Equity Derivatives (PDF)

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Guillaume BLACHERReech Capital

Advanced Pricing and Risk

Management for Equity Derivatives

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Slide 2

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Outline

➢ Presentation of products most actively studied in

Equity Derivatives

➢ Evolution in pricing exotics

➢ Choosing a proper diffusion model

➢ Numerical methods: complexity and performance

➢ Generic product description tools

➢ The Future of quantitative analysis



products most actively studied inEquity Derivatives



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GuillaumeBLACHER

ReechCapital

EquityDerivatives

Products• High variety of products in Equity

Derivatives• Always new products tailored to:

– new market conditions– new tax regimes– new corporate situation

• Among most popular:– Cliquets– Structured notes– Baskets and multi-underlying products– Hybrid Products



Slide 5

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Cliquets• Pays the equity positive performance each

year, defined as:

• Pricing:– By Monte-Carlo: most natural but slow– By Tree/PDE: requires a second dimension for

the strike

• Problems:– Almost a pure volatility product– Strong impact of smile and dividends

• Solution: pricing by static replication

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Slide 6

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Structured notes• Client gives up risk-free return, dividends

and/or x% of capital to get some equityexposure via embedded option.

• Often combines many features into oneproduct:– Bond features (coupons, redemption)– Exotic features (digitals, barriers…)– Swap features (equity swaps)– Multi-underlying (best of, basket)– Path dependent features (average)– American style features (puttable)



Slide 7

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Exotics on basket• Still a lot of business on baskets, mainly with

path-dependent features (asian baskets) andcapped

• Example: 5Y basket option of 3 indices, with aknock-in in the first 2.5Y at 175% to get an extra25% of gearing.

• Modeling correlation realistically is crucial

• Correlation cannot be hedged on a trade by tradebasis (no exchange traded correlation sensitiveoptions): need for creativity to build up acorrelation - diversified portfolio



Slide 8

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Hybrid products

• More and more products across assetclasses:– Best of equity / interest rate– Interest swaps triggered by equity level– Debt products with FX components

• Requires– Combination of skills for modeling

heterogeneous assets together– Very high structuring and pay-off description

flexibility– Consistent risk management across all asset

classes



Slide 9

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Volatility products

• Swaps or options on:– realised volatility from start to maturity– Implied volatility

• Example:In 6M, option pays the 1M at-the-money implied

volatility.

• Pricing mechanism differs from standardstructure:– Mix of semi-static option hedge and dynamic

spot hedge– Several un-hedgeable elements that need to be

taken into account.



Pricing Exotics



Slide 11

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Pricing components

• Several components in a pricing engine:

Numericalmethod

Diffusionmodel

Productdescription

Modelparameters /Calibration



Diffusion models (I)



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GuillaumeBLACHER

ReechCapital

EquityDerivatives

Diffusion models

• All pricings should be consistent withmarket prices of hedging instruments:– Current yield– Smile– Credit curve

• The least it should do to achieve this withdeterministic models:– short term rate: r(time)– instantaneous volatility σ(spot, time)– default probability p(spot, time)

• Those functions can be calibrated tomarket instruments



Slide 14

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Example: volatility modeling

• The simplest diffusion consistent with amarket skew is:

• Always exists if market prices are notarbitrageable

• Would be unique if a continuum of priceswere given

• Significant price impact on:– Barriers– Cliquets– Basket options

( ) ( ) tdWtSdtdivreporSdS ,σ+−−=



Slide 15

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Example

• Pricing with smile can be critical when theright price cannot be reach with Black-Scholes formula

Up and Out Call option

0

0.5

1

1.5

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2.5

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0% 20% 40% 60%

Black-Scholes volatility

Bar

rier p

rice

Black-Scholes priceSmile price

25

27

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35

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Market smile

Strike Maturity

25

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Flat smile

Strike Maturity



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GuillaumeBLACHER

ReechCapital

EquityDerivatives

Hedging

• What if the smile disappears?

Black-Scholes assumption would then apply. Our

mark-to-market position would be showing a

big loss (if we bought the barrier) or a big

profit (if we sell the barrier)

• Only makes sense if a consistent vega

hedge is used.



Risk Management



Slide 18

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Vega analysis

Volatility risk can be measured at 3 levels:

1) Single vega number:– Parallel shift of the

whole smile surface

– No protection againstdeformations

– Which Europeanoptions to buy as ahedge?

-1%

0%

1%

2%

Strike Maturity

Classic Vega perturbation



Slide 19

GuillaumeBLACHER

ReechCapital

EquityDerivatives

-1%

0%

1%

2%

Strike Maturity

Bucket Vega perturbation

Vega analysis (2)

2) Vega buckets:– Parallel shift of the skew at each tenor

– Protection against time deformations only

– Which strike should we buy as a hedge?Vega bucket for an exotic

1M 2M 3M 4M 5M 6M 7M

Tenor

Vega



Slide 20

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Vega

StrikeMat

Superbucket: Up&Out Call

-1%

0%

1%

2%

Strike Maturity

Superbucket perturbation

Vega analysis (3)

3) Superbuckets:– Shift of each individual point on the surface

– Protection against any deformation

– Provides strikes and maturities to hedge with.

barrier

strike



Slide 21

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Global risk management

• Each individual exotic has got its superbuckettopography.

