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

LSV : « when statistics and derivatives meet »

2nd March 2012

Adil REGHAI Joint W ork w ith Vincent KLAEYLE & Amine BOUKHAFFA – NATIXIS Equity & Commodity QUANT TEAM

2 nd March 2 0 1 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

2 nd March 2 0 1 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

3

2 nd March 2 0 1 2

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

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2 nd March 2 0 1 2

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

• f 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

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2 nd March 2 0 1 2

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) ,

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∆ − ∆ ∆ ∂ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∆ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∆ ∂ ∂ = ∆ t S S S S t t S S S S PnL

Model Model Model Model Model

α σ ρ σ σ σ σ π α σ σ σ σ π σ π

2 2 2 2 2 2 2 2 2 2 2

2 1 2 1

Volga realization Vanna realization Usual Gam m a explanation

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2 nd March 2 0 1 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

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

7

2 nd March 2 0 1 2

Local Volatility

Local Volatility m odels

• Dynam ic:
• Pricing by replication : Vega Bucket
• Sm ile Dynam ic ( sticky local volatility)
• The m odel hides Vanna/ Volga costs into Gam m a

t t t t

dW F t F dF ) , ( σ =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Σ ≈ Σ T F F K F T K F , , ) , , (

1 1

( 3 ) ( 4 )

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2 nd March 2 0 1 2

Local Volatility

Local Volatility Sm ile dynam ic

(very strong correlation between spot and RR)

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2 nd March 2 0 1 2

Stochastic Volatility

Stochastic Volatility m odels

• Dynam ic :
• Sm ile Dynam ic ( sticky stochastic volatility)
• 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)

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Σ ≈ Σ T F F K F T K F , , ) , , (

1 1

dt B W d dB dt d dW F dF

t t t t t t t t

, ln ) ln( ρ ν σ σ λ σ σ = > < + = =

∞

Controls RR Controls SS Steepness

( 5 ) ( 6 )

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2 nd March 2 0 1 2

Stochastic Volatility

Stochastic Volatility Sm ile dynam ic

(very low correlation between spot and RR)

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2 nd March 2 0 1 2

Sum m ary

Pros Cons

Local Volatility

• Allow Vega Bucket
• Low dim ension
• Dynam ic not
• bserved in practice
• Does not account

for Vanna-Volga

• Too m uch RR/ Spot

correlation

Stochastic Volatility

• Account for Vanna

Volga

• Provides a dynam ic

process for volatility

• No Vega Bucket

since sm ile is not perfectly fitted

• Too low RR/ Spot

correlation

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2 nd March 2 0 1 2

Quant Sum m ary

Local Volatility Stochastic Volatility Dynam ic Pricing by dynam ic Hedge Pricing by Replication Significance of Vega KT Vanna Volga Pricing Sm ile Dynam ic

( )

σ

α σ λ θ σ σ σ

t t

dW dt d dW S dS + − = = ln / /

t

dW S t S dS ) , ( / σ =

( )

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − + =

∫

dt S P S t E P P

BS BS BS LV τ

σ σ

2 2 2 2 ,

2 1

( )

dt P dt S P S dt P P P

BS BS BS BS SV

∫ ∫ ∫

∂ ∂ − + ∂ ∂ ∂ + ∂ ∂ + =

τ τ τ

σ σ λ θ σ ρασ σ σ α

2 2 2 2 2 2

ln 2 1

( ) ( ) ( )

∫ ∫

∞

− Σ Σ ∂ ∂ + = , ,

T BS BS BS LV

dkdt t k t k P P P σ

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Σ = Σ

1 1

, , S S K S K S

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Σ = Σ

1 1

, , S S K S K S

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LSV : Best of tw o w orlds : New asym ptotic Result

2 nd March 2 0 1 2

Mixing them together gives LSV m odel:

• Dynam ic
• Sm ile Dynam ic ( tune the RR vs Spot correlation)
• Allow Vega Bucket thanks to LV’s perfect fit on vanillas

2 2

MKT ) 1 ( MKT LSV

, , , :

γ γ

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Σ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Σ → Σ

−

F F K T F F K T F T

dt B W d dB dt d dW S t f S dS

t t t t t t t t t

, ln ) ln( ) , ( ρ ν γ σ σ λ σ σ = > < + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = =

∞

( 7 ) ( 8 )

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2 nd March 2 0 1 2

LSV : capture the best of tw o w orlds

Local Stochastic Volatility Sm ile dynam ic

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2 nd March 2 0 1 2

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:

– Calibrate Local Volatility on Vanillas (using 2D PDE on joint

density)

( )

LV SV 2 LV LSV

P P P P − + ≈

−

γ

γ

( )

% LSV % 100 LSV 2 % LSV LSV − − − −

− + ≈ P P P P γ

γ

( 9 )

[ ]

K S S t f IE K t

t t t

= = | ) , ( ) , (

2 2 2 Dupire

σ σ

( 1 0 )

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2 nd March 2 0 1 2

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)

% 5 . 38 7 . 9 = =

∞

σ λ

Strong Mean reversion Half-life less that

• ne month

) , ( ln ln x N X X t

ATM ATM ATM

≈ + + − ≈ ∆ ∆

∞

σ λ σ λ σ σ

( 1 1 )

