A Bayesian non parametric estimation of the conditional physical measure and its use for the investigation of the pricing kernel puzzle (Draft)

Transcription

1 A Bayesian non parametric estimation of the conditional physical measure and its use for the investigation of the pricing kernel puzzle (Draft) Sala Carlo, Barone Adesi Giovanni, Mira Antonietta University of Lugano and Swiss Finance Institute January 9, 5

2 Table of Contents 4 5 Daily estimates Yearly estimates Robustness check 6

3 Main problem: the pricing kernel puzzle Despite its key role in asset pricing, there is still not a cut and clear agreement among financial researchers and pratictioners on the best procedure to estimate the pricing kernel Pricing kernel puzzle: the estimated pricing kernel is not general enough to properly explain the whole cross section of option data non-monotonic function Main problems: at the extremes (U shape, over/under estimation) or in the central area (flex) Puzzling PK Literature: huge for measures estimation (RN>Phyiscal). Since Jackwerth () a lot of attention on pricing kernel puzzle The literature so far Non-monotonic PK invalid economic results

4 The measures problem - Three empirical facts Joint use of time series of stock returns and cross section of option data provides more precise parameter estimates than using only past stock returns (since Chernov and Ghysel ()) The physical measure estimated using historical stock returns is a poor and unconditional measure (Jackwerth & Brown (4)). Agents beliefs: unobservable and fwd looking. Hist data: bwd looking Pricing kernel puzzles present in literature feature a common problem of non-homogeneity bias: compare a forward looking conditional measure with a backward looking unconditional measure

5 Proposal and research questions Proposal We propose a time varying, flexible, non-parametric, conditional physical measure that empirically adds part of the forward looking information embedded into the options volatility into the unconditional historical denominator so that it can be homogeneously related with the already fully conditional risk neutral measure and produce an unbiased ratio Is a fully conditional PK still non-monotonic? What is the statistical behaviour of the new physical measure wrt the old and the risk neutral ones? Implications? Daily and yearly estimations, why do they matter? 4 How does it behave during market turmoils? 5 Is it a robust measure?

6 Proposal and anticipated conclusions Given the problem, two key points to concretize our proposal: Daily volatility estimation GJR GARCH + FHS ˆσ t estimation Empirical mixing Bayesian non-parametric methodology: Dirichlet process Dirichlet process Exploiting, by simulation, we obtain monotonically decreasing daily and yearly empirical pricing kernels Anticipated conclusion Given our findings, we claim that one possible reason behind the pricing kernel non-monotonicity is an econometric issue related to an improper conditioning of the information

7 Table of Contents 4 5 Daily estimates Yearly estimates Robustness check 6

8 The pricing kernel is the characteristic function or the sufficient statistic of asset pricing. By theory, it embeds all necessary info required for pricing any financial asset: P t = e rt,τ τ E Q t [ψ T (S T ) F t ] () = M t,t (S T ) ψ T (S T )p t,t (S T S t )ds T () = E P t [M t,t (S T ) ψ T (S T ) F t ] () While the conditioning in equation () is true by construction, the ones in () and () are often violated in the pricing kernel. This leads to an inequality

9 The empirical pricing kernel is defined as: M t,t = e rt,τ (T t) q t,t (S T S t ) p t,t (S T S t ) (4) q t,t : Cond. risk neutral measure (usually estimated from cross section of option prices) p t,t :Cond. physical measure (usually estimated from the past history of stock returns) Present value of two conditional densities, which depends on all information given by the time t information set (F t )

10 The advantage of joining the measures If the joint measure is properly calibrated: Better calibration of the risk premium One-to-one stock-option relation use shorter samples for estimation Stock-option parameters consistency Model misspecification diagnostic Statistically, a natural approach to exploit simultaneously different data and provide statistical inference is the Bayesian approach (Berger (985), Bernardo and Smith (994))

