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Options for Managing Foreign Exchange Risk

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Options for Managing Foreign Exchange Risk
- by Dr Zili Zhu, Principal Research Scientist, Business and Financial Engineering, CSIRO

Published in: Economy & Finance, Business
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Options for Managing Foreign Exchange Risk

  1. 1. Options for Managing Foreign Exchange Dr Zili Zhu Quantitative Risk Management Mathematics, Informatics & Statistics 26th March 2010
  2. 2. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Background of CSIRO Organization: • Commonwealth Scientific and Industrial Research Organization (7200 staff members) • Division of Mathematics, Informatics and Statistics (150 Scientists) • Quantitative Risk Management Group (25 scientists) Commercial activities • CSIRO Exotic math for FX markets • Consulting assignments for major banks • Development of new options models for hedge funds. • Development of major risk-management software. • Rea-options valuation in energy industries.
  3. 3. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Content An introduction to common derivative products in FX Understanding the key components of pricing derivatives. How reliable are the pricing models given recent and excessive volatility Other risk valuation methods
  4. 4. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Financial Derivatives Exchange markets: standardised Futures, swaps and options are actively traded on exchanges. Over-the-counter (OTC) market: forwards, exotic options are traded directly among institutions and outside of exchanges. Derivative – financial instrument whose value depends on other more basic variables (stocks, futures, FXs, interest rates), e.g. Vanilla call/put options on traded shares.
  5. 5. Simple standard derivatives: Call/Put vanilla options Digital payoff Forwards Averaged rate ..... CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au
  6. 6. Multi-leg structure • Zero cost • Tailor-made risk profile • Multiple expiries • Flexible CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au
  7. 7. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Some exotic options used in FX Window barrier options (KO, KI, Touches, Digital) Basket options Range accrual Target-redemption notes B
  8. 8. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Example: Reverse Knockout Call Up and Out Call Payoff is: V(S,T) = (S – K) if S < B V(S,T) = 0 if S ≥ B Barrier is: V(S,t) = 0 if S ≥ B t B K t S K B
  9. 9. CSIRO Mathematical & Information Sciences www.cmis.csiro.au A Typical Exotic Option: Two-Asset No-Touch FENICS FX Pricing Page:
  10. 10. CSIRO Mathematical & Information Sciences www.cmis.csiro.au Other Exotic Options Compound Call/Put Quanto options Lookbacks Asian average options Trans-Atlantic options Holder Extendible options Knock-out and Knock-in barrier options Multiple window barrier options One-touch/No-touch options Best/Worst options Basket options Beta Basket options. Two-asset digitals Two-asset Knock-out/in options …….
  11. 11. CSIRO Mathematical & Information Sciences www.cmis.csiro.au Exotic option: Beta Basket
  12. 12. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au How to Price Derivatives in FX The price of a derivative should be the hedging cost of the derivative over its life cycle. Financial mathematics is well established. Option-pricing formula and numerical methods are available. Industry conventions need to be considered. B 0)( ))()((PutVanilla 0)( ))()((CallVanilla 1 102 1 210 <−−= ∂ ∂ =∆ −−−= >= ∂ ∂ =∆ −= − − dN S Q dNeSdKNeprice dN S Q dKNdNeSeprice P P rTrT C C rTrT
