Pricing Foreign Exchange Risk - by Glen Dixon, Associate Lecturer, Griffith University

1. Currency Hedging in Turbulent times,Currency Hedging in Turbulent times,
Executive Briefing Seminar, Grace Hotel, SydneyExecutive Briefing Seminar, Grace Hotel, Sydney
1010thth
November 2003November 2003
““Pricing Foreign Exchange Risk”Pricing Foreign Exchange Risk”
By Glen DixonBy Glen Dixon
Acting LecturerActing Lecturer
School of Accounting,School of Accounting,
Banking & Finance,Banking & Finance,
Faculty of Commerce andFaculty of Commerce and
ManagementManagement

2. OVERVIEW :OVERVIEW :
An introduction to common derivative productsAn introduction to common derivative products
Understanding the key components of the BlackUnderstanding the key components of the Black
Scholes pricing methodologyScholes pricing methodology
Constructing and using a forward price curveConstructing and using a forward price curve

3. An introduction to common derivative productsAn introduction to common derivative products
Understanding the key components of the Black Scholes pricing methodologyUnderstanding the key components of the Black Scholes pricing methodology
Constructing and using a forward price curveConstructing and using a forward price curve

4. ““Introduction plus History of FX”Introduction plus History of FX”

5. Introduction
(Overview of FX)
People have been borrowing, lending and exchanging money for centuries.
The foreign exchange market exists to facilitate this conversion
of one currency into another.
As a result, the foreign exchange market today is the largest
and most truly global financial market in the world.

6. Volatility in FX Markets
Interest Rates
Foreign Exchange
Commodities
Electricity

7. Shape of FX Forward Price CurveShape of FX Forward Price Curve
Time (Minutes, Days, Weeks & Months)
Price
A$/US$
Time (Weeks)
Time (Minutes)
Time (Months)
Time (Days)

8. • 1983 March1983 March - (5th
, 8th
), OctoberOctober - (28th
), DecemberDecember - (8th
, 9th
,12th
and 13th
)
• 1984 March1984 March - (5th
) February till early 1985February till early 1985
• 1985 February1985 February - (6th
till 8th
)
• 1986 May1986 May - (13th
, 14th
), JulyJuly - (2nd
, 4th,
25th
and 28th
), AugustAugust - (19th
)
• 1987 October1987 October - (20th
), November till DecemberNovember till December
• 1988 April1988 April - till December 1989till December 1989
• 1989 February1989 February - Late February till AprilLate February till April - MayMay - AugustAugust
• 1990 January1990 January – (23rd
), AugustAugust
• 1991 June1991 June - (3rd
) – December (December (1919thth
))
• 1992 January1992 January – February (26February (26thth
) – February till March, June, October,) – February till March, June, October,
DecemberDecember
History of FX in Australia
Source: Securities Institute Education

9. • 1993 January1993 January –– April till June, August (17– April till June, August (17thth
), October, Late 1993 till), October, Late 1993 till
early 1994early 1994
• 1994 February, April1994 February, April- (7th
- 8th
), Early Mary, Late June, July tillEarly Mary, Late June, July till
OctoberOctober
• 1995 June till December1995 June till December
• 1996 January till February1996 January till February, March, May, November till DecemberMarch, May, November till December
• 1997 February1997 February
• 2003 October2003 October
History of FX in Australia (Cont.)
Source: Securities Institute Education

10. ““Currency Futures and Options Market“Currency Futures and Options Market“

11. The Currency Futures and Options Markets
• Foreign Currency Options
– History and Size of Market
– Options - General
– Currency Options
– Quotations
• Foreign Currency Speculations

12. The Currency Futures and Options Markets (2)
Foreign Exchange Contracts
FX Portfolio
FX Contracts
AUS/US
AUS/DEM
AUS/SF
FX Profiles

13. Foreign Currency Options
History and Size of Market
Attention will be focused on plain-vanilla European puts and calls on
foreign exchange as well as on some of the more popular exotic
varieties of currency options.
The currency option market can rightfully claim to be the world’s
only truly global, 24-hour option market.
The underlying asset for currency options is foreign exchange.

