Open issues in equity derivatives modelling

(1)

Open issues in equity derivatives modelling

(2)

Talk Outline

Equity derivatives at SG

A brief history of equity derivative products

Prehistory –

1997

History

1997 –

2003

Modern times 2003 –

Modelling issues, algor

ithmic issues

Risk m

easurem

ent and management

Conclusion

g

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Equity derivatives at SG

SG regarded by industry particip

ants as No 1 in

equity derivatives

g

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A brief history of equity derivative products

Prehistory –

1997

Products

Risks

Barrier options / Digitals

Skew

: level / dynamics (little)

Max / Min options

same

Asian options

Smile

Basket options

Correlation (level)

Vo

latility swaps

Smile, V

olOfV

ol

Simple cliquets

Forward smile

Models / algos

-Black S

choles / local vol

-PDE / straight Monte Carlo

gomi

+ ⎟ ⎠⎞ ⎜ ⎝ ⎛ − ∑ K t S N i 1 + ⎟ ⎠⎞ ⎜ ⎝ ⎛ − ∑ K i T S N 1

()

+ ⎟ ⎠⎞ ⎜ ⎝ ⎛ − K t S t max K σ k S k S T k ˆ 2 1 ln 1 − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∑ + _⎞ _⎟ _⎟ _⎠ ⎜ ⎜ ⎝ ⎛ − K S S T T2 1

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A brief history of equity derivative products

History

-1 -1997 –

2005

Capital-guaranteed products di

stributed by retail networks

Everest 1997

5 y

ears

/ 12 s

tocks

¸

Emerald 2004

10 years / 20 stocks

Every year,

the stock whose perfomance since

t = 0

is the largest gets f

rozen and

removed from t

he bas

ket, and its level is

floored a t 200% ot its initial value. ¸ 100% + maximum perormance of yearly ba

sket values since

t = 0

, floored at 0.

… and many, many, many, other variations

¸ tryin g to fin d closed -form fo rmu

las for specific exo

tic payo ffs n ow ir relevant and u seless g omi ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + j S j _T S 0 min % 100

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g

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A brief history of equity derivative products

History

-2 1997 –

2005

Variance Swaps

3 months

¸

5 years

Pays realize d varianc e – usually mea

sured using daily returns

stocks / indices

Napoleon

5 y

ears

/ 1 index

Every year, pays coupon reduced by worst

of 12 monthly performanc es of the index.

Accum

ulat

or

3 years / 1 i

ndex

At maturity pays the sum –

if it is positiv e – of the mont hly performances, cappe d and and floored. g omi + _⎞ _⎟ _⎟⎟ _⎠ ⎜⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + 1 min k S k S k C + _⎞ _⎟ _⎟⎟ _⎠ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − −

∑

k k S k S %1 , %1, 1 1 min max T σ k S k S k 2 2 ˆ 1 ln − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −

∑

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Modern times

Corridor variance swaps

Daily varia nce only c ou nte d whe n un

derlying is inside given int

erval

Correlation swaps

Pays realize

d correlation over 3 yea

rs by stocks

of an index

Gap notes

Maturity = 1 year, a series of daily

puts on daily returns of

an

index

with strikes 85%, 90%

Options on realized variance

On indic

es, maturities: 3 months to 2 years

Timer options

Va nilla pa yo ff, paid wh en realized variance Qt reaches se t level:

Hybrids

Equities / Ra tes / Forex / Commodities A rbitrary payoffs g omi []

∑

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

[]

[

]

[

]

H L H L ,0 , , , , ∞ + + _⎞ _⎟ _⎟ _⎠ ⎜ ⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

∑

− 2 2 1 ˆ ln 1 K τ τ τ σ S S T

∑

= + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = τ _i i i τ S S Q 1 2 1 ln

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Modelling issues

–

1 Why not just delta-hedge ?

Variance of residual P&

L too l

ar

ge

¸

use other options

¸

Options are hedged with options

Once we start using options

as hedging instr

uments

Less sensit

ivity to

h

isto

ric

al parameters,

mo

re sensi

st

ivity to

imp

lied

parameters

¸

Model the dynamics of implied parameters

Example of simple cliquet

g omi t 1 T 2 T 12 ˆσ

()

L,

,

ˆ

12

r

σ

P

+ _⎞ _⎟ _⎟ _⎠ ⎜ ⎜ ⎝ ⎛ − 1 1 2 _T T S S 200 8 2007 2006 2005 200 4 2003 2002 S m ile 3 m o is K = 9 5 - K = 1 0 5 0 1 2 3 4 5 6 7 2/5 /20 01 2 /5/2 0 0 2 2/5 /20 03 2/5 /2 0 0 4 2 /4 /20 05 2/4/2 0 0 6 2 /4 /20 07 2/4/2 0 0 8

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Modelling issues

–

2 How should calibration be

done ? Do we really

need to calibrate ?

• N ot compulsory: char ge a h edg

ing cost. We hedg

e par ameter p by tr ad ing ins trument O so that s ensit ivity to p •M od el p ri ce P is adjusted so as to includ e h edging cost:

• Then what is the po

int in calibrating ?

•

Ensures price factors in he

dging c

osts incurred at

t = 0

–

not future costs !

¸

Necessary to

calib

rate mo

del on

relevant set of

hedging instruments

¸

Useless if one is unabl

e to specify how to hedge the ex

otic with the he

dge instruments

g omi dp dO λ dp dP =

()

(

)

()

Market Market Price p p P p O p O λ p P = ≈ − + = ˆ ˆ

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Modelling issues

–

3 Volatility risk –

m

odels

• «

O

ld models

»

• L ocal volatility •H es to n •S A B R • M odels b as ed on process of inst antaneou s variance: •J u m p / L év y

• Challenge: Build models that give

control on joint dynamics of

implied volat

ilities and spot:

•First step: model dynamics of

curve of forward variances

•Next step: model dynamics of

the implied volatility surface

•

Dir ect mod el li ng of d ynamics of im

plied volatilities is a dead end

• L ow-dimens ional M arkov r epr es entat ion d esir abl e • H ow muc h freedom are we al lowed ? g omi V S dW dt dV dW S V dt dS ) ( + = + = K K E u ro st oxx 5 0 - m at 1 an 10 15 20 25 80 90 10 0 1 10 120 S m ile o rig in a l S m ile S = 1 0 5% S m ile S = 9 5 %

(12)

Modelling issues

–

4 Hybrids

•

E quities • Interest rates • Forex • C ommoditi es

•Hyb

rid

mo

dels are no

t built

by simp

ly g

lueing together models f

or each asset

class

• Passive hybrids: payoff in

volves one asset class only

•

Long-dat ed equit y, For ex options • C

redit / Equity: convertibl

e bonds

• Act

ive hyb

rids

:payof

f invo

lves all asset

classes

• R equir e st at e-of-the a rt mo d els fo r ea ch a sse t cla ss • E

ven local vol calibration for

equity smil

es not easy when

interest rates are stochastic

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Modelling issues

–

5 Correlation –

how de we put to

gether correlation matrices ?

•

How do we bu ild th e l arg e corr el at ion ma trices n eed ed in hybrid model ling ? • Simpler qu est ion: imagine a 1-f actor s toch. vo l mo d el and a pa yo ff in vo lv in g 2 se cu ri ties • H ow do we s et the cr oss-corr el ations ? •E ve n s im p le r q u es ti on –h ow d o we me asure c orrela ti on s ? • E xa mple of Europ ean / Jap anes e stocks – n o over la p

Open issues in equity derivatives modelling