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2 nd March 2 0 1 2

LSV : Best of tw o w orlds

Full LSV :

Tim e consum ption of 2 D pricing m ethods

dt B W d dB dt d dW S t f dt S dS

t t t t t t t t t t

ρ γω σ σ λ σ σ µ = > < + = + =

∞

, ln ln ) , (

( 1 2 )

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2 nd March 2 0 1 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

19

2 nd March 2 0 1 2

Fast Mean Reverting LSV m odel

Vanna Volga Proxy

• One the m ost popular Stochastic Volatility pricing tricks
• Trading
• riented

( effective decom position

• f

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)

Compute xi so that we vega/ vanna/ volga hedge the option when smile tends to flatten (remaining stochastic)

( )

∑

− + = ) ( ) (

i i i

K K x

BS BS

Call Call Price Price

( 1 3 )

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2 nd March 2 0 1 2

Fast Mean Reverting LSV m odel

Vanna Volga Proxy

• Using perturbation techniques, w e can find the sam e

vanna/ volga cost ( w ith lim it to a flattening sm ile) :

where

• W hich is also a lim ited developm ent of pricing form ula

( )

2 2 BS

Price ε ε

• VV

P + + =

) , ( Vanna 2 1 ) , ( Volga 3 1 2 1

BS 2 BS 2 2

S TS S T VV ω ρσ σ ω + =

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = Σ = =

− − T Y T T X T

e Fe E

2 2

2 1 2 1

, (

ω ω β β

σ Fwd Black Price

( 1 4 ) ( 1 5 ) ( 1 6 )

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Fast Mean Reverting LSV m odel

2 nd March 2 0 1 2

Herm ite Pricing ( Fast Stochastic Volatility)

• I f w e take a close look at m odel:
• I n case of Fast m ean-reversion on both Volatility and

Drift

t t t t t t t t t t t

dB dt d dZ dt d dW dt S dS ω σ σ λ σ β µ µ λ µ σ µ + = + − = + =

∞

ln ln ) (

2 ref 1 2 2 2 2 1 2 2 1

2 1 2 2 1 2 ν λ ω α λ β ω λ β λ → → ∞ → ∞ → ∞ → ∞ → with

( 1 7 ) ( 1 8 )

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Fast Mean Reverting LSV m odel

2 nd March 2 0 1 2

Herm ite Pricing ( Fast Stochastic Volatility)

• Then, both Volatility and Drift reaches their stationary

regim es w hich w e suppose independents

• Now , if w e consider those regim es as state variables

( relevant if vega duration is w ell sam pled around one particular date) , then pricing any option under those assum ptions gives:

T T Y t t

e T X

2

2 1 ref law 2 ref law

2 1

ν ν

σ σ α α µ µ

−

≈ − + ≈

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = Σ = =

− − T Y T T X T

e Fe E

2 2

2 1 2 1

, (

ω ω β β

σ Fwd Black Price

( 1 9 ) ( 2 0 )

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Fast Mean Reverting LSV m odel

2 nd March 2 0 1 2

Herm ite Pricing ( Fast Stochastic Volatility)

• Sm ile generated com pared to Vanna Volga

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Fast Mean Reverting LSV m odel

2 nd March 2 0 1 2

Herm ite Pricing ( Fast Stochastic Volatility)

• Sm ile generated

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Fast Mean Reverting LSV m odel

2 nd March 2 0 1 2

Fast Local Stochastic Volatility

• W e can generate a very fast/ robust Stochastic Volatility

pricing ( integration is perform ed w ith 2 D-Gauss Herm ite)

• Now w e can extend our m odel into a LSV m odel ( w ith

stochastic drift)

• And thus, under fast m ean reversion w e have:

t t t t t t t t t t t t

dB dt d dZ dt d dW S t f dt S dS γω σ σ λ σ γβ µ µ λ µ σ µ + = + − = + =

∞

ln ln ) ( ) , (

2 ref 1

( 2 1 ) ( 2 2 )

t t t t

dW S t f T T Y dt T X S dS ) , ( ) 2 1 exp( 2 1

2 2 2 2 ref

ω γ γω σ β γ γβ µ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + =

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Fast Mean Reverting LSV m odel

2 nd March 2 0 1 2

Fast Local Stochastic Volatility

• On W hich w e can com pute local volatility redress using

Dupire’s generalized form ula: W here conditional expectations are com puted w ith a few 1 D forw ard PDE for each Gauss-Herm ite integration nodes

[ ]

) , ( | ) , ( ) 2 exp(

2 2 2 2 2

K t K S S t f T T Y E

t t Dupire

σ ω γ γω σ = = −

( 2 3 )

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Fast Mean Reverting LSV m odel

2 nd March 2 0 1 2

Calibration pattern:

• Calibrate the pure Herm ite pricing m odel
• Tem per the param eters w ith a m ixing w eight value
• Launch the Local Volatility calibration thanks to the

quick com putation

• f

conditional expectations in Generalized Dupire Form ula

• Rem arks:

– If

: Pure LV pricing.