14. Foreign Currency Options (2)
Option: A contract that gives the option buyer (holder) the
right (not obligation) to buy or sell a given amount of the
underlying asset at a fixed price (exercise price) over a
specified period of time (or at a specified date).
• Underlying asset: e.g stock, commodities, stock indices,
foreign currency etc.
• Rule for exercise:
– American - exercisable anytime until expiration
– European - exercisable only at expiration
• Types of option:
– Call option: option to buy the underlying asset (e.g. foreign
currency)
– Put option: option to sell the underlying asset

15. Foreign Currency Options (3)
Consider the following option on
dollar/yen:
USD call/JPY put
Face amount in dollars
$10,000,000
Option put/call Yen put
Option expiry 90 days
Strike 120.00
Exercise European

16. Foreign Currency Options (4a)
An exotic currency option is an option that has some nonstandard feature
that sets it apart from ordinary vanilla currency options.
The most popular exotic currency options are the:
1) Barrier Option
2) Binary Option
3) Basket Option
4) Asian Option

17. Foreign Currency Options (4b)
A Stock Simulation for the Barrier Option
Source: Griffith University & Kerr 2000

18. Foreign Currency Options (5)
Example: a $60 call (expiration in 3 months) on an
ABC stock; option premium $1
Holder exercises if the spot price > $60
Payoff Profile
S X=60 Premium Payoff
$50 (60) (1) -1 out of the money
55 (60) (1) -1
60 (60) (1) -1
61 (60) (1) 0 at the money
62 (60) (1) 1
67 (60) (1) 6 in the money
Payoff
-1
Payoff
-1
X=60
61 S

19. Foreign Currency Options (6)
• Payoff Profile - Call option on DM
– 1 option is for purchase of DM62,500
– exercise price $0.5850/DM
– Option Premium $0.0050/DM or $312.50
• option in the money for spot > 0.5850
• option at the money for spot = 0.5850
• out of the money for spot < 0.5850
• Breakeven price = $0.5900/DM
• Payoff Profile - Put Option on DM
– exercise price $0.5850/DM
– option premium $0.0050/DM
• option in the money for spot < 0.5850
• at the money for spot = 0.5850
• out of the money for spot > 0.5850

20. Foreign Currency Options (7)

21. Foreign Currency Options (8)

22. Foreign Currency Options (9)
An option hedge
• A currency option is like one-half of a
forward contract
• An option to buy pound sterling at the
current exchange rate
– the option holder gains if pound sterling rises
– the option holder does not lose if pound sterling
falls

23. Foreign Currency Options (10)
Currency option quotations
British pound (CME)
£62,500; cents per pound
Strike Calls-Settle Puts-Settle
Price Oct Nov Dec Oct Nov Dec
1430 2.38 . . . . 2.78 0.39 0.61 0.80
1440 1.68 1.94 2.15 0.68 0.94 1.16
1450 1.12 1.39 1.61 1.12 1.39 1.61
1460 0.69 0.95 1.17 1.69 1.94 2.16
1470 0.40 0.62 0.82 2.39 . . . . 2.80

24. Foreign Currency Options (11)
• The time value of an option is the difference between
the option’s market value and its intrinsic value if
exercised immediately.
• The time value of a currency option is a function of the
following six determinants:
– Underlying exchange rate
– Exercise price
– Riskless rate of interest in currency d
– Riskless rate of interest in currency f
– Time to expiration
– Volatility in the underlying exchange rate

25. Foreign Currency Options (12)
• Foreign Currency Speculation - Trading on the basis of
expectations about future prices
• Speculation in Spot Markets
• Speculation in Forward Markets
– occurs if one believes that the forward rate differs from the future spot rate
– if expect Forward < future spot, buy currency forward
– if expect Forward > future spot, sell currency forward
• Speculation using options
– call options
– put options
• Speculation via Borrowing and Lending: Swaps
• Speculation via Not Hedging Trade
• Speculation on Exchange-Rate Volatility

26. An introduction to common derivative productsAn introduction to common derivative products
Understanding the key components of the BlackUnderstanding the key components of the Black
Scholes pricing methodologyScholes pricing methodology
Constructing and using a forward price curveConstructing and using a forward price curve

27. ““Overview of Black Scholes (1973) , MertonOverview of Black Scholes (1973) , Merton
((1973) and Garman Kohlhagen (1983)”((1973) and Garman Kohlhagen (1983)”

28. Black, Fischer and Myron S. Scholes (1973).Black, Fischer and Myron S. Scholes (1973).
The pricing of options and corporate liabilities,The pricing of options and corporate liabilities,
Journal of Political EconomyJournal of Political Economy, 81, 637-654., 81, 637-654.
Good Journals

29. Black Scholes (1973) Options Pricing Formula
Values for a call price c or put price p are:
where:

30. The Five Greeks
• DELTA measures first order (linear) sensitivity to an underlier;
• GAMMA measures second order (quadratic) sensitivity to an underlier;
• VEGA measures first order (linear) sensitivity to the implied
Volatility of an underlier;
• THETA measures first order (linear) sensitivity to the passage of time;
RHO measures first order (linear) sensitivity to an applicable interest rate.

