Smile in the low moments

L. De Leo, T.-L. Dao, V. Vargas, S. Ciliberti, J.-P. Bouchaud10 jan 2014

Outline

1 The Option Smile: staticsA trading styleThe cumulant expansionA low-moment formula: the moneyness expansion

2 Smile from historical data: the Hedged Monte Carlo

3 The Option Smile: dynamicsA different trading styleSkew Stickiness Ratio in non-linear models

4 Conclusions

Outline

1 The Option Smile: staticsA trading styleThe cumulant expansionA low-moment formula: the moneyness expansion

2 Smile from historical data: the Hedged Monte Carlo

3 The Option Smile: dynamicsA different trading styleSkew Stickiness Ratio in non-linear models

4 Conclusions

Context

Let’s suppose a trader wants to buy/sell an option and hedge it until expiry. Whatdoes she/he need to evaluate to take a decision?

Simply the price of the option, i.e. its implied volatility, and compare it to the market

No need to know the evolution of the option price! (No dynamics needed)

Context

The option smile is the sign that the Black-Scholes model does not provide anadequate description of the underlying dynamics

“Stylized facts” about underlying dynamics that make the Gaussian model fail:– Fat tails → non-trivial kurtosis κT– Volatility is not constant → non-trivial kurtosis term structure– Volatility depends on past returns → anomalous skewness ςT

Different possible approaches:– Model driven: jumps and Lévy processes, GARCH and stochastic volatility models (e.g.

Heston or SABR), multifractal models, etc.– Phenomenological: start from Gaussian behavior and include “corrections”

The cumulant expansion

Take an additive stock price process: St+1 = (1 + r)St + δSt , where δSt are iid.Let uT = (ST /S0 − 1)/σ0

√T be the normalized return

If T is large but finite, the central limit theorem reads

P(ST |S0) = N(

S0(1 + r)T , σ0√

T)[

1 +ςT

6H3(uT ) +

κT

24H4(uT ) + . . .

]where Hn(·) are Hermite polynomials

The cumulant expansion

Plugging this into the option pricing formula C = E [(ST − K )+] yields

σBS ' σ0

[1 +

ςT

6M+

κT

24(M2 − 1)

]whereM = (K/S0 − 1)/σ0

√T is the moneyness

[Backus et al., 1997, Bouchaud et al., 1998]

Main disadvantages:– The expansion assumes that higher order moments are finite and small (not the case

usually)– Even if finite, the estimation of moments of order 3 and 4 is subject to huge errors

A new expansion: moneyness

Moneyness expansion: rigorous and general [De Leo et al., 2013]

It involves moments of order <= 2

It lends itself to analytical treatment

The coefficients of the expansion can be estimated with different methods. Forexample with Hedged Monte Carlo (see later)

The moneyness expansion

If we look for a smile expansion of the form σBS = σ0(α+ βM+ γM2) we

obtain:

α =

√π

2E [|uT |] +σ0

√T√π

2

(E[u2

T 1uT>0

]− P(uT > 0)

)β =

√2π

[12− P(uT > 0)+

σ0√

T2

(pT (0)−

E [|uT |]2

)]

γ =

√π

2pT (0)−

1√

2πE [|uT |]+

√π

2σ0√

T(

12− P(uT > 0)

)+

+σ0√

T√

2π

E [u2T 1uT>0]− P(uT > 0)

E [|uT |]2

The at-the-money volatility is related to the mean absolute deviation

The slope is a consequence of the asymmetry of the stock return distribution

The curvature is fixed by the probability density in zero (an indirect tail effect)

The expansion coincides with the cumulant expansion when skewness andkurtosis are small and cumulants of higher order can be neglected

Outline

1 The Option Smile: staticsA trading styleThe cumulant expansionA low-moment formula: the moneyness expansion

2 Smile from historical data: the Hedged Monte Carlo

3 The Option Smile: dynamicsA different trading styleSkew Stickiness Ratio in non-linear models

4 Conclusions

Numerical estimate of the smile parameters

At short maturity the ATM volatility is σ(M = 0) = σ0

√π2 E [|uT |]

Note thatC(K = S0,T ) + P(K = S0,T ) = E [|ST − S0|]

i.e., E [|uT |] can be calculated as the fair price of an at-the-money straddle

The first coefficient of the smile expansion can be calculated by pricing an (exotic)option, for example with Monte Carlo

The other coefficients can be calculated using other “exotic” payoffs

Idea: Delta-hedge the Monte Carlo to reduce the error and remove the drift[Bouchaud et al., 2001]

Hedged Monte Carlo: the general idea

For an arbitrary process St , determine both the price Ct and the optimal hedge φtby optimizing locally a risk function, e.g.

