure 3.1: Excess kurtosis of

X ₁

^for ^dierent ^values ^of

α

(β, δ, µ)

^insuring ^the ^same ^three ^rst

moments.

dening, for any real

b ∈ (0, 1]

N

rebalancing dates such that

t ^b,N _k = T − T (1 − k

N ) ^1/b

^for ^all

k ∈ {0, · · · , N − 1} .

^(5.39)

Notethat

π ^1,N

^coincides^withequidistantrebalancingdateswhereaswhen

b

^converges^to^zero,

the rebalancingdatesconcentrate nearthe maturity. Tovisualizethe impactofparameter

b

on the rebalancing dates grid, we have reported onFigure 3.2the sequences of rebalancing

dates generated by

π _N ^b

^for ^dierent^values ^of

b

1 2 3 4 5 6 7 8 9 10 11 12

0 0.0208 0.0417 0.0625 0.0833 0.1042 0.125 0.1458 0.1667 0.1875 0.2083 0.2292 0.25

Rebalancing moment

Time (year)

π ^0.1 π ^0.3 π ^0.5 π ^0.7 π ¹

Figure 3.2: Sequencesof rebalancingdates for dierent values of

b

^,^for

N = 12

We have reported on Figure 3.3 the standard deviation of the Variance Optimal hedging

error for dierent values of coecient

C

^and ^dierent ^choices ^of rebalancing grids. More precisely, wehaveconsideredthree typesof rebalancinggrids,for

N = 12

rebalancingdates.

1. Equidistant rebalancingdates (corresponding to

π ^1,N

^);

π ^b ^∗ ^,N

^where

b ^∗

îsôbtained^by^minimizing^the ^VarianceÔptimal^hedgingêrror ^w.r.t. ^to

parameter

b

3. The non parametric optimal grid

π ^∗

^obtained ^by ^minimizing ^the ^Variance ^Optimal

hedgingerror w.r.t. the

N

rebalancing dates.

Notice that inboth cases the optimal(parametricand non parametric)grid is estimatedby

anoptimization algorithmbased onNewton's method.

First, one can notice that for any choice of rebalancing grid, the hedging error increases

when

C

^decreases. ^Hence, ône ^can ^conclude, âsêxpected, ^that ^the ^degree ôf incompleteness increases when the tails of log-returnsdistribution get heavier.

Besides, one can notice that the parametrization (5.39) of the rebalancing grid seems

re-markably relevant since the optimal parametric grid

π ^b ^∗

^achieves ^similar performances as the optimal non-parametricgrid

π ^∗

Moreover, we observe that the hedging error can be noticeably reduced by optimizing the

rebalancingdatesessentiallyfor

C ≥ 1

î.e. âround^the^Gaussian^case. În^these^cases,ône^can

observeonFigure3.4that the optimalrebalancing gridis noticeablydierent fromthe

uni-formgrid since rebalancingdatesare much moreconcentrated nearmaturity. This conrms

the result of [39] that shows that, in the Gaussian case, taking a non uniform rebalancing

grid (corresponding to

b = 0.5

⁾âllows ^toôbtain â^hedging êrror ^with ^the convergence order for the

L ²

^norm ^of

N ^−1/2

^(up ^to ^a ^log ^factor) ^improving ^the ^rate

N ^−1/4

^achieved ^with ^a

uniform rebalancing grid (i.e.

b = 1

^), ôbtained ⁱⁿ ^[43]. ^However, ît îs interesting to notice that this phenomenon is less pronounced when the tails of the log-returns distribution get

heavier. Inparticular,onecanobserveonFigure3.5thatthehedgingerrorgetslesssensitive

tothe rebalancing grid when

C

^decreases êven îf^the ôptimal^grid ^seems^to ^get ^closer^to ^the

uniform grid.

C = 2 C = 1 C = 0.2 C = 0.14 10 × ST D V O(π ^∗ ) 1.483 (30.82) 1.652 (34.33) 2.663 (54.80) 3.017 (61.53) 10 × ST D V O(π ^b∗ ) 1.520 (31.58) 1.685 (35.01) 2.665 (54.84) 3.017 (61.53) 10 × ST D V O(π ¹ ) 1.892 (39.32) 1.952 (40.56) 2.691 (55.38) 3.028 (61.76)

V 0 (π ¹ ) 0.4903 0.4859 0.4813 0.4812

V 0 (π ^∗ ) 0.4903 0.4860 0.4814 0.4813

b ^∗ 0.4078 0.4394 0.6106 0.6710

Figure 3.3: Standard deviation of the Variance Optimal hedging error (

×10

⁾ ^(reported ^within

parenthesis in percent of the option value

V ₀ (π ¹ )

^), ^initial ^capitals, ^optimal ^grid parameters, for dierent choices of parameters

C

^and

b

^with

N = 12

^and

K = 99

^(Digital^option).