31. The Five Greeks for Black Scholes (1973)
Options Pricing Formula for a Call
The Greeks—delta, gamma, vega, theta and rho—for a call are:
delta = Φ(d1
)
gamma =
vega =
theta =

32. The Five Greeks for Black Scholes (1973)
Options Pricing Formula for a Put
where
denotes the standard normal probability density function. For a put, the Greeks are:
delta = Φ(d1
) – 1
gamma =
vega =
theta =

33. Good Journals
Merton, Robert C. (1973).Merton, Robert C. (1973).
Theory of rational option pricing,Theory of rational option pricing,
Bell Journal of Economics and Management ScienceBell Journal of Economics and Management Science, 4 (1), 141-183., 4 (1), 141-183.

34. Merton (1973) Options Pricing Formula
Values for a call price c or put price p are:
where:

35. The Five Greeks for Merton (1973)
Options Pricing Formula for a Call
The Greeks—delta, gamma, vega, theta and rho—for a call are:

36. The Five Greeks for Merton (1973)
Options Pricing Formula for a Put
where denotes the standard normal probability density function. For a put, the Greeks are:

37. Good Journals
Garman, Mark B. and Steven W. Kohlhagen (1983).Garman, Mark B. and Steven W. Kohlhagen (1983).
Foreign currency option values,Foreign currency option values,
Journal of International Money and FinanceJournal of International Money and Finance, 2, 231-237., 2, 231-237.

38. Garman and Kohlhagen (1983) FX Options Pricing Formula
Values for a call price c or put price p are:
where:

39. The Five Greeks for Garman and Kohlhagen (1983) FX Options
Pricing Formula for a Call
The Greeks—delta, gamma, vega, theta and rho—for a call are:

40. The Five Greeks for Garman and Kohlhagen (1983) FX Options
Pricing Formula for a Put
where denotes the standard normal probability density function. For a put, the Greeks are:

41. An introduction to common derivative productsAn introduction to common derivative products
Understanding the key components of the Black Scholes pricing methodologyUnderstanding the key components of the Black Scholes pricing methodology
Constructing and using a forward price curveConstructing and using a forward price curve

42. ““Overview of Interest Rate Markets: Bachelior (1901),Overview of Interest Rate Markets: Bachelior (1901),
including Single Factor Models like Vasicek (1977) ,including Single Factor Models like Vasicek (1977) ,
Cox Ingersoll and Ross (1985)”Cox Ingersoll and Ross (1985)”

43. Stochastic Differential Equation
(Wiener processes or Random Walk)
Geometric Brownian Motion (Stock Markets) –Geometric Brownian Motion (Stock Markets) –
BacheliorBachelior
ddistributenormally,incrementstindependen
Noise),(WhitemotionBrownian:)(
yvolatilitthe:
ratelincrementathe:
where
)(
)(
)(
followsSDEgeometricThe
price.assetunderlyingtheas)(Define
tW
tdWdt
tS
tdS
tS
σ
µ
σµ +=

44. Models: (Foreign Exchange, Interest Rate and Energy Markets)
• Single factor models Vasicek (1977), Cox Ingersoll and
Ross (1985), Clewlow and Strickland (2000)
• Two factor models Brennan and Schwartz (1982),
Kennedy (1997), Pilipovic (1997) and
Kerr and Dixon (2002)
Price Spikes
Long Term Mean
Mean Reversion

45. Interest Rate Markets

46. Interest Rate Markets (1)

47. Interest Rate Markets (2)

48. Interest Rate Markets (3)

49. Stochastic Differential Equation
Single Factor Models (Interest Rate Markets)
The Vasicek (1977) ModelThe Vasicek (1977) Model
motionBrownian:
yvolatilitthe:σ
alueinterest vmeanthe:μ
ratereverting-meanthe:κ
where
followsSDEVasicekThe
rate.shorttheasDefine
W(t)
σdW(t)r(t))dtκ(μdr(t)
r(t)
+−=

