tremor model validation by wystup
TRANSCRIPT
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 1 of 41
Go Back
Full Screen
Close
Quit
The Tremor Stochastic-Local-VolatilityModel
Independent Validation byMathFinance
Uwe Wystup
June 2011
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 2 of 41
Go Back
Full Screen
Close
Quit
Abstract. In this document we summarize the hybridstochastic-local volatility model, called the Tremor-modelimplemented by Murex, and validate the implementationof the values and Greeks of vanilla and first generationexotic options using a Monte Carlo engine. The imple-mentation by Murex is based on finite difference methodof the model partial differential equation. The calibrationof the model parameters is done by Murex. What is vali-dated in this report are the derivatives values and Greeksbased on the calibrated model parameters delivered byMurex. The calibration as such is not part of this report.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 3 of 41
Go Back
Full Screen
Close
Quit
1. The Tremor Model
Model a the forward as ( [7], [8])
dfTt =√vt · g(fTt )dW
(1)t , (1)
dvt = κ(θ − vt)dt + σ√vtdW
(2)t (2)
g(f ) = min(a + bf + cf 2, cap · f ) (3)
Cov(dW(1)t , dW
(2)t ) = ρdt (4)
aThe name “Tremor” originates from Pascal Tremoureux, a senior mem-ber of the quant team at Murex
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 4 of 41
Go Back
Full Screen
Close
Quit
• fTt forward to maturity T seen from t
• vt variance process
• v0 initial instantaneous variance
• θ long-term variance set to v0
• κ > 0 rate of mean-reversion
• ρ instantaneous correlation between fTt and vt
• σ volatility of variance (vol of vol)
• a, b, c parameters of g the local volatility function
• cap: 500%
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 5 of 41
Go Back
Full Screen
Close
Quit
1.1. Parameter Choices in Murex and the MathFinance Implementa-tion
Local volatility parameters displayed in the Murex screen are
• a′: Slope
• b′: Parabola Minimum
• c′: Normalization Forward
These are shown in the MathFinance test results.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 5 of 41
Go Back
Full Screen
Close
Quit
1.1. Parameter Choices in Murex and the MathFinance Implementa-tion
Local volatility parameters displayed in the Murex screen are
• a′: Slope
• b′: Parabola Minimum
• c′: Normalization Forward
These are shown in the MathFinance test results.
They are related by (Source:[9])
a = c′(1− b′ + a′), (5)
b = b′ − 2a′, (6)
c = a′/c′. (7)
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 6 of 41
Go Back
Full Screen
Close
Quit
2. Implementation of the Tremor model byMathFinance
2.1. Discretization
We sample the distribution of the variance process di-rectly following the quadratic-exponential (QE) schemedescribed by Andersen in [1].
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 6 of 41
Go Back
Full Screen
Close
Quit
2. Implementation of the Tremor model byMathFinance
2.1. Discretization
We sample the distribution of the variance process di-rectly following the quadratic-exponential (QE) schemedescribed by Andersen in [1].
N = 224 simulated paths (Euler scheme in spot)
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 6 of 41
Go Back
Full Screen
Close
Quit
2. Implementation of the Tremor model byMathFinance
2.1. Discretization
We sample the distribution of the variance process di-rectly following the quadratic-exponential (QE) schemedescribed by Andersen in [1].
N = 224 simulated paths (Euler scheme in spot)
Equidistant time steps dt = 0.5 days = 0.00137, for alltest deals.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 7 of 41
Go Back
Full Screen
Close
Quit
2.2. Continuity Correction
• weights starting with value 1 at time zero.
• u: upper barrier
• d: lower barrier
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 7 of 41
Go Back
Full Screen
Close
Quit
2.2. Continuity Correction
• weights starting with value 1 at time zero.
• u: upper barrier
• d: lower barrier
• In any time step we check if the simulated path is notbeyond the barrier, and in such a case the weightsare multiplied with
wNTUp = 1− e−2 (u−s(t))(u−s(t+dt))
v(t)s(t)2dt
wNTDown = 1− e−2 (s(t)−d)(s(t+dt)−d)
v(t)s(t)2dt
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 7 of 41
Go Back
Full Screen
Close
Quit
2.2. Continuity Correction
• weights starting with value 1 at time zero.
• u: upper barrier
• d: lower barrier
• In any time step we check if the simulated path is notbeyond the barrier, and in such a case the weightsare multiplied with
wNTUp = 1− e−2 (u−s(t))(u−s(t+dt))
v(t)s(t)2dt
wNTDown = 1− e−2 (s(t)−d)(s(t+dt)−d)
v(t)s(t)2dt
• If the barrier is touched, then we set w = 1.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 7 of 41
Go Back
Full Screen
Close
Quit
2.2. Continuity Correction
• weights starting with value 1 at time zero.
• u: upper barrier
• d: lower barrier
• In any time step we check if the simulated path is notbeyond the barrier, and in such a case the weightsare multiplied with
wNTUp = 1− e−2 (u−s(t))(u−s(t+dt))
v(t)s(t)2dt
wNTDown = 1− e−2 (s(t)−d)(s(t+dt)−d)
v(t)s(t)2dt
• If the barrier is touched, then we set w = 1.
• If it is not touched explicitly, we compute the touchprobability as 1 minus the product of the weightsabove.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 8 of 41
Go Back
Full Screen
Close
Quit
2.3. Feller Condition
The CIR process for the variance (2), remains strictlypositive if
4κθ
σ2≥ 2. (8)
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 8 of 41
Go Back
Full Screen
Close
Quit
2.3. Feller Condition
The CIR process for the variance (2), remains strictlypositive if
4κθ
σ2≥ 2. (8)
42% of the examined markets yield calibrated parametersthat do not satisfy the Feller condition.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 8 of 41
Go Back
Full Screen
Close
Quit
2.3. Feller Condition
The CIR process for the variance (2), remains strictlypositive if
4κθ
σ2≥ 2. (8)
42% of the examined markets yield calibrated parametersthat do not satisfy the Feller condition.
Therefore: use fine time grid in PDEs or Monte Carlo.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 9 of 41
Go Back
Full Screen
Close
Quit
3. Validation Experiment
• Implement independent pricing engine (Monte Carloand PDE)
• Calibration: parameter delivered by Murex for agiven maturity based on spot, rates, ATM vol, 25delta risk reversals and butterflies
• Communication: Murex produces files irsm-form.xml and irsmout.xml containing all thenecessary information on market input, calibratedparameters, contract data and results
• MathFinance compares prices and Greeks
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 10 of 41
Go Back
Full Screen
Close
Quit
3.1. Deals under Consideration
EUR/CHF, 5
EUR/TRY 5EUR/HUF, 3
USD/MXN, 3 EUR/PLN, 3USD/BRL, 3 AUD/CAD, 1
AUD/NZD, 1GBP/JPY, 1
NZD/USD, 8EUR/GBP, 7
USD/TRY, 6
EUR/TRY, 5 AUD/JPY, 2/ ,
GBP/CHF, 1USD/SGD, 1
USD/CHF, 14
USD/CAD, 11
USD/JPY, 170
GBP/USD 23
EUR/JPY, 14
AUD/USD, 26
GBP/USD, 23
EUR/USD, 80EUR/USD, 80
388 deals in 22 currency pairs considered388 deals in 22 currency pairs considered
Figure 1: Distribution of currency pairs.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 11 of 41
Go Back
Full Screen
Close
Quit
55%
44%
33%
22%
11%
matur
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
10…
89%
78%
67%
55%
44%
33%
22%
11%
maturity
Figure 2: Distribution of maturities.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 12 of 41
Go Back
Full Screen
Close
Quit
3.2. Test Results
delta in per cent
price Mkt Given market price provided by Murex
price Calib Analytic approximation computed by Murex after calibration
price Comp Price computed by MathFinance with Monte Carlo simulation
price Comp D 4 standard deviations of the simulated price
vols Mkt Given market Black-Scholes implied vols provided by Murex
vols Mkt i Implied vols computed by MathFinance based on “price Mkt”
vols Calib Implied vols provided by Murex based on “price Calib”
vols Calib i Implied vols computed by MathFinance based on “price Calib”
vols Comp i Implied vols computed by MathFinance based on “price Comp”
Table 1: Explanations of Test Results
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 13 of 41
Go Back
Full Screen
Close
Quit
3.3. Vanilla Consistency Check
Detailed test results for many vanilla markets showed thatthe pricing of vanilla options consistently agrees with oursimulation based values.
