what can the risk neutral moments tell us about future returns?
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What can the risk neutral moments tell us about futurereturns?
Juan ImbetNuria Mata
Barcelona GSE
juan.imbet@barcelonagse.eunuria.mata@barcelonagse.eu
June 29, 2015
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 1 / 27
Agenda
1 Objectives
2 Motivation
3 Literature Review
4 Methodolody
5 Results
6 Conclusions and future work
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 2 / 27
Objectives
Objectives
Estimate periodically the risk neutral distribution for several timehorizons
Test the predictability power of the statistical moments of thedistribution with respect to future returns
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 3 / 27
Motivation
Motivation
Return predictability has an important implication both forpractitioners and for financial models of risk and return.
We want to determine if investors’ expectations reflected in the riskneutral distribution contain information about future prices.
Investors in the options’ market seem to be more sophisticated thaninvestors in the stock market. Black (1975)
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 4 / 27
Literature Review
Return Predictability
Kendall (1953): Prices move in a random fashion
Use aggregate economic variables to predict future returnsFama and Schwert (1977), Keim and Stambaugh (1986), Fama andFrench (1989) and Kothari and Shanken (1997)
Use financial ratios to predict future returnsFama and French (1988), Campbell and Shiller (1988) and Cochrane(1991):
Do options’market lead the stock market, or vice versa?Diltz and Kim (1996), Finucane (1999), Manaster and Rendleman(1982)
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Literature Review
Return Predictability (cont.)
Common regressors from the options’ market are the implied volatility(VIX) and the trading volume.
We test the predictability power of the moments of the risk neutraldistribution.
The common methodology consists of regressing returns on past lagsof the regressors
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 6 / 27
Literature Review
Return Predictability (cont.)
Inference is problematic because predictors are highly persistent
Standard inference analysis leads to overestimation of thepredictability power
Stambaugh (1999) and Lewellen (2004) propose a methodology tocorrect these issues.
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 7 / 27
Literature Review
Estimation of the risk neutral distribution
We will focus on the branch of non parametric estimations of the riskneutral distribution considering the following papers:
Ait-Sahalia and Lo (1998): Seminar paper on the estimation of theunconditional risk neutral distribution using the same variables as theBlack Scholes model.
Ait-Sahalia and Duarte (2003): Propose a methodology to estimatethe conditional distribution using information at the end of eachtrading day
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 8 / 27
Methodology
Methodology
Under no arbitrage conditions, the fundamental theorem of asset pricingmust hold:
pt = e−r(T−t)∫ ∞−∞
XT f (XT )dXT (1)
It is well known that we can recover the risk neutral distribution from calloption prices: Breeden and Litzenberger (1978)
f (x) = er(T−t)∂2C ()
∂K 2|K=x (2)
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 9 / 27
Methodology
Methodology (cont.)
At the end of each trading day what differs between call options is theirtime-to-maturity and the strike price.
It is coherent to estimate the call option function C () as a function ofthe strike price and time-to-maturities.
Instead of estimating the function itself, we will estimate all of itsderivatives jointly.
We can achieve this goal by a non parametric local polynomial fitting.
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 10 / 27
Methodology
Methodology (cont.)
As noted by Ait-Sahalia and Duarte (2003) using daily observationslead to non-convex estimators of the price function with respect tothe strike price.
We propose a variant of their constrained least squares program(CLS) to transform observed prices to correct this problem.
minm
N∑n=1
(mn − Cn)2
s.t.mk −mj
Kk − Kj≥
mj −mi
Kj − Ki
∀i , j , k = 1, . . . ,N : Kk ≥ Kj ≥ Ki and Ti = Tj = Tk
(3)
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Methodology
Methodology (cont.)
Assuming that the function C () is smooth enough, we approximate itaround a point (K0,T0) using a two dimensional Taylor expansion:
C (K0,T0) ≈2∑
k=0
(∑
i+j=k
βij(K − K0)i (T − T0)j) (4)
Where:
(i !)(j!)βij =∂ i+j
∂K i∂T jC (K0,T0) (5)
The risk neutral distribution can be estimated directly from β20
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Methodology
Methodology (cont.)
