Sum m ary 1 . Rapid review of literature 2 . Local Stochastic - PowerPoint PPT Presentation

LSV : « when statistics and derivatives meet » 2 nd March 2012 Adil REGHAI Joint W ork w ith Vincent KLAEYLE & Amine BOUKHAFFA – NATIXIS Equity & Commodity QUANT TEAM

Sum m ary 1 . Rapid review of literature 2 . Local Stochastic Volatility : The best of tw o w orlds • Local Volatility characteristics • Stochastic Volatility characteristics • Mixed LSV diffusion • Full LSV 3 . Fast Mean Reverting LSV m odel for Precious Metals • Vanna Volga Proxy • Fast Stochastic Volatility • Fast LSV 4 . Num erical results – Com parison w ith Full LSV 2 nd March 2 0 1 2 2

Sum m ary 1 . Rapid review of literature 2 . Local Stochastic Volatility : The best of tw o w orlds Local Volatility characteristics • • Stochastic Volatility characteristics • Mixed LSV diffusion • Full LSV 3 . Fast Mean Reverting LSV m odel for Precious Metals • Vanna Volga Proxy • Fast Stochastic Volatility • Fast LSV 4 . Num erical results – Com parison w ith Full LSV 2 nd March 2 0 1 2 3

Review of literature Since Dupire ( 9 4 ) Local Stochastic Volatility diffusion has been w ell studied in literature. • Dupire 1 9 9 4 : Constrain over the expectation of local variance • Bergom i 2 0 0 9 : Spot/ Sm ile dynam ic • Labordere 2 0 0 9 : Speeding up LV calibration in LSV fram ew orks • Blacher 2 0 0 1 : Calibration schem e for a particular LSV process • Madan et al. 2 0 0 7 : Com parisons LSV vs LV pricings • Andreasen 2 0 1 0 : Consistency for calibration/ pricing under LSV fram ew ork • Reghai Klaeyle Boukhaffa 2 0 1 2 LSV w ith a m ixing w eight LSV m odels are m ore an engineering issue than a theoretical one. • Draw backs of LSV m odel are know n : – Tim e consum ptions – I nstabilities These results w ere possible thanks to the w ork of Philippe Balland, Ben Nasatyr and Adil Reghaï w ho began this kind of analysis back in 2 0 0 4 for the FX m arkets 2 nd March 2 0 1 2 4

General Rem ark Tw o types of quants 1 . Derivatives Pricing : calibration / hedging Fit the photography and prices non liquid instrum ents as a function of liquid ones, typically vanilla options Extrapolate the present 2 . Forecast Quants : statistics Data crunching, strategies optim ization Predict the Future Our w ork com bines the tw o approaches • Som e calibration : consistence w ith the hedging instrum ents • Som e statistics : consistence w ith the cost of the hedge unw ind • And a bit of engineering 2 nd March 2 0 1 2 5

How to judge a m odel? The perform ance of a m odel is m easured w ith it ability to explain the PnL I n a context of delta vega hedging ( ignoring financing term s) , ⎛ ⎞ Usual Gam m a 2 ∂ π ⎛ ∆ ⎞ 2 1 S ⎜ ⎟ ∆ = − σ ∆ ⎜ ⎟ 2 2 explanation PnL S t ⎜ ⎟ ∂ Model 2 ⎝ ⎠ 2 S S ⎝ ⎠ Volga realization ⎛ ⎞ 2 ∂ π ⎛ ∆ σ ⎞ 2 1 ⎜ ⎟ + σ − α ∆ ⎜ ⎟ 2 2 t ⎜ ⎟ σ ∂ σ ⎝ ⎠ Model 2 2 ⎝ ⎠ Vanna realization ∂ π ∆ ∆ σ 2 ⎛ ⎞ S + σ − ρ σ α ∆ ⎜ ⎟ S t ∂ σ ∂ σ Model Model Model ⎝ ⎠ S S 2 nd March 2 0 1 2 6

