View
11
Download
0
Category
Preview:
Citation preview
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
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
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
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
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
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
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 =
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
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
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
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
dσ
σ
±−+
=±
,21log 2
BS0
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
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
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.
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
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)=?
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
Recommended