• What is really relevant is the superbucket of thewhole book, aggregating exotics and vanillas on thesame underlying:

Vega

StrikeMat

Portfolio superbucket

Sensitivities tobe hedged out(by tradingcorrespondingEuropeans)



Slide 22

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Risk Reports

Spot

Time Vega

Classic Report:

Strike

Mat

urity

Vega

CORRECT Report:

Traders will try to flatten the Classic Report where

they should be trying to flatten the Correct Report:

• Equivalent for European options

• Different for Exotics



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GuillaumeBLACHER

ReechCapital

EquityDerivatives

Global risk management (2)

• This (correct) analysis should be available at

portfolio level on all model parameters, whether

they be:

– hedgeable (credit term structure, smile surface)

– un-hedgeable (instantaneous forward correlation)

• It gives today’s hedge but may not be static: need

for readjusting in the future

• Question: if readjusting is necessary, what is the

cost?



Diffusion models (II)



Slide 25

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Cost of hedging

Some options exhibit a vega that varies with:

• Spot

• Volatility itself

Rebalancing vega hedge will produce systematic

cost/profit that can only be estimated will a

stochastic volatility model.

To price the cost of having... we need to model...

dSpotdVega

dVoldVega

Correlation (Spot, Vol)

Volatility of volatility



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GuillaumeBLACHER

ReechCapital

EquityDerivatives

Stationarity

• Many different ways to match current smile

• Choosing the wrong model means having non-stationary parameters:– Even when market does not change, model parameters

do

– Forward smiles not realistic

• Although the reason for choosing a model isconsistency and not ability to forecast, the cost ofrehedge will be wrongly estimated

• Conclusion: try to find the model with most stableparameters



Model parameters / Calibration



Slide 28

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Auto-calibrated models

• Some model can be considered as auto-calibrated:“calibration” is analytical, quasi-instantaneous andhappens automatically inside the numericalmethod

• Examples:– Local volatility in a Black-Scholes framework

– Local volatility in a deterministic volatility model

– Drift of the short term rate in an Heath-Jarrow-Mortonframework

– Instantaneous default probability if deterministic

– etc...



Slide 29

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Calibration

• More and more advanced models are not auto-calibrated

• Calibration is a long process

• On the other hand, it is done once and for all (untilmarket conditions change significantly)

• Examples:– Parameters of stochastic volatility / jump models

– Volatility in interest rate models

• Systems have to be flexible enough to let thecalibration process happen independently frompricing



Numerical methods



Slide 31

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Trees• Just a way to compute an expectation• Binomial trees obsolete (except for quick

approximations)• Trinomial trees do the job, although they are less

efficient than PDEs.• 2-dimensional trees technology well mastered

Binomial Trinomial 2D Trinomialtime

spot

time

spot

time

S1

S2



Slide 32

GuillaumeBLACHER

ReechCapital

EquityDerivatives

PDEs• Allows for implicit discretisation• More flexibility on the grid• Better performance in general, much better for some

products (barriers for example)• 2-dimensional PDEs technology well mastered• Provides option price at any time for any spot

Optionprice Time Intrinsic value

Spot

Pay-off for anUp&Out CallPDE gridPrice

profiletoday



Slide 33

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Monte Carlo• Low discrepancy series (ex: Sobol) provide very high

convergence rate compared to standard Monte Carlo• Unfortunately biased in high dimension• Dimension reduction techniques have to be used in high

dimension (spectral truncation)• Several very efficient techniques to price american style

options within simulations

0

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

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Standard MC sampling Sobol sampling



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GuillaumeBLACHER

ReechCapital

EquityDerivatives

Complexity vs Performance

• More accurate models usually mean greatercomputation time.

• Becomes an issue when thousands of positionshave to be revalued.

• Better understanding of models and numericalspeed-ups should precede the purchase of high-end hardware.

• Using those instead of brutal calculation powerneeds structural changes in trading systemsarchitecture.



Product description



Slide 36

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Equity Derivatives richness

• Structured products and Derivatives have 3 mainproperties:1) Extreme diversity of features

2) All features have to be priced together

3) Product popularity does not last forever

• This means:– A lot of work for quants to implement new features in

the library

– Library contain high quantity of obsolete products (oreven products that have never been dealt)



Slide 37

GuillaumeBLACHER

ReechCapital

EquityDerivatives

Generic product description

• Quants should:– spend no time implementing new products in a Library

– spend more time modeling the market parameters

– spend more time improving their numerical methods

• Structurers/Marketers should:– spend no time waiting for quants to come up with a new

product in the Library

– spend more time studying the risks of the product

– spend more time trying new ideas

• What’s the solution?

A generic product description language



Slide 38

GuillaumeBLACHER

ReechCapital

EquityDerivatives

The future of quantitative analysis

• Main challenge:

model more sources of risk while maintainingreasonable computation speed.

• How is it possible?– Better understanding of the products helps identifying

most important factors

– Some risks can be taken into account without increasingthe dimension

– More advanced numerical methods and numericalspeed-ups are to be used


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