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Page 1: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 1

Local volatility model

Nikos Skantzos ULB 2011 2

Local Vol As Instantaneous Vol (1)

� Local Volatility means that the value of the vol depends on time (and spot)

� It reminds closely of the instantaneous vol

� To get an idea, one can always calculate instantaneous volsdirectly from the time series of the underlying using the log-returns:

)()(log1)(

tSttS

tt ∆+

∆=σ

Page 2: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 3

Local Vol As Instantaneous Vol (2)

� Annualisedinstantaneous volsof S&P500

� Regression 3rd order

� “Local” vols are different in shape, showing skew, frown, smirk, smile

Nikos Skantzos ULB 2011 4

Dupire Local Vol (1)

� Comes from a need to price path-dependent options while reproducing the vanilla mkt prices

� Assumes: Underlying follows lognormal process, but� Vol depends on underlying at each time and time itself� It is therefore indirectly stochastic

� Local vol is a time- and spot-dependent vol(something the BS implied vol is not!)

� No-arbitrage fixes drift µ to risk-free rate

( ) ttttt dWtSSdtSdS ⋅⋅+⋅⋅= ,σµ

Page 3: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 5

Dupire Local Vol (2)

( ) ( )tTSK

KK

KTt tCK

CrrKCrCtS ==⋅⋅−⋅+⋅+

= ,221

1212 ,σ

� Technology invented independently by:� B. Dupire Risk (1994) v.7 pp.18-20 � E. Derman and I. Kani Financial Analysts Journ (1996) v.53 pp.25

� They expressed local vol in terms of market-quoted vanillasand its time/strike derivatives

� Or, equivalently, in terms of BS implied-vols:

( )( )

( )

tTSKt tddKKtTK

dKtTK

K

KrrK

TtTtS ==

∂∂

+∂∂

+−∂

∂+

−⋅

∂∂⋅−⋅+

∂∂+

−= ,

BS

212

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21,

σσσ

σσ

σ

σσσ

σ

Nikos Skantzos ULB 2011 6

Dupire Local Vol (3)

� The Dupire Local Vol is a “non-parametric” model which means that it does not introduce parameters into the modeling

� No calibration is needed to match the vanilla prices. The fit is done by simple computations on the market vanillas.

� Another advantage is that the model remains one-dimensionaland therefore analytically and numerically tractable

� It is a non-arbitrageable model

Page 4: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 7

Derivation (1)� Starting point is the Kolmogorov forward equation

� Multiply with call-payoff and discount-factor DF=e-rT and integrate:

� To proceed we use the identities

obtained from simple differentiation of a call price

( ) ( )222

2

),(21),()( σ⋅⋅∂∂

+⋅⋅∂∂

⋅−−=∂∂ StSP

SStSP

Sqr

tP

( )

( )∫

∫∫∞

∞∞

⋅⋅

∂∂

⋅−⋅+

∂∂

⋅−⋅−⋅−=

∂∂⋅−⋅

KTT

TK

TK

TT

StSPS

KSdS

StSPS

KSdSqrtPKSdS

222

2

),()(DF21

),()()(DF)(DF

σ

CallCall)(DF ⋅+∂∂

=

∂∂⋅−⋅∫

rTt

PKSdSK

TT ),(DFCall mkt2

2

KTPK

⋅=∂∂

Nikos Skantzos ULB 2011 8

Derivation (2)� Integration by parts gives for the first term in the r.h.s.

(assuming boundary terms vanish)

� For the boundary terms we assume that S·P(t,S) goes to 0 sufficiently fast as S goes to infinity

( )

( )( )

KK

KKStSPdS

StSPdSStSPS

KSdS

K

KK

∂∂

−−=

+−⋅⋅⋅−=

=⋅⋅⋅−=

∂∂

⋅−⋅⋅

∫∫∞

∞∞

CallCall

),(DF

),(DF),()(DF

Page 5: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 9

Derivation (3)� Integration twice by parts for the second term in the r.h.s. gives

(assuming boundary terms vanish)

( )

( ) ( )

( )

),(),(

),(),(

),(),(

),(),()(

22

22

22

222

2

TKKtKP

TSSTSPS

dS

KSS

TSSTSPS

dS

TSSTSPS

KSdS

K

K

K

σ

σ

σ

σ

⋅⋅=

⋅⋅∂∂

⋅−=

−∂∂

⋅⋅⋅∂∂

⋅−=

=

⋅⋅

∂∂

⋅−⋅

Nikos Skantzos ULB 2011 10

Derivation (5)

� The Fokker-Plank derivation assumes “small growth” which implies that µ and s should be sufficiently small