Sum m ary 1 . Rapid review of literature 2 . Local Stochastic Volatility : The best of tw o w orlds • Local Volatility characteristics • Stochastic Volatility characteristics • Mixed LSV diffusion • Full LSV 1 . Fast Mean Reverting LSV m odel for Precious Metals 4 . Vanna Volga Proxy 5 . Fast Stochastic Volatility 6 . Fast LSV Num erical results – Com parison w ith Full LSV 2 nd March 2 0 1 2 7

Local Volatility Local Volatility m odels • Dynam ic: dF = σ t ( , ) t F dW ( 3 ) t t F t • Pricing by replication : Vega Bucket • Sm ile Dynam ic ( sticky local volatility) ⎛ ⎞ F ⎜ ⎟ Σ ≈ Σ 1 ( , , ) , , ( 4 ) F K T F K T ⎜ ⎟ 1 0 ⎝ ⎠ F 0 • The m odel hides Vanna/ Volga costs into Gam m a 2 nd March 2 0 1 2 8

Local Volatility Local Volatility Sm ile dynam ic (very strong correlation between spot and RR) 2 nd March 2 0 1 2 9

Stochastic Volatility Stochastic Volatility m odels • Dynam ic : Steepness dF = σ t dW t t F t ( 5 ) σ Controls SS σ = λ + ν ∞ ln( ) ln d dt dB σ t t t Controls RR < > = ρ , d W B dt t • Sm ile Dynam ic ( sticky stochastic volatility) ⎛ ⎞ F ⎜ ⎟ Σ ≈ Σ 0 ( , , ) , , F K T F K T ⎜ ⎟ ( 6 ) 1 0 ⎝ ⎠ F 1 The m odel account for stand alone Vanna/ Volga risks • due to Vega re-hedges • Most often w ell studied ( sm ooth and fast calibration) 2 nd March 2 0 1 2 10

Stochastic Volatility Stochastic Volatility Sm ile dynam ic (very low correlation between spot and RR) 2 nd March 2 0 1 2 11

Sum m ary Pros Cons -Dynam ic not observed in practice Local Volatility - Allow Vega Bucket -Does not account for Vanna-Volga - Low dim ension -Too m uch RR/ Spot correlation Stochastic -No Vega Bucket -Account for Vanna since sm ile is not Volatility Volga perfectly fitted -Provides a dynam ic -Too low RR/ Spot process for volatility correlation 2 nd March 2 0 1 2 12

Quant Sum m ary Local Volatility Stochastic Volatility = σ / dS S dW = σ t Dynam ic / ( , ) dS S t S dW ( ) σ σ = θ − λ σ + α σ / ln d dt dW t t τ ∂ 2 1 P Pricing by ∫ = + α σ 2 2 BS P P dt ∂ σ SV BS 2 ⎛ ∂ ⎞ ( ) 2 2 1 τ ( ) P dynam ic ∫ 2 , = + ⎜ σ − σ ⎟ 0 2 BS P P E t S dt ⎜ ⎟ τ τ ∂ ∂ ∂ LV BS BS 2 2 ⎝ 2 ⎠ 0 S P ( ) P ∫ ∫ Hedge + ρασ + θ − λ σ 2 BS BS ln S dt dt ∂ σ ∂ ∂ σ S 0 0 ∂ ∞ P ( ( ) ) ∫ ∫ T = + Σ − σ BS , Pricing by P P k t dkdt ( ) ∂ Σ LV BS BS , 0 0 k t Vanna Volga Pricing Replication Significance of Vega KT ⎛ ⎞ ⎛ ⎞ S S ( ) ( ) ⎜ ⎟ ⎜ ⎟ Σ = Σ Σ = Σ Sm ile Dynam ic 1 0 , , , , S K S K S K S K ⎜ ⎟ ⎜ ⎟ 1 0 1 0 ⎝ ⎠ ⎝ ⎠ S S 0 1 2 nd March 2 0 1 2 13

