From Spot Volatilities to Implied Volatilities...Julien Guyon Global Markets Quantitative Research...

Motivation A link between spot and implied volatilities Previous results Our new estimates of implied volatilities Numerical experiments Conclusion

From Spot Volatilitiesto Implied Volatilities

Julien Guyon

Global Markets Quantitative Research

Research In Options 2010 conference, Rio de JaneiroDecember 1st, 2010

Joint work with Pierre Henry-Labordère

[email protected]@sgcib.com

Julien Guyon Société Générale

From Spot Volatilities to Implied Volatilities


Outline

Motivation

A link between spot volatilities and implied volatilities

Previous results

Our new estimates of implied volatilities

Numerical experiments

Conclusion




Motivation

One knows how to infer the local volatility function from the impliedvolatility surface (Dupire, 1994):

σ(T, K)2 =∂C∂T

12K2 ∂

2C∂K2

How to invert this formula?

Classical solution: solve numerically the forward equation for call prices.Alternative solutions?






Let σt be the spot volatility of the T -forward price ftT of some asset St:

dftT = σtftT dWTt

No assumption is made on σt: it is a general stochastic process. WT is

a (Ft)-Brownian motion under the T -forward measure PT , where (Ft)denotes the filtration generated by the processes σt and ftT .

Let B denote the Black price:

B(t, f ; σ) = fN`d+`f, σ2(T − t), K

´´−KN

`d−`f, σ2(T − t), K

´´d± (f, v, K) =

ln f/K√v

± 12

√v





(ST −K)+ = (fTT −K)+ = B(T, fTT ; σ) for any σ, so by Itô’s formula,

ETˆ(ST −K)+

˜−B(0, f0T ; σ) = ET

»Z T0

„∂tB +

1

2σ2t f

2tT ∂

2fB

«(t, ftT ; σ)dt

–Besides, ∂tB +

12σ2f2∂2fB = 0.

⇒ σBS(T, K) is the value of the constant parameter σ such that

ET»Z T

0

`σ2t − σ2

´f2tT ∂

2fB(t, ftT ; σ)dt

–= 0 (1)

The time integral is the P&L generated between inception and maturity bya delta-hedged position using constant volatility σ.

⇒ σBS(T, K) obeys the self-consistence equation:1

σBS(T, K)2 =

EThR T

0σ2t f

2tT ∂

2fB(t, ftT ; σBS(T, K))dt

iEThR T

0f2tT ∂

2fB(t, ftT ; σBS(T, K))dt

i (2)1As far as we know, this representation of implied volatility was first used by Dupire in the late

1990’s [2].




Comparison with Gatheral’s formula


Gatheral [3] states that

σBS(T, K)2 =

1

T

Z T0

ETˆσ2t f

2tT ∂

2fB

v(t, ftT )˜

EThf2tT ∂

2fB

v(t, ftT )i dt

Bv slightly differs from B: it is the Black price of the call option withtime-dependent deterministic volatility

pv(t), where v is the so-called

“forward implied variance” function, implicitly defined by

v(t) =ETˆσ2t f

2tT ∂

2fB

v(t, ftT )˜

EThf2tT ∂

2fB

v(t, ftT )i





Both expressions for σBS(T, K) can be seen as special cases of a moregeneral result: For any positive function w(t),

σBS(T, K)2 =

1

T

Z T0

w(t)dt ⇐⇒Z T

0

ETˆ`

σ2t − w(t)´f2tT ∂

2fB

w(t, ftT )˜dt = 0

The implied forward variance v is the only function w such that, the timeintegrand on the r.h.s. is zero for each time slice t.

The implied volatility σBS(T, K) is the only constant function w such thatthe equation holds.




Our general strategy for estimating implied volatilities


Let σ2loc(t, ftT ) = ETˆσ2t |ftT

˜be the local variance of the forward.

Since

ETˆσ2t f

2tT ∂

2fB(t, ftT ; σ)

˜= ET

ˆσ2loc(t, ftT )f

2tT ∂

2fB(t, ftT ; σ)

˜one can replace σt by σloc(t, ftT ) in (1)-(2).

We will build estimates of the implied volatility by estimating the fixedpoint of the mapping

σ 7→

vuuutEThR T

0σ2loc(t, ftT )f

2tT ∂

2fB(t, ftT ; σ)dt

iEThR T

0f2tT ∂

2fB(t, ftT ; σ)dt

i =sZ T0

Z ∞0

σ2loc(t, f) qσ(t, f) dtdf

qσ(t, f) =f2∂2fB(t, f ; σ)p(t, f)R T

0

R∞0

f ′2∂2fB(t′, f ′; σ)p(t′, f ′)df ′dt′





Note that

f2∂2fB(t, f ; σ)p(t, f) =f

σp

2π(T − t)exp

„−d+(f, σ

2(T − t), K)2

2

«p(t, f).

p(t, f), hence qσ(t, f), is not known explicitly.