market expiry fwd comp fwd factor spot rd rf v0
USD/JPY 0.99178 103.32762 103.32762 105.28181 0.01284 0.03173 0.10129
κ θ ξ ρ c′ b′ a′
1 0.01026 0.11674 -0.41524 103.32762 -2.48952 10.44604
delta -10 -25 50 25 10
strike 84.58869 95.17271 102.7913 109.33172 114.97983
price Mkt 0.75692 1.84591 4.18465 1.34143 0.40213
price Calib 0.79372 1.79034 4.20102 1.37707 0.3534
price Comp 0.81574 1.81104 4.20651 1.36733 0.34323
price Comp D 0.00479 0.00661 0.00487 0.00283 0.00141
vols Mkt 0.16375 0.12768 0.10234 0.08714 0.08366
vols Mkt i 0.16375 0.12768 0.10234 0.08714 0.08366
vols Calib 0.16598 0.12582 0.10277 0.08825 0.08085
vols Calib i 0.16598 0.12582 0.10277 0.08825 0.08085
vols Comp i 0.16729 0.12652 0.10291 0.08795 0.08024
Table 2: USD-JPY vanilla market check: exampledata irsm201008/02052008/irsmform_hp222srv_28945_6684515_premium_133.xml
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 14 of 41
Go Back
Full Screen
Close
Quit
3.4. Exotic Options
24 KID knock-in down
13 KIU knock-in up
21 KOD knock-out down
72 KOU knock-out up
49 OTD one-touch down
53 OTU one-touch up
153 DNT double-no-touch
3 DOT double-one-touch
Table 3: Number of exotic option types under consider-ation with abbreviations used by Murex
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 15 of 41
Go Back
Full Screen
Close
Quit
3.5. Test Results
type payCcy strike call/put down up maturity delivery
KID 0 101 1 101 0 0.991781 0.99726
delta -10 -25 50 25 10
vanilla (FDM) 0.81898 1.81489 4.20619 1.36789 0.34419
price MF(MC) dev price MF (FDM) price MX
2.18414 0.00386 2.20382 2.21282
Table 4: Example: USD-JPY Down-and-in Call
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 16 of 41
Go Back
Full Screen
Close
Quit
folder date deal no market fwd comp spot rd rf ‐10 ‐25 50 25 10 25 RR 25 BF vol0 kappa theta xi rho a b c cursor Feller ok type payCcy strike call/put down up maturity delivery price MF dev price MX mkt‐calib mkt‐mc../../../exam 9052008 hp222srv_ USD/CAD 1.00634 1.00622 0.02908 0.02273 0.10737 0.10464 0.10361 0.10465 0.10735 0.00002 0.00103 0.10499 1 0.01102 0.49224 0.02794 1.00634 0.37658 77.58444 0.9 0 DNT 1 0 0 0.965 1.035 0.01918 0.03014 0.93307 0.00024 0.92899 0 0../../../exam23052008 hp222srv_ EUR/USD 1.57731 1.579 0.02258 0.04119 0.10066 0.09604 0.09435 0.09603 0.10067 0 0.00168 0.09498 1 0.00902 0.26109 ‐0.00333 1.57731 0.96847 117.0344 0.65 0 DNT 1 0 0 1.539 1.591 0.05753 0.06849 0.10989 0.0003 0.11797 0 0../../../exam 6052008 hp222srv_ GBP/CHF 2.06857 2.07214 0.02177 0.05174 0.15004 0.13627 0.12686 0.12178 0.12074 ‐0.01449 0.00216 0.12654 1 0.01601 0.29029 ‐0.3337 2.06857 ‐1.37946 75.71854 0.5 0 DNT 1 0 0 1.92 2.14 0.05753 0.06301 0.7055 0.00044 0.70499 0.00001 0.00001../../../exam21052008 hp222srv_ EUR/JPY 162.2243 162.5849 0.00552 0.04235 0.13866 0.12029 0.10748 0.09951 0.09541 ‐0.02077 0.00242 0.10657 1 0.01136 0.20719 ‐0.2806 162.2243 ‐6.97593 117.6013 0.35 0 DNT 1 0 0 150 175 0.06027 0.07123 0.96787 0.00017 0.96094 0.00002 0.00002../../../exam14052008 hp222srv_ GBP/USD 1.93926 1.9436 0.0253 0.05345 0.10184 0.09442 0.08919 0.08718 0.08881 ‐0.00724 0.00161 0.09155 1 0.00838 0.28843 ‐0.34699 1.93926 2.37821 63.08655 0.85 0 DNT 1 0 0 1.905 1.985 0.07945 0.09041 0.26088 0.00043 0.26791 0 0.00001../../../exam15052008 hp222srv_ USD/CAD 1.00137 1.00093 0.03043 0.02491 0.10567 0.10205 0.10068 0.10206 0.10567 0.00001 0.00138 0.10279 1 0.01057 0.28317 0.02194 1.00137 0.70589 20.12226 0.9 0 DNT 1 0 0 0.985 1.025 0.07945 0.08767 0.14424 0.00034 0.15527 0 0../../../exam28052008 hp222srv_ USD/JPY 104.4931 104.6586 0.00454 0.02446 0.15557 0.13189 0.11607 0.10668 0.10245 ‐0.0252 0.00322 0.11526 1 0.01328 0.25521 ‐0.35866 104.4931 ‐5.01944 85.53857 0.4 0 DNT 1 0 0 102 106 0.07945 0.09041 0.04445 0.0002 0.05333 0.00005 0.00006../../../exam28052008 hp222srv_ USD/JPY 104.4931 104.6586 0.00454 0.02446 0.15557 0.13189 0.11607 0.10668 0.10245 ‐0.0252 0.00322 0.11526 1 0.01328 0.25521 ‐0.35866 104.4931 ‐5.01944 85.53857 0.4 0 DNT 1 0 0 102 106 0.07945 0.09041 0.04445 0.0002 0.05333 0.00005 0.00006../../../exam 6052008 hp222srv_ USD/JPY 104.5263 104.7173 0.0041 0.02631 0.14276 0.12227 0.10905 0.1018 0.09939 ‐0.02048 0.00298 0.10815 1 0.0117 0.22868 ‐0.32413 104.5263 ‐4.37677 104.8127 0.4 0 DNT 1 0 0 102.5 106.5 0.08219 0.09315 0.07452 0.00026 0.08622 0.00004 0.00004../../../exam16052008 hp222srv_ EUR/USD 1.55536 1.55796 0.02498 0.04345 0.10138 0.0954 0.09228 0.09251 0.09579 ‐0.00289 0.00167 0.09297 1 0.00864 0.20994 ‐0.15554 1.55536 1.18979 80.51514 0.65 0 DNT 1 0 0 1.51 1.565 0.09041 0.09589 0.03548 0.00018 0.04007 0.00001 0.00001../../../exam25042008 hp222srv_ USD/JPY 104.3067 104.5194 0.00708 0.02961 0.16185 0.13979 0.12486 0.11586 0.11191 ‐0.02393 0.00297 0.12394 1 0.01536 0.23663 ‐0.33289 104.3067 ‐3.76938 60.91757 0.4 0 DNT 1 0 0 102.5 106.25 0.09041 0.09589 0.01643 0.00012 0.02307 0.00004 0.00005../../../exam25072008 hp222srv_ USD/JPY 107.7252 107.9219 0.00476 0.02494 0.15503 0.12787 0.10859 0.09593 0.08843 ‐0.03194 0.00332 0.10831 1 0.01173 0.25004 ‐0.43247 107.7252 ‐7.33594 58.27808 0.4 0 DNT 1 0 0 104.5 109.5 0.09041 0.09589 0.10014 0.00029 0.11231 0.00008 0.00009../../../exam29042008 hp222srv_ USD/JPY 103.7217 103.99 0.00671 0.02915 0.15971 0.13642 0.12078 0.11128 0.10698 ‐0.02514 0.00307 0.11989 1 0.01437 0.21624 ‐0.35107 103.7217 ‐3.6745 52.27585 0.4 0 DNT 1 0 0 101.5 107 0.11507 0.12055 0.10839 0.0003 0.12131 0.00006 0.00007../../../exam15052008 hp222srv_ USD/JPY 104.4681 104.74 0.00699 0.02635 0.14622 0.12381 0.10943 0.10133 0.09842 ‐0.02248 0.00314 0.10852 1 0.01178 0.19336 ‐0.34278 104.4681 ‐3.5253 64.36268 0.4 0 DNT 1 0 0 98.55 108 0.13425 0.14521 0.42601 0.00048 0.4319 0.00007 0.00008../../../exam12112008 hp222srv_ AUD/USD 0.63585 0.6388 0.02281 0.05051 0.44674 0.38454 0.34016 0.31399 0.3039 ‐0.07055 0.00911 0.35033 1 0.12273 0.7683 ‐0.48752 0.63585 0.28169 3.08732 