METHODS AND SYSTEMS FOR ESTIMATING OPTION GREEKS
Methods determine a representation of the option Greek delta Δ which expresses a dependence of an expected value V of a financial contract on one or more underlyings of the financial contract/ The method comprises obtaining: a complete set of algorithmic differentiation (AD) sensitivities of the expected value of the financial contract to a set of N input parameters {right arrow over (a)} in a form ∇ → V = [ ∂ V ∂ a 1 , ∂ V ∂ a 2 , … ∂ V ∂ a N ] T ; and a complete set of AD sensitivities of the expected value of the one or more underlyings F j for j=1 . . . M, where M<N and M is a number of the one or more underlyings to the set of N input parameters {right arrow over (a)} in a form ∇ → F j = [ ∂ F j ∂ a 1 , ∂ F j ∂ a 2 , … ∂ F j ∂ a N ] T for each j=1 . . . M. The method then reprojects the full set of AD sensitivities {right arrow over (∇)}V onto the full set of AD sensitivities {right arrow over (∇)}F j for j=1 . . . M to obtain reprojected sensitivity vectors and determines the parameter delta Δ from the reprojected sensitivity vectors.
1 . A method for determining a parameter delta Δ which expresses a dependence of an expected value V of a financial contract on one or more underlyings of the financial contract, the method comprising:
obtaining, by a processor, a computer representation of a complete set of algorithmic differentiation (AD) sensitivities of the expected value V of the financial contract to a set of N input parameters {right arrow over (a)} in a form
∇
→
V
=
[
∂
V
∂
a
1
,
∂
V
∂
a
2
,
…
∂
V
∂
a
N
]
T
or a mathematical equivalent thereof;
obtaining, by a processor, a computer representation of a complete set of AD sensitivities of the expected value of the one or more underlyings F j for j=1 . . . M, where M<N and M is a number of the one or more underlyings, to the set of N input parameters {right arrow over (a)} in a form
∇
→
F
j
=
[
∂
F
j
∂
a
1
,
∂
F
j
∂
a
2
,
…
∂
F
j
∂
a
N
]
T
for each j=1 . . . M or a mathematical equivalent thereof;
reprojecting, by the processor, the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract onto the full set of AD sensitivities {right arrow over (∇)}F j for j=1 . . . M of the one or more underlyings to obtain a computer representation of reprojected sensitivity vectors; and
determining, by the processor, the parameter delta Δ based on the computer representation of the reprojected sensitivity vectors.
2 . A method according to claim 1 wherein reprojecting the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract onto the full set of AD sensitivities {right arrow over (∇)}F j for j=1 . . . M of the one or more underlyings to obtain the computer representation of reprojected sensitivity vectors comprises decomposing, by the processor, the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract into a computer representation of a pair of orthogonal reprojected sensitivity vectors.
3 . A method according to claim 2 wherein decomposing the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract into the computer representation of the pair of orthogonal reprojected sensitivity vectors comprises:
decomposing, by the processor, the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract into the computer representation of the pair of orthogonal reprojected sensitivity vectors comprising J T {right arrow over (Δ)}=Σ j=1 M Δ j {right arrow over (∇)}F j and {right arrow over (ν)}, where the j th column of J T is {right arrow over (∇)}F j ; and
selecting, by the processor, the coefficients Δ j to minimize |{right arrow over (ν)}|.
4 . A method according to claim 3 wherein selecting the coefficients Δ j to minimize |{right arrow over (ν)}| comprises performing, by the processor, linear regression which minimizes {right arrow over (ν)}·{right arrow over (ν)}.
5 . A method according to claim 3 wherein determining the parameter delta Δ based on the computer representation of the reprojected sensitivity vectors comprises determining, by the processor, the parameter delta Δ in accordance with Δ=Σ j=1 M Δ j .
6 . A method according to claim 2 wherein the number M of underlyings is M=1 and wherein decomposing the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract into the computer representation of the pair of orthogonal reprojected sensitivity vectors comprises:
decomposing, by the processor, the the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract into the computer representation of the pair of orthogonal reprojected sensitivity vectors comprising Δ 1 {right arrow over (∇)}F 1 and {right arrow over (ν)}; and
determining, by the processor,
Δ
1
=
∇
→
F
1
·
∇
→
V
∇
→
F
1
2
.
7 . A method according to claim 6 wherein determining the parameter delta Δ based on the computer representation of the reprojected sensitivity vectors comprises determining, by the processor, the parameter delta Δ in accordance with Δ=Δ 1 .
8 . A method according to claim 3 comprising determining, by the processor, a direction of the reprojected sensitivity vector J T {right arrow over (Δ)}=Σ j=1 M Δ j {right arrow over (∇)}F j .