Our suggestion: approximate the fixed point using an estimate p̂(t, f)of p(t, f).

Of course, p̂(t, f) also leads to an estimate of the forward price of the calloption, by numerical integration of the payoff against p̂(t, f).But our method proves much more accurate, as we directly average σ2loc,rather than the payoff.





Figure: Graph of qσ(t, f) when the forward is lognormal; σ = 30%, f0 = 100,K = 80, T = 1.




The effective path estimator

Previous results

Several authors have already suggested estimators of the implied volatilitybased on approximations of qσ(t, f).

p(0, f) = δf0(f) and ∂2fB(T, f ; σ) = δK(f), qσ(t, f)

⇒ Double Dirac Property: qσ(t, f) reduces to a multiple of a Dirac massat f = f0 when t = 0, and to a multiple of a Dirac mass at f = K whent = T .

Inspired by this property, Gatheral [3], then Reghai [6] have suggestedapproximating qσ(t, f) by q̂

EPσ (t, f) =

1T

δEP(t)(f) where δEP(t) is a Diracmass that singles out one particular effective path EP(t) with fixed endpoints f0 at t = 0 and K at t = T .

The authors call this approximation the “most likely path” technique.





Examples of effective paths

Common examples are

EP(t) = f0 +t

T(K − f0) , EP(t) = f0 exp

„t

Tln

K

f0

«Alternatively, one may use

EP(t) = f0 exp

„ν(t)

ν(T )ln

K

f0+

1

2ν(T )

ν(t)

ν(T )

„1− ν(t)

ν(T )

««ν(t) =

Z t0

σ(s)2ds, σ(t) = σloc (t, EP(t))

EP(t) is the expected value of a lognormal forward f̄t having volatilityσloc (t, EP(t)), given f̄T = K:

EP(t) = ET [f̄t˛̨f̄T = K ], df̄t = σloc (t, EP(t)) f̄tdW

Tt

One can build this effective path using a fixed point algorithm. Verysimilar to Reghai.





Properties of the effective path approximation

The estimated density q̂EPσ (t, f) does not depend on σ.

The estimated implied volatility squared reads explicitly

σ̂EPBS (T, K)2 =

1

T

Z T0

σ2loc(t, EP(t)) dt

One estimates the implied volatility squared by the time average of thesquare of the local volatility of the forward, taken at some effective path.

The average of the local volatility over all possible values for the forwardhas been purely and simply replaced by the value of the local volatility atsome effective path.





Some drawbacks of the effective path estimator

A first drawback. Imagine that for each time t the local volatility is U-shapedwith a minimum at f = f0, and that K = f0. For the usual choices (2), theeffective path is constant at f0, so σ̂

EPBS (T, f0)

2 is just the time average ofσ2loc(t, f0), the minimum value of σ

2loc at time t. Hence

σBS(T, f0)2 =

1

T

Z T0

ETˆσ2loc (t, ftT ) f

2tT ∂

2fB

v(t, ftT )˜

EThf2tT ∂

2fB

v(t, ftT )i dt

≥ 1T

Z T0

ETˆσ2loc (t, f0) f

2tT ∂

2fB

v(t, ftT )˜

EThf2tT ∂

2fB

v(t, ftT )i dt

=1

T

Z T0

σ2loc (t, f0) dt = σ̂EPBS (T, f0)

2

σ̂EPBS (T, f0) surely underestimates σBS(T, f0).





Some drawbacks of the effective path estimator

A second drawback. Assume EP (t) = f0 exp“

tT

ln Kf0

”:

σ̂EPBS (T, K)2 =

Z 10

σ2loc

„uT, f0 exp

„u ln

K

f0

««du

⇒ Under a mild domination assumption:

σ̂EPBS (0, K)2 =

Z 10

σ2loc

„0, f0 exp

„u ln

K

f0

««du

This is not in line with Berestycki-Busca-Florent:

σBS(0, K)−1 =

Z 10

σ−1loc

„0, f0 exp

„u ln

K

f0

««du




The conditional distribution estimator

The conditional distribution estimator

In [6], Reghai suggests another heuristic approximation for the impliedvolatility (CD stands for conditional distribution):

σBS(T, K)2 ' σ̂CDBS (T, K)2 =

1

T

Z T0

ETˆσ2loc(t, ftT ) |fTT = K

˜dt

This boils down to estimating qσ(t, f) by1T

times the pdf of ftTconditional on fTT = K under PT :

q̂CD(t, f) =1

Tp (t, f |fTT = K )

Computation of q̂CD cannot be performed exactly. We suggest to use

Law (ftT |fTT = K) ' Law`f̄t|f̄T = K

´= Law (f0 exp (Gt))

where Law(Gt) = N“

ν(t)ν(T )

ln Kf0

, ν(T ) ν(t)ν(T )

“1− ν(t)

ν(T )

””. One can then

approximate ETˆσ2loc(t, ftT ) |fTT = K

˜thanks to a Hermite quadrature.