50. Stochastic Differential Equation
Single Factor Models (Interest Rate Markets)
The Cox, Ingersoll and RossThe Cox, Ingersoll and Ross
(1985) Model (CIR)(1985) Model (CIR)
motionBrownian:)(
yvolatilitthe:
alueinterest vmeanthe:
ratereverting-meanthe:
where
)()())(()(
followsSDECIRThe
rate.shorttheas)(Define
tW
tdWtrdttrtdr
tr
σ
µ
κ
σµκ +−=

51. ““Comparison of Currencies: Australian, US,Comparison of Currencies: Australian, US,
Asian , Latin American”Asian , Latin American”

52. Comparisons of Currencies: Australian Dollar (Daily)

53. Comparisons of Currencies: Australian Dollar (Monthly)

54. Comparisons of Currencies: US Dollar (Daily)

55. Comparisons of Currencies: US Dollar (Monthly)

56. Comparisons of Asian Currencies (Daily)

57. Comparisons of Asian Currencies (Monthly)

58. Comparisons of Latin American Currencies (Daily)

59. Comparisons of Latin American Currencies (Monthly)

60. ““Overview of Monte Carlo, ScenarioOverview of Monte Carlo, Scenario
Development and Stress Testing”Development and Stress Testing”

61. Iterative Procedure
(Euler Method)
(CIR).1/2or(Vasicek)0where
0atrateinterestinitialwith the
)()())(()()(
0
=
=
∆+∆−+=∆+
τ
σµκ τ
tr
tWtrttrtrttr
1.0.ofdeviationstandard
andzeroofmeanon withdistributinormala
fromsamplerandomaiswhere)( εε ttW ∆=∆

62. Monte Carlo Simulation
(Generate Random Numbers)
Box Muller:
Marsaglia:
Note: Box Muller and Marsaglia will generateNote: Box Muller and Marsaglia will generate
standard Gaussian random variables basedstandard Gaussian random variables based
on two independent uniformly distributedon two independent uniformly distributed
random variables from [0, 1].random variables from [0, 1].




=
=
)2sin()log(2
)2cos()log(2
212
211
XXY
XXY
π
π




=<+=
−=−=
−
V
V
UXXV
UXYXY
)log(22
2
2
1
2211
;1
;*)12(U;*)12(

63. Monte Carlo Simulation
(Generate Random Numbers from [0, 1])
Pseudo-Random use seed,
convergence rate
(M is the number of iterations).
E.g. Pseudo-Random (400)
M
1
Quasi-Random (low discrepancy):
use a uniformed sequence,
e.g., Van der Corput sequence at
every points (k=1,2,…).
E.g. Quasi-Random (400)
k
2

64. Using Monte Carlo for FX Market
GENERATE 1 RANDOM
SAMPLE for FX
GENERATE 1 RANDOM
SAMPLE for FX
FX(5): $A/$US
4:30 pm
FX(4): $A/$US
12:30 pm
FX(3): $A/$US
8:30 am
FX(2): $A/$US
4:30 am
FX(1): $A/$US
0:30 am Time
$A/
$US

65. Monte Carlo for FX Market (cont.)
GENERATE MULTIPLE
RANDOM SAMPLES for
FX
GENERATE MULTIPLE
RANDOM SAMPLES for
FX
Time
$A/
$US

66. Number of Samples0
$0
1 A$/$US
1A/$ 0.5US
STABILISE?
-USE STOPPING
RULES, I.E.
Tolerance-
STABILISE?
-USE STOPPING
RULES, I.E.
Tolerance-
when the change between two consecutive
average monthly fx prices becomes insignificant
then the process is said to have stabilised.
Estimated Average
Monthly Prices
In the FX Market
The Accuracy of Estimates is related
to the number of Simulations

67. Using Monte Carlo for Sensitivity Analysis on the
FX Forward Curve
• Construct scenarios
– High, medium and low, forecast
FX levels
• Perform Monte Carlo
Simulation
– generate fx price paths for each
scenario using different sets of
sensitivity analysis

68. Sensitivity Analysis for FX

69. Scenario Development for FX
• Scenario analysis
– Is a strategic technique which enables a firm to
evaluate the potential impact on its earnings stream
of various different eventualities.
– It uses multidimensional projections, and helps the
firm to assess its longer term strategic
vulnerabilities.