Page 6: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 11

Local Vol as a Conditional Expectation (1)

� We can interpret the local variance as the conditional expectation of the instantaneous (and, possibly, stochastic) variance given that the final underlying is at ST

� In other words: if we assume that the underlying follows

� where is the instantaneous volatility at time t depending on e.g. the spot and other stochastic elements parametrized by (as is the case for a stochastic vol model) then

( ) ( )[ ]KSaaTSTK TnTLV == ,,;,E, 122 Kσσ

( ) tntttt dWaaStSdtSdS ⋅⋅+⋅⋅= ,,,; 1 Kσµ

( )nt aaSt ,,,; 1 Kσ

naa ,,1 K

Nikos Skantzos ULB 2011 12

Local Vol as a Conditional Expectation (2)

� Proof� First, directly from the call price we have the useful identities

� For simplicity we will use the notation: max(x,y)=(x,y)+

( )[ ]0,maxEDF KSC T −⋅=

( )[ ]KSKC

T −Θ⋅−=∂∂ EDF

( )[ ]KSKC

T −⋅=∂∂ δEDF2

2

( )[ ]0,maxEDF KST

rCTC

T −∂∂

⋅+−=∂∂

( )[ ] ( )[ ] ( )[ ]KSKKSKSS TTTT −Θ⋅+−=−Θ⋅ E0,maxEE

Page 7: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 13

Local Vol as a Conditional Expectation (3)

� Apply Ito’s formula to the terminal payoff

� With sT the instantaneous vol. Use the SDE dST=… and take expectations

� Integrate from 0 to T

� Differentiate with respect to T

( ) ( ) ( ) dTKSSdSKSKSd TTTTTT ⋅−⋅⋅+⋅−Θ=− + δσ 22

21

( )[ ] ( )[ ] ( )[ ] dTKSSdTKSSKSd TTTTTT ⋅−⋅⋅+⋅−Θ⋅⋅=− + δσµ 22E21EE

( )[ ] ( )[ ] ( )[ ] ( )[ ]∫∫ ⋅−⋅⋅+⋅−Θ⋅⋅=−−− ++T

0

22

00 E

21EEE dtKSSdtKSSKSKS ttt

T

ttT δσµ

( )[ ] ( )[ ] ( )[ ]KSSKSSTKS

TTTTTT −⋅⋅+−Θ⋅⋅=∂−∂ +

δσµ 22E21EE

Nikos Skantzos ULB 2011 14

Local Vol as a Conditional Expectation (4)� Notice that we can write

� And also from Bayes’ rule

� Now use this and the general identities from p.(2) to obtain

� where

( )[ ] ( )[ ]KSKKSS TTTTT −⋅⋅=−⋅⋅ δσδσ 2222 EE

( )[ ] [ ] ( )[ ]KSKSKS TTTTT −⋅==−⋅ δσδσ EEE 22

[ ] ),(

21

)(E 2

2

22

2 KT

KCK

CqKCKqr

TC

KS LVTT σσ =

∂∂⋅⋅

⋅+∂∂⋅−+

∂∂

==

( )NTT aaST ,,,; 1 Kσσ =

Page 8: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 15

Local Vol as a Conditional Expectation (5)

� The significance of the formula

� is that it applies to ANY stochastic volatility model

� The conditional expectation of the stochastic variance must equal the Dupire Local variance

( ) ( )[ ]KSaaTSTK TnTLV == ,,;,E, 122 Kσσ

Nikos Skantzos ULB 2011 16

Stochastic Local Volatility (1)� This formula relating the Dupire Local Vol with the expected variance of

stochastic-vol models has led to significant modeling developments

� “Stochastic Local Volatility” model Y.Ren, D.Madan and M.Qian Qian Risk Sept.2007 p.138

� “Stochastic Local Vol” = s (St,t)·Z(t), no longer deterministic � lnZ(t) follows mean-reverting process, with speed ?, long-term mean value

?(t) (observed from market) and ? the vol-of-vol� We choose uncorrelated Wiener because S and Z are already correlated

through s(S,t)

Sttttt dWStZtSdtSqrdS ⋅⋅⋅+⋅⋅−= )(),()( σ

( ) Ztttt dWdtZZd ⋅+⋅−= λθκ lnln

[ ] 0E =⋅ Zt

St WW

Page 9: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 17

Stochastic Local Volatility (2)

� The conditional expectation formula implies that

� To calculate the conditional expectation over Z2 we need the joint transition density p(S,Z;t) of S and Z at time t given that at time t=0 S(0)=S0 and Z(0)=1