0.6 0 DNT 1 0 0 0.615 0.725 0.16712 0.1726 0.0814 0.00027 0.08626 0.00066 0.00087/ / /exam17042008 hp222srv EUR/USD 1 58457 1 58843 0 02947 0 044 0 1216 0 11157 0 10641 0 10715 0 1136 0 00443 0 00295 0 11001 1 0 0121 0 28763 0 15282 1 58457 1 30382 27 18037 0 8 0 DNT 1 0 0 1 55 1 7 0 16712 0 1726 0 34853 0 00047 0 34961 0 0 00001../../../exam17042008 hp222srv_ EUR/USD 1.58457 1.58843 0.02947 0.044 0.1216 0.11157 0.10641 0.10715 0.1136 ‐0.00443 0.00295 0.11001 1 0.0121 0.28763 ‐0.15282 1.58457 1.30382 27.18037 0.8 0 DNT 1 0 0 1.55 1.7 0.16712 0.1726 0.34853 0.00047 0.34961 0 0.00001../../../exam28052008 hp222srv_ GBP/USD 1.97065 1.9801 0.02603 0.05419 0.097 0.0886 0.08338 0.08223 0.08575 ‐0.00636 0.00203 0.08604 1 0.0074 0.21502 ‐0.35719 1.97065 2.86402 54.71836 0.80167 0 DNT 1 0 0 1.93 2.03 0.16986 0.17534 0.20697 0.00039 0.21083 0 0.00001../../../exam28052008 hp222srv_ GBP/USD 1.97065 1.9801 0.02603 0.05419 0.097 0.0886 0.08338 0.08223 0.08575 ‐0.00636 0.00203 0.08604 1 0.0074 0.21502 ‐0.35719 1.97065 2.86402 54.71836 0.80167 0 DNT 1 0 0 1.93 2.03 0.16986 0.17534 0.20697 0.00039 0.21083 0 0.00001../../../exam22052008 hp222srv_ EUR/HUF 246.6435 245.0201 0.08315 0.04489 0.09044 0.0866 0.09053 0.10403 0.12898 0.01743 0.00478 0.09133 1 0.00834 0.23836 0.39997 246.6435 2.20144 111.5049 0.5 0 DNT 1 0 0 237 252 0.1726 0.18356 0.2968 0.00044 0.29228 0.00015 0.00029../../../exam12062008 hp222srv_ EUR/USD 1.53746 1.5421 0.02736 0.04481 0.11131 0.10617 0.10431 0.10617 0.11132 0 0.00186 0.10544 1 0.01112 0.1892 ‐0.00179 1.53746 0.93344 27.9297 0.7 0 DNT 1 0 0 1.5135 1.5935 0.1726 0.18356 0.06725 0.00024 0.07583 0.00001 0.00001../../../exam15052008 hp222srv_ USD/JPY 104.3868 104.7345 0.00757 0.02683 0.14787 0.12394 0.10859 0.09971 0.09621 ‐0.02423 0.00324 0.10773 1 0.0116 0.17956 ‐0.36098 104.3868 ‐3.33727 50.18435 0.4 0 DNT 1 0 0 102.5 107.5 0.1726 0.1863 0.0453 0.0002 0.05496 0.00011 0.00013../../../exam22052008 hp222srv_ USD/JPY 104.0332 104.3653 0.00752 0.02598 0.16454 0.13713 0.11912 0.10814 0.10309 ‐0.02898 0.00352 0.11828 1 0.01399 0.20054 ‐0.37964 104.0332 ‐3.38493 37.71477 0.4 0 DNT 1 0 0 99 105 0.1726 0.18356 0.01154 0.0001 0.01367 0.00014 0.00016../../../exam14052008 hp222srv_ AUD/USD 0.9257 0.93455 0.02712 0.07534 0.13618 0.12378 0.11665 0.11519 0.12077 ‐0.00859 0.00284 0.12005 1 0.01441 0.27339 ‐0.47018 0.9257 3.26003 37.01666 0.75 0 DNT 1 0 0 0.8875 0.9575 0.19726 0.20822 0.16111 0.00036 0.17039 0 0.00008../../../exam28052008 hp222srv_ EUR/GBP 0.79147 0.78961 0.05488 0.04541 0.0846 0.08057 0.08159 0.08733 0.09689 0.00677 0.00236 0.08439 1 0.00712 0.19381 0.37344 0.79147 ‐0.99464 42.42097 0.8 0 DNT 1 0 0 0.77 0.815 0.24932 0.2548 0.17821 0.00037 0.18056 0 0.00003../../../exam17042008 hp222srv_ EUR/USD 1.58215 1.5883 0.02975 0.04531 0.12133 0.11115 0.10607 0.10669 0.11316 ‐0.00446 0.00285 0.10837 1 0.01174 0.21365 ‐0.21479 1.58215 1.60992 27.55763 0.7 0 DNT 1 0 0 1.55 1.7 0.24932 0.26027 0.23031 0.00041 0.23073 0 0.00001../../../exam17042008 hp222srv_ EUR/USD 1.58215 1.5883 0.02975 0.04531 0.12133 0.11115 0.10607 0.10669 0.11316 ‐0.00446 0.00285 0.10837 1 0.01174 0.21365 ‐0.21479 1.58215 1.60992 27.55763 0.7 0 DNT 1 0 0 1.55 1.7 0.24932 0.26027 0.23031 0.00041 0.23073 0 0.00001../../../exam 3062008 hp222srv_ EUR/USD 1.53952 1.5467 0.02733 0.04578 0.10926 0.10123 0.09774 0.09927 0.10559 ‐0.00196 0.00251 0.09957 1 0.00991 0.18868 ‐0.08672 1.53952 1.08479 28.86466 0.7 0 DNT 1 0 0 1.505 1.595 0.25206 0.25753 0.09279 0.00028 0.09666 0.00001 0.00001../../../exam13112008 hp222srv_ GBP/USD 1.46635 1.4677 0.02253 0.02618 0.31961 0.28276 0.25826 0.24871 0.25587 ‐0.03405 0.00748 0.26975 1 0.07277 0.60502 ‐0.4471 1.46635 1.2505 4.7711 0.75 0 DNT 1 0 0 1.39 1.59 0.25206 0.26301 0.06631 0.00025 0.07185 0.00029 0.00066../../../exam15052008 hp222srv_ GBP/USD 1.93085 1.945 0.02859 0.05756 0.10492 0.09606 0.08973 0.08697 0.08829 ‐0.00909 0.00178 0.0927 1 0.00859 0.18804 ‐0.39433 1.93085 1.80777 21.97632 0.85 0 DNT 1 0 0 1.855 1.995 0.25206 0.26301 0.22274 0.0004 0.2268 0.00002 0.00003../../../exam11062008 hp222srv_ USD/CAD 1.02072 1.0196 0.03279 0.02843 0.10745 0.10245 0.10115 0.10422 0.11133 0.00177 0.00218 0.10257 1 0.01052 0.1736 0.04034 1.02072 1.17286 23.10635 0.7 0 DNT 1 0 0 0.9725 1.04 0.25206 0.2548 0.10087 0.00029 0.10449 0.00001 0.00001../../../exam29042008 hp222srv_ USD/TRY 1.33744 1.2916 0.16653 0.02967 0.14103 0.14429 0.15821 0.18707 0.22272 0.04278 0.00747 0.17053 1 0.02908 0.51 0.40982 1.33744 2.28403 ‐0.68884 0.9 0 DNT 1 0 0 1.1 1.35 0.2548 0.25753 0.34569 0.00044 0.3484 0.00002 0.00014../../../exam 7112008 hp222srv_ EUR/GBP 0.81184 0.81177 0.03591 0.03554 0.15389 0.1468 0.1468 0.15433 0.16811 0.00752 0.00377 0.15 1 0.0225 0.29245 0.28743 0.81184 0.19713 11.63782 0.75 0 DNT 1 0 0 0.745 0.835 0.26027 0.26575 0.15923 0.00035 0.15949 0 0.00004/ / / p _ /../../../exam25042008 hp222srv_ USD/JPY 103.9347 104.5185 0.0087 0.03022 0.16375 0.1361 0.11811 0.10693 0.1018 ‐0.02917 0.00341 0.11725 1 0.01375 0.17039 ‐0.38101 103.9347 ‐2.66435 25.1264 0.4 0 DNT 1 0 0 100 110 0.26027 0.26575 0.2523 0.00043 0.25672 0.00019 0.00023../../../exam28042008 hp222srv_ EUR/USD 1.55629 1.5642 0.03038 0.0469 0.11578 0.10595 0.1008 0.10085 0.10619 ‐0.0051 0.0026 0.10294 1 0.0106 0.18386 ‐0.23253 1.55629 1.46976 24.1345 0.7 0 DNT 1 0 0 1.49 1.625 0.30685 0.31233 0.22295 0.0004 0.22379 0.00001 0.00002../../../exam28042008 hp222srv_ AUD/USD 0.92481 0.9395 0.03046 0.078 0.14602 0.12973 0.12121 0.12103 0.1315 ‐0.00869 0.00417 0.12694 1 0.01611 0.28785 ‐0.39197 0.92481 2.64478 21.57236 0.75 0 DNT 1 0 0 0.895 0.975 0.33151 0.33699 0.13792 0.00033 0.13828 0.00001 0.00019../../../exam21052008 hp222srv_ EUR/USD 1.56792 1.57803 0.02743 0.04683 0.11441 0.10541 0.10137 0.10266 0.10924 ‐0.00276 0.00267 0.10305 1 0.01062 0.17231 ‐0.13777 1.56792 1.24341 23.44427 0.66593 0 DNT 1 0 0 1.5 1.63 