QUANTITATIVE RESEARCH FOR INFORMATION ONLY
16-17 April , 2018
Fair Pricing in the presence of Local Stochastic Volatility
Francfort 16th-17th April
Adil Reghaรฏ
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The Fair price is the cost of Hedge of an instrument.
We motivate our study on the most actively traded instrument in the equity derivatives space i.e. Autocall Structure.
We present a classical model and highlight the right level of fine tuning parameter in order to price and hedge it in a consistent way.
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Many thanks for the members of my team (Luc Mathieu, Camille Brossette, Marouen Messaoud, Sebastien Mollaret, Claude Muller, who supported this work).
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The risk dynamic for the Autocall:
vega spot ladder
-90,00%
-80,00%
-70,00%
-60,00%
-50,00%
-40,00%
-30,00%
-20,00%
-10,00%
0,00%
0% 20% 40% 60% 80% 100% 120% 140% 160% 180%
spot
Vega Spot Ladder
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[1] Hagan, P., D. Kumar, A. Lesniewski, and D. Woodward (2002, September). โManaging Smile Riskโ Wilmott magazine, 84โ108.
[2] H. Berestycki, J. Busca, I Florent. Asymptotics and calibration of local volatility models.
[3] L. Andersen, R. Brotherton-Ratcliffe, Extended Libor Market Models with Stochastic Volatility, December 2001.
[4] B. Dupire, Pricing with a smile, Risk Magazine, 1994.
[5]A.Reghaรฏ, V. Klaeyle, A. Boukhaffa, โLSV with a mixing weightโ, SSRN 2012.
[6]Adil Reghaรฏ : Quantitative Finance : Back to Basics, Palgrave Mac Millan 2015,
[7]Lorenzo Bergomi : Stochastic Volatility, Wiley 2015,
And many more recent literature.
Litterature
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โข Most papers discuss the calibration of the local volatility with the presentation of several numerical techniques: Forward PDE, Fixed Point Algorithm, Particular Method
โข In [7] the properties of the Skew Stickiness Ratio (SSR) are presented as a key property of the dynamic of the volatility
โข In [6] a mix between implied market and statistics is done to establish a fair pricing approach
Litterature
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The objectives of this presentation are to :
1. Link a dynamic property of the market SSR with its equivalent mixing weight to pick the right model out of a group of Models
2. Derive a formula for the dynamic of the volatility in a LSV model
3. Propose a historical estimation of the parameter4. Share Some other useful techniques in practice
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Part I
Model Description
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We need to depart from the Black & Scholes model when there is a changinggamma sign. We have two alternatives :
1. Local volatility modela) Fits the volatility smile : vanilla options
b) Permits to perform a vega transparisation on all strikes
c) Takes into account the cost of static hedge with vanilla
2. Stochastic Volatilitya) Fits a parametric smile
b) Permits to perform a factor transparisation
c) Takes into account the cost of dynamic vega Hedge
Model Exploration
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The general dynamic :๐๐๐ก
๐๐ก= ๐ ๐ก, ๐๐ก ๐๐๐ก
The pricing equation obtained through the PnL equation :
๐๐๐ฃ โ ๐๐๐ +1
2๐ผ๐ธ
0
๐
๐2๐2๐๐๐
๐๐2(๐2 ๐ก, ๐๐ก โ ๐0
2) ๐๐ก
The result from a static hedging argument :
๐๐๐ฃ โ ๐๐๐ + 0
๐
0
โ ๐๐๐๐
๐ฮฃ ๐พ, ๐ฮฃ ๐พ, ๐ โ ๐0 ๐๐พ๐๐
The induced dynamic of the local volatility : invariant strike x spot
ฮฃ๐1๐พ, ๐ โ ฮฃ๐0
๐พ๐1
๐0, ๐
Local Volatility : key concepts
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FX : low volatility, little skew, mostly curvature
