QUANTITATIVE RESEARCH FOR INFORMATION ONLY

16-17 April , 2018

Fair Pricing in the presence of Local Stochastic Volatility

Francfort 16th-17th April

Adil Reghaï

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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). “Managing Smile Risk” Wilmott magazine, 84–108.

[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.Reghaï, V. Klaeyle, A. Boukhaffa, “LSV with a mixing weight”, SSRN 2012.

[6]Adil Reghaï : 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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• Most papers discuss the calibration of the local volatility with the presentation of several numerical techniques: Forward PDE, Fixed Point Algorithm, Particular Method

• In [7] the properties of the Skew Stickiness Ratio (SSR) are presented as a key property of the dynamic of the volatility

• 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 :𝑑𝑆𝑡

𝑆𝑡= 𝜎 𝑡, 𝑆𝑡 𝑑𝑊𝑡

The pricing equation obtained through the PnL equation :

𝜋𝑙𝑣 ≈ 𝜋𝑏𝑠 +1

2𝔼𝑸

0

𝑇

𝑆2𝜕2𝜋𝑏𝑠

𝜕𝑆2(𝜎2 𝑡, 𝑆𝑡 − 𝜎0

2) 𝑑𝑡

The result from a static hedging argument :

𝜋𝑙𝑣 ≈ 𝜋𝑏𝑠 + 0

𝑇

0

∞ 𝜕𝜋𝑏𝑠

𝜕Σ 𝐾, 𝑇Σ 𝐾, 𝑇 − 𝜎0 𝑑𝐾𝑑𝑇

The induced dynamic of the local volatility : invariant strike x spot

Σ𝑆1𝐾, 𝑇 ≈ Σ𝑆0

𝐾𝑆1

𝑆0, 𝑇

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 : 𝑑𝑆𝑡

𝑆𝑡= 𝜎𝑡𝑑𝑊𝑡 ,

𝑑𝜎𝑡

𝜎𝑡= 𝛼𝑑 𝑊𝑡 , < 𝑑𝑊𝑡 , 𝑑 𝑊𝑡 > = 𝜌𝑑𝑡

The pricing equation obtained through the PnL equation :

𝜋𝑠𝑣 = 𝜋𝑏𝑠 +1

2𝔼𝑄

0

𝑇

𝛼2𝜎2𝜕2𝜋𝑏𝑠

𝜕𝜎2𝑑𝑡 + 𝔼𝑄

0

𝑇

𝜌𝛼𝜎2𝑆𝜕2𝜋𝑏𝑠

𝜕𝜎𝜕𝑆𝑑𝑡

The result from a static hedging argument :

𝜋𝑠𝑣 = 𝜋𝑏𝑠 +1

6𝑇𝛼2

𝜕2𝜋𝑏𝑠

𝜕𝑙𝑛𝜎2+

1

2𝑇 𝛼𝜌𝜎

𝜕2𝜋𝑏𝑠

𝜕𝑙𝑛𝑆𝜕𝑙𝑛𝜎The induced dynamic of the local volatility : invariant strike / spot

Σ𝑆1𝐾, 𝑇 = Σ𝑆0

𝐾𝑆0

𝑆1, 𝑇

Stochastic Volatility : key concepts

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Equivalent Local Volatility :

𝔼𝑄(𝜎𝑡2|𝑆𝑡 = 𝑆) =

𝔼𝑄(𝜎𝑡

2𝑒−

𝐾2

2 1−𝜌2 0𝑇

𝜎𝑠2𝑑𝑠

0

𝑇𝜎𝑠

2𝑑𝑠

)

𝔼𝑄(𝑒

−𝐾2

2 1−𝜌2 0𝑇

𝜎𝑠2𝑑𝑠

0

𝑇𝜎𝑠

2𝑑𝑠

)

