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FX Derivatives: Stochastic-Local-Volatility Model Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary FX Derivatives: Stochastic-Local-Volatility Model Uwe Wystup Fr´ed´ericBossens, 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 Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary Agenda 1 Review Vanna Volga Early Days and Drawbacks Versions of Vanna-Volga Design and Consistency Issues 2 Stochastic-Local-Volatility LV and SV vanilla smile fit SLV Step by Step SLV Pricing / Validation 3 Mixture Local Volatility Model MLV Main Features Vol Process Comparison MLV vs. SLV Granular Model Marking 4 Summary Product/Model Matrix Key Take-Aways FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 2 / 37 Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary Agenda 1 Review Vanna Volga Early Days and Drawbacks Versions of Vanna-Volga Design and Consistency Issues 2 Stochastic-Local-Volatility LV and SV vanilla smile fit SLV Step by Step SLV Pricing / Validation 3 Mixture Local Volatility Model MLV Main Features Vol Process Comparison MLV vs. SLV Granular Model Marking 4 Summary Product/Model Matrix Key Take-Aways FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 3 / 37 2 For a long time it matched market prices for barrier options and touch contracts quite good. 3 Market fit at least better than the alternatively available Local Volatility (LV) or Stochastic Volatility (SV) models. 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 1 Pricing in Vanna Volga (VV) model is popular in FX as it still allows to 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 3 Market fit at least better than the alternatively available Local Volatility (LV) or Stochastic Volatility (SV) models. 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 1 Pricing in Vanna Volga (VV) model is popular in FX as it still allows to compute prices nearly as fast as in the BS model, using analytical formulas. 2 For a long time it matched market prices for barrier options and touch contracts quite good. FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 4 / 37 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 1 Pricing in Vanna Volga (VV) model is popular in FX as it still allows to compute prices nearly as fast as in the BS model, using analytical formulas. 2 For a long time it matched market prices for barrier options and touch contracts quite good. 3 Market fit at least better than the alternatively available Local Volatility (LV) or Stochastic Volatility (SV) models. FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 4 / 37 2 Single product type rules of thumb 3 Fits the hedging instruments (plain vanilla options) only reasonably well in the 25-delta range 4 Not clear how to apply to more exotic products 5 Volatility too flat in the wings 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 1 Consistent pricing for different option types not guaranteed FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 5 / 37 3 Fits the hedging instruments (plain vanilla options) only reasonably well in the 25-delta range 4 Not clear how to apply to more exotic products 5 Volatility too flat in the wings 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 1 Consistent pricing for different option types not guaranteed 2 Single product type rules of thumb FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 5 / 37 4 Not clear how to apply to more exotic products 5 Volatility too flat in the wings 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 1 Consistent pricing for different option types not guaranteed 2 Single product type rules of thumb 3 Fits the hedging instruments (plain vanilla options) only reasonably well in the 25-delta range FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 5 / 37 5 Volatility too flat in the wings 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 1 Consistent pricing for different option types not guaranteed 2 Single product type rules of thumb 3 Fits the hedging instruments (plain vanilla options) only reasonably well in the 25-delta range 4 Not clear how to apply to more exotic products FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 5 / 37 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 1 Consistent pricing for different option types not guaranteed 2 Single product type rules of thumb 3 Fits the hedging instruments (plain vanilla options) only reasonably well in the 25-delta range 4 Not clear how to apply to more exotic products 5 Volatility too flat in the wings FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 5 / 37 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 = × OH RR (2) vannaRR 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 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 X c(K; σK ) = c(K; σBS ) + xi (K)[c(Ki ; σi ) − c(Ki ; σBS )] (6) i=1 with @c(K,σBS ) K2 K3 @σ ln K ln K x1(K) = @c(K1,σBS ) ln K2 ln K3 @σ K1 K1 K @c(K,σBS ) ln K ln 3 @σ K1 K x2(K) = @c(K2,σBS ) ln K2 ln K3 @σ K1 K2 @c(K,σBS ) ln K ln K @σ K1 K2 x3(K) = (7) @c(K3,σBS ) ln K3 ln K3 @σ K1 K2 FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 7 / 37 Include vega Include third order Greeks (volunga, vanunga) One-touch probability: which measure? domestic, foreign, physical, mixes? Apply different weights to vega, vanna and volga, e.g. vanna weight = 1+p No-Touch-Probability p, vega and volga weight = 2 , [Fisher, 2007] Not clear how to translate to Double-Touch products Which volatility to use for the touch probability: ATM, average of ATM and barrier vol, derived from equilibrium condition NTvv=NTbs+NTvv*...[Bossens et al., 2010] 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 Include third order Greeks (volunga, vanunga) One-touch probability: which measure? domestic, foreign, physical, mixes? Apply different weights to vega, vanna and volga, e.g. vanna weight = 1+p No-Touch-Probability p, vega and volga weight = 2 , [Fisher, 2007] Not clear how to translate to Double-Touch products Which volatility to use for the touch probability: ATM, average of ATM and barrier vol, derived from equilibrium condition NTvv=NTbs+NTvv*...[Bossens et al., 2010] 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 One-touch probability: which measure? domestic, foreign, physical, mixes? Apply different
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