Rt =⟨

[Ct+1(St+1)− Ct (St ) + φt (St )(St+1 − St )]2⟩

Linear parametrization of price and hedge using Nf variational functions:

Ct (S) =

Nf∑a=1

γ(a)t y (a)(S) φt (S) =

Nf∑a=1

γ(a)t y (a)(S)

Start from the known payoff at expiry and work backwards in t

Hedged Monte Carlo vs Control Variates

A first possible approximation is to replace optimal hedge by ∆-hedge(y (a)(S) = ∂y (a)/∂S) to reduce the computational cost

A second approximation consists in using a Black-Scholes ∆-hedge with acarefully chosen volatility

Given N realizations of the process S(n)t , the option price is then

Ct =1N

N∑n=1

(S(n)T − K )+ −

T−1∑u=t

∆u(S(n)u , σ(n))(S(n)

u+1 − S(n)u )

which is the Black-Scholes version of the classical variance reduction technique

Hedged Monte Carlo: advantages

Substantial variance reduction, for the same reason for which hedged options areless risky than unhedged ones

Provide a numerical estimate of:– the price of the derivative– the optimal hedge– the residual risk

Construct the adequate risk neutral measure for a given risk objective and for anarbitrary model of the underlying

It allows to use purely historical data to price derivatives, short-circuiting themodeling phase

Theoretical smile for US stocks

−3 −2 −1 0 1 2 3Moneyness

0.2

0.3

0.4

0.5

0.6

0.7

0.8

HM

C v

ola

tilit

ySmall capMid capLarge cap

Data: US stocks in SPX + MID, 1996-2012

Estimation of the smile parameters

�1.0 �0.5 0.0 0.5 1.0 1.5 2.0Price (dollars)

0

50000

100000

150000

200000

250000Asym bin 0.30-0.50 - T 60 days

Non-hedgedHedged

0 10 20 30 40 50 60Days

1.4

1.6

1.8

2.0

2.2

2.4

�NH/�H

Asymmetry

bin 0.10-0.15bin 0.15-0.20bin 0.20-0.30bin 0.30-0.50bin 0.50-1.00

Using Hedged Monte Carlo to obtain the skew of the smile

β coefficient related to the price of a binary option (pay $1 if ST > S0eµT , 0otherwise)

Delta-hedging reduces the error by a factor 2 at 30 days

Estimation of the smile parameters - SP500 index

0 5 10 15 20T (days)

0.20

0.15

0.10

0.05

0.00

βT

ςT /6

0 5 10 15 20T (days)

0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

γT

T /24

Data: 1970-2011. Hedge crucial to reduce the noise

|ςT |/6 is systematically different from |βT |The kurtosis overestimates dramatically the smile curvature

Outline

1 The Option Smile: staticsA trading styleThe cumulant expansionA low-moment formula: the moneyness expansion

2 Smile from historical data: the Hedged Monte Carlo

3 The Option Smile: dynamicsA different trading styleSkew Stickiness Ratio in non-linear models

4 Conclusions

Context

Let’s suppose another trader wants to hold a dynamic position on an hedgedoption

In this case the impact of the option price move is important: smile dynamicsneeded

More complicated problem!

Typical question: how does the at-the-money volatility change if the underlyingmoves?

Implied leverage

Assuming a linear smile σBS,T ' σATM,T (1 + SkewTM) define the SkewStickiness Ratio (SSR) RT : [Bergomi, 2009]

δσATM,T = −RT SkewTδS

S√

T

“Popular” values:– RT = 0: Sticky Delta– RT = 1: Sticky Strike– RT = 2: Short T limit for stochastic volatility models

Implied leverage beyond linear models

Modeling the forward variance curve {v i+li }l≥0

ri := lnSi+1

Si= σiεi , v i+l

i+1 − v i+li = νλi+l

i ({vui }u≥i )f (εi )

where v i+li = E [σ2

i+l |Fi−1] and E [f (εi )] = 0

For linear models (f (x) = x) we have SkewT ≡ ςT /6 [Bergomi and Guyon, 2011]

For general models we can parametrize RT as

RT = RT

∣∣∣lin×

ςT /6SkewT

where RT

∣∣∣lin

is the SSR result in the linear model framework that saturates to 2 in

the short T limit

Short term limit of RT exceeds 2 [Vargas et al., 2013]

Skew stickiness ratio

1 21 41 61 81 101.0

T

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0SSR

Historical estimator

Option data (DAX_IDX)

Garch model

Data: DAX index, 2002-2013

Outline

1 The Option Smile: staticsA trading styleThe cumulant expansionA low-moment formula: the moneyness expansion

2 Smile from historical data: the Hedged Monte Carlo

3 The Option Smile: dynamicsA different trading styleSkew Stickiness Ratio in non-linear models

4 Conclusions

Conclusions

Avoid large moments: smile expansion in moneyness is more reliable and able tocapture non-linear effects

The coeffients of the expansion can be calculated with Hedged Monte Carlo: themodeling phase can be bypassed using historical prices

The smile dynamics for indexes shows features compatible with a non-linear origin

Backus, D., Foresi, S., Lai, K., and Wu, L. (1997).Accounting for biases in black-scholes.Working paper of NYU Stern school of Business.

Bergomi, L. (December 2009).Smile dynamics iv.Risk Magazine, pages 94–100.

Bergomi, L. and Guyon, J. (2011).The smile in stochastic volatility models.http://ssrn.com/abstract=1967470.

Bouchaud, J.-P., Cont, R., and Potters, M. (1998).Financial markets as adaptive systems.Europhysics Letters, 41(3):239–244.

Bouchaud, J.-P., Potters, M., and Sestovic, D. (March 2001).Hedge your monte carlo.Risk Magazine, pages 133–136.

De Leo, L., Vargas, V., Ciliberti, S., and Bouchaud, J.-P. (July 2013).One of these smiles.Risk Magazine, pages 64–67.

Vargas, V., Dao, T.-L., and Bouchaud, J.-P. (2013).Skew and implied leverage effect: smile dynamics revisited.arXiv:1311.4078v1[q-fin.ST].