0.33151 0.34247 0.16326 0.00036 0.16485 0.00001 0.00001../../../exam21042008 hp222srv_ EUR/USD 1.58314 1.59146 0.02977 0.04546 0.12208 0.1111 0.10561 0.1061 0.11307 ‐0.005 0.00299 0.10826 1 0.01172 0.19934 ‐0.26495 1.58314 1.84776 22.83253 0.7 0 DNT 1 0 0 1.53 1.73 0.33425 0.34521 0.35299 0.00047 0.35156 0 0.00003../../../exam 3062008 hp222srv_ USD/JPY 104.0725 104.88 0.00978 0.02884 0.16112 0.1298 0.11019 0.09798 0.092 ‐0.03182 0.0037 0.10896 1 0.01187 0.13546 ‐0.37869 104.0725 ‐2.98419 20.63508 0.35 1 DNT 1 0 0 98 107.5 0.40548 0.41096 0.06657 0.00024 0.06811 0.00038 0.00045../../../exam21042008 hp222srv_ EUR/USD 1.58098 1.59146 0.03049 0.04646 0.12272 0.11101 0.10514 0.10547 0.11297 ‐0.00554 0.00311 0.1082 1 0.01171 0.1923 ‐0.33657 1.58098 2.24163 21.35668 0.7 0 DNT 1 0 0 1.54 1.69 0.4137 0.42466 0.15704 0.00036 0.15667 0 0.00007../../../exam21042008 hp222srv_ EUR/USD 1.58098 1.59146 0.03049 0.04646 0.12272 0.11101 0.10514 0.10547 0.11297 ‐0.00554 0.00311 0.1082 1 0.01171 0.1923 ‐0.33657 1.58098 2.24163 21.35668 0.7 0 DNT 1 0 0 1.42 1.62 0.4137 0.42466 0.15362 0.00034 0.15337 0 0.00007../../../exam 2052008 hp222srv_ EUR/USD 1.52995 1.5416 0.02962 0.04772 0.11915 0.10661 0.10032 0.10066 0.10833 ‐0.00594 0.00331 0.10356 1 0.01073 0.19489 ‐0.28087 1.52995 1.83734 22.13073 0.7 0 DNT 1 0 0 1.35 1.605 0.41918 0.43014 0.44637 0.00047 0.44282 0 0.00005../../../exam18112008 hp222srv_ USD/JPY 95.715 96.42 0.006 0.0208 0.33733 0.23908 0.18038 0.14316 0.12659 ‐0.09592 0.01074 0.18237 1 0.03326 0.39414 ‐0.55706 95.715 ‐2.12966 3.01075 0.35 0 DNT 1 0 0 90 105 0.49589 0.50137 0.21308 0.00041 0.2088 0.00352 0.00343../../../exam 7112008 hp222srv_ USD/JPY 97.40974 98.05355 0.00812 0.02133 0.32341 0.23254 0.17798 0.14291 0.12681 ‐0.08964 0.00974 0.17894 1 0.03202 0.3596 ‐0.53609 97.40974 ‐2.13905 3.43402 0.35 0 DNT 1 0 0 85 105 0.49863 0.50959 0.25013 0.00042 0.25182 0.00302 0.00303../../../exam 7052008 hp222srv_ GBP/USD 1.92727 1.95252 0.02946 0.05543 0.10951 0.09965 0.09302 0.09025 0.09174 ‐0.00941 0.00193 0.09606 1 0.00923 0.15018 ‐0.41227 1.92727 1.6036 13.05363 0.8 0 DNT 1 0 0 1.9 2.1 0.50137 0.51233 0.17121 0.00037 0.16986 0.00006 0.0001../../../exam21042008 hp222srv_ NZD/USD 0.76969 0.7928 0.03129 0.09029 0.16941 0.14813 0.13813 0.13693 0.14531 ‐0.0112 0.0044 0.1391 1 0.01935 0.19113 ‐0.38135 0.76969 1.27958 15.84212 0.5 1 DNT 1 0 0 0.73 0.845 0.50137 0.50685 0.19472 0.00038 0.19437 0.00015 0.00054../../../exam11062008 hp222srv_ USD/JPY 105.7794 106.9175 0.01007 0.03142 0.1607 0.13015 0.11152 0.10007 0.09499 ‐0.03008 0.00359 0.11013 1 0.01213 0.12651 ‐0.35705 105.7794 ‐2.31899 17.5673 0.35 1 DNT 1 0 0 100 112 0.50137 0.51233 0.1612 0.00036 0.16148 0.00039 0.00049../../../exam21042008 hp222srv_ USD/JPY 102.0235 103.12 0.00996 0.03129 0.18022 0.14089 0.11771 0.10346 0.09705 ‐0.03743 0.00446 0.11645 1 0.01356 0.15517 ‐0.40783 102.0235 ‐2.4651 15.79213 0.35 1 DNT 1 0 0 101 111 0.50137 0.50685 0.08327 0.00027 0.08359 0.00038 0.00054../../../exam23042008 hp222srv_ USD/JPY 102.4356 103.5082 0.01055 0.03132 0.17972 0.14038 0.11721 0.10296 0.09653 ‐0.03742 0.00446 0.11597 1 0.01345 0.15498 ‐0.40945 102.4356 ‐2.4896 15.90576 0.35 1 DNT 1 0 0 89.5 105.5 0.50137 0.51233 0.09433 0.00028 0.09504 0.00039 0.00055/ / /exam28042008 hp222srv EUR/USD 1 5508 1 5642 0 03103 0 0481 0 11711 0 1063 0 10064 0 10042 0 1061 0 00588 0 00273 0 10327 1 0 01066 0 16365 0 29961 1 5508 1 69753 16 56696 0 7 0 DNT 1 0 0 1 485 1 665 0 50411 0 50959 0 2509 0 00042 0 24968 0 00001 0 00005../../../exam28042008 hp222srv_ EUR/USD 1.5508 1.5642 0.03103 0.0481 0.11711 0.1063 0.10064 0.10042 0.1061 ‐0.00588 0.00273 0.10327 1 0.01066 0.16365 ‐0.29961 1.5508 1.69753 16.56696 0.7 0 DNT 1 0 0 1.485 1.665 0.50411 0.50959 0.2509 0.00042 0.24968 0.00001 0.00005../../../exam16052008 hp222srv_ USD/CHF 1.04695 1.04761 0.02822 0.02945 0.12361 0.11075 0.10408 0.10373 0.11055 ‐0.00702 0.00316 0.10691 1 0.01143 0.1853 ‐0.2152 1.04695 1.20537 14.90141 0.7 0 DNT 1 0 0 1.015 1.095 0.50685 0.51233 0.06538 0.00024 0.06656 0.00001 0.00004../../../exam16052008 hp222srv_ EUR/HUF 251.6484 246.758 0.08699 0.04847 0.07614 0.07582 0.082 0.0982 0.12629 0.02238 0.00501 0.08366 1 0.007 0.15891 0.45303 251.6484 2.76179 38.10052 0.5 0 DNT 1 0 0 244 272 0.50959 0.51507 0.11334 0.00031 0.1146 0.00053 0.00105../../../exam 8052008 hp222srv_ EUR/USD 1.52544 1.54026 0.02896 0.04793 0.12059 0.10856 0.10254 0.10285 0.11018 ‐0.00571 0.00316 0.10572 1 0.01118 0.18054 ‐0.29057 1.52544 1.80212 17.09168 0.7 0 DNT 1 0 0 1.46 1.6 0.50959 0.51507 0.12361 0.00032 0.12412 0 0.00006../../../exam 7032008 hp222srv_ USD/JPY 101.7619 102.6294 0.0104 0.02705 0.14667 0.13588 0.12116 0.11028 0.09995 ‐0.0256 0.00192 0.12004 1 0.01441 0.03277 0.08501 101.7619 ‐2.61224 4.85146 0.35 1 DNT 1 0 0 90.5 105 0.50959 0.51507 0.07665 0.00025 0.07813 0.03428 0.03438../../../exam25072008 hp222srv_ USD/JPY 106.6887 107.9234 0.00955 0.03213 0.1723 0.13348 0.109 0.09309 0.08431 ‐0.04039 0.00428 0.10824 1 0.01172 0.1475 ‐0.45381 106.6887 ‐3.43821 12.69486 0.35 1 DNT 1 0 0 102 114 0.50959 0.51507 0.21657 0.0004 0.21677 0.00069 0.00078../../../exam 9052008 hp222srv_ EUR/CHF 1.59511 1.61156 0.02809 0.04812 0.0806 0.06419 0.05493 0.05066 0.05088 ‐0.01353 0.00249 0.05568 1 0.0031 0.10086 ‐0.44532 1.59511 ‐1.52126 84.60618 0.5 0 DNT 1 0 0 1.55 1.65 0.51233 0.51781 0.26775 0.00043 0.2656 0.00014 0.00028../../../exam17042008 hp222srv_ EUR/HUF 259.6129 253.6063 0.0863 0.04798 0.08572 0.08644 0.09284 0.10881 0.1351 0.02237 0.00478 0.09408 1 0.00885 0.14586 0.4144 259.6129 2.63704 22.04448 0.5 0 DNT 1 0 0 230 270 0.61096 0.61644 0.4496 0.00046 0.44805 0.00059 0.00103../../../exam12112008 hp222srv_ EUR/USD 1.24475 1.251 0.01965 0.02635 0.23455 0.20124 0.18551 0.18908 0.21266 ‐0.01216 0.00966 0.20028 1 0.04011 0.4522 ‐0.20159 1.24475 1.06937 2.98187 0.75 0 DNT 1 0 0 1.14 1.435 0.74795 0.75343 