11 Table of Contents 4 5 Daily estimates Yearly estimates Robustness check 6

12 Proposal Inputs: finance market data non-parametric models Problem: x i P i =,..., n (5) given a set of observations x i, with unknown distribution, we wish to infer given some prior info. Goal: estimate the unknown probability distribution P (RV), with: No constraints on the function (but the FTAP) Exploiting prior info from RN Problem + Bayesian non-parametric density estimation

13 Proposal Solution: Place a prior over P get posterior over P given x i. Non-parametric approach: prior over all possible distributions infinite-dimensional function space widest support Prior: Dirichlet Process (Ferguson 97) P DP(α, G) DP has a conjugate family of priors over distributions that is closed under posterior updates given observations Posterior: after collecting x i the posterior mean is still DP: P x,..., x n = (n + α) n α δ(x i ) + (n + α) G (6) i=

14 The conditional measure through the Dirichlet Process Given our problem, (6) has to be properly adjusted s.t. the conditional physical distribution (P ) is: P x,..., x n = = (n + α) (n + α ) n α δ(x i ) + (n + α) G (7) i= n α δ(s i ) + (n + α ) G (8) i= = + α p + α + α q (9) Inputs: simulated financial data S i with i 5. For homogeneity n is normalized to Being DP a discrete process, we smooth the obtained function

15 The modified pricing kernel We go from: the Classic (unconditional) pricing kernel estimation: ( ) Risk neutral M t,t = e rt,τ Maturities Risk physical ( ) = e rt,τ τ qt,t p ( ) qt,t = PV p () () () where the conditioning of p t,t is omitted since violated by construction

16 The modified pricing kernel to: a Modified pricing kernel estimation: ( ) M t,t = τ Risk neutral e rt,τ +α Risk physical + α +α Modified risk neutral = e rt,τ τ ( = PV q t,t +α p + α ( ) q t,t p t,t +α q t,t ) () (4) (5) where the measure p t,t is now fully conditional M t,t : discounted Radon Nikodym derivative of q t,t relative to p t,t

17 Frequentist justification - Sketch The denominator misspecification arises from the numerical nature of the problem q p = EP (q) E P (p) ( ) q E P p ( ) q + q = E P p + p ( ) = qe p + p (6) (7) (8) (9) where (q ) and (p ) are probability estimates and (p) and (q) are probabilities Numerator: unbiased and fixed Denominator: biased and to be approximated numerically playing with α

18 Frequentist justification Assuming (p + p) C n for n and, we approximate the expectation of the ratio through a second order Taylor expansion: ( ) q E p ( ) q =E p + p () = q p + f p p + f pp p + O(n ) () = q p q p p + q p p + O(n ) () q p + q Var( p) () p We obtain an inequality which can be fixed by simulation

19 Datasets S&P 5 index and the SPX perfect correlation with aggregate wealth Interpolated risk free rate and the daily dividend rates Data duly filtered to reduce misleading results.trade-off: richness and quality of data; tails issue All data from OptionMetrics Semi-synthetic datasets (forthcoming)

20 Table of Contents 4 5 Daily estimates Yearly estimates Robustness check 6

21 Estimation of the P parameters - θ Proposal The model under the physical probability P has the form: log S t S t = r t = µ + ɛ t (4) ɛ t = σ t z t (5) z t F t f(, ) (6) σt = ω + βσt + αɛ t + γ t ɛ t (7) {, if ɛ t <, t = (8) if ɛ t PML: optimizing the negative of the LH (5 obs.), we obtain a set of objective GARCH parameters θ = (ω, α, β, γ)

22 FHS and information set The scaled return innovation, z t, is defined as a filtered historical simulation (FHS) (Barone Adesi et al. 999): ẑ t = ˆɛ t ˆσt (9) The information set: The (z t ) t {...T } are real random variables defined on a complete filtered space (Ω, F, P, (F t ) t {...T } ) and their distributions lie in a parametric class of Lebesgue densities (F t ) t {...T } : information filtration, increasing family of σ algebra. Extremes are F = {, Ω} and F = Ω ; at each time t (F t = σ(z u ;, u t)) t {...T }