� For convenience define x=lnS, y=lnZ

( ) ( ) ( )[ ]KSTZTKTK TLV =⋅= 222 E,, σσ

( )[ ] ( )KSZZSpdSdZKSTZ TTTTTTT −⋅⋅⋅== ∫ δ22 ),(E

Nikos Skantzos ULB 2011 18

Stochastic Local Volatility (3)

� For this coupled process the Forward Kolmogorov Equation is

� Boundary conditions:

� We have assumed no correlation, thus no cross-term

( )[ ] 02

)(

2),(

2),(

2

2

2

22

2

222

=

∂∂

+−⋅∂∂

∂∂

+

−−

∂∂

−∂∂

px

pyty

pteex

pteeqrxt

p xyxy

λθκ

σσ

0)0(ln)0(ln)0( 0 === ZySx

Page 10: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 19

Stochastic Local Volatility (4)

� Derivation of the Forward Kolmogorov for the SLV model

Nikos Skantzos ULB 2011 20

Stochastic Local Volatility (5)

� We write the conditional expectation in terms of x,y

� For numerical reasons it is a good idea to ensure normalization by

[ ] ( )∫∫

⋅⋅=

−⋅⋅⋅===

yK

KxyxKxy

eyepdy

eeeyepdydxeee2

22

),(

),(E δψ

∫∫

⋅⋅=

),(

),( 2

yepdy

eyepdyK

yK

ψ

Page 11: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 21

Stochastic Local Volatility (6)

∫∫

⋅⋅=

),(

),( 2

yepdy

eyepdyK

yK

ψ

� t=0:� the solution is known (boundary conditions):� x(0)=lnS0 and y(0)=0 ? ? (K,0)=E[e0]=1 ? s2(S0,0)=s2LV(S0,0)� with ? , s known calculate p(t=1) by solving the PDE

� t=1:� with p(t=1) known, calculate ? (K,1)� with ? (K,1) known, calculate s2(S,1)=s2LV(S0,1) / ? (K,1)� with ? , s known calculate p(t=2) by solving the PDE

� …

� How do we find the vanilla price from these three equations?

[ ] [ ] [ ] [ ] 0)(),(),( 2

2

2

2

=∂∂

+∂∂

−∂∂

+∂∂

−∂∂

− Dpx

pyCy

pyBx

pyAxt

p σσ

),(),(),(

22

TKTKTK LV

ψσσ =

Nikos Skantzos ULB 2011 22

Using market quotes in Dupire expression (1)

� Market often quotes vanilla options in terms of implied vols sBSinstead of call prices

� where sBS(K,T) denotes the smile surface and CBS the Black-Scholes formula

� It is more convenient to express local-vol in terms of implied volsthan option prices

( ) ( )( )TKTKCTKC ,,,, BSBSMKT σ=

( ) ( )−−+

− ⋅⋅−⋅⋅= dNKedNSeC rTqT0

BS( )

T

TTKqrKS

σ

±−+

,21log 2

BS0

Page 12: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 23

Using market quotes in Dupire expression (2)

� This can be done with some computations � First, notice that

� since the market price CMKT=CMKT(K,T) does not have indirect dependencies on strike and maturity time

� Therefore we can use the chain rule and write

2

MKT2

2

MKT2MKTMKTMKTMKT

dKCd

KC

dKdC

KC

dTdC

TC

=∂∂

=∂

∂=

∂∂

TC

TC

TC

∂∂

∂∂

+∂∂

=∂

∂ BS

BS

BSBSMKT σσ K

CKC

KC

∂∂

∂∂

+∂∂

=∂

∂ BS

BS

BSBSMKT σσ

∂∂

∂∂

+∂∂

∂∂

∂∂

+∂∂

=∂

∂K

CKC

KKKC BS

BS

BSBS

BS

BS2

MKT2 σσσ

σ

Nikos Skantzos ULB 2011 24

Using market quotes in Dupire expression (3)

� Doing the computations leads to

� Inserting the above into the Dupire formula results in

� The Local-Vol is now written fully in terms of market observables

BS

BS

BS2BS

2

BS

BS

BSBS

2

BS

BS

2BS

2

2 1σσσσσσσσ ∂∂

=∂∂

∂∂

=∂∂∂

∂∂

=∂∂ −++ CddCC

TKd

KCC

TKKC

( )( )

tTSKt tddKKTK

dKTK

K

KqrK

TTtS ==

∂∂

+∂∂

+∂∂

+⋅

∂∂⋅−⋅+

∂∂

+= ,

BS

212

BS2BS

2

BS

1BS2

BS

221

BSBSBS21

2

21,

σσσ

σσ

σ

σσσ

σ

Page 13: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 25

Benchmark tests of the Dupire vol (1)

