risk math week 2001 full

8/8/2019 Risk Math Week 2001 Full

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Modeling and hedging volatility


Laurent Nguyen-Ngoc

Deutsche Bank, Global Equities

Global Quantitative Research

Risk Math week, Nov. 2001


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Outline

Volatility as activity: time changes and the impliedvolatility

Using the models I

Some examples

Using the models II

An example: barrier and lookback options in a skewedimplied volatility model


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Volatility as activity

An empirical study by An and Geman (1999) shows that

returns are Gaussian in an appropriate time scale:

are Gaussian, where the instants T(i) are

random

No volatility without transactions! T(i) are determined bythe number of trades

According to this observation, returns are not Gaussian,but conditionally Gaussian

)(

)()1(

iT

iTiT

S

SS +


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Diffusion models

Local / Stochastic volatility models are time-changed

Brownian motion: the martingale part can bewritten as whereB is another BM and

(t) is the aggregated square volatility.

In the Black-Scholes model, (t)=2t.

ttdW)(tdB

dstt

s= 02)(


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Jump models

Mertons model is Black-Scholes overlaid by jumps

The Variance-Gamma model of Madan & Seneta (1990),is a time-changed Brownian motion. It can be

generalized as the CGMY model.

Jump processes can capture large, sudden marketmoves, but also the infinitesimal, usual moves

They can also be combined with diffusion models in

order to take advantage of the good features in each ofthem


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Implied volatility in equity markets

In equity / indices markets, the implied volatility is

skewed with respect to the strike. This indicatesdeparture from the log-normal assumption.

The skew is much more pronounced for short termmaturities than for longer ones.

This can be modeled with a conditionally Gaussian

process


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Equity implied volatility

STOXX50 implied volatility

30 8092.5

105117.5

160210

1m

1.5y

7y

0

20

40

60

80

100

120


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Using the models I

We always put and assume that

is a martingale

The price of a call can always be written as

P is the risk-neutral measure; PS is the risk-neutral

measure relative to the numraire S

In all the following models, we know

][][ KSPKeKSSPC TrT

T

S >>=

][ tiuX

eE

)exp( tt XSS =

t

rt

Se


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Pricing and hedging vanilla options

We use Fourier inversion to obtain the probabilities,

either a numerical integration or the FFT algorithm (seeCarr & Madan, 1999)

These methods lead not only to the price, but also to thedelta of the option

More general payoffs can be valued using the same

Fourier inversion technology

][ KSPT

S >=


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Fitting market data

Calibration to market data can be done in an efficient

fashion

The models have specific strengths and weaknesses

It is important to use a model that is adapted to the

product. Quite a few products depend strongly on thevolatility structure, e.g. cliquets, volatility swaps


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Examples of models

A Stochastic volatility model: Heston (1993)

Jump models

Merton (1976)

Variance-Gamma (1990), CGMY (1999)

Combination of jumps and stochastic volatility

CGMY with stochastic activity (2001)

Heston-Merton (2001)


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Heston model

The volatility is a square-root process

and the stock is given by St=eX(t), where

Wand W have a correlation

In this case, is well known thanks toPitman & Yor, 1982

tttt dWvdtvdv += )(

=t

sdsvt

0)(

( ) '2 tt

v

t dWvdtrdXt +=


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Heston implied volatility

3070 87.5 97.5

107.5 117.5150

190230

1m

1.5y

7y0

20

40

60

80

100

120


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Merton model

In this and the following models, X is a Lvy process,

i.e.has independent and stationary increments.