0.2558 0.00043 0.2525 0 0.00036../../../exam20052008 hp222srv_ USD/CAD 0.99337 0.99221 0.03141 0.02986 0.11451 0.1065 0.10392 0.10699 0.11561 0.00048 0.00283 0.1061 1 0.01126 0.14945 0.10754 0.99337 0.5063 8.59089 0.7 1 DNT 1 0 0 0.93 1.07 0.75616 0.76438 0.22834 0.0004 0.22772 0.00002 0.00006../../../exam19052008 hp222srv_ USD/TRY 1.3647 1.229 0.16895 0.03044 0.14222 0.14749 0.16043 0.18866 0.22688 0.04117 0.00765 0.16678 1 0.02782 0.27578 0.49121 1.3647 1.59739 2.3616 0.7 0 DNT 1 0 0 1.17 1.53 0.75616 0.7589 0.27438 0.00042 0.27264 0.0006 0.00133../../../exam12052008 hp222srv_ GBP/USD 1.91786 1.95652 0.03019 0.05649 0.11276 0.10077 0.09375 0.09158 0.09464 ‐0.00918 0.00243 0.09666 1 0.00934 0.13939 ‐0.40538 1.91786 1.53478 13.00344 0.7 0 DNT 1 0 0 1.86 2.1 0.7589 0.76986 0.20195 0.00039 0.20065 0.00016 0.0003../../../exam17042008 hp222srv_ USD/SGD 1.33962 1.35219 0.01935 0.03166 0.07428 0.06386 0.0595 0.06049 0.06764 ‐0.00336 0.00268 0.06011 1 0.00361 0.08992 ‐0.30345 1.33962 2.49506 64.44449 0.5 0 DNT 1 0 0 1.285 1.385 0.7589 0.76438 0.1727 0.00036 0.17481 0 0.00015../../../exam 2052008 hp222srv_ EUR/USD 1.52135 1.54175 0.0304 0.04788 0.12029 0.10705 0.10048 0.10055 0.10829 ‐0.0065 0.00332 0.1044 1 0.0109 0.17194 ‐0.32607 1.52135 1.8129 12.87369 0.7 0 DNT 1 0 0 1.45 1.65 0.76164 0.76712 0.21941 0.0004 0.21802 0.00001 0.00015../../../exam23052008 hp222srv_ USD/JPY 101.6337 103.0906 0.01207 0.03075 0.16936 0.13061 0.10777 0.09336 0.08596 ‐0.03725 0.00422 0.10606 1 0.01125 0.11684 ‐0.39066 101.6337 ‐2.73081 12.53405 0.3 1 DNT 1 0 0 91 110 0.76164 0.76712 0.37023 0.00047 0.36939 0.00118 0.00132../../../exam23042008 hp222srv_ USD/JPY 101.8045 103.52 0.01222 0.03255 0.18434 0.13861 0.11327 0.09778 0.09061 ‐0.04083 0.00492 0.11151 1 0.01243 0.13664 ‐0.41279 101.8045 ‐2.33067 11.75301 0.3 1 DNT 1 0 0 95.1 110.25 0.82192 0.8274 0.20159 0.00039 0.20142 0.00124 0.00145../../../exam16052008 hp222srv_ EUR/USD 1.5349 1.55795 0.03127 0.04911 0.1125 0.10095 0.09557 0.09631 0.10396 ‐0.00464 0.00306 0.09898 1 0.0098 0.15679 ‐0.25147 1.5349 1.58181 11.71255 0.7 0 DNT 1 0 0 1.415 1.6 0.83562 0.8411 0.11676 0.0003 0.11744 0.00001 0.00008/ / / p _ /../../../exam12052008 hp222srv_ USD/CAD 1.00631 1.0048 0.03236 0.03074 0.11573 0.10694 0.10422 0.1077 0.11748 0.00076 0.0031 0.1068 1 0.01141 0.15434 0.11826 1.00631 0.50628 7.30813 0.7 0 DNT 1 0 0 0.96 1.13 0.92603 0.92877 0.20482 0.00039 0.20511 0 0.00004../../../exam29042008 hp222srv_ USD/JPY 102.0082 103.9845 0.014 0.03324 0.17222 0.13138 0.10788 0.09281 0.0851 ‐0.03857 0.00422 0.10624 1 0.01129 0.1193 ‐0.40755 102.0082 ‐2.30197 9.27385 0.3 1 DNT 1 0 0 99.25 111.5 0.99726 1.00548 0.08756 0.00028 0.08794 0.00168 0.00186../../../exam21042008 hp222srv_ AUD/JPY 90.7273 97.15966 0.01275 0.08125 0.27209 0.20321 0.16932 0.14901 0.14379 ‐0.0542 0.0068 0.16587 1 0.02751 0.22937 ‐0.39436 90.7273 ‐0.70247 5.9953 0.3 1 DNT 1 0 0 85 108 1 1.00548 0.23821 0.00041 0.23614 0.01136 0.01199../../../exam11062008 hp222srv_ EUR/JPY 159.9301 166.4536 0.01274 0.05272 0.16331 0.12754 0.10765 0.09492 0.08851 ‐0.03262 0.00358 0.10521 1 0.01107 0.08898 ‐0.23708 159.9301 ‐2.54801 11.77291 0.25 1 DNT 1 0 0 144 173.5 1 1.01096 0.18275 0.00037 0.18152 0.0015 0.00151../../../exam15052008 hp222srv_ EUR/PLN 3.44612 3.39564 0.06481 0.05006 0.06627 0.06084 0.06077 0.06661 0.07776 0.00577 0.00296 0.06363 1 0.00405 0.11755 0.31462 3.44612 ‐0.14676 26.25276 0.7 0 DNT 1 0 0 3.18 3.61 1 1.01096 0.44547 0.00046 0.44194 0.00006 0.00026../../../exam20052008 hp222srv_ EUR/TRY 2.19888 1.94497 0.17216 0.04946 0.14966 0.15526 0.16947 0.20042 0.23798 0.04516 0.00837 0.18087 1 0.03271 0.3322 0.56104 2.19888 1.08493 0.63124 0.9 0 DNT 1 0 0 1.75 2.45 1 1.00274 0.40646 0.00044 0.39995 0.00015 0.00057../../../exam 3062008 hp222srv_ EUR/USD 1.51265 1.5365 0.0277 0.04334 0.09852 0.09585 0.09427 0.09464 0.0974 ‐0.00121 0.00097 0.09557 1 0.00913 0.08077 ‐0.3541 1.51265 1.9219 4.97286 0.75 1 DNT 1 0 0 1.25 1.6 1 1.01096 0.32024 0.00042 0.31953 0.00645 0.00654../../../exam 6032008 hp222srv_ EUR/USD 1.51265 1.5365 0.0277 0.04334 0.09852 0.09585 0.09427 0.09464 0.0974 ‐0.00121 0.00097 0.09557 1 0.00913 0.08077 ‐0.3541 1.51265 1.9219 4.97286 0.75 1 DNT 1 0 0 1.25 1.6 1 1.01096 0.32024 0.00042 0.31953 0.00645 0.00654../../../exam13112008 hp222srv_ EUR/USD 1.2666 1.2692 0.02139 0.02344 0.23065 0.19631 0.17996 0.18329 0.20729 ‐0.01302 0.00984 0.19657 1 0.03864 0.44414 ‐0.22704 1.2666 1.09701 2.35634 0.75 0 DNT 1 0 0 1.1 1.4 1 1.01096 0.21001 0.00039 0.21182 0 0.00054../../../exam15052008 hp222srv_ EUR/USD 1.51925 1.5453 0.03306 0.05006 0.11474 0.103 0.0976 0.09831 0.1061 ‐0.00469 0.00306 0.10126 1 0.01025 0.15606 ‐0.26696 1.51925 1.59471 9.51353 0.7 0 DNT 1 0 0 1.3175 1.6175 1 1.01096 0.30092 0.00042 0.30188 0.00001 0.00011../../../exam19052008 hp222srv_ EUR/USD 1.52433 1.5512 0.03222 0.04969 0.11224 0.1005 0.0951 0.09581 0.1036 ‐0.00469 0.00306 0.09878 1 0.00976 0.15468 ‐0.26719 1.52433 1.61454 10.12568 0.7 0 DNT 1 0 0 1.305 1.605 1 1.00548 0.24907 0.0004 0.25043 0.00001 0.00011../../../exam24062008 hp222srv_ EUR/USD 1.53221 1.5575 0.03663 0.05301 0.11977 0.10801 0.1027 0.1036 0.11153 ‐0.00441 0.0031 0.1063 1 0.0113 0.16013 ‐0.22615 1.53221 1.39438 8.0279 0.7 0 DNT 1 0 0 1.35 1.6 1 1.00548 0.13145 0.00031 0.13244 0.00003 0.0001../../../exam29042008 hp222srv_ EUR/USD 1.53238 1.55627 0.03327 0.04873 0.11713 0.10462 0.09861 0.09838 0.10501 ‐0.00624 0.00289 0.10175 1 0.01035 0.14332 ‐0.41675 1.53238 2.03874 12.42715 0.65 1 DNT 1 0 0 1.415 1.645 1 1.0137 0.20022 0.00038 0.20044 0.00001 0.00045../../../exam12112008 hp222srv_ GBP/USD 1.49077 1.49607 0.01967 0.02322 0.23884 0.2017 0.18047 0.17637 0.19242 ‐0.02533 0.00857 0.19914 1 0.03966 0.42192 ‐0.47574 1.49077 1.53491 3.42801 0.75 0 DNT 1 0 0 1.3 1.8 1 1.01096 0.39046 0.00049 0.38749 0.00041 0.00463../../../exam21042008 hp222srv_ USD/CAD 