With 𝐾 = ln𝑆

𝑆0+

1

2 0

𝑇𝜎𝑠

2𝑑𝑠 − 𝜌 0

𝑇𝜎𝑠𝑑 𝑊𝑠𝑑𝑠

Short Term Asymptotics 𝑇 → 0:

𝛼2 ln2𝑆𝑡

𝑆0+ 2𝜌𝜎0𝛼 ln

𝑆𝑡

𝑆0+ 𝜎0

2

Stochastic Volatility : key concepts (II)

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The general dynamic :

𝑑𝑆𝑡

𝑆𝑡= 𝜎𝑡𝑑𝑊𝑡 ,

𝑑𝜎𝑡

𝜎𝑡= 𝛼𝑑 𝑊𝑡 , < 𝑑𝑊𝑡 , 𝑑 𝑊𝑡 > = 𝜌𝑑𝑡,

𝜎0+ = 𝜎0

−𝑒𝛽𝑍−12𝛽2

, 𝑍 = 𝑁(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:

𝑅𝑇 =𝔼 𝑑𝛴𝑡 𝐹, 𝑇 𝑑𝑙𝑛𝑆𝑡

𝑑𝛴𝑡 𝐾, 𝑇

𝑑𝑙𝑛𝐾 𝐾=𝐹𝔼 𝑑𝑙𝑛𝑆𝑡

2

Where 𝛴𝑡 𝐾, 𝑇 is the implied volatility at time 𝑡 for residual maturity 𝑇 andstrike 𝐾

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:

𝑅𝑇 = 0 corresponds to the sticky-delta regime,

𝑅𝑇 = 1 corresponds to the sticky-strike regime,

𝑅𝑇 = 2 corresponds to the sticky-Dupire regime.

Market Exploration : key concept

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Stochastic Volatility : 𝑅𝑇 = 0

Local Volatility : 𝑅𝑇 = 2

Rough Volatility : 𝑅𝑇 =3

2+ H

Shifted Log Normal : 𝑅𝑇 = 0

Fast Stochastic Volatility : 𝑅𝑇 = 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:

𝜙 =|𝜋𝑠𝑣 − 𝜋𝑙𝑣|

𝜋𝑠𝑣 + 𝜋𝑙𝑣

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 𝝐 mixing weight

Model Evolution

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The general dynamic LSV (P1):𝑑𝑆𝑡

𝑆𝑡= 𝜎𝑡𝑓𝜖 𝑡, 𝑆𝑡 𝑑𝑊𝑡

,𝑑𝜎𝑡

𝜎𝑡= ⋯ 𝑑𝑡 + 𝜖𝛼𝑒−𝜆 𝑇−𝑡 𝑑 𝑊𝑡 , < 𝑑𝑊𝑡 , 𝑑 𝑊𝑡 > = 𝜌𝑑𝑡

The pricing PDE equation obtained through the PnL equation :

𝜋𝑙𝑠𝑣 = 𝜋𝑙𝑣 + 𝜖 (𝜋𝑠𝑣 − 𝜋𝑙𝑣)

The induced dynamic of the local volatility : geometric average

Σ𝑆1(𝐿𝑆𝑉) 𝐾, 𝑇 = Σ𝑆0

𝐾𝑆0

𝑆1, 𝑇

𝜖

Σ𝑆0

𝐾𝑆1

𝑆0, 𝑇

1−𝜖

Stochastic Local Volatility : key concepts

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The general dynamic LSV (P2):𝑑𝑆𝑡

𝑆𝑡= 𝜎𝑡𝑓𝜖 𝑡, 𝑆𝑡 𝑑𝑊𝑡

,𝑑𝜎𝑡

𝜎𝑡= ⋯ 𝑑𝑡 + 𝜖𝛼𝑒−𝜆 𝑇−𝑡 𝑑 𝑊𝑡 , < 𝑑𝑊𝑡 , 𝑑 𝑊𝑡 > = 𝝐𝜌𝑑𝑡

The pricing equation obtained through the PnL equation :