Our new estimates of implied volatilities

q̂CD is a very natural candidate among all the functions satisfying theDouble Dirac Property.

However, this property is stable by multiplication.

It is not clear why q̂CD would be a good proxy for q, in the sense that itwould produce a good proxy for the implied volatility.

⇒ We will build two new estimates q̂LNσ (t, f) and q̂HKEσ (t, f) of

qσ(t, f) =f2∂2fB(t, f ; σ)p(t, f)R T

0

R∞0

f ′2∂2fB(t′, f ′; σ)p(t′, f ′)df ′dt′

(3)

based on estimations of the pdf p(t, f) of ftT .

In the first estimate, we replace p(t, f) by a lognormal density.

In the second one, we replace p(t, f) by its heat kernel expansion.




The lognormal estimator


The estimate q̂LNσ is defined by (3), where p(t, f) is simply replaced by alognormal density

p̂LN(t, f) =1

fp

2πvLNtexp

„−d+(f, v

LNt , f0)

2

2

«where

vLNt =

Z t0

σLN(u)2du

for some deterministic function of time σLN(t), the Black volatility.

If the forward has volatility σLN(t), p̂LN(t, f) = p(t, f), q̂LNσ = qσ (andσ2BS(T, K) is simply v

LNT /T , independently of K).

Provided σloc(t, f) has little dependence on f , it can be well approximatedby some σLN(t) and we expect σ̂LNBS (T, K) ' σBS(T, K).





Some straightforward computations lead to

σ̂LNBS (T, K)2 =

R T0

a(t)R

R σ2loc

`t, fLN(t, z)

´1√2π

exp“− z

2

2

”dzdtR T

0a(t)dt

where

fLN(t, z) = f0 exp

vLNt

vLNt + σ̂LNBS (T, K)

2(T − t)ln

K

f0+

pvLNt σ̂

LNBS (T, K)

√T − tp


2(T − t)z

!

a(t) =1p


2(T − t)exp

−

d+`f0, v

LNt + σ̂

LNBS (T, K)

2(T − t), K´2

2

!





One has

a(t) =

√2π

f0ET»“

fLNtT

”2∂2fB(t, f

LNtT ; σ̂

LNBS (T, K))

–where fLNtT has lognormal dynamics df

LNtT = σ

LN(t)fLNtT dWTt .

Ex: pick σLN(t) = σ̂LNBS (T, K). Then`fLNtT

´2∂2fB

`t, fLNtT ; σ̂

LNBS (T, K)

´is

the Black-Scholes Gamma notional “ΓS2”, it is a PT -martingale, a(t)does not depend on t and

σ̂LNBS (T, K)2 =

1

T

Z T0

ZR

σ2loc

t, f0 exp

t

Tln

K

f0+ σ̂LNBS (T, K)

√T

st

T

„1− t

T

«z

!!1√2π

exp

„−z

2

2

«dzdt.

This is exactly what one would get at first order in δσloc when oneexpands the local volatility around some constant value σ̄, provided that inthe r.h.s. σ̂LNBS (T, K) is replaced by σ̄.





Equation (2) improves the effective path estimator σ̂EPBS (T, K)2: we do

not only integrate σ2loc along the effective path

EPLN(t) = fLN(t, 0) = f0 exp

„vLNt


2(T − t)ln

K

f0

«but also around it.

The fluctuation around EPLN(t) is given by a (time-changed) Brownianbridge. It is Gaussian, with standard deviation

STDEVLN(t) =

pvLNt σ̂

LNBS (T, K)

p(T − t)p


2(T − t)

In the effective path estimator, this standard deviation is taken to be zero,or, equivalently, the Gaussian pdf 1√

2πexp

`− 1

2z2´

is replaced by a Diracmass at point z = 0.





We do not give the same weight dt/T to each infinitesimal interval[t, t + dt].

Rather, we attach a varying weight a(t)dt which is an estimate ofETˆf2tT ∂

2fB(t, ftT ; σBS(T, K))

˜dt, and depends on the particular function

σLN(t) chosen.