Market exploration
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Commodity : high volatility, positive skew (refuge), moderate curvature
Market exploration
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Equity : moderate volatility, negative skew, moderate curvature
Market exploration
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The general dynamic : ๐๐๐ก
๐๐ก= ๐๐ก๐๐๐ก ,
๐๐๐ก
๐๐ก= ๐ผ๐ ๐๐ก , < ๐๐๐ก , ๐ ๐๐ก > = ๐๐๐ก
The pricing equation obtained through the PnL equation :
๐๐ ๐ฃ = ๐๐๐ +1
2๐ผ๐
0
๐
๐ผ2๐2๐2๐๐๐
๐๐2๐๐ก + ๐ผ๐
0
๐
๐๐ผ๐2๐๐2๐๐๐
๐๐๐๐๐๐ก
The result from a static hedging argument :
๐๐ ๐ฃ = ๐๐๐ +1
6๐๐ผ2
๐2๐๐๐
๐๐๐๐2+
1
2๐ ๐ผ๐๐
๐2๐๐๐
๐๐๐๐๐๐๐๐The induced dynamic of the local volatility : invariant strike / spot
ฮฃ๐1๐พ, ๐ = ฮฃ๐0
๐พ๐0
๐1, ๐
Stochastic Volatility : key concepts
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Equivalent Local Volatility :
๐ผ๐(๐๐ก2|๐๐ก = ๐) =
๐ผ๐(๐๐ก
2๐โ
๐พ2
2 1โ๐2 0๐
๐๐ 2๐๐
0
๐๐๐
2๐๐
)
๐ผ๐(๐
โ๐พ2
2 1โ๐2 0๐
๐๐ 2๐๐
0
๐๐๐
2๐๐
)
With ๐พ = ln๐
๐0+
1
2 0
๐๐๐
2๐๐ โ ๐ 0
๐๐๐ ๐ ๐๐ ๐๐
Short Term Asymptotics ๐ โ 0:
๐ผ2 ln2๐๐ก
๐0+ 2๐๐0๐ผ ln
๐๐ก
๐0+ ๐0
2
Stochastic Volatility : key concepts (II)
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The general dynamic :
๐๐๐ก
๐๐ก= ๐๐ก๐๐๐ก ,
๐๐๐ก
๐๐ก= ๐ผ๐ ๐๐ก , < ๐๐๐ก , ๐ ๐๐ก > = ๐๐๐ก,
๐0+ = ๐0
โ๐๐ฝ๐โ12๐ฝ2
, ๐ = ๐(0,1)
This can be seen as the limit of a fast mean reverting model (proof in [5])
A recent paper in Risk magazine applies it for Heston Model
Hot Start Stochastic Volatility : key concepts
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Market exploration : implied Volatility
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Market exploration : local Volatility
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The general dynamic : it is a special 2 Factor Bergomi Model with:
1st mean reversion to infinity super fast short term
2nd mean reversion to zero super slow term
Strategy to fit the parameters from Bergomi approach!
Fits the market quite nicely with parsimony.
This model is simpler and numerics are more robust.
It is easily extendible with a local volatility component.
Hot Start Stochastic Volatility : key concepts
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Neither the local volatility nor the Stochastic volatility have the right volatility dynamic (we shall quantify this with the SSR)
We shall identify in a group of LSV model the one that is the closest to the observed market dynamic
The Choice of the pure stochastic volatility model needs to be close enough to the market volatility smile in order to make the adjustment due to local volatility as small as possible
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Part II
Market Description
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In the beginning we had two models that match the market. However, they differon the smile dynamic. Stochastic Volatility keeps the moneyness constant whereasthe Local Volatility keeps the product strike by maturity constant. Where doesreality stand?
We measure reality thanks to the SSR Skew Stickiness Ratio defined as follows:
๐ ๐ =๐ผ ๐๐ด๐ก ๐น, ๐ ๐๐๐๐๐ก
๐๐ด๐ก ๐พ, ๐
๐๐๐๐พ ๐พ=๐น๐ผ ๐๐๐๐๐ก
2
Where ๐ด๐ก ๐พ, ๐ is the implied volatility at time ๐ก for residual maturity ๐ andstrike ๐พ
Market Exploration : key concept
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This ratio quantifies the variation of the ATMF volatility when the spot moves, in units of ATMF skew.
The SSR allows classifying the stochastic volatility models:
๐ ๐ = 0 corresponds to the sticky-delta regime,
๐ ๐ = 1 corresponds to the sticky-strike regime,
๐ ๐ = 2 corresponds to the sticky-Dupire regime.
