Open issues in equity derivatives modelling - CMAP

Equity Derivatives Quantitative Research Société Générale [email protected]

Lorenzo Bergomi

Open issues in equity derivatives modelling

Lorenzo Bergomi

1997 – 2003

ƒ Conclusion

ƒ Risk measurement and management

ƒ Modelling issues, algorithmic issues

Modern times 2003 –

History

Prehistory – 1997

ƒ A brief history of equity derivative products

ƒ Equity derivatives at SG

Talk Outline

1

Lorenzo Bergomi

ƒ SG regarded by industry participants as No 1 in equity derivatives

Equity derivatives at SG

2

Lorenzo Bergomi

- PDE / straight Monte Carlo

- Black Scholes / local vol

ƒ Models / algos

ƒ Simple cliquets

ƒ Volatility swaps

⎛ 1 ⎜ ⎝N

ƒ Basket options

S

⎛ S ⎜ ⎜ S ⎝ T1

T 2

−

+

− σˆ

+

⎞ K ⎟ ⎟ ⎠

2

+

+

⎞ − K⎟ ⎠

⎞ − K ⎟ ti ⎠

∑ S Ti

∑

⎛ S ⎞ 1 ln⎜ k ⎟ ∑ ⎟ T k ⎜S ⎝ k −1 ⎠

⎛ 1 ⎜ ⎝ N

ƒ Asian options

ƒ Max / Min options

( )

K

Forward smile

Smile, VolOfVol

Correlation (level)

Smile

same

Skew: level / dynamics (little)

ƒ Barrier options / Digitals ⎛ max S − K ⎞ ⎜ ⎟ t ⎝ t ⎠

Risks

Prehistory – 1997

Products

A brief history of equity derivative products

3

10 years / 20 stocks

ƒ Emerald 2004

Lorenzo Bergomi

1997 – 2005

¸ 100% + maximum perormance of yearly basket values since t = 0, floored at 0.

Every year, the stock whose perfomance since t = 0 is the largest gets frozen and removed from the basket, and its level is floored at 200% ot its initial value.

¸

⎞ ⎟ ⎟ ⎟ ⎠

History - 1

¸ trying to find closed-form formulas for specific exotic payoffs now irrelevant and useless

… and many, many, many, other variations

5 years / 12 stocks

ƒ Everest 1997

⎛Sj ⎜ 100% + min⎜ T ⎜Sj ⎝ 0

Capital-guaranteed products distributed by retail networks

A brief history of equity derivative products

4

Lorenzo Bergomi

5

3 years / 1 index

ƒ Accumulator

Lorenzo Bergomi

5 years / 1 index

ƒ Napoleon

stocks / indices

ƒ Variance Swaps 3 months ¸ 5 years

History - 2

1997 – 2005

⎞ ⎟ ⎟⎟ ⎠

+

⎛ ⎛ ⎞⎞ ⎛ S ⎞ ⎜ ⎟ ⎜ ⎜ ⎟ k − 1,1% ,−1% ⎟⎟ ⎟ ⎜ ∑ max⎜⎜ min⎜ S ⎟ ⎟⎟ ⎜k ⎝ k −1 ⎠ ⎝ ⎠⎠ ⎝

+

6

At maturity pays the sum – if it is positive – of the monthly performances, capped and and floored.

⎛ ⎛ ⎞ ⎜ C + min ⎜ S k ⎟ ⎜⎜ k ⎜⎝ S k − 1 ⎟⎠ ⎝

Every year, pays coupon reduced by worst of 12 monthly performances of the index.

⎛ S ⎞ ∑k ln⎜⎜ S k ⎟⎟ − σˆ 2T ⎝ k −1 ⎠

2

Pays realized variance – usually measured using daily returns

A brief history of equity derivative products

Lorenzo Bergomi

ƒ Hybrids

ƒ Timer options

ƒ Options on realized variance +

2

Equities / Rates / Forex / Commodities Arbitrary payoffs

τ

⎛S ⎞ Qτ = ∑ ln⎜⎜ i +1 ⎟⎟ i =1 ⎝ Si ⎠

Vanilla payoff, paid when realized variance Qt reaches set level:

2 ⎞ ⎛1 ⎛ Sτ ⎞ 2⎟ ⎜ ⎜ ⎟ ˆ ln σ − K ⎜S ⎟ ⎟ ⎜T ∑ ⎠ ⎝ τ ⎝ τ −1 ⎠

On indices, maturities: 3 months to 2 years

with strikes 85%, 90%

Maturity = 1 year, a series of daily puts on daily returns of an index

ƒ Gap notes

[L, H ], [L, + ∞], [0, H ]