Because of the shape of qσ(t, f), a natural choice for σLN(t) is

σloc(t, EP(t)) for some convenient effective path.




The Heat Kernel Expansion estimator


The estimate q̂HKEσ is defined by (3), where p(t, f) is replaced by its expansionat first order in the potential Q, the so-called heat kernel expansion

p̂HKE(t, f) =1

f

σloc(t, f0)1/2

σloc (t, f)3/2

sf0f

eR ff0

df′C(t,f′)

R f′f0

∂t1

C(t,f′′) df′′P0(t, s)

`1 + tQ(t, s) + · · ·

´C(t, f) = fσloc(t, f), s =

Z ff0

df ′

C(t, f ′)

P0(t, s) =1√2πt

exp

„−s

2

2t

«Q(t, s) =

Z 10

du

ZR

dz Q“ut, us +

√tp

u (1− u)z” 1√

2πexp

„−1

2z2«

Q(t, s) = −12

`∂sb + b

2´ (t, s)− Z s0

∂tb(t, s′)ds′

b(t, s) =

Z ff0

∂t1

C(t, f ′)df ′ − 1

2∂fC(t, f)





After some calculations, we get:

σ̂HKEBS (T, K)2 =

R T0

RR σloc

“t, f0e

x(t,z)”1/2

w(t, z)dzdtR T0

RR σloc (t, f0e

x(t,z))−3/2

w(t, z)dzdt

with

w(t, z) = π (t, x(t, z)) exp

„−1

2z2«

1 + tQ

t,

Z x(t,z)0

dv

σloc (t, f0ev)

!!

x(t, z) =t

Tln

K

f0+ σ̂HKEBS (T, K)

√T

st

T

„1− t

T

«z

π(t, x) = σloc(t, f0)1/2 exp (λ(t, x)− µ(t, x)/2)

λ(t, x) =

Z x0

dv

σloc (t, f0ev)

Z v0

∂tdw

σloc(t, f0ew)

µ(t, x) = −14σ̂HKEBS (T, K)

2t +1

t

„ε(t, x)2 +

2xε(t, x)

σ̂HKEBS (T, K)

«ε(t, x) =

Z x0

dv

σloc (t, f0ev)− x

σ̂HKEBS (T, K)





Similarly to the log-normal estimator, we do not only integrate the localvolatility along the effective path

EPHKE(t) = f0 exp(x(t, 0)) = f0 exp

„t

Tln

K

f0

«but also around it.

The fluctuation around EPHKE(t) is not Gaussian.

Two spatial integrations are needed:

σ1/2loc in the numerator,

σ−3/2loc in the denominator.




Another version of the lognormal and heat kernel expansion estimates

Another version of the lognormal and heat kernel expansion estimates

Recall Gatheral’s representation of implied volatility:

σBS(T, K)2 =

1

T

Z T0

Z ∞0

σ2loc(t, f)ht(f) dfdt

ht(f) =f2∂2fB

v(t, f)p(t, f)R∞0

f ′2∂2fBv(t′, f ′)p(t′, f ′)df ′

We can replace p(t, f) by our estimate p̂LN(t, f) or p̂HKE(t, f) into theabove formula to get an estimate ĥt(f) of ht(f), and then get an estimate

σ̂BS(T, K)2 =

1

T

Z T0

Z ∞0

σ2loc(t, f)ĥt(f) dfdt

However, ĥt(f) is not known explicitly: it depends on v, and v is

unknown: we are precisely looking for 1T

R T0

v(t)dt = σBS(T, K)2

⇒ Also replace v(t) by some v̂(t) such that v̂(0) = σ2loc(0, f0) andv̂(T ) = σ2loc(T, K). For instance, one may pick v̂(t) = σ

2loc(t, EP(t)) for

some convenient effective path.




Numerical comparison of estimators

Numerical experiments

We have picked a parametric local volatility f 7→ σloc(t, f). To give the“effective path estimator” a chance to be competitive, we deliberatelypicked a smooth local volatility.

We have fitted the (time-dependent) parameters to the implied volatilitysurface of the Eurostoxx 50, as of November 28, 2007.

To compute the EP estimator, we have used the effective path

EP(t) = f0 exp“

tT

ln Kf0

”. For the lognormal estimator, we have used

σLN(t) = σloc(t, EP(t)).