70. Scenario Development for FX (2)
• Scenario analysis
– Distinguish between scenario analysis and stress testing.
– Both are forward looking techniques which seek to
quantify the potential loss which might arise as a
consequence of unlikely events.
– Stress testing is designed to evaluate the short-term impact
on a given portfolio of a series of predefined moves, in
particular market variables.
– Scenario analysis on the other hand seeks to assess the
broader impact on the firm of more complex and inter-
related developments. Huge losses often occur due to a
sequence of several adverse events. Scenario analysis can
help to identify such potential problems in advance.

71. Scenario Development for FX (3)
• Scenario analysis
– The purpose of scenario analysis is to help the firm’s
decision makers think about and understand the impact
of unlikely, but catastrophic, events before they happen.
A management team that learns its lessons from previous
catastrophic situations is more likely to avoid losses in the
future. Scenario analysis is an effective tool to assist
management in that process.

72. Scenario Development for FX (4)
Risk
Political Risk
Operational Risk
Legal Risk
Credit Risk
Reputational Risk

73. Scenario Development for FX (5)
• The Scenario analysis process:
Step 1: Scenario definition
 Description of the starting scenario
 Basic assumptions
 Definition of the time horizon

74. Scenario Development for FX (6)
Step 2: Scenario-field analysis
 Identification of the scenario fields, the risk
dimensions and risk factors which are affected
and relevant for this scenario analysis

75. Scenario Development for FX (7)
Step 3: Scenario projections
 Estimate the likely movements of the identified scenario factors and
determine the potential loss in that case

76. Scenario Development for FX (8)
Step 4: Scenario consolidation
Consolidate the results
Check for consistency errors, doubling
counting
Independent validation checks

77. Scenario Development for FX (9)
Step 5: Scenario presentation and follow-up
 Summarise results
 Analyse and evaluate
 next steps: eg, put on a hedge

78. Stress Testing for FX
 In financial markets where 4-standard-deviation events happen
approximately once per year, the October 1987 crash was a 25-standard
deviation event.
 Stress testing deals with these “outlier” events. It addresses the large
moves in key market variables that lie beyond day-to-day risk monitoring
but that could potentially occur.

79. Stress Testing for FX (2)
 Low probability extreme market events;
 Hidden assumption in models;
 Structural breakdowns in the market environment;
 Robustness of risk management systems.
Stress Testing is another form of
risk management which tests exposure to:

80. Stress Testing for FX (3)
Steps in Stress Testing
• Step 1: Picking what to stress
 Choice of market variables
 Range of stress
 Usefulness of stress information vs data overload

81. Stress Testing for FX (4)
Step 2: Identifying assumptions
 Will correlations hold or break?
 For correlations that break, what are the new
assumptions?
 Does the underlying financial model still hold?

82. Stress Testing for FX (5)
Step 3: Revaluing the portfolio
 Back of the envelope vs
 sophisticated modeling
 Adjusting for market liquidity
Trading
Settlements
Portfolio
Management
Contract
Management

83. Stress Testing for FX (6)
Step 4: Deciding on action steps
 Reporting
 Cross-checks on model and pricing validity
 Action plan for dealing with actual catastrophe
situation

84. ““Overview of Interest Rate Markets including TwoOverview of Interest Rate Markets including Two
Factor Models like Brennan and Schwartz (1982), KerrFactor Models like Brennan and Schwartz (1982), Kerr
and Dixon (2003)”and Dixon (2003)”

85. Stochastic Differential Equations
Two Factor Models (Interest Rate Markets) + Monte Carlo Simulation
The Brennan and Schwartz (1982) Stochastic VolatilityThe Brennan and Schwartz (1982) Stochastic Volatility
ModelModel
( )
( )
1/2)or0(motions.Brownian
tindependenare)(and)(variance,itsand
rateshortebetween thncorrelatiotheiswhere
))(1)((
)()()(
)()()()()(
variancetheas)(andrateshorttheas)(Define
21
2
2
1
1
=






−+
+−=
+−=
τ
ρ
ρρ
βνναν
νµκ
ν
τ
tWtW
tdWtdW
tdttmtd
tdWtrtdttrtdr
ttr
Iterative ProcedureIterative Procedure
(Euler Method) with(Euler Method) with
1/2)( =τ
( )
( )
variance.initialtheandrateinitialwith the
))(1)((
)()()()(
)()()()()()(
2
2
1
1






∆−+∆
+∆−+=∆+
∆+∆−+=∆+
tWtW
tttmttt
tWtrtttrtrttr
ρρ
βννανν
νµκ τ

86. Stochastic Differential Equations
Monte Carlo Simulation (Two Factor Models-CIR)
The Brennan and Schwartz (1982) Stochastic VolatilityThe Brennan and Schwartz (1982) Stochastic Volatility
ModelModel
with Iterative Procedure (Euler Methowith Iterative Procedure (Euler Metho
withwith 1/2)( =τ