23 Estimation of the Q parameters - Option pricing with FHS For a given date t, a GJR GARCH model (4) - (8) is calibrated to the cross section of filtered OTM call and put options on the S&P 5, to capture the index dynamic under the risk neutral probability density function Q The set of risk neutral GARCH parameters θ = f( ω, α, β, γ) are obtained through a numerical search using the Neadler-Mead simplex method (or Quasi-Newton): min ( ω, α, β, γ) [ Mkt P.(K, τ) Model P.(K, τ) ] () K τ Model P. (K, τ) option pricing: rescaled MC (Duan et al.) simulation + FHS and Gauss. innovations

24 Extracting the densities To extract the p, q and q densities: simulate (n) log-prices using the set of parameters θ and θ with both innovations: S i = Ŝi S corr. i,t () where : () [( ) Ŝ i = S exp Drift ˆσ sim dt + ˆσ simdw ] t i =,..., n () Si,t corr. = S e rt,τ τ n (4) n i= Ŝi,t Then apply a smoothed non-parametric kernel density estimation at n equally spaced points (n = ; ; 5)

25 The drift inputs To preclude any arbitrage, the drifts of equation () are: Physical measure (p): p = 65 Risk neutral measure (q): q = 65 ( T ) r t d t + φ t t= ( T ) r t d t t= Modified Risk neutral measure (q ): ( q = T ) r t d t + φ t 65 t= (5) (6) (7)

26 Table of Contents Yearly estimates Robustness check 4 5 Daily estimates Yearly estimates Robustness check 6

27 Summary Yearly estimates Robustness check Daily estimates of PDFs and PKs Single day single τ PDFs and PKs Yearly PDFs and PKs Robustness checks: rolling window, GJR GARCH, numerical noise errors

28 Daily PDFs - PKs, t=6, 9-Feb- Daily, one τs τ=59 τ=.... PDF, Date9 Feb R.N. dens. Unc. Obj. dens. Cond Obj. dens. Pricing Kernel. Date9 Feb S&P5 price Alpha= N.Sim= 5 RP= τ= τ= τ= PDFs and Mod.PKs M t,t+τ, α, (t = 6), all τs

29 Daily PDFs - PKs, t=4, -Jan- Key difference: left Skewness and Kurtosis Daily, one τs τ=4 τ=5 τ=87 τ=5 τ=4 τ= FHS PDF, Date Jan Pricing Kernel. Date Jan R.N. dens..5 Unc. Obj. dens. Cond Obj. dens Gauss. S&P5 price Alpha= N.Sim= 5 RP=.8 PDFs and Mod.PKs M t,t+τ, α, (t = 4), all τs

30 Left tails focus, t=4, -Jan-, short (τ)..8 x Left tail PDFs. Date: Jan 5.8 x Left tail PDFs. Date: Jan 5.8 x Left tail PDFs. Date: Jan 5 alpha=.5 alpha=.6 N.Sim= N.Sim= 5 RP=.4 RP=.4 alpha=.5 N.Sim= 5 RP= τ=4 days.8 τ=4 days.8 τ=4 days x 5 4 Left tail PDFs. Date: Jan alpha= N.Sim= 5 RP= x 5 4 Left tail PDFs. Date: Jan alpha=9 N.Sim= 5 RP= x 5 4 Left tail PDFs. Date: Jan alpha= N.Sim= 5 RP= τ=5 days.5 τ=5 days.5 τ=5 days LEFT: single day (t = 4), short (τ), α, R.P.= 4%-8%

31 Left tails focus, t=4, -Jan-, medium (τ) x Left tail PDFs. Date: Jan 4 x Left tail PDFs. Date: Jan 4 x Left tail PDFs. Date: Jan 4 alpha=.5 alpha= N.Sim= 5 N.Sim= 5 RP=.4 RP= alpha=.5 N.Sim= 5 RP= τ=87 days.5 τ=87 days.5 τ=87 days x 4. Left tail PDFs. Date: Jan alpha=.5 N.Sim= 5 RP=.4.4 x 4. Left tail PDFs. Date: Jan alpha=7 N.Sim= 5 RP=.4.4 x 4. Left tail PDFs. Date: Jan alpha=.5 N.Sim= 5 RP=.8 τ=5 days.8.6 τ=5 days.8.6 τ=5 days LEFT: single day (t = 4), medium (τ), α, R.P.= 4%-8%