Market Exploration : key concept
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Stochastic Volatility : ๐ ๐ = 0
Local Volatility : ๐ ๐ = 2
Rough Volatility : ๐ ๐ =3
2+ H
Shifted Log Normal : ๐ ๐ = 0
Fast Stochastic Volatility : ๐ ๐ = 0
How does it compare with reality?
SSR Model Results
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Market Exploration : SSR estimation
Underlying SSR
SX5E 130,38%
KOSPI 107,52%
SPX 87,50%
EURUSD 115,50%
GOLD 72,00%
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We could stop here!!
Why are we going to use another level of complexity ?
We take the risk of slower implementation, unstable greeks?
We take the risk to increase our regulatory capital!
We propose a test to decide on increasing complexity or not?
Do we really need to use complex models?
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We introduce a measure that gives us how sensitive is a Product to the difference between two models Local Volatility and Stochastic Volatility which price the same market vanilla but differ in the smile dynamic:
๐ =|๐๐ ๐ฃ โ ๐๐๐ฃ|
๐๐ ๐ฃ + ๐๐๐ฃ
1. When this index is zero, it tells us that the payoff is simply replicable by vanillaEuropean options. In other words, this product depends only on marginaldistributions.
2. When the index approaches one, we have a very toxic product. Indeed, notonly does it tell us that the cheapest model sees no value in the product, butalso that the other models (the most expensive model) sees a lot of value.Typically, the cheapest model is the local volatility model and the mostexpensive the stochastic volatility.
Product x Model Toxicity Index
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Part III
Model Extension
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In the case of non zero toxicity index, we need to build a model that matches vanilla options and the dynamic of the volatility surface?
We introduce a group of Local Stochastic Volatility Models with a ๐ mixing weight
Model Evolution
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The general dynamic LSV (P1):๐๐๐ก
๐๐ก= ๐๐ก๐๐ ๐ก, ๐๐ก ๐๐๐ก
,๐๐๐ก
๐๐ก= โฏ ๐๐ก + ๐๐ผ๐โ๐ ๐โ๐ก ๐ ๐๐ก , < ๐๐๐ก , ๐ ๐๐ก > = ๐๐๐ก
The pricing PDE equation obtained through the PnL equation :
๐๐๐ ๐ฃ = ๐๐๐ฃ + ๐ (๐๐ ๐ฃ โ ๐๐๐ฃ)
The induced dynamic of the local volatility : geometric average
ฮฃ๐1(๐ฟ๐๐) ๐พ, ๐ = ฮฃ๐0
๐พ๐0
๐1, ๐
๐
ฮฃ๐0
๐พ๐1
๐0, ๐
1โ๐
Stochastic Local Volatility : key concepts
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The general dynamic LSV (P2):๐๐๐ก
๐๐ก= ๐๐ก๐๐ ๐ก, ๐๐ก ๐๐๐ก
,๐๐๐ก
๐๐ก= โฏ ๐๐ก + ๐๐ผ๐โ๐ ๐โ๐ก ๐ ๐๐ก , < ๐๐๐ก , ๐ ๐๐ก > = ๐๐๐๐ก
The pricing equation obtained through the PnL equation :
๐๐๐ ๐ฃ = ๐๐๐ฃ + ๐2(๐๐ ๐ฃ โ ๐๐๐ฃ)
The induced dynamic of the local volatility : geometric average
ฮฃ๐1(๐ฟ๐๐) ๐พ, ๐ = ฮฃ๐0
๐พ๐0
๐1, ๐
๐2
ฮฃ๐0
๐พ๐1
๐0, ๐
1โ๐2
Stochastic Local Volatility : key concepts
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Start with a PDE description
๐๐ก๐ข +1
2๐2๐๐
2๐2๐๐๐๐ข +1
2๐2๐ผ2๐๐๐๐ข + ๐2๐๐ผ๐2๐๐๐๐๐๐ข = 0
P2 assumpation makes a balance between volga and vanna
Perturbation theory : we search ๐ข = ๐ข0 + ๐2๐ข1 + โฏ ๐๐๐ ๐๐ = ๐0 + ๐2๐1 + โฏ
๐๐ก๐ข0 +1
2๐0