87. Foreign Currency
 Stochastic Modeling
rateinterestforeignthe:
rateinterestdomesticthe:where
)()())(()(
asmodelmotionBrownian
geometricafollowsrateexchangespotthat the
assumeWerate.exchangespottheas)(Define
f
f
r
r
tdWtSdtrrtStdS
tS
σ+−=
Stochastic Differential Equations

88.  Stochastic Interest Rates
( )
rateinterestforeigntheandrate
interestdomesticebetween thncorrelatiotheiswhere
))(1)((
)())(()(
)()())(()(
)()()()()(
rateforeigntheas)(andratedomestictheas)(Define
3
2
2
2
1
ρ
ρρ
βα
βα
σ








−+
+−=
+−=
+−=
tdWtdW
trdttrmtdr
tdWtrdttrmtdr
tdWtSdttrtrStdS
trtr
ffffff
ft
f
Stochastic Differential Equations

89. Monte Carlo Simulation
 Iterative Procedure (Euler Method)
( )
( )
( )








∆−+∆
+∆−+=∆+
∆+∆−+=∆+
∆+∆−+=∆+
))(1)((
)()()()(
)()()()()(
)()()()()()(
3
2
2
2
1
tWtW
trttrmtrttr
tWtrttrmtrttr
tWtSttrtrStSttS
fffffff
ft
ρρ
βα
βα
σ

90. Do we need a Crystal Ball inDo we need a Crystal Ball in
Weather Modelling to see the application forWeather Modelling to see the application for
Foreign Exchange Forward CurvesForeign Exchange Forward Curves

91. Pricing MethodologiesPricing Methodologies
• Historical simulation by Hunter (1999), Garman, Blanco and
Erickson (2000), Zeng (2000a)
• Indirect modeling of the underlying variable’s distribution (via a
Monte Carlo technique as this involves simulating a sequence of
data), by Pilipovic (1997), Rookley (2000), Garman, Blanco and
Erickson (2000), Zeng (2000b) and Dornier and Queruel (2000).
• Direct modeling of the underlying variable’s distribution (short
and long term forecasting) by Dischel (1999), Torro, Meneu and
Valor (2000), Davis (2001), Alaton, Djehiche and Stillberger (2001),
Diebold and Campbell (2002), Cao and Wei (2002) and Brody,
Syroka and Zervos (2002).

92. Figure 5.7 Histogram of Sydney Temperature
in °C for Whole Season
9.00 11.25 13.50 15.75 18.00 20.25 22.50 24.75 27.00 29.25 31.50
AvgT
0.00
0.02
0.04
0.06
0.08
Figure 5.8 Histogram of Sydney Temperature
in °C for Winter Season
9.00 10.64 12.28 13.92 15.56 17.20 18.84 20.48 22.12 23.76 25.40
AvgT
0.00
0.05
0.10
0.15
Figure 5.9 Histogram of Sydney Temperature
in °C for Summer Season
13.450 15.255 17.060 18.865 20.670 22.475 24.280 26.085 27.890 29.695 31.500
AvgT
0.00
0.04
0.08
0.12

93. MMathematicalathematical FFormulation oformulation of MMeanean FFunctionunction
...)
365
6
sin()
365
4
sin()
365
2
sin( 321
2
+++++++++= ϕ
π
ϕ
π
ϕ
π
θ
t
f
t
e
t
dctbtat
...effectyearthirdoneeffectyearhalfeffectyearonetrendmean ++++=
Table 5.1 Frequency for Summer Season in Sydney
One Year (153 days) Half Year (76.5 days) One-3rd Year (51 days)
0.006535948 0.0130719 0.01960784
Figure 5.10 Periodogram for Summer Spectrum in Sydney
Freq
Spectrum
0.0 0.1 0.2 0.3 0.4 0.5
-20-100102030
Summer Spectrum
Freq
Spectrum
0.0 0.005 0.010 0.015 0.020 0.025 0.030
0102030