1.01075 1.0069 0.03704 0.03321 0.11913 0.1094 0.10609 0.10992 0.12002 0.00052 0.00357 0.11246 1 0.01265 0.20962 0.08154 1.01075 0.58752 2.4405 0.9 0 DNT 1 0 0 0.94 1.19 1 1.00274 0.41935 0.00048 0.41765 0.00001 0.00007../../../exam27052008 hp222srv_ USD/CAD 0.9967 0.9944 0.03401 0.03169 0.1156 0.107 0.10415 0.10702 0.11562 0.00002 0.00286 0.1065 1 0.01134 0.14622 0.13308 0.9967 0.37484 6.51431 0.7 1 DNT 1 0 0 0.945 1.115 1 1.00274 0.20404 0.00039 0.20417 0.00003 0.00008/ / / 6112008 h 222 USD/JPY 96 3495 97 77 0 00648 0 02111 0 31168 0 21609 0 16126 0 12555 0 11008 0 09054 0 00956 0 16409 1 0 02692 0 32495 0 54792 96 3495 1 71208 1 03303 0 3 0 DNT 1 0 0 93 133 1 1 01096 0 32558 0 00049 0 32364 0 00432 0 00368../../../exam 6112008 hp222srv_ USD/JPY 96.3495 97.77 0.00648 0.02111 0.31168 0.21609 0.16126 0.12555 0.11008 ‐0.09054 0.00956 0.16409 1 0.02692 0.32495 ‐0.54792 96.3495 ‐1.71208 1.03303 0.3 0 DNT 1 0 0 93 133 1 1.01096 0.32558 0.00049 0.32364 0.00432 0.00368../../../exam10112008 hp222srv_ USD/JPY 96.2575 97.465 0.00702 0.01949 0.31648 0.21806 0.16212 0.12624 0.11126 ‐0.09182 0.01003 0.1656 1 0.02742 0.34044 ‐0.55884 96.2575 ‐1.59234 1.22339 0.3 0 DNT 1 0 0 77.5 105 1 1.00548 0.25848 0.00042 0.25916 0.0046 0.00394../../../exam11112008 hp222srv_ USD/JPY 96.2575 97.465 0.00702 0.01949 0.31648 0.21806 0.16212 0.12624 0.11126 ‐0.09182 0.01003 0.1656 1 0.02742 0.34044 ‐0.55884 96.2575 ‐1.59234 1.22339 0.3 0 DNT 1 0 0 77.5 105 1 1.00548 0.25848 0.00042 0.25916 0.0046 0.00394../../../exam12112008 hp222srv_ USD/JPY 94.29129 95.53314 0.00658 0.01967 0.32863 0.22927 0.17276 0.1354 0.11967 ‐0.09388 0.00957 0.17516 1 0.03068 0.34067 ‐0.53899 94.29129 ‐1.4455 1.2017 0.3 0 DNT 1 0 0 92 119 1 1.01096 0.22743 0.00043 0.22729 0.00457 0.00392../../../exam13112008 hp222srv_ USD/JPY 94.865 96.39 0.00544 0.02139 0.33127 0.22939 0.17175 0.13417 0.11872 ‐0.09522 0.01003 0.17506 1 0.03064 0.35613 ‐0.54914 94.865 ‐1.42707 1.17222 0.3 0 DNT 1 0 0 85.75 105.5 1 1.01096 0.19299 0.00039 0.19347 0.00494 0.00421../../../exam18112008 hp222srv_ USD/JPY 94.945 96.42 0.00497 0.02038 0.3294 0.22603 0.16715 0.12926 0.11376 ‐0.09677 0.01049 0.17165 1 0.02946 0.36719 ‐0.56496 94.945 ‐1.53478 0.8344 0.3 0 DNT 1 0 0 92.75 128.25 1 1.00548 0.26016 0.00045 0.25945 0.00481 0.00401../../../exam24062008 hp222srv_ USD/JPY 105.404 107.9 0.01323 0.03663 0.16439 0.12583 0.10345 0.08893 0.08118 ‐0.0369 0.00393 0.10192 1 0.01039 0.11184 ‐0.40954 105.404 ‐2.48941 9.57065 0.3 1 DNT 1 0 0 102 114 1 1.00548 0.07616 0.00026 0.07573 0.00149 0.00165../../../exam10062008 hp222srv_ USD/MXN 10.9128 10.4283 0.08195 0.03653 0.07447 0.06973 0.07097 0.07837 0.0913 0.00864 0.00308 0.07479 1 0.00559 0.13663 0.50105 10.9128 ‐1.13826 21.90776 0.75 0 DNT 1 0 0 10.15 11.25 1 1.00548 0.18622 0.00037 0.18401 0.00003 0.00078../../../exam23052008 hp222srv_ USD/JPY 101.1335 103.0908 0.01259 0.03166 0.17276 0.13034 0.10626 0.09122 0.08376 ‐0.03913 0.00452 0.10474 1 0.01097 0.12398 ‐0.41744 101.1335 ‐2.32378 9.94407 0.3 1 DNT 1 0 0 98 118 1.00548 1.01096 0.3231 0.00047 0.32153 0.00193 0.00212../../../exam22052008 hp222srv_ EUR/USD 1.54265 1.57003 0.03272 0.05016 0.11655 0.10514 0.10009 0.10118 0.10929 ‐0.00396 0.00307 0.10362 1 0.01074 0.15703 ‐0.22134 1.54265 1.43918 8.44999 0.7 0 DNT 1 0 0 1.26 1.63 1.00822 1.0137 0.28649 0.00041 0.28811 0.00001 0.00007../../../exam28052008 hp222srv_ GBP/USD 1.91806 1.9801 0.0331 0.05864 0.1086 0.09338 0.08641 0.08519 0.09007 ‐0.00819 0.00288 0.08684 1 0.00754 0.09896 ‐0.30686 1.91806 0.72562 16.65889 0.45027 1 DNT 1 0 0 1.685 2.155 1.24658 1.25206 0.53438 0.00045 0.5328 0.00053 0.00087../../../exam28052008 hp222srv_ GBP/USD 1.91806 1.9801 0.0331 0.05864 0.1086 0.09338 0.08641 0.08519 0.09007 ‐0.00819 0.00288 0.08684 1 0.00754 0.09896 ‐0.30686 1.91806 0.72562 16.65889 0.45027 1 DNT 1 0 0 1.685 2.155 1.24658 1.25206 0.53438 0.00045 0.5328 0.00053 0.00087../../../exam13112008 hp222srv_ USD/JPY 94.33162 96.39 0.00546 0.02177 0.32334 0.22226 0.16493 0.12746 0.11311 ‐0.09479 0.00993 0.17246 1 0.02974 0.3962 ‐0.5705 94.33162 ‐1.071 0.31968 0.3 0 DNT 1 0 0 84 126 1.32329 1.33425 0.57123 0.00052 0.56861 0.00699 0.00607../../../exam28042008 hp222srv_ USD/JPY 101.357 104.2655 0.01373 0.03289 0.17111 0.12911 0.10527 0.08933 0.08085 ‐0.03977 0.00395 0.104 1 0.01082 0.12334 ‐0.43736 101.357 ‐1.87056 5.56022 0.3 1 DNT 1 0 0 95 125 1.47671 1.48219 0.48371 0.0005 0.47917 0.00286 0.00311../../../exam28052008 hp222srv_ EUR/USD 1.53088 1.56338 0.03349 0.04751 0.11604 0.10367 0.09816 0.09877 0.10687 ‐0.00489 0.00306 0.1026 1 0.01053 0.15915 ‐0.29513 1.53088 1.58642 6.5163 0.7 0 DNT 1 0 0 1.305 1.705 1.49863 1.50959 0.42538 0.00045 0.42758 0.00003 0.0003../../../exam28052008 hp222srv_ EUR/USD 1.53088 1.56338 0.03349 0.04751 0.11604 0.10367 0.09816 0.09877 0.10687 ‐0.00489 0.00306 0.1026 1 0.01053 0.15915 ‐0.29513 1.53088 1.58642 6.5163 0.7 0 DNT 1 0 0 1.305 1.705 1.49863 1.50959 0.42538 0.00045 0.42758 0.00003 0.0003../../../exam28042008 hp222srv EUR/USD 1.53433 1.5642 0.03285 0.04569 0.11666 0.10371 0.0976 0.09712 0.1034 ‐0.00659 0.00282 0.10129 1 0.01026 0.14156 ‐0.41177 1.53433 1.78293 8.21063 0.65 1 DNT 1 0 0 1.48 1.74 1.50137 1.50685 0.14067 0.00034 0.13857 0.00005 0.00075../../../exam28042008 hp222srv_ EUR/USD 1.53433 1.5642 0.03285 0.04569 0.11666 0.10371 0.0976 0.09712 0.1034 0.00659 0.00282 0.10129 1 0.01026 0.14156 0.41177 1.53433 1.78293 8.21063 0.65 1 DNT 1 0 0 1.48 1.74 1.50137 1.50685 0.14067 0.00034 0.13857 0.00005 0.00075../../../exam21052008 hp222srv_ USD/JPY 98.99303 103.0287 0.01168 0.03169 0.17014 0.12678 0.10115 0.08411 0.07508 ‐0.04268 0.00429 0.09841 1 0.00968 0.08653 ‐0.33914 98.99303 ‐2.36089 5.37989 0.2 1 DNT 1 0 0 78.25 108.25 1.99726 2.00822 0.20585 0.00037 0.2016 0.00358 0.00334../../../exam27052008 hp222srv_ EUR/PLN 3.49189 3.39962 0.05875 0.04536 0.06543 0.06015 0.06033 0.06701 