𝜋𝑙𝑠𝑣 = 𝜋𝑙𝑣 + 𝜖2(𝜋𝑠𝑣 − 𝜋𝑙𝑣)

The induced dynamic of the local volatility : geometric average

Σ𝑆1(𝐿𝑆𝑉) 𝐾, 𝑇 = Σ𝑆0

𝐾𝑆0

𝑆1, 𝑇

𝜖2

Σ𝑆0

𝐾𝑆1

𝑆0, 𝑇

1−𝜖2

Stochastic Local Volatility : key concepts

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Start with a PDE description

𝜕𝑡𝑢 +1

2𝜎2𝑓𝜖

2𝑆2𝜕𝑆𝑆𝑢 +1

2𝜖2𝛼2𝜕𝜎𝜎𝑢 + 𝜖2𝜌𝛼𝜎2𝑓𝜖𝜕𝜎𝑆𝑢 = 0

P2 assumpation makes a balance between volga and vanna

Perturbation theory : we search 𝑢 = 𝑢0 + 𝜖2𝑢1 + ⋯ 𝑎𝑛𝑑 𝑓𝜖 = 𝑓0 + 𝜖2𝑓1 + ⋯

𝜕𝑡𝑢0 +1

2𝜎0

2𝑓02𝑆2𝜕𝑆𝑆𝑢0 = 0 classical local volatility model

𝜕𝑡𝑢1 +1

2𝜎2𝑓0

2𝑆2𝜕𝑆𝑆𝑢1 +1

2𝜎2𝑓1

2𝑆2𝜕𝑆𝑆𝑢0 +1

2𝛼2𝜕𝜎𝜎𝑢0 + 𝜌𝛼𝜎2𝑓0𝜕𝜎𝑆𝑢0 = 0

Feymann-Kacc equation

𝑢1 = 𝔼 0

𝑇 1

2𝜎2𝑓1

2𝑆2𝜕𝑆𝑆𝑢0 +1

2𝛼2𝜕𝜎𝜎𝑢0 + 𝜌𝛼𝜎2𝑓0𝜕𝜎𝑆𝑢0

Matching the extreme point with a mixing weight of 100%

Deduce that for any option : 𝑢 = 𝑢𝐿𝑉 + 𝜖2 𝑢𝑆𝑉 − 𝑢𝐿𝑉

Apply previous to VanillaΔ𝐿𝑆𝑉 = Δ𝐿𝑉 + 𝜖2 Δ𝑆𝑉 − Δ𝐿𝑉

Use general formula for delta of Vanilla Δ(K, T) = Δ𝐵𝑆(𝐾, 𝑇) + 𝑉𝑒𝑔𝑎 𝐾, 𝑇 𝜕𝑆ΣS0(𝑆0, 𝐾, 𝑇)

Deduce 𝜕SΣ𝐿𝑆𝑉,𝑆0(𝐾, 𝑇) = 𝜕SΣ𝐿𝑉,𝑆0

(𝐾, 𝑇) + 𝜖2 𝜕SΣ𝑆𝑉,𝑆0𝐾, 𝑇 − 𝜕SΣ𝐿𝑉,𝑆0

(𝐾, 𝑇)

Divide by common term Σ𝑀𝑎𝑟𝑘𝑒𝑡(𝑆0, 𝐾, 𝑇)

Integrate and obtain the dynamic formula

Σ𝑆1(𝐿𝑆𝑉) 𝐾, 𝑇 = Σ𝑆0

𝐾𝑆0

𝑆1, 𝑇

𝜖2

Σ𝑆0

𝐾𝑆1

𝑆0, 𝑇

1−𝜖2

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 𝑅𝑇 and a particular LSV model with a parameter𝜖 is given by the following formula:

𝜖2 = 1 −1

2𝑅𝑇

Stochastic Local Volatility : key formula

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𝜖 = 1 corresponds to the full stochastic volatility,