� The Dupire formula should take us back to the Black-Scholes world of no-smile by requiring that the implied vol is constant over time and strike

� Indeed, setting in the Dupire formula

� gives

� as it should

000 2BS

2BSBS =

∂∂

=∂∂

=∂∂

KKTσσσ

( ) BS, σσ =tStLV

Nikos Skantzos ULB 2011 26

Benchmark tests of the Dupire vol (2)� Let us now consider an implied vol that is time-dependent but

constant in strike

� Setting these conditions in the Dupire formula gives

� To check if this is result makes sense we argue that in a Black-Scholes world a time-dependent (instantaneous) volatility has the interpretation

� Now differentiate:

000 2BS

2BSBS =

∂∂

=∂∂

≠∂∂

KKTσσσ

( )t

ttLV ∂∂

+= BSBS

2BS 2 σσσσ

( ) ττσσ dt

t

⋅= ∫0

22BS

1

( ) ( ) ( ) dttdtt

tdtttd ⋅=⋅

+

∂∂

⇒⋅=⋅ 22BS

BSBS

22BS 2 σσσσσσ

Page 14: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 27

Numerical issues (1)

� The Dupire formula involves numerical differentiations and therefore requires that there is a continuity of market quotes for all maturities

� In practise, there are limited quotes

� The practitioner is required to find the missing market values by interpolating in a reasonable way between tenors and strikes� Strike interpolation: spline� Time interpolation: linear in variance

Nikos Skantzos ULB 2011 28

Numerical issues (2)

� The local vol surface depends on the current spot level (explicit dependence in numerator) and the current “moneyness” level S0/K (through the variables d1,d2)

� This implies that every time the spot changes the local-volsurface must be re-computed.

� In practise this is unfeasible, especially in the Forex world where the spot changes almost continuously.

� One computes the local-vol surface once a day.

Page 15: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 29

Numerical issues (3)

� The denominator CKK can cause numerical problems if

� CKK < 0 (smile is locally concave) which implies s 2(K,t)<0 and therefore s (K,t) is imaginary

� Normally the convexity CKK must be non-zero because it is equivalent to the payoff of a Butterfly strategy which is non-zero by no-arbitrage

0)()(2)(0 ≥∆−+−∆+→≥ KKCKCKKCCKK

)()(2)(BF KKCKCKKC ∆−+−∆+=

Butterfly

Nikos Skantzos ULB 2011 30

How to use the Local Vol?� The Local-vol can be seen as an instantaneous volatility that depends on

where is the spot at each time step

� It offers a convenient recipe for pricing path-dependent options

� Precompute the entire Local-Vol surface for all tenors and strikes before beginning the simulation

� At every time-step of the simulation check where is the spot level, obtain the corresponding local-vol and use it into the lognormal process

� In this way, obtain the spot path till maturity

( ) ( ) ( )T

tStStS SSS TT → → → −− 112211 ,,2

,1

σσσ L

( ) ( ) ttLVtLV WttSttS

ttt eSS∆⋅∆+∆⋅

∆+ ⋅=,,

21 2 σσµ

Page 16: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 31

Local Vol rule of thumb

Rule of thumb:Local vol varies with index level twice as fast as implied vol varies with strike(Derman & Kani)

Sinitial

Sfinal

Nikos Skantzos ULB 2011 32

Local Vol interpolation (1)� In practice implied vols are quoted on tenors

1w, 2w, 3w, 1m, 2m, 4m, 6m, 9m, 1y, 2y, …

� This means that the Local vol will be built on a grid of discrete points

� The Monte Carlo simulation must interpolate local-vols for time steps that fall within the grid points

t1 t2

s LV(t1)s LV(t2)

t

s LV(t)=?

Page 17: itf.fys.kuleuven.beitf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf · ˇ ) *+"˘ ˛˜˚ ˜ f ˘ ˘ c 1+ 29 g 0 ˆ˘ ˘ $ˇ 2 ˘ ˝ ˘ ˇ ˙ ˝ ˇˆ ˇˆ ˙ ˇˆ ˝ ˇˆ ˇˆ

Nikos Skantzos ULB 2011 33

Local Vol interpolation (2)

� A simple interpolation scheme is to take an average between the local vols at grid points

� Average is linear-in-variance in time

( )222121

2

2

12

1211

12

2

31

)(1 2

1

σσσσσ

σσσσ

++=

−−

−+−

= ∫t

t

dstt

tstt

Nikos Skantzos ULB 2011 34

Brainteaser question

� Consider

� What is the value of x?

2=Nxxxx