32 Left tails focus, t=4, -Jan-, long (τ) x Left tail PDFs. Date: Jan 4 x Left tail PDFs. Date: Jan 4 x 4 alpha=.9 alpha=6.9.9 N.Sim= 5 N.Sim= 5 RP=.4 RP= Left tail PDFs. Date: Jan alpha= N.Sim= 5 RP= τ=4 days τ=4 days τ=4 days x 4 Left tail PDFs. Date: Jan alpha= N.Sim= 5 RP=.4 x 4 Left tail PDFs. Date: Jan alpha=6 N.Sim= 5 RP=.4 x 4 Left tail PDFs. Date: Jan alpha= N.Sim= 5 RP=.8 τ= days τ= days τ= days LEFT: single day (t = 4), long (τ), α, R.P.= 4%-8%

33 Right tails focus, t=4, -Jan-, short (τ) 9 x RIGHT tail PDFs: Date Jan 6 9 x RIGHT tail PDFs: Date Jan 6 9 x RIGHT tail PDFs: Date Jan 6 alpha=.5 alpha= 8 8 N.Sim= 5 8 N.Sim= 5 RP=.4 RP= alpha=.5 N.Sim= 5 RP= τ=4 days 5 4 τ=4 days 5 4 τ=4 days x 6 RIGHT tail PDFs: Date Jan alpha= N.Sim= 5 RP= x 6 RIGHT tail PDFs: Date Jan alpha=9 N.Sim= 5 RP= x 6 RIGHT tail PDFs: Date Jan alpha= N.Sim= 5 RP= τ=5 days.5 τ=5 days.5 τ=5 days RIGHT: single day (t = 4), short (τ), α, R.P.= 4%-8%

34 Right tails focus, t=4, -Jan-, medium (τ).4 x RIGHT tail PDFs: Date Jan 6.4 x RIGHT tail PDFs: Date Jan 6.4 x RIGHT tail PDFs: Date Jan 6 alpha=.5 alpha=8 N.Sim= 5 N.Sim= 5... RP=.4 RP=.4 alpha=.5 N.Sim= 5 RP=.8 τ=87 days.8.6 τ=87 days.8.6 τ=87 days x 6 RIGHT tail PDFs: Date Jan alpha=.5 N.Sim= 5 RP= x 6 RIGHT tail PDFs: Date Jan alpha=7 N.Sim= 5 RP= x 6 RIGHT tail PDFs: Date Jan alpha=.5 N.Sim= 5 RP= τ=5 days.5 τ=5 days.5 τ=5 days RIGHT: single day (t = 4), medium (τ), α, R.P.= 4%-8%

35 Right tails focus, t=4, -Jan-, long (τ) x RIGHT tail PDFs: Date Jan 6.5 x RIGHT tail PDFs: Date Jan 6.5 x RIGHT tail PDFs: Date Jan 6 alpha= alpha=6.8 N.Sim= 5 N.Sim= 5 RP=.4 RP=.4.6 alpha= N.Sim= 5 RP= τ=4 days τ=4 days τ=4 days x 7 5 RIGHT tail PDFs: Date Jan alpha= N.Sim= 5 RP=.4 9 x 7 8 RIGHT tail PDFs: Date Jan alpha=6 N.Sim= 5 RP=.4 7 x 7 6 RIGHT tail PDFs: Date Jan alpha= N.Sim= 5 RP= τ= days τ= days 5 4 τ= days RIGHT: single day (t = 4), long (τ), α, R.P.= 4%-8%

36 Daily PK, t=9, 7-Sep-, τ = 8 Daily, all τs SPD per unit probability, Maturity(τ)= days Pricing Kernel. Date7 Sep FHS Gauss. S&P5 price Alpha= N.Sim= 5 RP= S&P 5 value Classical (M t,t+τ ) and modified (M t,t+τ ) PKs