94. Further AnalysisFurther Analysis
...)
365
6
sin()
365
4
sin()
365
2
sin( 321
2
+++++++++= ϕ
π
ϕ
π
ϕ
π
θ
t
f
t
e
t
dctbtat
...effectyearthirdoneeffectyearhalfeffectyearonetrendmean ++++=
Table 5.2 Frequency for Winter Season in Sydney
One Year (212 days) Half Year (106 days) One-3rd Year (70.67 days)
0.004716981 0.009433962 0.01415094
Figure 5.11 Periodogram for Winter Spectrum in Sydney
Freq
Spectrum
0.0 0.1 0.2 0.3 0.4 0.5
-20-1001020
Winter Spectrum
Freq
Spectrum
0.0 0.01 0.02 0.03 0.04
510152025

95. Figure 5.12 Mean and Variance over Time for Sydney
Year
Temperature
0.0 0.5 1.0 1.5 2.0 2.5 3.0
10152025
Year
Varaince
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0204060

96. ModelModel: 2 Factor Mean-Reverting Diffusion Process with Stochastic Volatility: 2 Factor Mean-Reverting Diffusion Process with Stochastic Volatility
Kerr Q. and G. Dixon (2002) ~ 2FMRDwithSVKerr Q. and G. Dixon (2002) ~ 2FMRDwithSV
where (kappa) and (alpha) are two constant mean-reverting rates and (beta) is a constant volatility
of the stochastic volatility process for simplicity. Time varying volatility (nu) based on the observed
temperature . (e.g. high temperature then high volatility)
The mean (mean temperature – theta) and (mean of volatility) are periodical functions which contain
sine and cosine functions.
and are two correlated Wiener processes, i.e., .
So is the temperature model and is the volatility model for temperature.
tX t
κ α β
tθ tm
1
tW
2
tW dtdWdWcor tt ρ=),(
21
Let denote the daily average temperature at time .
The daily average temperature is the arithmetic average of the maximum and minimum temperature recorded on
a day from mid-night to mid-night basis. Taking into account the seasonality and stochastic volatility, a
temperature model can be given as




+−=
+−=
2
1
)(
)(
ttttt
tttttt
dWdtmd
dWXdtXdX
νβναν
νθκ
γ
tν
X
tdX tdν

97. Markov Chain Monte Carlo MethodMarkov Chain Monte Carlo Method
In our SV model, we have to estimate the parameter set
and its time varying volatility based on the observed temperature ,that is a
complete joint distribution .
By Bayes Rule, we could possibly decompose the joint distribution
to .
This theorem implies that knowing the marginal distributions of and
would completely characterise the joint distribution .
Furthermore, the likelihood functions can be obtained as
and
),,,,,( ρβαθκ tttttt m=Θ
tν X
)|,( Xp νΘ
)|,( Xp νΘ )()|(),|()|,( ΘΘΘ∝Θ ppXpXp ννν
),|( νΘXp
)|( Θνp )|,( Xp νΘ
∏
−
=
∆+ Θ=Θ=Θ
1
0
0 ),,|(),|,...,(),|(
T
t
tttttT XXpXXpXp ννν
∏
−
=
∆+ Θ=Θ
1
0
),|()|(
T
t
tttpp ννν

98. Gibbs Sampling AlgorithmGibbs Sampling Algorithm
The iterative estimating procedure is defined by the following algorithm
1. Given initial values
2. Simulate based on the given distribution
where we can choose the prior distributions for different parameters
such as normal distribution or inverse gamma distribution.
3. Simulate based on the given distribution .
(Steps 2 and 3 will be repeated until it converges)
)(),|(),|( )0()0(
ΘΘ∝Θ pXpXp νν
)(Θp
)1(
ν ),|( )0(
Xp Θν
),( )0()0(
νΘ
)1(
Θ

99. Monte Carlo Simulations (Euler Method)Monte Carlo Simulations (Euler Method)
Given a real pay off function , we can define the derivative price as
A Monte Carlo approximation of can be expressed as
where is the number of simulations.
The discrete version of the dynamic process can be written as




∆+∆−+=
∆+∆−+=
∆+
∆+
2
1
)(
)(
ttttttt
tttttttt
Wtm
WXtXXX
νβνανν
νθκ γ
N
),( xtu
∑=
≈
N
i
i
TX
N
xtu
1
)(
)(
1
),( φ
),( xtφ
)|)((),( 00 xXXExtu T == φ