In the Merton model, where the jumps

Jare lognormal(g,d)

The marginals St are log-normal, conditionally on thenumber of jumpsNt

=

++=tN

i

ittJWtX

1


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Merton implied volatility

3070

87.597.5

107.5 117.5150

190230

1m

2y

10y

0

10

20

30

40

50

60

70

80


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VG and CGMY

In the VG model, is a time-changed

drifted Brownian motion. is a Gamma() process. Xis apure jump Lvy process, with Lvy measure

The CGMY model generalizes the VG model. Again, X isa pure jump Lvy process with Lvy measure

)(tt WtX +=

dxxx

x

+2

2

2

2exp

1

)0(),0( 11 xdxeCxxdxeCx xGYMxY


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CGMY implied volatility

3060 82.5 90 97.5 105 112.5 120 150

1m

1.5y

7y0

20

40

60

80

100

120


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Volatility of Lvy processes

IfXis a Lvy process, tt 2)( =

( ) caseCGMYin the)2(caseVGin the

caseMertonin the)(

222

222

2222

+=

+=

++=

YY GMYC


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Jumps and stochastic volatility combined

Idea: take a Lvy process, and try to make stochastic

to combine advantages of jumps and stochastic volatility

Geman, Madan, Yor, 2001 choose as an integratedsquare-root process.

The model is built in such a way that remains amartingale.

t

rtSe


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Application to CGMY

LetZbe a CGMY process, i.e. a Lvy process with Lvy

measure

Let v be an independent square-root process, and set

Now, set and

We obtain a process for which

See Geman, Madan and Yor, 2001

)0(),0( 11 xdxeCxxdxeCx xGYMxY

=t

s dsvtI0

)(

)(tIt ZX = )exp( ttt XS =

+=t

s

YY dsvGMYt0

22 ))(2()(


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Application to the Merton model

If we apply the same to the Merton model, we obtain:

a continuous martingale similar to the Heston model jump sizes are still independent and log-normal, but their

intensity is stochastic

In this case,( ) ++=

t

s dsvt0

222 )()(


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Heston-Merton implied volatility

3080

92.5105

117.5160

2101m

1y

5y

0

10

20

30

40

50

60

70

80

90

100


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Forward implied volatilities

Some products also depend on the forward implied

volatility (e.g. cliquets structures)

The forward-start European options can be valued

efficiently by Fourier inversion methods, ifis known

This is the case in the models presented here

][ tTivXiuX

eE +


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Heston forward implied volatility

40 80 90 100 110 120 160 200 240

1m

3y0

10

20

30

40

50

60

70

80

90

100

Fwd 1m


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Heston forward implied volatility

40 80 90 100 110 120 160 200 240

1m

3y0

10

20

30

40

50

60

70

80

90

100

Fwd 6m


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Example: forward volatility in the Heston model

The forward implied volatilities obtained from the Heston

model look very much like the implied volatilitiesobtained from the VG family.

WhenXis a Lvy process, the forward implied volatilitiesare exactly the same as todays implied volatilities


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Using the models II

Exotic options, in general, require Monte Carlo methods,

or finite difference. There is not a unique method of simulation

These methods are very demanding numerically, but are

applicable to simple products (e.g. barrier options)


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Risk-management

The VaR can then incorporate fat tails assumption, and

provisions be made accordingly

The risks related to the volatility exposure are asserted in

a better way.

Trading limits can be imposed consistently


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Example

We give an example of exotic options for which Monte

Carlo can be circumvented: digital barrier options

They can be valued semi-analytically, in the vein of the

methods developed for European options. Namely wecompute the Laplace transform of the price and are leftwith a numerical inversion problem.


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Digital barrier option

A digital Up and In Call pays off where

H>K and T(h)=inf{t, X(t)>H}, and therefore its price is

P[ST>K,T(lnH)H] > 0

{ } { }THTKST )(ln11


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Solution for price

We can obtain the time-space Laplace transform of the

distribution (see Nguyen-Ngoc and Yor, 2001)

where the function is given by

In order to get the price, we need to invert the Laplacetransform above numerically (this is not easy!)

[ ]),()(

),(),(

0

)()( )(

qq

qdxeEe

xXxTqx xT

=

( )

=

0 0)(

1exp),( dxXPee

tdt

t

xtt


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Extension

Note that a barrier option can be written as

A lookback option (here, floating strike put)

So the method extends to them, by taking expectationson both sides.

( ) { } { } { }

risk math week 2001 full

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