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What can the risk neutral moments tell us about future
returns?
Juan Imbet
Nuria Mata
Barcelona GSE
juan.imbet@barcelonags...
Agenda
1 Objectives
2 Motivation
3 Literature Review
4 Methodolody
5 Results
6 Conclusions and future work
Juan Imbet Nuri...
Objectives
Objectives
Estimate periodically the risk neutral distribution for several time
horizons
Test the predictabilit...
Motivation
Motivation
Return predictability has an important implication both for
practitioners and for financial models of...
Literature Review
Return Predictability
Kendall (1953): Prices move in a random fashion
Use aggregate economic variables t...
Literature Review
Return Predictability (cont.)
Common regressors from the options’ market are the implied volatility
(VIX...
Literature Review
Return Predictability (cont.)
Inference is problematic because predictors are highly persistent
Standard...
Literature Review
Estimation of the risk neutral distribution
We will focus on the branch of non parametric estimations of...
Methodology
Methodology
Under no arbitrage conditions, the fundamental theorem of asset pricing
must hold:
pt = e−r(T−t)
∞...
Methodology
Methodology (cont.)
At the end of each trading day what differs between call options is their
time-to-maturity ...
Methodology
Methodology (cont.)
As noted by Ait-Sahalia and Duarte (2003) using daily observations
lead to non-convex esti...
Methodology
Methodology (cont.)
Assuming that the function C() is smooth enough, we approximate it
around a point (K0, T0)...
Methodology
Methodology (cont.)
We use a non parametric Generalized Least Squares (GLS) to estimate the
derivatives of the...
Methodology
Methodology (cont.)
Where:
Ω =





ω1 0 . . . 0
0 ω2 . . . 0
...
...
...
...
0 0 . . . ωN





, β ...
Methodology
Methodology (cont.)
And
, X =





1 (K1 − K0) (T1 − T0) (K1 − T0)(K1 − T0) (K1 − K0)2
(T1 − T0)2
.
.
.
....
Methodology
Methodology (cont.)
For each time-to-maturity we estimate the risk neutral distribution
from the estimations o...
Methodology
Methodology (cont.)
Return predictability
The estimated moments are non stationary so we can not use the
direc...
Methodology
Methodology (cont.)
We follow the methodology from Lewellen (2004)
rt = α0 + αM∆Mt−1 + αV ∆Vt−1 + αS ∆St−1 + α...
Methodology
Methodology (cont.)
To test the significance of αM, αV , αs, αK we use the following adjusted
standard error es...
Methodology
Methodology (cont.)
Assuming γi ≈ 1 the distribution of ˆαi is:
ˆαi ∼ N(αi , σ2
υi
(X X)−1
ii ) (16)
Where X i...
Results
Results
Data Description
We use S & P 500 options from January 4th 1996 to April 14th 2014
as our study sample
We ...
Results
Results (cont.)
The risk neutral distribution is estimated daily, weekly and monthly with a
time horizon of 1, 8, ...
Results
Predicting Returns
Figure: Testing the predictability power of the risk neutral moments
Juan Imbet Nuria Mata (Bar...
Results
Predicting Results (cont.)
There is no statistical evidence to conclude that the sample moments
of the risk neutra...
Conclusions and future work
Conclusions
We made two main contributions: Methodologically, we extend the
framework of Ait-S...
Conclusions and future work
Suggestions for future research
Automatic data driven bandwidth selection algorithms
Use the r...
Conclusions and future work
THANK YOU!
Questions?
juan.imbet@barcelonagse.eu
nuria.mata@barcelonagse.eu
Juan Imbet Nuria M...
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What can the risk neutral moments tell us about future returns?

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Master project by Juan Imbet and Nuria Mata. Barcelona GrSE Master's Program in Finance

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What can the risk neutral moments tell us about future returns?