  13. 13. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Currency prices follow stochastic processes: i dZ i St i dt i S ii dS )(σµ += i=1,2,…..N j dZ i dZ ij =ρ Methodologies for Pricing Derivatives $0.6 $0.7 $0.8 $0.9 $1.0 $1.1 $1.2 $1.3 $1.4 $1.5 0 0.2 0.4 0.6 0.8 1 time stockprice,S(t)
  14. 14. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Example of Pricing a Call Option – delta hedging Portfolio= S0Δ – call option Stock price, S0 = $10 Strike=$11 Stock price, ST = $12 Option price = $1 Portfolio1 = 12Δ - 1 Stock price, ST = $8 Option price = $0 Portfolio2 = 8Δ - 0Time period T optionscall102Portfolio 2Portfolio2Portfolio1..25.0..8112...Portfolio2Portfolio1:wantWe −∆== ===∆∆=−∆= andsoei 5.02 4 1 10priceoptionCall =−×=
  15. 15. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Tree Methods Trinomial Tree 2222 3 2 3 333 4 3 4 444 5 4 5 555 6 5 6 666 7 6 7 777 8 7 8 888 9 8 9 999 10 9 10 101010 11 10 11 111111 12 11 12 121212 3333 4 3 4 444 5 4 5 555 6 5 6 666 7 6 7 777 8 7 8 888 9 8 9 999 10 9 10 101010 11 10 11 111111 4444 5 4 5 555 6 5 6 666 7 6 7 777 8 7 8 888 9 8 9 999 10 9 10 101010 5555 6 5 6 666 7 6 7 777 8 7 8 888 9 8 9 999 6666 7 6 7 777 8 7 8 888 777 4.7050653 2.9506730 1.8504464 7.5025729 11.9634046 19.0765291 30.4189297 48.5052222 77.3451467 123.3325289 196.6627944 313.5932998 500.0475965 7.5018801 4.7046308 2.9504005 11.9622999 19.0747677 30.4161209 48.5007434 77.3380049 123.3211408 196.6446352 313.5643436 11.9611854 7.5011811 4.7041925 19.0729904 30.4132870 48.4962245 77.3307992 123.3096507 196.6263135 19.0711974 11.9600610 7.5004760 30.4104279 48.4916655 77.3235295 123.2980587 30.4075437 19.0693887 11.9589266 48.4870664 77.3161960 30.4017000 48.4824273 30.4046344 19.0675642 S SL SU SM pm pL pu S Time S SdZtSdtqrdS )()( σ+−=
  16. 16. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Using Monte-Carlo Simulations: ]|]0,[max[Pr 0SKSEeAmountemium T Trd −×= − Simulated future carbon prices 1 10 100 1000 10000 2008 2012 2016 2020 2024 2028 2032 2036 2040 2044 2048 Year Carbonprice dtdZdZEtdZtdtttSd ijjiiii ii t ρσσµ =+−= ][);()()](5.0)([ln 2)()( ]ˆ)ˆˆexp[( 2 2 1)()( iiii i t i tt ZttXX δσδσµδ +−=+
  17. 17. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Finite Difference, Element, Volume Methods 0),()( ),( ])()([ ),( 2 ),(),( 2 222 =− ∂ ∂ −+ ∂ ∂ + ∂ ∂ tSVtr S tSV Stqtr S tSVStS t tSV σ S0 S
  18. 18. CSIRO Mathematical & Information Sciences www.cmis.csiro.au An Exotic Option: two-asset options • 2 asset Black-Scholes equation: • Payoff function )max( 2,1 SSPayoff = S2 ∂ ∂ σ ∂ ∂ σ ∂ ∂ ρσ σ ∂ ∂ ∂ µ ∂ ∂ µ ∂ ∂ V t S V S S V S S S V S S S V S S V S rV+ + + + + − = 1 2 1 2 01 2 2 1 2 1 2 2 2 2 2 2 2 2 1 2 1 2 2 1 2 1 1 1 2 2 2 S2 S2S1 )0,max( 21 11 KSwSwPayoff −+= )min( 2,1 SSPayoff =
  19. 19. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au How Reliable are Pricing Models? All models are constructed under certain assumptions. All models have their limitations. Model implementations can also have their own limitations. Computer code can often have bugs. Market data may not be arbitrage-free. Market data may be inconsistent. Models and pricing functions should have been tested for extreme market conditions. On-going updates and maintenance are needed. Market is evolving, and models should too.
  20. 20. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Practical Issues in Pricing Derivatives Volatility is not constant, vol skew/smile exists. Correlation is dependent on ATM price. Correlation should be dependent on strike levels? How to price basket options with skew. How much correction is needed to get market price? Compromise between speed and accuracy.