9 . A method according to claim 8 wherein determining the direction of the reprojected sensitivity vector J T {right arrow over (Δ)}=Σ j=1 M Δ j {right arrow over (∇)}F j comprises determining, by the processor, a computer representation of a unit vector {right arrow over (e)} Δ in the direction of the reprojected sensitivity vector J T {right arrow over (Δ)}=Σ j=1 M Δ j {right arrow over (∇)}F j .
10 . A method according to claim 1 further comprising determining, by the processor, a parameter vega ν which expresses a dependence of the expected value V of the financial contract to any volatilities which may be present in the one or more underlyings F j for j=1 . . . M based at least in part on the computer representation of the reprojected sensitivity vectors.
11 . A method according to claim 3 further comprising determining, by the processor, a parameter vega ν which expresses a dependence of the expected value V of the financial contract to any volatilities which may be present the one or more underlyings F j for j=1 . . . M according to ν=({right arrow over (ν)} ·{right arrow over (ν)}) 1/2 .
12 . A method according to claim 10 comprising determining, by the processor, that the parameter vega ν is zero and outputting, by the processor, an indication that the financial contract does not have optionally.
13 . A method according to claim 10 comprising determining, by the processor, that the parameter vega ν is non-zero and outputting, by the processor, an indication that the financial contract does have optionally.
14 . A method according to claim 1 further comprising determining, by the processor, a parameter gamma Γ which expresses a dependence of the parameter delta Δ on the one or more underlyings F j for j=1 . . . M, wherein determining the parameter gamma Γ comprises applying, by the processor, a finite difference technique using the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract.
15 . A method according to claim 3 further comprising determining, by the processor, a parameter gamma Γ which expresses a dependence of the parameter delta Δ on the one or more underlyings F j for j=1 . . . M, wherein determining the parameter gamma Γ comprises applying, by the processor, a finite difference technique using the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract and wherein applying the finite difference technique using the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract comprises:
forming, by the processor, a computer representation of a displaced market vector {right arrow over (a)}′ according to {right arrow over (a)}′={right arrow over (a)}+δa{right arrow over (e)} Δ where {right arrow over (a)} is an original market vector, δa is a finite difference magnitude and {right arrow over (e)} Δ is a unit vector having a direction of the reprojected sensitivity vector J T
Δ
→
=
∑
j
=
1
M
Δ
j
∇
→
F
j
(
e
→
Δ
=
J
T
Δ
→
J
T
Δ
→
≡
∑
j
=
1
M
Δ
j
∇
→
F
j
∑
j
=
1
M
Δ
j
∇
→
F
j
)
;
determining, by the processor, the parameter delta Δ for the expected value of the financial contract at both the original market vector {right arrow over (a)} and for the displaced market vector {right arrow over (a)}′;
determining, by the processor, the parameter gamma Γ according to
Γ
≈
1
δ
a
(
Δ
(
a
→
+
δ
a
e
Δ
→
)
-
Δ
(
a
→
)
)
.
16 . A method according to claim 4 wherein determining the parameter delta Δ from the computer representation of the reprojected sensitivity vectors comprises determining, by the processor, the parameter delta Δ in accordance with Δ=Σ j=1 M Δ j .
17 . A method according to claim 4 comprising determining, by the processor, a direction of the reprojected sensitivity vector J T {right arrow over (Δ)}=Σ j=1 M Δ j {right arrow over (∇)}F j .
18 . A method according to claim 1 wherein some or all of the steps are performed by one or more suitably configured processors.
19 . A system for determining a parameter delta Δ which expresses a dependence of an expected value V of a financial contract on one or more underlyings of the financial contract, the system comprising a processor configured, by execution of suitable software, to:
obtain a computer representation of a complete set of algorithmic differentiation (AD) sensitivities of the expected value V of the financial contract to a set of N input parameters {right arrow over (a)} in a form
∇
→
V
=
[
∂
V
∂
a
1
,
∂
V
∂
a
2
,
…
∂
V
∂
a
N
]
T
or a mathematical equivalent thereof;
obtain a computer representation of a complete set of AD sensitivities of the expected value of the one or more underlyings F j for j=1 . . . M, where M<N and M is a number of the one or more underlyings, to the set of N input parameters {right arrow over (a)} in a form
∇
→
F
j
=
[
∂
F
j
∂
a
1
,
∂
F
j
∂
a
2
,
…
∂
F
j
∂
a
N
]
T
for each j=1 . . . M or a mathematical equivalent thereof;
reproject the full set of AD sensitivities {right arrow over (∇)}V of the expected value of the financial contract onto the full set of AD sensitivities {right arrow over (∇)}F j for j=1 . . . M of the one or more underlyings to obtain a computer representation of reprojected sensitivity vectors; and
determine the parameter delta Δ based on the computer representation of the reprojected sensitivity vectors.
20 . A computer program product comprising a non-transitory computer-readable medium having instructions stored thereon, the instructions, when executed by a processor causing the processor to perform the method of claim 1 .