100. UsingUsing
)
365
2
sin(2
ϕ
π
θ ++++=
t
dctbtat )
365
2
sin(and φ
π
++=
t
vumt
Table 5.3 Parameter Estimation for Winter in Sydney: the Mean Functions
for Temperature and Volatility processes in our fitting
Prior PosteriorParameter List
Mean Std Mean Std
a 10 20 13.93033 0.02291
b 1 1 0.6902098 0.3422
c 1 1 -0.1012509 0.1883
d 1 2 2.259497 0.2131
t
θ
ϕ 1 2 -1.570796 0.2360
κ 100 25 110.0321 0.1532
α 80 25 68.0563 0.1001
β 0.5 1 0.3243 0.0490
u 5 10 11.99307 0.0232
v 1 2 2.105039 0.1673
tm
φ 1 2 1.560437 0.0669
ρ 0.5 1 0.09213 0.0115

101. Average Temperature in New York and Philadelphia for 22 years
Figure 5.13 Average Daily Temperature in
New York (LGA) from 1980-2002 (22 years) in °F
Figure 5.14 Average Daily Temperature in
Philadelphia (PHI) from 1980-2002 (22 years) in °F

102. Mean Fitting Curves in New York and Philadelphia for 3 years
Figure 5.15 Mean Fitting Curve in
New York (LGA) from 1998-2000 (3 years) in °F
Figure 5.16 Mean Fitting Curve in
Philadelphia (PHI) from 1998-2000 (3 years) in °F

103. Figure 5.17 Standard Deviation Fitting of Temperature in
New York (LGA) for 2000 (1 year) in °F

104. Figure 5.18 Standard Deviation Fitting of Temperature in
Philadelphia (PHI) for 2000 (1 year) in °F

105. Figure 5.19 Average Temperature Simulation vs Average Observed
Temperature in New York (LGA) from 1998-2000 (3 years) in °F

106. Figure 5.20 Average Temperature Simulation vs Average Observed
Temperature in Philadelphia (PHI) from 1998-2000 (3 years) in °F

107. Energy Derivative Price Comparison
(Using 2002 Calender Year – Weekly)
2002 Weeklyswaps for QLD
0
50
100
150
200
250
300
350
1
5
9
13
17
21
25
29
33
37
41
45
49
MRJD BS MCLP Actual Price
2002 Cap with the strike price $50 for QLD
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
1
5
9
13
17
21
25
29
33
37
41
45
49
BS MRJD MCLP Actual Payoff
Average Price - BS (47.91) MRJD ($46.08)
MCLP ($38.58) Actual ($40.78)
Average Price - BS ($9840.1) MRJD ($7804.5)
MCLP ($6419.78.58) Actual ($6767.63)
Swap Price ComparisonSwap Price Comparison Cap Price ComparisonCap Price Comparison

108. Good Industry Books
Baird, Allen J. (1993).Baird, Allen J. (1993). Option Market MakingOption Market Making should be theshould be the secondsecond
book you read on options trading.book you read on options trading.
Boyle, Phelim and Feidhlim Boyle (2001).Boyle, Phelim and Feidhlim Boyle (2001). DerivativesDerivatives containscontains
intriguing details about the historical origins of the Black-Scholesintriguing details about the historical origins of the Black-Scholes
formula.formula.
Chriss, Neil A. (1997).Chriss, Neil A. (1997). Black-Scholes and BeyondBlack-Scholes and Beyond is the definitiveis the definitive
non-technical introduction to option pricing theory and financialnon-technical introduction to option pricing theory and financial
engineering.engineering.
Haug, Espen G. (1997).Haug, Espen G. (1997). Option Pricing FormulasOption Pricing Formulas is an encyclopediais an encyclopedia
of published option pricing formulas.of published option pricing formulas.
Hull, John C. (2002).Hull, John C. (2002). Options, Futures and Other DerivativesOptions, Futures and Other Derivatives is theis the
standard introduction to financial engineering.standard introduction to financial engineering.
Merton, Robert C. (1992).Merton, Robert C. (1992). Continuous Time FinanceContinuous Time Finance is an editedis an edited
collection of Merton's most important papers. It includes Mertoncollection of Merton's most important papers. It includes Merton
(1973).(1973).
Natenberg, Sheldon (1994).Natenberg, Sheldon (1994). Option Volatility and PricingOption Volatility and Pricing. Most. Most
introductions to options trading are brief.introductions to options trading are brief. This one isn't.

109. Thank You - Mr Glen DixonThank You - Mr Glen Dixon
Email: g.dixon@griffith.edu.auEmail: g.dixon@griffith.edu.au
“Foreign ExchangeForeign Exchange Markets are Key Research Areas for Griffith University”Markets are Key Research Areas for Griffith University”