Pricing Foreign Exchange Risk

Currency Hedging in Turbulent times,Currency Hedging in Turbulent times,Executive Briefing Seminar, Grace Hotel, SydneyExecutive Briefing Seminar, Grace Hotel, Sydney

1010thth November 2003 November 2003

““Pricing Foreign Exchange Risk” Pricing Foreign Exchange Risk”

By Glen DixonBy Glen DixonActing Lecturer Acting Lecturer School of Accounting, School of Accounting, Banking & Finance,Banking & Finance,Faculty of Commerce and Faculty of Commerce and ManagementManagement

OVERVIEW :OVERVIEW :An introduction to common derivative An introduction to common derivative

products products

Understanding the key components of the Understanding the key components of the Black Scholes pricing methodology Black Scholes pricing methodology

Constructing and using a forward price Constructing and using a forward price curvecurve

An introduction to common derivative An introduction to common derivative products products

Understanding the key components of the Black Scholes pricing Understanding the key components of the Black Scholes pricing methodology methodology

Constructing and using a forward price curveConstructing and using a forward price curve

““Introduction plus History of FX” Introduction plus History of FX”

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.

Volatility in FX Markets

Interest Rates

Foreign ExchangeCommodities

Electricity

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)

• 1983 March 1983 March - (5th, 8th), OctoberOctober - (28th), DecemberDecember - (8th, 9th,12th and 13th)

• 1984 March1984 March - (5th) February till early 1985 February till early 1985

• 1985 February1985 February - (6th till 8th)

• 1986 May1986 May - (13th, 14th), July July - (2nd, 4th, 25th and 28th), August August - (19th)

• 1987 October1987 October - (20th), November till DecemberNovember till December

• 1988 April1988 April - till December 1989till December 1989

• 1989 February1989 February - Late February till April Late February till April - May May - August August

• 1990 January 1990 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

• 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 till Early 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

““Currency Futures and Options Currency Futures and Options Market“Market“

The Currency Futures and Options Markets

• Foreign Currency Options– History and Size of Market – Options - General– Currency Options– Quotations

• Foreign Currency Speculations

The Currency Futures and Options Markets (2)

Foreign Exchange Contracts

FX Portfolio

FX Contracts

AUS/US

AUS/DEM

AUS/SF

FX Profiles

Foreign Currency Options

History and Size of Market

Attention will be focused on plain-vanilla European puts and calls onforeign 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.

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


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

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

Foreign Currency Options (4b) A Stock Simulation for the Barrier Option

Source: Griffith University & Kerr 2000


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


• 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




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


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


• 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


• 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

An introduction to common derivative products An introduction to common derivative products

Understanding the key components of the Understanding the key components of the Black Scholes pricing methodology Black Scholes pricing methodology

Constructing and using a forward price curveConstructing and using a forward price curve

““Overview of Black Scholes (1973) , Overview of Black Scholes (1973) , Merton ((1973) and Garman Merton ((1973) and Garman

Kohlhagen (1983)” Kohlhagen (1983)”

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

Black Scholes (1973) Options Pricing Formula

Values for a call price c or put price p are:

where:

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.

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 =

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 =

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.

Merton (1973) Options Pricing Formula


where:

The Five Greeks for Merton (1973) Options Pricing Formula for a Call


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:

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.

Garman and Kohlhagen (1983) FX Options Pricing Formula


where:

The Five Greeks for Garman and Kohlhagen (1983) FX Options Pricing Formula for a Call


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:

An introduction to common derivative products An introduction to common derivative products

Understanding the key components of the Black Scholes pricing Understanding the key components of the Black Scholes pricing methodology methodology

Constructing and using a forward price Constructing and using a forward price curvecurve

““Overview of Interest Rate Markets: Overview of Interest Rate Markets: Bachelior (1901), including Single Factor Bachelior (1901), including Single Factor Models like Vasicek (1977) , Cox Ingersoll Models like Vasicek (1977) , Cox Ingersoll

and Ross (1985)” and Ross (1985)”

Stochastic Differential Equation (Wiener processes or Random Walk)

Geometric Brownian Motion (Stock Markets) – Geometric Brownian Motion (Stock Markets) – BacheliorBachelior

ddistributenormally ,incrementst independen

Noise), (Whitemotion Brownian :)(

y volatilit the:

rate lincrementa the:

where

)()(

)(

follows SDE geometric The

price.asset underlying theas )( Define

tW

tdWdttS

tdS

tS

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 MeanMean Reversion

Interest Rate Markets

Interest Rate Markets (1)



Stochastic Differential Equation Single Factor Models (Interest Rate Markets)

The Vasicek (1977) ModelThe Vasicek (1977) Model

motionBrownian :

y volatilit the:σ

alueinterest vmean the:μ

rate reverting-mean the:κ

where

follows SDEVasicek The

rate.short theas Define

W(t)

σdW(t)r(t))dtκ(μdr(t)

r(t)