0.07953 0.00686 0.00324 0.06391 1 0.00408 0.12342 0.3671 3.49189 ‐0.08901 13.18394 0.7 0 DNT 1 0 0 3.1 3.6 2 2.0137 0.28601 0.0004 0.28233 0.00016 0.0008../../../exam 6052008 hp222srv_ EUR/USD 1.51941 1.55311 0.03152 0.04248 0.11559 0.10265 0.09588 0.09596 0.10334 ‐0.00669 0.00342 0.1012 1 0.01024 0.15701 ‐0.43655 1.51941 1.93547 6.20451 0.7 0 DNT 1 0 0 1.45 1.75 2 2.01096 0.16202 0.00035 0.15802 0 0.0012../../../exam 3062008 hp222srv_ USD/JPY 100.7655 104.8685 0.01237 0.03232 0.16876 0.12654 0.10112 0.08401 0.07511 ‐0.04253 0.00415 0.09922 1 0.00984 0.10545 ‐0.42395 100.7655 ‐1.94455 4.17636 0.25 1 DNT 1 0 0 76 111.25 2 2.01096 0.30115 0.00042 0.29847 0.00297 0.00303../../../exam19052008 hp222srv_ USD/JPY 100.3289 104.36 0.01198 0.03167 0.16438 0.12313 0.09891 0.08303 0.07481 ‐0.04011 0.00417 0.09623 1 0.00926 0.08207 ‐0.29544 100.3289 ‐2.3872 6.13738 0.2 1 DNT 1 0 0 91 109 2 2.00548 0.06565 0.00023 0.06412 0.00328 0.00307../../../exam20052008 hp222srv_ USD/JPY 99.73927 103.6591 0.01156 0.03083 0.16533 0.12393 0.09911 0.08237 0.07331 ‐0.04156 0.00404 0.09648 1 0.00931 0.08164 ‐0.34262 99.73927 ‐2.43759 5.28063 0.2 1 DNT 1 0 0 101 131 2 2.01096 0.14011 0.00035 0.14113 0.00311 0.00295../../../exam15052008 hp222srv_ AUD/USD 0.86509 0.94122 0.03231 0.07436 0.14551 0.12683 0.11618 0.11426 0.1219 ‐0.01257 0.00437 0.12587 1 0.01584 0.21799 ‐0.53169 0.86509 1.90115 4.51991 0.75 0 DNT 1 0 0 0.71 0.99 2.00548 2.01096 0.23309 0.00037 0.23898 0.0001 0.00361../../../exam23052008 hp222srv_ AUD/USD 0.8785 0.96174 0.03223 0.07737 0.14316 0.12448 0.11383 0.11191 0.11955 ‐0.01257 0.00437 0.1235 1 0.01525 0.21679 ‐0.53213 0.8785 1.92518 4.73391 0.75 0 DNT 1 0 0 0.89 1.19 2.00548 2.01096 0.17887 0.00037 0.17369 0.0001 0.00356../../../exam13112008 hp222srv_ GBP/USD 1.45814 1.46773 0.0232 0.02647 0.24631 0.20867 0.18535 0.17867 0.18918 ‐0.03 0.00832 0.2083 1 0.04339 0.41638 ‐0.50756 1.45814 1.26036 1.37738 0.75 0 DNT 1 0 0 1.18 1.85 2.00548 2.01096 0.33289 0.00046 0.33303 0.00154 0.00989../../../exam 9052008 hp222srv_ USD/JPY 99.11916 102.9808 0.01099 0.03004 0.17632 0.13118 0.10454 0.08673 0.07734 ‐0.04445 0.00441 0.10167 1 0.01034 0.09088 ‐0.34597 99.11916 ‐2.25112 4.94482 0.2 1 DNT 1 0 0 93 117 2.00548 2.01096 0.31304 0.00046 0.31187 0.00391 0.00365../../../exam15052008 hp222srv_ USD/JPY 100.5546 104.7517 0.01192 0.03231 0.16309 0.12204 0.098 0.08234 0.07428 ‐0.0397 0.00419 0.09536 1 0.00909 0.08176 ‐0.2903 100.5546 ‐2.41378 6.34701 0.2 1 DNT 1 0 0 92 115 2.00548 2.01096 0.28788 0.00043 0.28531 0.0033 0.00308../../../exam 7112008 hp222srv_ USD/JPY 94.67948 98.06407 0.00661 0.0241 0.29257 0.20137 0.14725 0.11104 0.09646 ‐0.09033 0.00896 0.15314 1 0.02345 0.33973 ‐0.55741 94.67948 ‐1.16947 ‐0.47872 0.25 0 DNT 1 0 0 92.5 117.5 2.00822 2.0137 0.22255 0.00042 0.22328 0.00807 0.0086../../../exam29042008 hp222srv_ USD/JPY 99.81544 103.9788 0.01241 0.03207 0.16978 0.12731 0.10232 0.08569 0.07714 ‐0.04161 0.00418 0.09948 1 0.0099 0.08569 ‐0.29844 99.81544 ‐2.21868 5.35251 0.2 1 DNT 1 0 0 94.5 114.5 2.07945 2.09041 0.17044 0.00036 0.16972 0.00354 0.00331../../../exam14052008 hp222srv_ USD/JPY 99.44543 105.3068 0.01224 0.03516 0.1645 0.12286 0.09752 0.07979 0.07002 ‐0.04307 0.0038 0.09498 1 0.00902 0.08663 ‐0.39928 99.44543 ‐2.18294 3.43594 0.2 1 DNT 1 0 0 87.75 110.75 2.49863 2.50959 0.12189 0.00031 0.11714 0.00368 0.00348../../../exam29042008 hp222srv_ USD/JPY 98.8109 103.9784 0.0128 0.0332 0.17033 0.12722 0.10186 0.08466 0.07583 ‐0.04256 0.00408 0.099 1 0.0098 0.09173 ‐0.32896 98.8109 ‐1.95011 4.14977 0.2 1 DNT 1 0 0 93 118 2.49863 2.50959 0.27488 0.00043 0.27308 0.00438 0.00405../../../exam25042008 hp222srv_ USD/JPY 98.91557 104.5189 0.0122 0.03418 0.17325 0.12955 0.1036 0.08571 0.07638 ‐0.04384 0.00402 0.10071 1 0.01014 0.09301 ‐0.3471 98.91557 ‐1.9166 3.70065 0.2 1 DNT 1 0 0 89.75 110.25 2.50685 2.51233 0.07423 0.00025 0.07256 0.00433 0.00405
Figure 3: Screenshot of Detailed Test Results.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 17 of 41
Go Back
Full Screen
Close
Quit
100%Price Comparison of all FOR paying DNT
Murex (PDE)
80%
90%Murex (PDE)
60%
70%
50%
60%
30%
40%
10%
20%
0%
10%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
MathFinance (Monte Carlo)
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Figure 4: Prices of all FOR-paying DNT compared in aQ-Q-Plot.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 18 of 41
Go Back
Full Screen
Close
Quit
4%
Normalized Differences for all FOR paying DNT
3%
1%
2%
0%
.02
.08
.08
.12
.17
.17
.25
.25
.26
.33
.42
.50
.50
.51
.75
.76
.84
.00
.00
.00
.00
.00
.00
.25
.50
.00
.01
.01
.50
.53
.00
.00
.18
.00
.51
‐2%
‐1%
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 1. 1. 1. 1. 1. 1. 1. 2. 2. 2. 2. 2. 3. 3. 3. 4. 4.
‐3%
2%
‐5%
‐4%
5%
Figure 5: Difference of MathFinance and Murex Pricenormalized by
√T as a function of maturity T .
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 19 of 41
Go Back
Full Screen
Close
Quit
4. Greeks
• Finite quotients with a default of forward differences
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 19 of 41
Go Back
Full Screen
Close
Quit
4. Greeks
• Finite quotients with a default of forward differences
• Symmetric differences for gamma and volga
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 19 of 41
Go Back
Full Screen
Close
Quit
4. Greeks
• Finite quotients with a default of forward differences
• Symmetric differences for gamma and volga
• hybrid Greeks: only the spot changes; all other pa-rameters are fixed. Only available for spot sensitivi-ties.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 19 of 41
Go Back
Full Screen
Close
Quit
4. Greeks
• Finite quotients with a default of forward differences
• Symmetric differences for gamma and volga
• hybrid Greeks: only the spot changes; all other pa-rameters are fixed. Only available for spot sensitivi-ties.