We use a non parametric Generalized Least Squares (GLS) to estimate thederivatives of the function:
minβij
N∑n=1
[(mn −2∑
k=0
(∑
i+j=k
βij(Kn − K0)i (Tn − T0)j))κ(Ki−K0
hK)κ(Ti−T0
hT)
hKhT]
(6)
The weighted least squares estimator is therefore:
β = (X′ΩX)−1X′Ωm
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Methodology
Methodology (cont.)
Where:
Ω =
ω1 0 . . . 00 ω2 . . . 0...
.... . .
...0 0 . . . ωN
, β =[(βij)i+j=k∀k=0,...3
](7)
ωn =κ(
Kn−K0hK
)κ(Tn−T0
hT)
hKhT
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 14 / 27
Methodology
Methodology (cont.)
And
, X =
1 (K1 − K0) (T1 − T0) (K1 − T0)(K1 − T0) (K1 − K0)
2 (T1 − T0)2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1 (KN − K0) (TN − T0) (KN − K0)(TN − T0) (KN − K0)2 (TN − T0)
2
(8)
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 15 / 27
Methodology
Methodology (cont.)
For each time-to-maturity we estimate the risk neutral distributionfrom the estimations of β20 around each strike price
We scale the distribution to have an integral of one
The four moments are estimated numerically using Riemann sums
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Methodology
Methodology (cont.)
Return predictability
The estimated moments are non stationary so we can not use thedirectly to predict stationary variables
We use the first differences of the moments to test their predictabilitypower of future returns
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Methodology
Methodology (cont.)
We follow the methodology from Lewellen (2004)
rt = α0 + αM∆Mt−1 + αV ∆Vt−1 + αS∆St−1 + αK∆Kt−1 + εt (9)
∆Mt = γ0M + γM∆Mt−1 + εMt (10)
∆Vt = γ0V + γV ∆Vt−1 + εVt (11)
∆St = γ0S + γS∆St−1 + εSt (12)
∆Kt = γ0K + γK∆Kt−1 + εKt (13)
Where M,V ,S ,K are the mean, volatility, skewness and kurtosis of thedistribution. ∆Xt = Xt − Xt−1 We assume that each one of theseregressors follows an AR(1) process.
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Methodology
Methodology (cont.)
To test the significance of αM , αV , αs , αK we use the following adjustedstandard error estimator:
αi = αOLSi − θi (γi − γi ) i ∈ M,V , S ,K (14)
Where θi is estimated from the following regression:
εt = θiεit + υit (15)
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 19 / 27
Methodology
Methodology (cont.)
Assuming γi ≈ 1 the distribution of αi is:
αi ∼ N (αi , σ2υi
(X ′X )−1ii ) (16)
Where X is the matrix of regressors and σ2υi the variance of υi .
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Results
Results
Data Description
We use S & P 500 options from January 4th 1996 to April 14th 2014as our study sample
We cleaned the data using the same conditions as in Ait-Sahalia andLo (1998)
Since in-the-money options are less liquid than out-of-the-money weuse put call parity to complete the spectrum of strike prices.
We use bandwidths proportional to the standard deviation of eachvariable
We use the standard Gaussian kernel
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Results
Results (cont.)
The risk neutral distribution is estimated daily, weekly and monthly with atime horizon of 1, 8, and 30 days. Figure 1 shows as an example the dailyrisk neutral distribution with an horizon of 30 days during 2008:
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Results
Predicting Returns
Figure: Testing the predictability power of the risk neutral moments
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Results
Predicting Results (cont.)
There is no statistical evidence to conclude that the sample momentsof the risk neutral distribution predict future returns at any timehorizon considered.
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Conclusions and future work
Conclusions
We made two main contributions: Methodologically, we extend theframework of Ait-Sahalia and Duarte (2003) to consider bothtime-to-maturities and strike prices as variables of the call optionprice function.
The empirical contribution consists of considering the first fourmoments of the risk neutral distribution as regressors to predictreturns.
Our results suggest that these statistical moments do not predictfuture returns
As far as we can tell this is the paper with the largest amount of riskneutral distributions estimated
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Conclusions and future work
Suggestions for future research
Automatic data driven bandwidth selection algorithms
Use the risk neutral distribution to estimate the likelihood of eventsand use them to anticipate changes in the stock market.
Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 26 / 27
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