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 :)(

y volatilit the:

alueinterest vmean the:

rate reverting-mean the:

where

)()())(()(

follows SDE CIR The

rate.short theas )( Define

tW

tdWtrdttrtdr

tr

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

Comparisons of Currencies: Australian Dollar (Daily)

Comparisons of Currencies: Australian Dollar (Monthly)

Comparisons of Currencies: US Dollar (Daily)

Comparisons of Currencies: US Dollar (Monthly)

Comparisons of Asian Currencies (Daily)

Comparisons of Asian Currencies (Monthly)

Comparisons of Latin American Currencies (Daily)

Comparisons of Latin American Currencies (Monthly)

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

Iterative Procedure (Euler Method)

(CIR). 1/2or (Vasicek) 0 where

0at rateinterest initial with the

)()())(()()(

0

tr

tWtrttrtrttr

1.0. ofdeviation standard

and zero ofmean on with distributi normal a

from sample random a is where)( ttW

Monte Carlo Simulation(Generate Random Numbers)

Box Muller:

Marsaglia:

Note: Box Muller and Marsaglia will generate Note: Box Muller and Marsaglia will generate

standard Gaussian random variables based standard Gaussian random variables based

on two independent uniformly distributed on two independent uniformly distributed

random variables from [0, 1].random variables from [0, 1].

)2sin()log(2

)2cos()log(2

212

211

XXY

XXY

VVUXXV

UXYXY

)log(222

21

2211

;1

;*)12( U;*)12(

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)

M1

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)

k2

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

Monte Carlo for FX Market (cont.)

GENERATE MULTIPLE RANDOM

SAMPLES for FX

GENERATE MULTIPLE RANDOM

SAMPLES for FX

Time

$A/$US

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 AverageMonthly Prices

In the FX Market

The Accuracy of Estimates is related to the number of Simulations

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

Sensitivity Analysis for FX

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.

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.


• 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.


Risk

Political Risk

Operational Risk

Legal Risk

Credit Risk

Reputational Risk

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


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


Step 3: Scenario projections

Estimate the likely movements of the identified scenario factors and determine the potential loss in that case


Step 4: Scenario consolidation

Consolidate the results

Check for consistency errors, doubling counting

Independent validation checks


Step 5: Scenario presentation and follow-up

Summarise results Analyse and evaluate next steps: eg, put on a hedge

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.

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:


Steps in Stress Testing

• Step 1: Picking what to stress

Choice of market variables Range of stress Usefulness of stress information vs data overload


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?


Step 3: Revaluing the portfolio Back of the envelope vs sophisticated modeling Adjusting for market liquidity

Trading

SettlementsPortfolio

Management

ContractManagement


Step 4: Deciding on action steps

Reporting Cross-checks on model and pricing validity Action plan for dealing with actual catastrophe

situation

““Overview of Interest Rate Markets Overview of Interest Rate Markets including Two Factor Models like Brennan including Two Factor Models like Brennan

and Schwartz (1982), Kerr and Dixon and Schwartz (1982), Kerr and Dixon (2003)” (2003)”

Stochastic Differential Equations Two Factor Models (Interest Rate Markets) + Monte Carlo Simulation

The Brennan and Schwartz (1982) Stochastic Volatility The Brennan and Schwartz (1982) Stochastic Volatility ModelModel

1/2)or 0( motions.Brownian

t independen are )( and )( variance,its and

rateshort ebetween thn correlatio theis where

))(1)((

)()()(

)()()()()(

variance theas )( and rateshort theas )( Define

21

22

1

1

tWtW

tdWtdW

tdttmtd

tdWtrtdttrtdr

ttr

Iterative Procedure Iterative Procedure (Euler Method) with(Euler Method) with

1/2) (

variance.initial theand rate initial with the

))(1)((

)()()()(

)()()()()()(

22

1

1

tWtW

tttmttt

tWtrtttrtrttr

Stochastic Differential Equations Monte Carlo Simulation (Two Factor Models-CIR)

The Brennan and Schwartz (1982) Stochastic Volatility The Brennan and Schwartz (1982) Stochastic Volatility ModelModel

with Iterative Procedure (Euler Method) with Iterative Procedure (Euler Method) withwith 1/2) (

Foreign Currency Stochastic Modeling

rateinterest foreign the:

rateinterest domestic the: where

)()())(()(

as modelmotion Brownian

geometric a follows rate exchangespot that the

assume Werate. exchangespot theas )( Define

f

f

r

r

tdWtSdtrrtStdS

tS

Stochastic Differential Equations

Stochastic Interest Rates

rateinterest foreign theand rate

interest domestic ebetween thn correlatio theis where

))(1)((

)())(()(

)()())(()(

)()()()()(

rateforeign theas )( and rate domestic theas )( Define

32

2

2

1

tdWtdW

trdttrmtdr

tdWtrdttrmtdr

tdWtSdttrtrStdS

trtr

ffffff

ft

f

Stochastic Differential Equations

Monte Carlo Simulation

Iterative Procedure (Euler Method)

))(1)((

)()()()(

)()()()()(

)()()()()()(

32

2

2

1

tWtW

trttrmtrttr

tWtrttrmtrttr

tWtSttrtrStSttS

fffffff

ft

Do we need a Crystal Ball in Do we need a Crystal Ball in Weather Modelling to see the application for Weather Modelling to see the application for

Foreign Exchange Forward CurvesForeign Exchange Forward Curves

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).