Pays realized correlation over 3 years by stocks of an index

⎞ ⎛ ⎛ S ⎞2 ⎟ ⎜ ⎜ k ⎟ 2 ˆ 1 ln σ t − ∆ ∑k Sk ∈[L, H ] ⎜ ⎜ S ⎟ ⎟ ⎟ ⎜ ⎝ k −1 ⎠ ⎠ ⎝

Daily variance only counted when underlying is inside given interval

ƒ Correlation swaps

ƒ Corridor variance swaps

Modern times

7

Variance of residual P&L too large

¸ use other options

2002

Lorenzo Bergomi

10

15

20

25

30

35

40

45

50

55

60

2003

2004

2005

ƒ Example of simple cliquet

2006

1

2

⎛ ST ⎞ ⎜ − 1⎟ ⎜S ⎟ ⎝ T ⎠

2007

+

t

2008

T1 σˆ12

T2

Smile 3 mois K = 95 - K= 105

0 2/5/2001 2/5/2002 2/5/2003 2/5/2004 2/4/2005 2/4/2006 2/4/2007 2/4/2008

1

2

3

4

5

6

7

P(σˆ12 , r , L )

Less sensitivity to historical parameters, more sensistivity to implied parameters

¸ Model the dynamics of implied parameters

ƒ

ƒ Once we start using options as hedging instruments

¸ Options are hedged with options

ƒ

ƒ Why not just delta-hedge ?

Modelling issues – 1

8

≈ P( p = pMarket )

= P( pˆ ) + λ(O( pMarket ) − O( pˆ ))

Lorenzo Bergomi

¸ Useless if one is unable to specify how to hedge the exotic with the hedge instruments

¸ Necessary to calibrate model on relevant set of hedging instruments

• Ensures price factors in hedging costs incurred at t = 0 – not future costs !

• Then what is the point in calibrating ?

Price

• Model price P is adjusted so as to include hedging cost:

dP dO = λ dp dp

• Not compulsory: charge a hedging cost. We hedge parameter p by trading instrument O so that sensitivity to p vanishes:

ƒ How should calibration be done ? Do we really need to calibrate ?

Modelling issues – 2

9

80

+ ( ) dWV

= K dt

dV

• Jump / Lévy

Lorenzo Bergomi

• How much freedom are we allowed ?

• Low-dimensional Markov representation desirable

• Direct modelling of dynamics of implied volatilities is a dead end

• Next step: model dynamics of the implied volatility surface

• First step: model dynamics of curve of forward variances

• Challenge: Build models that give control on joint dynamics of implied volatilities and spot:

+

= K dt

dS

V S dWS

10

15

20

25

• Models based on process of instantaneous variance:

• SABR

• Heston

• Local volatility

• « Old models »

ƒ Volatility risk – models

Modelling issues – 3

90

100

110

Smile S = 95%

Smile original Smile S = 105%

Eurostoxx 50 - mat 1 an

120

10

• Commodities

• Forex

• Interest rates

• Equities

Lorenzo Bergomi

• Even local vol calibration for equity smiles not easy when interest rates are stochastic

• Require state-of-the art models for each asset class

• Active hybrids : payoff involves all asset classes

• Credit / Equity: convertible bonds

• Long-dated equity, Forex options

• Passive hybrids: payoff involves one asset class only

• Hybrid models are not built by simply glueing together models for each asset class

ƒ Hybrids

Modelling issues – 4

11

o

C

o o

C C

o o

C C

o o

C

Lorenzo Bergomi

ƒ Corrrelation – how to model correlation smile ?

ƒ Correlation – how de we measure correlation risk ?

Japan

Europe C

• Example of European / Japanese stocks – no overlap

o

C

• Even simpler question – how do we measure correlations ?

• How do we set the cross-correlations ?

• Simpler question: imagine a 1-factor stoch. vol model and a payoff involving 2 securities

• How do we build the large correlation matrices needed in hybrid modelling ?

ƒ Correlation – how de we put together correlation matrices ?

Modelling issues – 5

?

?

12

Lorenzo Bergomi

•Parameters of dynamics (volatilities / correlations, etc..)

•Initial conditions

• Computing sensitivies to

• Callable / putable options

• Discretization of SDEs ?

• Quasi-random numbers

• How can we speed up pricing ?

ƒ Monte Carlo

Algorithmic issues

13

Lorenzo Bergomi

ƒ Rich mathematical toolbox from which to pick

ƒ Lots of new instruments / product / algorithmic issues

ƒ These are exciting times for doing quantitative finance

Conclusion

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