– If

: Full SV pricing with LV minute adjustment for perfect vanilla match

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = ) 2 1 exp( ), 2 1 exp( ) (

2 2 2 2

T Y T T X T F E ω γ γω σ β γ γβ γ Black Hermite

% = γ % 100 = γ

( 2 4 )

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Fast Mean Reverting LSV m odel

2 nd March 2 0 1 2

Mixing W eight calibration:

• I f you have access to m arket prices: im plicit the m ixing

w eight param eter should be straightforw ard

– Market Quotes – Totem Consensus

• I f not, just use statistical estim ation

– the correlation between RR25 and Spot. – Then build synthetic market DNT quotes with Full LSV – Proceed Fast LSV mixing weight calibration through synthetic

quotes

29

Mixing W eight Estim ation Equity W orld

2 nd March 2 0 1 2

I ndices

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Mixing W eight Estim ation Equity W orld

2 nd March 2 0 1 2

Stocks

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Mixing W eight Estim ation Com m odities

2 nd March 2 0 1 2

Gold : February 2 0 0 3 Gold : February 2 0 1 2

Volatility Prediction Weight to minimize distance between Realised Implied volatility and predicted one

0.4 0.5 0.6 0.7 0.8 0.9 0.5 1 1.5 2 2.5

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Mixing W eight Estim ation Forex

2 nd March 2 0 1 2

EURUSD : February 2 0 1 2

33

2 nd March 2 0 1 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

34

2 nd March 2 0 1 2

Num erical Application

Managing the Risk of Am erican Digital options

TV = Black-Scholes w ith ATM Volatility Fast LSV w ith m ixing w eight 8 0 % Full LSV w ith m ixing w eight 8 0 % Fast LSV and Full LSV returns pretty m uch sam e prices and delta

35

2 nd March 2 0 1 2

Num erical Application

TV = Black-Scholes w ith ATM Volatility Fast LSV w ith m ixing w eight 8 0 % Full LSV w ith m ixing w eight 8 0 % Fast LSV and Full LSV returns pretty m uch sam e greeks but Fast LSV is m uch sm oother

36

2 nd March 2 0 1 2

Num erical Application

TV = Black-Scholes w ith ATM Volatility Fast LSV w ith m ixing w eight 8 0 % Full LSV w ith m ixing w eight 8 0 % Fast LSV and Full LSV returns pretty m uch sam e greeks but Fast LSV is m uch sm oother Rem ark: Greeks a price w ith re-calibration of m arket data

37

2 nd March 2 0 1 2

Num erical Application

Tim e consum ption for both LSV m odels

DNT 1 y ( Price/ Delta/ Gam m a/ Vega/ Vanna/ Volga) ( 9 calibrations – 9 pricings)

Full LSV 1 9 0 s Fast LSV 8 .5 s Fast LSV returns pretty m uch the sam e prices and greeks that Full LSV but under 1 sec per pricing

38

2 nd March 2 0 1 2

Num erical Application

Managing the Risk of Am erican Digital options

Spot ladders of DNT 1y (XAU/ USD) 85% -140% within Fast LSV models Our quote at MW = 80%

39

2 nd March 2 0 1 2

Num erical Application

Managing the Risk of Am erican Digital options

Vega Bucket in LV for a DNT 1y (XAU/ USD) 85% -140% (as we see no fly exposure)

40

2 nd March 2 0 1 2

Num erical Application

Managing the Risk of Am erican Digital options

Vega Bucket in Fast LSV for a DNT 1y 85% -140% (as expected strong fly exposure)

41

2 nd March 2 0 1 2

Conclusion

Fast LSV w as designed as a production tool

• Pricing/ Hedging and relevant PnL explanation
• Gather a financial dynam ic m odel w ith the practice rule

( vanna volga proxy)

• Enhance m any tim es stability of business constrains to

produce a reliable tool

• Allow tim e consum ing LSV fram ew ork for flow business

as w ell.

• Mapping betw een Full LSV and Fast LSV m odel allow

to use either of them for estim ating system ic param eter Mixing W eight.

• W hich is a param eter you expect not to change on intra

day basis

42

2 nd March 2 0 1 2

Conclusion

Thank you for your attention

43

2 nd March 2 0 1 2

References

Dupire – Pricing w ith a sm ile – 1 9 9 4 Bergom i – Sm ile Dynam ic I V – 2 0 0 9 Labordere – Calibration of local stochastic volatility m odels to m arket sm iles – 2 0 0 9 Blacher – A new approach for designing and calibrating stochastic volatility m odels for optim al delta-vega hedging of exotics – 2 0 0 1 Ren, Madan, Qian Qian – Calibrating and pricing w ith em bedded local volatility m odels – 2 0 0 7 Andreasen, Huge – Finite Difference Based Calibration and Sim ulation - 2 0 1 0 Mercurio, Castagna – The Vanna-Volga Method for I m plied Volatilities – 2 0 0 7 More details on this work can be found in SSRN Reghaï-Klaeyle-Boukhaffa – SSRN-LSV Models w ith a Mixing W eight - 2 0 1 2

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