• sticky delta Greeks: all the parameters of the modelare recalibrated.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 20 of 41
Go Back
Full Screen
Close
Quit
4.1. Test Results for Greeks
value spot+ spot- vol+ vol- rd+ rf+ s+v+ s+v- s-v+ s-v-
mc 2.31554 2.30138 2.32966 2.62364 2.01903 2.33777 2.29133 2.60775 2.00662 2.63964 2.03142
mx 2.32886 2.31491 2.34290 2.63886 2.03159 2.35127 2.30464 2.62223 2.01957 2.65373 2.04385
diff value delta delta2 gamma vega vega2 volga vanna rhod rhof
mc 2.31554 -0.13456 -0.13433 -0.00458 0.30809 0.30230 0.01158 -0.01681 0.22224 -0.24210
mx 2.32886 -0.13257 -0.13292 0.00712 0.31000 0.30363 0.01273 -0.01715 0.22405 -0.24222
cash value delta delta2 delta PI delta2 PI gamma vega vega2 volga vanna rhod rhof
mc -21994 134560 134330 156553 156324 4581 -2926 -2871 -110 16813 -2111 2300
mx -22120 132567 132923 154687 155044 -7124 -2944 -2884 -121 17146 -2128 2301
shifts spot vol rates
0.105281 0.01 0.001
Table 5: Sticky Delta Greeks for the USD-JPY Down-and-in Call: example data irsm201008/02052008/irsmform_hp222srv_28945_6684515_premium_133.xml
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 21 of 41
Go Back
Full Screen
Close
Quit
4.2. Stability of Greeks
Spot rd rf -10 -25 50 25 10 25 RR 25 BF
1.5416 0.02962 0.04772 0.11915 0.10661 0.10032 0.10066 0.10833 -0.00594 0.00331
v0 κ θ ξ ρ c′ b′ a′
0.10356 1 0.01073 0.19489 -0.28087 1.52995 1.83734 22.13073
down up maturity delivery price MF std dev price MX
1.35 1.605 0.41918 0.43014 0.44637 0.00047 0.44282
Table 6: Greeks Stability: Example EUR-USD DNT trade6684518 paying EUR
Generally watch out: Greeks depend on currency, nu-merous conventions, shift size, symmetric vs. forwarddifferences, absolute vs. relative
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 22 of 41
Go Back
Full Screen
Close
Quit
400500600700800900
1000
Thou
sand
P&L
0100200300400500600700800900
1000
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sand
Spot
P&L
Figure 6: Spot Ladder: Value.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 23 of 41
Go Back
Full Screen
Close
Quit
0
5000
10000
15000
Thou
sand
Delta
‐15000
‐10000
‐5000
0
5000
10000
15000
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sand
Spot
Delta
Figure 7: Spot Ladder: Delta.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 24 of 41
Go Back
Full Screen
Close
Quit
‐1000
‐500
0
500
1000
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sand
Gamma
‐2500
‐2000
‐1500
‐1000
‐500
0
500
1000
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sand
Spot
Gamma
Figure 8: Spot Ladder: Gamma.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 25 of 41
Go Back
Full Screen
Close
Quit
‐60
‐40
‐20
0
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sand
Vega
‐120
‐100
‐80
‐60
‐40
‐20
0
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sand
Spot
Vega
Figure 9: Spot Ladder: Vega.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 26 of 41
Go Back
Full Screen
Close
Quit
2468
101214
Thou
sand
Volga
‐6‐4‐202468
101214
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sand
Spot
Volga
Figure 10: Spot Ladder: Volga.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 27 of 41
Go Back
Full Screen
Close
Quit
500
1000
1500
2000
2500
Thou
sand
Vanna
‐1500
‐1000
‐500
0
500
1000
1500
2000
2500
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sand
Spot
Vanna
Figure 11: Spot Ladder: Vanna.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 28 of 41
Go Back
Full Screen
Close
Quit
10‐505
101520
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sandRho USD
‐30‐25‐20‐15‐10‐505
101520
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sand
Spot
Rho USD
Figure 12: Spot Ladder: Rhod.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 29 of 41
Go Back
Full Screen
Close
Quit
50510152025
Thou
sandRhoF EUR
‐25‐20‐15‐10‐50510152025
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
Thou
sand
Spot
RhoF EUR
Figure 13: Spot Ladder: Rhof.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 30 of 41
Go Back
Full Screen
Close
Quit
600.00
800.00
1,000.00
1,200.00
1,400.00
1,600.00
Thou
sand
P&L
0.00
200.00
400.00
600.00
800.00
1,000.00
1,200.00
1,400.00
1,600.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Thou
sand
Vol
P&L
Figure 14: Vol Ladder: Value.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 31 of 41
Go Back
Full Screen
Close
Quit
8 000 00
‐6,000.00
‐4,000.00
‐2,000.00
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Delta
‐12,000.00
‐10,000.00
‐8,000.00
‐6,000.00
‐4,000.00
‐2,000.00
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Vol
Delta
Figure 15: Vol Ladder: Delta.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 32 of 41
Go Back
Full Screen
Close
Quit
‐3,000.00‐2,500.00‐2,000.00‐1,500.00‐1,000.00‐500.00
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Gamma
‐5,000.00‐4,500.00‐4,000.00‐3,500.00‐3,000.00‐2,500.00‐2,000.00‐1,500.00‐1,000.00‐500.00
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Vol
Gamma
Figure 16: Vol Ladder: Gamma.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 33 of 41
Go Back
Full Screen
Close
Quit
‐80.00
‐60.00
‐40.00
‐20.00
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Vega
‐140.00
‐120.00
‐100.00
‐80.00
‐60.00
‐40.00
‐20.00
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Vol
Vega
Figure 17: Vol Ladder: Vega.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 34 of 41
Go Back
Full Screen
Close
Quit
15 00‐10.00
‐5.000.00
5.00
10.0015.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Volga
‐30.00
‐25.00‐20.00
‐15.00‐10.00
‐5.000.00
5.00
10.0015.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Vol
Volga
Figure 18: Vol Ladder: Volga.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 35 of 41
Go Back
Full Screen
Close
Quit
‐1,000.00
‐500.00
0.00
500.00
1,000.00
1,500.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Vanna
‐2,500.00
‐2,000.00
‐1,500.00
‐1,000.00
‐500.00
0.00
500.00
1,000.00
1,500.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Vol
Vanna
Figure 19: Vol Ladder: Vanna.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 36 of 41
Go Back
Full Screen
Close
Quit
‐10.00
0.00
10.00
20.00
30.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Rho
Rh USD
‐40.00
‐30.00
‐20.00
‐10.00
0.00
10.00
20.00
30.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Vol
Rho
Rho USD
Rhof EUR
Figure 20: Vol Ladder: Rhos.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 37 of 41
Go Back
Full Screen
Close
Quit
5. Summary
• Vanilla pricing consistent
• First Generation Exotics pricing within typical MonteCarlo Simulation based ranges
• Stable Greeks
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 38 of 41
Go Back
Full Screen
Close
Quit
6. Contact Information
Uwe WystupMathFinance AGMainluststraße 460329 Frankfurt am MainGermany+49-700-MATHFINANCE
http://www.mathfinance.com
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 39 of 41
Go Back
Full Screen
Close
Quit
References
[1] Andersen, L. (2007). Efficient Simulation of the Heston StochasticVolatility Model. Bank of America Working paper, available at http:
//ssrn.com/abstract=946405.
[2] Black, F. and Scholes, M. (1973). The pricing of Options and Cor-porate Liabilities, Journal of Political Economy 81 (3): 637-654.
[3] Broadie, M and Kaya, O. (2006). Exact Simulation of StochasticVolatility and other Affine Jump Diffusion Processes. Operations Re-search, Vol.54, No.2, 217–231.
[4] Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering.Springer, Berlin.
[5] Jeanblanc, M., Yor, M. and Chesney, M. (2009). MathematicalMethods for Financial Markets, Springer.
[6] Matsumoto, M. and Nishimura, T. (1998). Mersenne Twister. A 623-dimensionally Equidistributed Uniform Pseudorandom Number Generator.ACM Transactions on Modeling and Computer Simulation, ISSN 1049-3301, 8, 3-30.
[7] Murex. (2008). Tremor Model Flyer.
[8] Murex. (2008). Tremor Model and Smile Dynamics.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 40 of 41
Go Back
Full Screen
Close
Quit
[9] Murex. (2010). Data Exchange Format - Tremor Validation Project, V0.2, March 2010.
[10] Murex. (2010). Greeks Validation - Tremor Validation Project, V 1.0,November 2010.
[11] Reiswich, D. and Wystup, U. (2010). The FX Options Quant Book.MathFinance, Waldems.
[12] Rogers, L.C.G. and Williams, D. (2000). Diffusions, Markov Pro-cesses, and Martingales – Volume Two: Ito Calculus, McGraw-Hill.
The Tremor Model
Implementation of . . .
Validation Experiment
Greeks
Summary
Contact Information
mathfinance.com
Title Page
JJ II
J I
Page 41 of 41
Go Back
Full Screen
Close
Quit
Index
continuity correction, 7
Feller condition, 8
hybrid Greeks, 19
local volatility function, 4
quadratic-exponential scheme, 6
sticky delta Greeks, 19