  1. 1. What can the risk neutral moments tell us about future returns? Juan Imbet Nuria Mata Barcelona GSE juan.imbet@barcelonagse.eu nuria.mata@barcelonagse.eu June 29, 2015 Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 1 / 27
  2. 2. 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
  3. 3. Objectives Objectives Estimate periodically the risk neutral distribution for several time horizons Test the predictability power of the statistical moments of the distribution with respect to future returns Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 3 / 27
  4. 4. Motivation Motivation Return predictability has an important implication both for practitioners and for financial models of risk and return. We want to determine if investors’ expectations reflected in the risk neutral distribution contain information about future prices. Investors in the options’ market seem to be more sophisticated than investors in the stock market. Black (1975) Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 4 / 27
  5. 5. Literature Review Return Predictability Kendall (1953): Prices move in a random fashion Use aggregate economic variables to predict future returns Fama and Schwert (1977), Keim and Stambaugh (1986), Fama and French (1989) and Kothari and Shanken (1997) Use financial ratios to predict future returns Fama 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) Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 5 / 27
  6. 6. 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 neutral distribution. The common methodology consists of regressing returns on past lags of the regressors Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 6 / 27
  7. 7. Literature Review Return Predictability (cont.) Inference is problematic because predictors are highly persistent Standard inference analysis leads to overestimation of the predictability power Stambaugh (1999) and Lewellen (2004) propose a methodology to correct these issues. Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 7 / 27
  8. 8. Literature Review Estimation of the risk neutral distribution We will focus on the branch of non parametric estimations of the risk neutral distribution considering the following papers: Ait-Sahalia and Lo (1998): Seminar paper on the estimation of the unconditional risk neutral distribution using the same variables as the Black Scholes model. Ait-Sahalia and Duarte (2003): Propose a methodology to estimate the conditional distribution using information at the end of each trading day Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 8 / 27
  9. 9. Methodology Methodology Under no arbitrage conditions, the fundamental theorem of asset pricing must hold: pt = e−r(T−t) ∞ −∞ XT f (XT )dXT (1) It is well known that we can recover the risk neutral distribution from call option prices: Breeden and Litzenberger (1978) f (x) = er(T−t) ∂2C() ∂K2 |K=x (2) Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 9 / 27
  10. 10. Methodology Methodology (cont.) At the end of each trading day what differs between call options is their time-to-maturity and the strike price. It is coherent to estimate the call option function C() as a function of the strike price and time-to-maturities. Instead of estimating the function itself, we will estimate all of its derivatives 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
  11. 11. Methodology Methodology (cont.) As noted by Ait-Sahalia and Duarte (2003) using daily observations lead to non-convex estimators of the price function with respect to the strike price. We propose a variant of their constrained least squares program (CLS) to transform observed prices to correct this problem. min m 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) Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 11 / 27
  12. 12. Methodology Methodology (cont.) Assuming that the function C() is smooth enough, we approximate it around 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 ∂Ki ∂Tj C(K0, T0) (5) The risk neutral distribution can be estimated directly from ˆβ20 Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 12 / 27
  13. 13. Methodology Methodology (cont.) We use a non parametric Generalized Least Squares (GLS) to estimate the derivatives of the function: min βij 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 ) hK hT ] (6) The weighted least squares estimator is therefore: ˆβ = (X ΩX)−1 X Ωm Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 13 / 27
  14. 14. Methodology Methodology (cont.) Where: Ω =      ω1 0 . . . 0 0 ω2 . . . 0 ... ... ... ... 0 0 . . . ωN      , β = (βij )i+j=k∀k=0,...3 (7) ωn = κ( Kn−K0 hK )κ( Tn−T0 hT ) hK hT Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 14 / 27
  15. 15. 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
  16. 16. Methodology Methodology (cont.) For each time-to-maturity we estimate the risk neutral distribution from the estimations of ˆβ20 around each strike price We scale the distribution to have an integral of one The four moments are estimated numerically using Riemann sums Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 16 / 27
  17. 17. Methodology Methodology (cont.) Return predictability The estimated moments are non stationary so we can not use the directly to predict stationary variables We use the first differences of the moments to test their predictability power of future returns Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 17 / 27
  18. 18. Methodology Methodology (cont.) We follow the methodology from Lewellen (2004) rt = α0 + αM∆Mt−1 + αV ∆Vt−1 + αS ∆St−1 + αK ∆Kt−1 + t (9) ∆Mt = γ0M + γM∆Mt−1 + M t (10) ∆Vt = γ0V + γV ∆Vt−1 + V t (11) ∆St = γ0S + γS ∆St−1 + S t (12) ∆Kt = γ0K + γK ∆Kt−1 + K t (13) Where M, V , S, K are the mean, volatility, skewness and kurtosis of the distribution. ∆Xt = Xt − Xt−1 We assume that each one of these regressors follows an AR(1) process. Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 18 / 27
  19. 19. Methodology Methodology (cont.) To test the significance of αM, αV , αs, αK we use the following adjusted standard error estimator: ˆαi = ˆαOLS i − θi (ˆγi − γi ) i ∈ {M, V , S, K} (14) Where θi is estimated from the following regression: t = θi i t + υi t (15) Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 19 / 27
  20. 20. Methodology Methodology (cont.) Assuming γi ≈ 1 the distribution of ˆαi is: ˆαi ∼ N(αi , σ2 υi (X X)−1 ii ) (16) Where X is the matrix of regressors and σ2 υi the variance of υi . Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 20 / 27
  21. 21. Results Results Data Description We use S & P 500 options from January 4th 1996 to April 14th 2014 as our study sample We cleaned the data using the same conditions as in Ait-Sahalia and Lo (1998) Since in-the-money options are less liquid than out-of-the-money we use put call parity to complete the spectrum of strike prices. We use bandwidths proportional to the standard deviation of each variable We use the standard Gaussian kernel Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 21 / 27
  22. 22. Results Results (cont.) The risk neutral distribution is estimated daily, weekly and monthly with a time horizon of 1, 8, and 30 days. Figure 1 shows as an example the daily risk neutral distribution with an horizon of 30 days during 2008: Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 22 / 27
  23. 23. Results Predicting Returns Figure: Testing the predictability power of the risk neutral moments Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 23 / 27
  24. 24. Results Predicting Results (cont.) There is no statistical evidence to conclude that the sample moments of the risk neutral distribution predict future returns at any time horizon considered. Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 24 / 27
  25. 25. Conclusions and future work Conclusions We made two main contributions: Methodologically, we extend the framework of Ait-Sahalia and Duarte (2003) to consider both time-to-maturities and strike prices as variables of the call option price function. The empirical contribution consists of considering the first four moments of the risk neutral distribution as regressors to predict returns. Our results suggest that these statistical moments do not predict future returns As far as we can tell this is the paper with the largest amount of risk neutral distributions estimated Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 25 / 27
  26. 26. Conclusions and future work Suggestions for future research Automatic data driven bandwidth selection algorithms Use the risk neutral distribution to estimate the likelihood of events and use them to anticipate changes in the stock market. Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 26 / 27
  27. 27. Conclusions and future work THANK YOU! Questions? juan.imbet@barcelonagse.eu nuria.mata@barcelonagse.eu Juan Imbet Nuria Mata (Barcelona GSE) Master Project June 29, 2015 27 / 27

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