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Pricing Foreign Exchange Risk

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

Published in: Business, Economy & Finance
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Pricing Foreign Exchange Risk

  1. 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. 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. 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. 4. ““Introduction plus History of FX”Introduction plus History of FX”
  5. 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. 6. Volatility in FX Markets Interest Rates Foreign Exchange Commodities Electricity
  7. 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. 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. 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. 10. ““Currency Futures and Options Market“Currency Futures and Options Market“
  11. 11. The Currency Futures and Options Markets • Foreign Currency Options – History and Size of Market – Options - General – Currency Options – Quotations • Foreign Currency Speculations
  12. 12. The Currency Futures and Options Markets (2) Foreign Exchange Contracts FX Portfolio FX Contracts AUS/US AUS/DEM AUS/SF FX Profiles
  13. 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. 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. 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. 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. 17. Foreign Currency Options (4b) A Stock Simulation for the Barrier Option Source: Griffith University & Kerr 2000
  18. 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. 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. 20. Foreign Currency Options (7)
  21. 21. Foreign Currency Options (8)
  22. 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. 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. 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. 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. 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. 27. ““Overview of Black Scholes (1973) , MertonOverview of Black Scholes (1973) , Merton ((1973) and Garman Kohlhagen (1983)”((1973) and Garman Kohlhagen (1983)”
  28. 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. 29. Black Scholes (1973) Options Pricing Formula Values for a call price c or put price p are: where:
  30. 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. 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. 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. 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. 34. Merton (1973) Options Pricing Formula Values for a call price c or put price p are: where:
  35. 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. 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. 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. 38. Garman and Kohlhagen (1983) FX Options Pricing Formula Values for a call price c or put price p are: where:
  39. 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. 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. 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. 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. 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. 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. 45. Interest Rate Markets
  46. 46. Interest Rate Markets (1)
  47. 47. Interest Rate Markets (2)
  48. 48. Interest Rate Markets (3)
  49. 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. 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. 51. ““Comparison of Currencies: Australian, US,Comparison of Currencies: Australian, US, Asian , Latin American”Asian , Latin American”
  52. 52. Comparisons of Currencies: Australian Dollar (Daily)
  53. 53. Comparisons of Currencies: Australian Dollar (Monthly)
  54. 54. Comparisons of Currencies: US Dollar (Daily)
  55. 55. Comparisons of Currencies: US Dollar (Monthly)
  56. 56. Comparisons of Asian Currencies (Daily)
  57. 57. Comparisons of Asian Currencies (Monthly)
  58. 58. Comparisons of Latin American Currencies (Daily)
  59. 59. Comparisons of Latin American Currencies (Monthly)
  60. 60. ““Overview of Monte Carlo, ScenarioOverview of Monte Carlo, Scenario Development and Stress Testing”Development and Stress Testing”
  61. 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. 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. 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. 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. 65. Monte Carlo for FX Market (cont.) GENERATE MULTIPLE RANDOM SAMPLES for FX GENERATE MULTIPLE RANDOM SAMPLES for FX Time $A/ $US
  66. 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. 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. 68. Sensitivity Analysis for FX
  69. 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. 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. 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. 72. Scenario Development for FX (4) Risk Political Risk Operational Risk Legal Risk Credit Risk Reputational Risk
  73. 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. 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. 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. 76. Scenario Development for FX (8) Step 4: Scenario consolidation Consolidate the results Check for consistency errors, doubling counting Independent validation checks
  77. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 103. Figure 5.17 Standard Deviation Fitting of Temperature in New York (LGA) for 2000 (1 year) in °F
  104. 104. Figure 5.18 Standard Deviation Fitting of Temperature in Philadelphia (PHI) for 2000 (1 year) in °F
  105. 105. Figure 5.19 Average Temperature Simulation vs Average Observed Temperature in New York (LGA) from 1998-2000 (3 years) in °F
  106. 106. Figure 5.20 Average Temperature Simulation vs Average Observed Temperature in Philadelphia (PHI) from 1998-2000 (3 years) in °F
  107. 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. 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. 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”

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