𝜖 =1

2corresponds to the in between model,

𝜖 = 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.𝑑𝑆𝑖

𝑆𝑖= … + 𝜎𝑖 𝑑𝑊𝑖

𝑆 , 𝑖 = 1 … 𝑚

For each stock price, volatility is assumed to follow a stochastic process as defined in the following:

𝑑𝜎𝑖

𝜎𝑖= … + 𝛼𝑖 𝑑𝑊𝑖

∗𝜎, 𝑖 = 1 … 𝑚

Driving Brownian Motions of both processes are assumed to be correlated, which allows us to write:

𝑑𝑊𝑖∗𝜎

= 𝜌𝑖𝜎𝑆 𝑑𝑊𝑖

𝑆 + 1 − 𝜌𝑖𝜎𝑆 2

𝑑𝑊𝑖𝜎

Where 𝑊𝑖𝜎 and 𝑊𝑖

𝑆 are independent random variables. The parameter 𝜌𝑖𝜎𝑆 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 – Vol x Spot – Vol we shall build a global correlation matrix which does not move the parameter 𝜌𝑖

𝜎𝑆. 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 𝑊𝑖𝜎, for i= 1…m

according to:

𝑑𝑊𝑖𝜎 = 𝜌𝜎𝑑𝑍 + 1 − 𝜌𝜎𝑑𝐵𝑖

𝜎 , 𝑖 = 1 … 𝑚

We have introduced only one new parameter 𝜌𝜎, with 𝐵𝑖𝜎 and 𝑍 independent.

𝑆𝜎

𝜌𝑗𝑖𝑆𝑆 𝜌𝑗

𝜎𝑆𝜌𝑗𝑖𝑆𝑆

𝜌𝑗𝜎𝑆𝜌𝑗𝑖

𝑆𝑆 𝜌𝑗𝜎𝑆𝜌𝑖

𝜎𝑆𝜌𝑗𝑖𝑆𝑆 + 1 − 𝜌𝑖

𝜎𝑆 21 − 𝜌𝑗

𝜎𝑆 2𝜌𝜎 + 1 − 𝜌𝜎 1 𝑖 = 𝑗

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.

EQUITY SOLUTIONS

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We use the lambda mechanism 𝜌𝑖,𝑗𝑆,𝑆 → 1 − 𝜖𝑆 𝜌𝑖,𝑗

𝑆,𝑆 + 𝜖𝑆 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:

𝜙 =|𝜋𝑠𝑣 − 𝜋𝑙𝑣|

𝜋𝑠𝑣 + 𝜋𝑙𝑣

2. Estimate Skew Stickiness Ratio:

𝑅𝑇 =𝔼 𝑑𝛴𝑡 𝐹, 𝑇 𝑑𝑙𝑛𝑆𝑡

𝑑𝛴𝑡 𝐾, 𝑇

𝑑𝑙𝑛𝐾 𝐾=𝐹𝔼 𝑑𝑙𝑛𝑆𝑡

2

3. Estimate the mixing weight:

𝜖2 = 1 −1

2𝑅𝑇

4. Pricing Proxy Formula:𝜋𝑙𝑠𝑣 = 𝜋𝑙𝑣 + 𝜖2(𝜋𝑠𝑣 − 𝜋𝑙𝑣)

EQUITY SOLUTIONS

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Conclusion Multi

1. Introduce idiosyncratic volatility factor 𝜌𝜎

2. Build a global correlation Matrix

3. Recorrelate to match ATM and remarkably basket skew

𝜌𝑖,𝑗𝑆,𝑆 → 1 − 𝜖𝑆 𝜌𝑖,𝑗

𝑆,𝑆 + 𝜖𝑆

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)1.58.55.55.12adil.reghai@natixis.com

http://www.palgrave.com/page/detail/quantitative-finance-/?isb=9781137414496

NATIXIS Banque de Financement et d’Investissement - 47, quai d’Austerlitz – BP 4 - 75060 Paris Cedex 02 www.natixis.com

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