2๐02๐2๐๐๐๐ข0 = 0 classical local volatility model
๐๐ก๐ข1 +1
2๐2๐0
2๐2๐๐๐๐ข1 +1
2๐2๐1
2๐2๐๐๐๐ข0 +1
2๐ผ2๐๐๐๐ข0 + ๐๐ผ๐2๐0๐๐๐๐ข0 = 0
Feymann-Kacc equation
๐ข1 = ๐ผ 0
๐ 1
2๐2๐1
2๐2๐๐๐๐ข0 +1
2๐ผ2๐๐๐๐ข0 + ๐๐ผ๐2๐0๐๐๐๐ข0
Matching the extreme point with a mixing weight of 100%
Deduce that for any option : ๐ข = ๐ข๐ฟ๐ + ๐2 ๐ข๐๐ โ ๐ข๐ฟ๐
Apply previous to Vanillaฮ๐ฟ๐๐ = ฮ๐ฟ๐ + ๐2 ฮ๐๐ โ ฮ๐ฟ๐
Use general formula for delta of Vanilla ฮ(K, T) = ฮ๐ต๐(๐พ, ๐) + ๐๐๐๐ ๐พ, ๐ ๐๐ฮฃS0(๐0, ๐พ, ๐)
Deduce ๐Sฮฃ๐ฟ๐๐,๐0(๐พ, ๐) = ๐Sฮฃ๐ฟ๐,๐0
(๐พ, ๐) + ๐2 ๐Sฮฃ๐๐,๐0๐พ, ๐ โ ๐Sฮฃ๐ฟ๐,๐0
(๐พ, ๐)
Divide by common term ฮฃ๐๐๐๐๐๐ก(๐0, ๐พ, ๐)
Integrate and obtain the dynamic formula
ฮฃ๐1(๐ฟ๐๐) ๐พ, ๐ = ฮฃ๐0
๐พ๐0
๐1, ๐
๐2
ฮฃ๐0
๐พ๐1
๐0, ๐
1โ๐2
Sketch of a Proof cf[5]
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Stochastic Local Volatility : Numerical Illustration
We compare the ATM volatility evolution for the 3M tenor based on the followingassumptions
1. Exact recomputation under P2 of pricing and then implying the volatility,
2. Proxy under assumption P2,
3. Verification of Proxy under assumption P1,
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Stochastic Local Volatility : Numerical Illustration
We compare the ATM volatility evolution for the 2Y tenor based on the followingassumptions
1. Exact recomputation under P2 of pricing and then implying the volatility,
2. Proxy under assumption P2,
3. Verification of Proxy under assumption P1,
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Under (P2) : Vanna and Volga are well balanced:
The link between the smile dynamic expressed asthe ๐ ๐ and a particular LSV model with a parameter๐ is given by the following formula:
๐2 = 1 โ1
2๐ ๐
Stochastic Local Volatility : key formula
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๐ = 1 corresponds to the full stochastic volatility,
๐ =1
2corresponds to the in between model,
๐ = 0 corresponds to the full local volatility model.
Mixing Weight
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Mixing Weight time series
0
0,2
0,4
0,6
0,8
1
1,2
1,4
13/03/2007 13/03/2008 13/03/2009 13/03/2010 13/03/2011 13/03/2012 13/03/2013 13/03/2014 13/03/2015 13/03/2016 13/03/2017
SX5E MW
0
0,2
0,4
0,6
0,8
1
1,2
31/07/2007 31/07/2008 31/07/2009 31/07/2010 31/07/2011 31/07/2012 31/07/2013 31/07/2014 31/07/2015 31/07/2016 31/07/2017
SPX MW
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Market Exploration : Mixing Weight estimation
Underlying MW
SX5E 59,00%
KOSPI 68,00%
SPX 75,00%
EURUSD 65,00%
GOLD 80,00%
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Mixing Weight Term Structure Time Series
1. Rather than being a universalconstant the mixing weight issomething like a useful statisticalinvariant that describes the right amount of smile dynamic with the volatility surface.
2. It is an essential element of pricingand hedging.
3. The graph represents the evolutionof the term structure of the mixingweight over time.
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Part IV
Multi Asset SV
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Building a multi-dimensional process from the particular mono building blocks needs to be performed with care.