30%

35%

40%

45%

50%

3 months implied volatilities

PDE

EP

CD

LN

HKE order 0

10%

15%

20%

25%

30%

35%

40%

45%

50%

60 70 80 90 100 110 120 130 140 150 160

3 months implied volatilities

PDE

EP

CD

LN

HKE order 0

HKE order 1

Figure:





1 year implied volatilities

48%


38%

43%

48%


28%

33%

38%

43%

48%


PDE

EP

CD

LN

HKE order 0

23%

28%

33%

38%

43%

48%


PDE

EP

CD

LN

HKE order 0

HKE order 1

13%

18%

23%

28%

33%

38%

43%

48%

40 60 80 100 120 140 160 180 200 220


PDE

EP

CD

LN

HKE order 0

HKE order 1

13%

18%

23%

28%

33%

38%

43%

48%

40 60 80 100 120 140 160 180 200 220


PDE

EP

CD

LN

HKE order 0

HKE order 1

13%

18%

23%

28%

33%

38%

43%

48%

40 60 80 100 120 140 160 180 200 220


PDE

EP

CD

LN

HKE order 0

HKE order 1

Figure:





25%

30%

35%

3 years implied volatilities

PDE

EP

CD

LN

HKE order 0

15%

20%

25%

30%

35%

40 60 80 100 120 140 160 180 200 220

3 years implied volatilities

PDE

EP

CD

LN

HKE order 0

HKE order 1

Figure:




Computational complexity

Computational complexity

Complexity of PDE solver: nK × nT .When one needs to compute a single implied volatility for a strike K and amaturity T , one has to compute a whole set of nK × nT call prices.Complexity of new HKE estimator at order 1: nI × n3L × n2H where

nI : number of iterations in the fixed point algorithm (3 or 4)nL: number of operations needed to estimate integrals on finite intervals,e.g., the number of points in a Legendre integrationnH : number of operations needed to estimate Gaussian integrals of the typeR

R ϕ(z) exp(−z2/2)dz, e.g., the number of points in a Hermite integration.

Complexity of new HKE estimator at order 0: nI × n2L × nH .




Conclusion

We have suggested new ways of computing implied volatilities from spotvolatilities, based on the fact that the implied volatility is the fixed pointof a function mapping a volatility σ to a weighted quadratic averageof the effective local volatility, with a weight qσ(t, f) depending on σ.

In previous works (Gatheral, Reghai), some heuristic q̂(t, f)’s, independentof σ, have been suggested.

Here, our estimates q̂σ(t, f) are explicitly built as small deviations fromthe true weight qσ(t, f), by replacing the exact pdf of the forward by anestimated pdf:

a lognormal density, orthe HKE of the exact pdf of the forward, at first order in the potential




Conclusion

The HKE estimator proves to be remarkably accurate, even for longmaturities, while the accuracy of other estimators is not sufficient forpractical trading purposes.

Computing the local volatility on an effective path only, such as inGatheral, is too crude.

Averaging the local volatility using Gaussian fluctuations around aneffective path, like in our LN estimator, is not enough.

We need to average the local volatility using non-Gaussianfluctuations around an effective path to get accurate results.

In terms of computational time:

Our new HKE estimator cannot beat a PDE solver, if one wants tocompute a whole implied volatility surface.

However, if one needs to compute implied volatilities for only a few strikesand maturities, it is competitive. Dropping the order 1 in the HKEexpansion makes our estimator slightly less accurate, but much faster.




Some references

Berestycki H., Busca J., Florent I., Asymptotics and calibration of localvolatility models, Quantitative Finance, volume 2, 61–69, 2002.

Dupire B., A new approach for understanding the impact of volatility onoption prices, slides of a presentation at the Risk Conference, October30th, 1998.

Gatheral J., The volatility surface, a practitioner’s guide, Wiley, 2006.

Henry-Labordère P., Analysis, Geometry, and Modeling in Finance,Chapman & Hall, CRC Financial Mathematics Series, 2008.

Lee R., Implied volatility: Statics, dynamics, and probabilisticinterpretation, in Ricardo Baeza-Yates, Joseph Glaz, Henryk Gzyl, JürgenHüsler and José Luis Palacios, eds, Recent Advances in AppliedProbability. Berlin: Springer-Verlag, 2005.

Reghai A., The hybrid most likely path, Risk Magazine, April 2006.



MotivationA link between spot and implied volatilitiesA link between spot volatilities and implied volatilitiesComparison with Gatheral's formulaOur general strategy for estimating implied volatilities

Previous resultsThe effective path estimatorThe conditional distribution estimator

Our new estimates of implied volatilitiesThe lognormal estimatorThe Heat Kernel Expansion estimatorAnother version of the lognormal and heat kernel expansion estimates

Numerical experimentsNumerical comparison of estimatorsComputational complexity

Conclusion

From Spot Volatilities to Implied Volatilities...Julien Guyon Global Markets Quantitative Research...

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