LSV : Best of tw o w orlds : New asym ptotic Result Mixing them together gives LSV m odel: • Dynam ic dS = σ t ( , ) f t S dW t t t S t ( 7 ) ⎛ ⎞ σ ⎜ ⎟ σ = λ ∞ + γ ν ln( ) ln d dt dB ⎜ ⎟ σ t t ⎝ ⎠ t < > = ρ , d W B dt t • Sm ile Dynam ic ( tune the RR vs Spot correlation) − γ 2 ( 1 ) γ 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ F F ⎜ ⎟ ⎜ ⎟ Σ → Σ ⎜ Σ ⎟ ⎜ ⎟ LSV MKT MKT 0 : , , , T F T K ⎜ T K ⎟ ( 8 ) ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ F F ⎝ ⎠ 0 • Allow Vega Bucket thanks to LV’s perfect fit on vanillas 2 nd March 2 0 1 2 14

LSV : capture the best of tw o w orlds Local Stochastic Volatility Sm ile dynam ic 2 nd March 2 0 1 2 15

LSV : Best of tw o w orlds : New asym ptotic Result Mixing them together gives LSV m odel: • Account for Vanna/ Volga re-hedging costs • Calibration schem e ( m ixing tw o m dls explaining sm ile) – Calibrate the pure Stochadtic Volatility model – Input the mixing weight parameter - Statistical calibration on RR vs Spot correlation - If market quotes for exotics: ( ) − γ − − − ≈ + γ − LSV LSV 0 % 2 LSV 100 % LSV 0 % P P P P ( ) ( 9 ) − γ ≈ + γ − LSV LV 2 SV LV P P P P – Calibrate Local Volatility on Vanillas (using 2D PDE on joint density) [ ] 2 σ = σ = 2 Dupire 2 ( , ) ( , ) | t K IE f t S S K ( 1 0 ) t t t 2 nd March 2 0 1 2 16

LSV : capture the best of tw o w orlds Facts over Stochastic Volatility Model • High level of m ean reversion Given 1y ATM volatility (XAU/ USD) we find a mean reversion parameter at (Bloomberg data Jul 2009-> Dec 2009) ∆ σ ≈ − λ σ + λ σ + ATM ln ln ( 1 1 ) X ∞ σ ∆ ATM t ATM ≈ ( 0 , ) X N x Strong Mean reversion Half-life less that one month λ = 9 . 7 σ = 38 . 5 % ∞ 2 nd March 2 0 1 2 17

LSV : Best of tw o w orlds Full LSV : dS = µ + σ t ( , ) dt f t S dW t t t t S t ( 1 2 ) σ σ = λ + γω ∞ ln ln d dt dB σ t t t < > = ρ , d W B dt t Tim e consum ption of 2 D pricing m ethods 2 nd March 2 0 1 2 18

Sum m ary 1 . Rapid review of literature 2 . Local Stochastic Volatility : The best of tw o w orlds • Local Volatility characteristics • Stochastic Volatility characteristics • Mixed LSV diffusion • Full LSV 3 . Fast Mean Reverting LSV m odel for Precious Metals • Vanna Volga Proxy • Fast Stochastic Volatility • Fast LSV 4 . Num erical results – Com parison w ith Full LSV 2 nd March 2 0 1 2 19

Fast Mean Reverting LSV m odel Vanna Volga Proxy • One the m ost popular Stochastic Volatility pricing tricks • Trading oriented ( effective decom position of vega hedge) • Not very stable outside the m ost com m on range of strikes ( problem on form er trades – 2 5 Call 1 y w ill becom e a 5 Delta at som e point) ( ) ∑ = + − BS BS ( ) ( ) Price Price x Call K Call K ( 1 3 ) i i i Compute xi so that we vega/ vanna/ volga hedge the option when smile tends to flatten (remaining stochastic) 2 nd March 2 0 1 2 20