  21. 21. CSIRO Mathematical & Information Sciences www.cmis.csiro.au Volatility Smile/Skew ),()0(),( 0 VXVXVX Π−=∆•−∆Π−= T loss f }),({min)()( αξξ ξαα ≥Ψ=≡ ∈ XXX R VaR ][)( tail lossfECVaR −=≡ ααφ X
  22. 22. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Hedging Principles • Hedging to eliminate risk due to market movements in asset prices, volatility, interest-rates and correlations. • The cost of hedging reflects the premium received from clients. • Limit large down-side risk to P/L. • Trading in derivatives without hedging is speculation. • The objective of hedging is to protect business from unpredictable market movements on a daily basis.
  23. 23. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Greek Hedging: Using Sensitivity Parameters • Delta: • Gamma: • Vega • Rho • Time decay ; 0 Pr S emium ∂ ∂ =∆ . 2 0 Pr2 100 0 S emiumS ∂ ∂ =Γ ; Pr 100 1 σ∂ ∂ = emium v );,,,,, 0 (Pr),,, 365 1,, 365/ 0 (Pr µσµσµθ rTKSemiumrTKeSemium −−= R emium ∂ ∂ = Pr 100 1ρ
  24. 24. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Example: Window No-Touch Option FENICS FX Pricing Page:
  25. 25. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Example: Window No-Touch Option
  26. 26. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Example: Greeks of Window No-Touch Option
  27. 27. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Greeks: Window No-Touch Option
  28. 28. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Example: Window No-Touch Option
  29. 29. CSIRO Mathematical & Information Sciences www.cmis.csiro.au Delta hedging is automatically set for each individual option through the purchase/sell of underlying assets. Other greek parameters such as gamma, vega, rho are balanced through the purchase/sell of vanilla and/or more liquid exotic options at portfolio level. For options with discontinuous risk profiles or path-dependency (e.g. barrier options), hedging is difficult. Portfolio Approach
  30. 30. CSIRO Mathematical & Information Sciences www.cmis.csiro.au Loss Distribution without Hedges Target portfolio loss distribution 0 10 20 30 40 50 60 70 80 90 100 -6 -3.5 -1 1.5 4 6.5 9 11.5 14 16.5 Loss Frequency
  31. 31. CSIRO Mathematical & Information Sciences www.cmis.csiro.au A Greek Delta-Gamma Hedge To Reduce Risk Delta-gamma hedge 0 100 200 300 400 500 600 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Loss Frequency
  32. 32. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Hedging Strategy • Risk can only be reduced but not eliminated via hedging through greeks even if the Black-Scholes model is appropriate. • Hedging through greeks is model dependent. • For commodities and energies (e.g. electricity), model dependency can make hedging ineffective.
  33. 33. CSIRO Mathematical & Information Sciences www.cmis.csiro.au Hedging Through CVaR Minimisation CVaR-minimising hedge 0 100 200 300 400 500 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Loss Frequency
  34. 34. CSIRO Mathematics, Informatics &Statistics www.cmis.csiro.au Other risk valuation methods Implied volatility of Black-Scholes model is used for quoting FX options. New valuation models are developed and implemented regularly. Every model has its drawbacks, and no model is perfect. Speed, accuracy and robustness need to be considered.
  35. 35. CSIRO Mathematics, Informatics &Statistics www.cmis.csiro.au Local volatility surface model functionyvolatilitlocal),( ).(),()]()([/ tS tdWtSdttqtrSdS tt σ σ+−=
  36. 36. CSIRO Mathematics, Informatics &Statistics www.cmis.csiro.au Stochastic volatility model dtdZdWE dZVdtVdV dWSVdtSqrdS tt tttt ttttt ρ ξθα = +−= +−= ][ )( )(
  37. 37. CSIRO Mathematical & Information Sciences www.cmis.csiro.au Summary  Introduction of derivatives in the FX market.  A large number of options are available to accommodate specific risk appetites and market views of end-users.  The hedging of options can be implemented as part of a structure.  Full understanding of down-side risk of options is paramount before trading.  Introduced key concepts in pricing derivatives in the FX market, and different pricing methods are available.  All models have limitations. Implementation also has limitations.  Market data can be problematic.  New and sophisticated models are created regularly. No model is perfect.
  38. 38. CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au Acknowledgments • Thanks to FENICS FX, the global standard in FX options pricing and analysis, for the use of their trading system. The screenshots of pricing pages and market data pages in this presentation are from FENICS FX.

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