Figure 5.7 Histogram of Sydney Temperaturein °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 Temperaturein °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 Temperaturein °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

MMathematical athematical FFormulation of ormulation of MMean ean FFunctionunction

...)365

6sin()

365

4sin()

365

2sin( 321

2 tf

te

tdctbtat

...effectyear thirdoneeffectyear halfeffectyear onetrendmean

Table 5.1 Frequency for Summer Season in SydneyOne 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

Sp

ectr

um

0.0 0.1 0.2 0.3 0.4 0.5

-2

0-1

00

10

20

30

Summer Spectrum

Freq

Sp

ectr

um

0.0 0.005 0.010 0.015 0.020 0.025 0.030

01

02

03

0

Further AnalysisFurther Analysis

...)365

6sin()

365

4sin()

365

2sin( 321

2 tf

te

tdctbtat

...effectyear thirdoneeffectyear halfeffectyear onetrendmean

Table 5.2 Frequency for Winter Season in SydneyOne 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

Sp

ectr

um

0.0 0.1 0.2 0.3 0.4 0.5

-2

0-1

00

10

20

Winter Spectrum

Freq

Sp

ectr

um

0.0 0.01 0.02 0.03 0.04

51

01

52

02

5

Figure 5.12 Mean and Variance over Time for Sydney

Year

Tem

pe

ratu

re

0.0 0.5 1.0 1.5 2.0 2.5 3.0

10

15

20

25

Year

Va

rain

ce

0.0 0.5 1.0 1.5 2.0 2.5 3.0

02

04

06

0

ModelModel: 2 Factor Mean-Reverting Diffusion Process with Stochastic Volatility: 2 Factor Mean-Reverting Diffusion Process with Stochastic VolatilityKerr 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

t tm

1tW

2tW dtdWdWcor tt ),( 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

tX

tdX td

Markov Chain Monte Carlo MethodMarkov Chain Monte Carlo Method In our SV model, we have to estimate the parameter setand 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

),,,,,( tttttt m

t X

)|,( Xp

)|,( Xp )()|(),|()|,( ppXpXp

),|( Xp

)|( p )|,( Xp

1

00 ),,|(),|,...,(),|(

T

ttttttT XXpXXpXp

1

0

),|()|(T

ttttpp

Gibbs Sampling AlgorithmGibbs Sampling AlgorithmThe 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( pXpXp

)(p

)1( ),|( )0( Xp

),( )0()0( )1(

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

N

),( xtu

N

i

iTXN

xtu1

)( )(1

),(

),( xt)|)((),( 00 xXXExtu T

UsingUsing)

365

2sin(2

tdctbtat )

365

2sin( and

t

vumt

Table 5.3 Parameter Estimation for Winter in Sydney: the Mean Functionsfor Temperature and Volatility processes in our fitting

Prior PosteriorParameter ListMean 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 0.2131t

1 2 -1.570796 0.2360

100 25 110.0321 0.1532

80 25 68.0563 0.1001

0.5 1 0.3243 0.0490

u 5 10 11.99307 0.0232

v 1 2 2.105039 0.1673 tm

1 2 1.560437 0.0669

0.5 1 0.09213 0.0115

Average Temperature in New York and Philadelphia for 22 yearsFigure 5.13 Average Daily Temperature in

New York (LGA) from 1980-2002 (22 years) in °F

Figure 5.14 Average Daily Temperature inPhiladelphia (PHI) from 1980-2002 (22 years) in °F

Mean Fitting Curves in New York and Philadelphia for 3 yearsFigure 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

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

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

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

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

Energy Derivative Price Comparison (Using 2002 Calender Year – Weekly)

2002 Weekly swaps for QLD

0

50

100

150

200

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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

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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

Good Industry Books

Baird, Allen J. (1993). Baird, Allen J. (1993). Option Market MakingOption Market Making should be the should 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 contains contains intriguing details about the historical origins of the Black-Scholes intriguing 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 definitive is the definitive non-technical introduction to option pricing theory and financial non-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 encyclopedia is 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 the is 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 edited is an edited collection of Merton's most important papers. It includes Merton collection 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.

Thank You - Mr Glen DixonThank You - Mr Glen DixonEmail: [email protected]: [email protected]

“Foreign Exchange Foreign Exchange Markets are Key Research Areas for Griffith University”Markets are Key Research Areas for Griffith University”

Business

Pricing Foreign Exchange Risk