Smile in the low moments - PDF Free Download

Smile in the low moments L. De Leo, T.-L. Dao, V. Vargas, S. Ciliberti, J.-P. Bouchaud 10 jan 2014

Outline 1 The Option Smile: statics A trading style The cumulant expansion A low-moment formula: the moneyness expansion 2 Smile from historical data: the Hedged Monte Carlo 3 The Option Smile: dynamics A different trading style Skew 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. What does 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 an adequate description of the underlying dynamics Stylized facts about underlying dynamics that make the Gaussian model fail: Fat tails non-trivial kurtosis κ T Volatility is not constant non-trivial kurtosis term structure Volatility depends on past returns anomalous skewness ς T 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: S t+1 = (1 + r)s t + δs t, where δs t are iid. Let u T = (S T /S 0 1)/σ 0 T be the normalized return If T is large but finite, the central limit theorem reads ( P(S T S 0 ) = N S 0 (1 + r) T ) [, σ 0 T 1 + ς T 6 H 3(u T ) + κ ] T 24 H 4(u T ) +... where H n( ) are Hermite polynomials

The cumulant expansion Plugging this into the option pricing formula C = E[(S T K ) + ] yields [ σ BS σ 0 1 + ς T 6 M + κ ] T 24 (M2 1) where M = (K /S 0 1)/σ 0 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. For example with Hedged Monte Carlo (see later)

The moneyness expansion ( If we look for a smile expansion of the form σ BS = σ 0 α + βm + γm 2 ) we obtain: π α = 2 E [ u π ( [ ] ) T ] +σ 0 T E ut 2 2 1 u T >0 P(u T > 0) β = [ 1 2π 2 P(u T > 0)+ σ ( 0 T p T (0) E [ u ) ] T ] 2 2 ( ) π γ = 2 p 1 π 1 T (0) 2πE[ ut ] + 2 σ 0 T 2 P(u T > 0) + + σ 0 T E[uT 2 1 u T >0] P(u T > 0) 2π E[ u T ] 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 and kurtosis are small and cumulants of higher order can be neglected

Numerical estimate of the smile parameters At short maturity the ATM volatility is σ(m = 0) = σ 0 π 2 E[ u T ] Note that C(K = S 0, T ) + P(K = S 0, T ) = E[ S T S 0 ] i.e., E[ u T ] 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 S t, determine both the price C t and the optimal hedge φ t by optimizing locally a risk function, e.g. R t = [C t+1 (S t+1 ) C t (S t ) + φ t (S t )(S t+1 S t )] 2 Linear parametrization of price and hedge using N f variational functions: N f N C t (S) = γ (a) f t y (a) (S) φ t (S) = ˆγ (a) t ŷ (a) (S) a=1 a=1 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 (ŷ (a) (S) = y (a) / S) to reduce the computational cost A second approximation consists in using a Black-Scholes -hedge with a carefully chosen volatility Given N realizations of the process S (n) t, the option price is then N C t = 1 (S (n) T K ) + N n=1 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 are less 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 an arbitrary model of the underlying It allows to use purely historical data to price derivatives, short-circuiting the modeling phase

Theoretical smile for US stocks 0.8 Small cap Mid cap Large cap 0.7 HMC volatility 0.6 0.5 0.4 0.3 0.2 3 2 1 0 1 2 3 Moneyness Data: US stocks in SPX + MID, 1996-2012

Estimation of the smile parameters 250000 200000 Asym bin 0.30-0.50 - T 60 days Non-hedged Hedged 2.4 2.2 Asymmetry bin 0.10-0.15 bin 0.15-0.20 bin 0.20-0.30 bin 0.30-0.50 bin 0.50-1.00 150000 2.0 NH /H 100000 1.8 50000 1.6 0 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Price (dollars) 1.4 0 10 20 30 40 50 60 Days Using Hedged Monte Carlo to obtain the skew of the smile β coefficient related to the price of a binary option (pay $1 if S T > S 0 e µt, 0 otherwise) Delta-hedging reduces the error by a factor 2 at 30 days

Estimation of the smile parameters - SP500 index 0.00 0.8 β T ς T /6 0.7 γ T T /24 0.05 0.6 0.5 0.10 0.4 0.3 0.2 0.15 0.1 0.0 0.20 0 5 10 15 20 T (days) 0.1 0 5 10 15 20 T (days) Data: 1970-2011. Hedge crucial to reduce the noise ς T /6 is systematically different from β T The kurtosis overestimates dramatically the smile curvature

Context Let s suppose another trader wants to hold a dynamic position on an hedged option In this case the impact of the option price move is important: smile dynamics needed More complicated problem! Typical question: how does the at-the-money volatility change if the underlying moves?

Implied leverage Assuming a linear smile σ BS,T σ ATM,T (1 + Skew T M) define the Skew Stickiness Ratio (SSR) R T : [Bergomi, 2009] δs δσ ATM,T = R T Skew T S T Popular values: R T = 0: Sticky Delta R T = 1: Sticky Strike R T = 2: Short T limit for stochastic volatility models

Implied leverage beyond linear models Modeling the forward variance curve {v i+l i } l 0 r i := ln S i+1 = σ i ɛ i, v i+l i+1 S v i+l i i where v i+l i = E[σ 2 i+l F i 1] and E[f (ɛ i )] = 0 = νλ i+l i ({vi u } u i )f (ɛ i ) For linear models (f (x) = x) we have Skew T ς T /6 [Bergomi and Guyon, 2011] For general models we can parametrize R T as R T = ˆR T lin ς T /6 Skew T where ˆR T lin is the SSR result in the linear model framework that saturates to 2 in the short T limit Short term limit of R T exceeds 2 [Vargas et al., 2013]

Skew stickiness ratio 3.0 2.8 2.6 Historical estimator Option data (DAX_IDX) Garch model 2.4 SSR 2.2 2.0 1.8 1.6 1.4 1 21 41 61 81 101.0 T Data: DAX index, 2002-2013

Conclusions Avoid large moments: smile expansion in moneyness is more reliable and able to capture non-linear effects The coeffients of the expansion can be calculated with Hedged Monte Carlo: the modeling phase can be bypassed using historical prices The smile dynamics for indexes shows features compatible with a non-linear origin

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