1 2 3 4 5 6 7 8 9 10 11 12

0 0.0208 0.0417 0.0625 0.0833 0.1042 0.125 0.1458 0.1667 0.1875 0.2083 0.2292 0.25

Rebalancing moment

Time (year)

C=0.14 π ^* C=0.14 π ^b*

C=0.2 π ^* C=0.2 π ^b*

C=1 π ^* C=1 π ^b*

C=2 π ^* C=2 π ^b*

Figure 3.4: Parametric and non parametric optimal rebalancing grids for dierent choices of

pa-rameter

C

^with

N = 12

^and

K = 99

^(Digital^option).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.15

0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32

Value of b

Standard Deviation (euro)

C=0.14 C=0.2 C=1 C=2

Figure3.5: StandarddeviationoftheVarianceOptimalhedgingerrorasafunctionof

b

^,^for^dierent

choices of parameter

C

⁽

b ^∗

^being^indicated ^by^the ^dashed^line ^abscissa) ^with

N = 12

^and

K = 99

(Digitaloption).

In document Variance Optimal Hedging in incomplete market for processes with independent increments and applications to electricity market (Page 121-127)

3.5 Numerical results

3.5.1 The case of a Digital option

f (s) = 1 [K,∞) (s)

T > 0

f (s) = lim

c→∞

1 2πi

Z R+ic R−ic

s z K −z

z dz ,

R > 0

Π

Π(dz) = 1

2πi K −z

z dz .

σ

S c

t ∈ [0, T ]

S t c = e X t c ,

X c

X 1 c ∼ NIG(α, β, δ, µ) .

N + 1

0 = t 0 < t 1 < · · · < t N = T

X = X N

X k = X t c k , k ∈ {0, . . . , N}

X

z 7→ m(z, k)

∆X k = X k − X k−1 = X t c k − X t c k−1

k ∈ {1, · · · N}

D = [ −α − β; α − β]

X c

m(z, k) = E[exp(z∆X k )] = exp 

∆t k

µz + δ p

α 2 − β 2 − p

α 2 − (β + z) 2 

z ∈ D, k ∈ {0, . . . , N}

2 ≤ α − β

s 0 = 100

T = 0.25

α = 38.46 , β = −3.85 , δ = 6.40 , µ = 0.64 .

41%

−0.02

0.01

α

C

β, δ, µ

C

C

V 0

V 0 ∗ = H 0

π b,N := {0 = t b,N 0 , t b,N 1 , · · · , t b,N N }

C = 0.14 C = 0.2 C = 1 C = 2

α 5.38 7.69 38.46 76.92

0.61 0.30 0.01 4. 10 −3

X 1

α

(β, δ, µ)

b ∈ (0, 1]

N

t b,N k = T − T (1 − k

N ) 1/b

k ∈ {0, · · · , N − 1} .

π 1,N

b

b

π N b

b

1 2 3 4 5 6 7 8 9 10 11 12

0 0.0208 0.0417 0.0625 0.0833 0.1042 0.125 0.1458 0.1667 0.1875 0.2083 0.2292 0.25

Rebalancing moment

Time (year)

π 0.1 π 0.3 π 0.5 π 0.7 π 1

b

N = 12

C

N = 12

π 1,N

π b ∗ ,N

s ^z K ^−z

2πi K ^−z

S ^c

S _t ^c = e ^X ^t ^c ,

X ^c

X ₁ ^c ∼ NIG(α, β, δ, µ) .

X = X ^N

X k = X _t ^c _k , k ∈ {0, . . . , N}

∆X k = X k − X k−1 = X _t ^c _k − X t ^c _k−1

X ^c

m(z, k) = E[exp(z∆X k )] = exp

α ² − β ² − p

α ² − (β + z) ²

V ₀ ^∗ = H 0

π ^b,N := {0 = t ^b,N 0 , t ^b,N ₁ , · · · , t ^b,N N }

0.61 0.30 0.01 4. 10 ⁻³

X ₁

t ^b,N _k = T − T (1 − k

N ) ^1/b

π ^1,N

π _N ^b

π ^0.1 π ^0.3 π ^0.5 π ^0.7 π ¹

π ^1,N

π ^b ^∗ ^,N

b ^∗

π ^∗

π ^b ^∗

π ^∗

L ²

N ^−1/2

N ^−1/4

C = 2 C = 1 C = 0.2 C = 0.14 10 × ST D V O(π ^∗ ) 1.483 (30.82) 1.652 (34.33) 2.663 (54.80) 3.017 (61.53) 10 × ST D V O(π ^b∗ ) 1.520 (31.58) 1.685 (35.01) 2.665 (54.84) 3.017 (61.53) 10 × ST D V O(π ¹ ) 1.892 (39.32) 1.952 (40.56) 2.691 (55.38) 3.028 (61.76)

V 0 (π ¹ ) 0.4903 0.4859 0.4813 0.4812

V 0 (π ^∗ ) 0.4903 0.4860 0.4814 0.4813

b ^∗ 0.4078 0.4394 0.6106 0.6710

V ₀ (π ¹ )

C=0.14 π ^* C=0.14 π ^b*

C=0.2 π ^* C=0.2 π ^b*

C=1 π ^* C=1 π ^b*

C=2 π ^* C=2 π ^b*

b ^∗