Must at least satisfy the following conditions:
Condition 1: be consistent with the mono dimensional world
Condition 2: be consistent with market prices of ATM basket vanillas
Condition 3: be consistent with mathematical constraints concerning semi definite positiveness of the global correlation matrix
Requirements
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In our setting, we consider m stock prices with their respective volatility.๐๐๐
๐๐= โฆ + ๐๐ ๐๐๐
๐ , ๐ = 1 โฆ ๐
For each stock price, volatility is assumed to follow a stochastic process as defined in the following:
๐๐๐
๐๐= โฆ + ๐ผ๐ ๐๐๐
โ๐, ๐ = 1 โฆ ๐
Driving Brownian Motions of both processes are assumed to be correlated, which allows us to write:
๐๐๐โ๐
= ๐๐๐๐ ๐๐๐
๐ + 1 โ ๐๐๐๐ 2
๐๐๐๐
Where ๐๐๐ and ๐๐
๐ are independent random variables. The parameter ๐๐๐๐ is
essentially driving the mono dimensional LSV calibration.
Mathematical Setting
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To fit the Condition 1 (be consistent with the mono dimensional case) we shall build a correlation matrix Spot โ Vol x Spot โ Vol we shall build a global correlation matrix which does not move the parameter ๐๐
๐๐. Also to fit the condition 3, we propose a parametric correlation matrix that is the result of process.
Our idea to fit both conditions 1 and 3 is to decompose all factors ๐๐๐, for i= 1โฆm
according to:
๐๐๐๐ = ๐๐๐๐ + 1 โ ๐๐๐๐ต๐
๐ , ๐ = 1 โฆ ๐
We have introduced only one new parameter ๐๐, with ๐ต๐๐ and ๐ independent.
๐๐
๐๐๐๐๐ ๐๐
๐๐๐๐๐๐๐
๐๐๐๐๐๐๐
๐๐ ๐๐๐๐๐๐
๐๐๐๐๐๐๐ + 1 โ ๐๐
๐๐ 21 โ ๐๐
๐๐ 2๐๐ + 1 โ ๐๐ 1 ๐ = ๐
Fitting conditions 1 and 3
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Multi SV decorrelates and makes the basket options cheaper than theircorresponding LV price.
How are we doing with Condition 2
Comparison of basket prices for a model satisfying just conditions 1 and 3.
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We use the lambda mechanism ๐๐,๐๐,๐ โ 1 โ ๐๐ ๐๐,๐
๐,๐ + ๐๐ to recorrelate the underlyings and here are the results:
Fitting Condition 2 by recorrelating the spots
Comparison of basket prices for a model
satisfying conditions 1, 2 and 3.
what is remarkable is that not only do we match the ATM basket prices but also the other strikes 90% and 110%.
This result is very interesting as it say that matching the ATM basket prices with this particular matrix of correlation ends up by matching the basket distribution;
In some cases, this could beapplied to worst of depending on the hedging instruments.
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Part V
Conclusion
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Conclusion Mono
1. In case of non zero toxicity index:
๐ =|๐๐ ๐ฃ โ ๐๐๐ฃ|
๐๐ ๐ฃ + ๐๐๐ฃ
2. Estimate Skew Stickiness Ratio:
๐ ๐ =๐ผ ๐๐ด๐ก ๐น, ๐ ๐๐๐๐๐ก
๐๐ด๐ก ๐พ, ๐
๐๐๐๐พ ๐พ=๐น๐ผ ๐๐๐๐๐ก
2
3. Estimate the mixing weight:
๐2 = 1 โ1
2๐ ๐
4. Pricing Proxy Formula:๐๐๐ ๐ฃ = ๐๐๐ฃ + ๐2(๐๐ ๐ฃ โ ๐๐๐ฃ)
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Conclusion Multi
1. Introduce idiosyncratic volatility factor ๐๐
2. Build a global correlation Matrix
3. Recorrelate to match ATM and remarkably basket skew
๐๐,๐๐,๐ โ 1 โ ๐๐ ๐๐,๐
๐,๐ + ๐๐
Build a multi SV model which is
1. consistent with Mono LSV
2. Fitting basket options or any other hedginginstruments
3. Mathematically basedon a semi definitepostive matrix
Adil REGHAINatixis47 Quai d'Austerlitz,75013 Paris FranceTel: +33(0)[email protected]
http://www.palgrave.com/page/detail/quantitative-finance-/?isb=9781137414496
NATIXIS Banque de Financement et dโInvestissement - 47, quai dโAusterlitz โ BP 4 - 75060 Paris Cedex 02 www.natixis.com
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