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Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary FX Derivatives: Stochastic-Local-Volatility Model Uwe Wystup Fr ed eric Bossens, Andreas Weber MathFinance AG uwe.wystup@mathfinance.com July 2020

  1. Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary FX Derivatives: Stochastic-Local-Volatility Model Uwe Wystup Fr´ ed´ eric Bossens, Andreas Weber MathFinance AG uwe.wystup@mathfinance.com July 2020 FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 1 / 37

  2. Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary Agenda Review Vanna Volga 1 Early Days and Drawbacks Versions of Vanna-Volga Design and Consistency Issues Stochastic-Local-Volatility 2 LV and SV vanilla smile fit SLV Step by Step SLV Pricing / Validation Mixture Local Volatility Model 3 MLV Main Features Vol Process Comparison MLV vs. SLV Granular Model Marking Summary 4 Product/Model Matrix Key Take-Aways FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 2 / 37

  3. Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary Agenda Review Vanna Volga 1 Early Days and Drawbacks Versions of Vanna-Volga Design and Consistency Issues Stochastic-Local-Volatility 2 LV and SV vanilla smile fit SLV Step by Step SLV Pricing / Validation Mixture Local Volatility Model 3 MLV Main Features Vol Process Comparison MLV vs. SLV Granular Model Marking Summary 4 Product/Model Matrix Key Take-Aways FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 3 / 37

  4. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Review Vanna Volga: Early Days Pricing in Vanna Volga (VV) model is popular in FX as it still allows to 1 compute prices nearly as fast as in the BS model, using analytical formulas. FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 4 / 37

  5. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Review Vanna Volga: Early Days Pricing in Vanna Volga (VV) model is popular in FX as it still allows to 1 compute prices nearly as fast as in the BS model, using analytical formulas. For a long time it matched market prices for barrier options and touch 2 contracts quite good. FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 4 / 37

  6. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Review Vanna Volga: Early Days Pricing in Vanna Volga (VV) model is popular in FX as it still allows to 1 compute prices nearly as fast as in the BS model, using analytical formulas. For a long time it matched market prices for barrier options and touch 2 contracts quite good. Market fit at least better than the alternatively available Local Volatility 3 (LV) or Stochastic Volatility (SV) models. FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 4 / 37

  7. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Review Vanna Volga: Drawbacks Consistent pricing for different option types not guaranteed 1 FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 5 / 37

  8. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Review Vanna Volga: Drawbacks Consistent pricing for different option types not guaranteed 1 Single product type rules of thumb 2 FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 5 / 37

  9. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Review Vanna Volga: Drawbacks Consistent pricing for different option types not guaranteed 1 Single product type rules of thumb 2 Fits the hedging instruments (plain vanilla options) only reasonably well in 3 the 25-delta range FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 5 / 37

  10. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Review Vanna Volga: Drawbacks Consistent pricing for different option types not guaranteed 1 Single product type rules of thumb 2 Fits the hedging instruments (plain vanilla options) only reasonably well in 3 the 25-delta range Not clear how to apply to more exotic products 4 FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 5 / 37

  11. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Review Vanna Volga: Drawbacks Consistent pricing for different option types not guaranteed 1 Single product type rules of thumb 2 Fits the hedging instruments (plain vanilla options) only reasonably well in 3 the 25-delta range Not clear how to apply to more exotic products 4 Volatility too flat in the wings 5 FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 5 / 37

  12. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Wystup/Traders’ Rule of Thumb 2003 (VV) [Wystup, 2003], [Wystup, 2006]: compute the cost of the overhedge of risk reversals (RR) and butterflies (BF) to hedge vanna and volga of an option EXO. VV-value = TV + p [cost of vanna + cost of volga] (1) with vannaEXO cost of vanna = vannaRR × OH RR (2) volgaEXO cost of volga = × OH BF (3) volgaBF p = no-touch probability or modifications (4) OH = overhedge = market price − TV (5) FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 6 / 37

  13. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Castagna/Mercurio 2007 (VV2) [Castagna and Mercurio, 2007], [Castagna and Mercurio, 2006]: portfolio of three calls hedging an option risk up to second order (in particular the vanna and volga of an option 3 � c ( K , σ K ) = c ( K , σ BS ) + x i ( K )[ c ( K i , σ i ) − c ( K i , σ BS )] (6) i =1 with ∂ c ( K ,σ BS ) ln K 2 K ln K 3 ∂σ K x 1 ( K ) = ∂ c ( K 1 ,σ BS ) ln K 2 K 1 ln K 3 K 1 ∂σ ∂ c ( K ,σ BS ) ln K K 1 ln K 3 K ∂σ x 2 ( K ) = ∂ c ( K 2 ,σ BS ) ln K 2 K 1 ln K 3 ∂σ K 2 ln K K 1 ln K ∂ c ( K ,σ BS ) K 2 ∂σ x 3 ( K ) = (7) ∂ c ( K 3 ,σ BS ) ln K 3 K 1 ln K 3 K 2 ∂σ FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 7 / 37

  14. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Vanna-Volga Design Issues Hedge with BF and RR vs Hedge with 3 Vanillas, and if vanillas then which ones? FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 8 / 37

  15. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Vanna-Volga Design Issues Hedge with BF and RR vs Hedge with 3 Vanillas, and if vanillas then which ones? Include vega FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 8 / 37

  16. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Vanna-Volga Design Issues Hedge with BF and RR vs Hedge with 3 Vanillas, and if vanillas then which ones? Include vega Include third order Greeks (volunga, vanunga) FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 8 / 37

  17. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Vanna-Volga Design Issues Hedge with BF and RR vs Hedge with 3 Vanillas, and if vanillas then which ones? Include vega Include third order Greeks (volunga, vanunga) One-touch probability: which measure? domestic, foreign, physical, mixes? FX Derivatives: Stochastic-Local-Volatility Model c � by MathFinance AG uwe.wystup@mathfinance.com 8 / 37

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