37 Daily PK, t=9, 7-Sep-, τ = 64 Daily, all τs SPD per unit probability, Maturity (τ)=94 days Pricing Kernel. Date7 Sep FHS Gauss. S&P5 price Alpha=.5 N.Sim= 5 RP= S&P 5 value Classical (M t,t+τ ) and modified (M t,t+τ ) PKs

38 Bearish/bullish market environment Exploiting part of the forward looking information given by the options volatility, the modified PKs can better capture possible market up/downturn 5 S&P 5 closing prices (57) 5 Closing Price Date, only Wed.

39 Market upturn. t=4, 9-Oct- τ=8 τ=7 τ=64 τ=55 τ=46.5 PDF, Date9 Oct FHS Pricing Kernel. Date9 Oct Gauss. R.N. dens. Unc. Obj. dens. Cond Obj. dens. S&P5 price Alpha= N.Sim= 5 RP= PDFs and all PKs, α, (t = 4), all τs

40 Market upturn - focus - t=4, 9-Oct- SPD per unit probability, Maturity (τ)=7 days Pricing Kernel. Date9 Oct Pricing Kernel. Date9 Oct FHS Gauss..5.5 S&P5 price Alpha= N.Sim= 5 RP= SPD per unit probability, Maturity (τ)=64 days FHS Gauss. S&P5 price Alpha=.5 N.Sim= 5 RP= SPD per unit probability, Maturity (τ)=55 days S&P 5 value.5.5 Pricing Kernel. Date9 Oct FHS Gauss. S&P5 price Alpha= N.Sim= 5 RP=.4 SPD per unit probability, Maturity (τ)=46 days S&P 5 value.5.5 Pricing Kernel. Date9 Oct FHS Gauss. S&P5 price Alpha= N.Sim= 5 RP= S&P 5 value S&P 5 value Modified PKs M t,t+τ, α decreasing, (t = 4), all τs

41 Yearly PK - fixed α Year Year Year Year Year Year Year Year 4 Year SPD per unit prob. SPD per unit prob. SPD per unit prob S T /S t S T /S t S T /S t SPD per unit prob. SPD per unit prob. SPD per unit prob. S T /S t S T /S t S T /S t SPD per unit prob. SPD per unit prob. SPD per unit prob S T /S t S T /S t S T /S t Yearly classical (M t,t+τ ) and modified (M t,t+τ ) PKs

42 Yearly PK - fixed α - Both PKs Year Short Maturity FHS Gauss S T /S t Year Medium Maturity Year Long Maturity FHS Gauss. FHS Gauss Year Short Maturity FHS Gauss S T /S t Year Medium Maturity FHS Gauss Year 4 Long Maturity FHS Gauss Year 4 Short Maturity S T /S t Year 4 Medium Maturity FHS Gauss. FHS Gauss Year Long Maturity FHS Gauss SPD per unit prob. SPD per unit prob. SPD per unit prob SPD per unit prob. SPD per unit prob. SPD per unit prob S T /S t S T /S t S T /S t SPD per unit prob. SPD per unit prob. SPD per unit prob S T /S t S T /S t S T /S t Yearly classical (M t,t+τ ) and modified (M t,t+τ ) PKs

43 Yearly estimates Robustness check Model robustness to simulation noise error To test the robustness of the models to possible noises that can arise during the simulations, we propose to dirty the second and fourth moment of the distributions by adding a small extra value ( noise ) For the second moment, we add a noise to the ω so that we change the long run mean of the unconditional variance: ω σ uncond = (8) α β γ/ while for the fourth moment, following Engle and Mustafa (99):..it is known that the the conditional kurtosis of a distribution of multi-steps returns depend upon alpha. Higher alpha implies higher conditional kurtosis and this has implications for option prices... ; so we dirty α

44 Yearly estimates Robustness check Model robustness - ω (5e7) - t=9.5 SPD per unit prob fhs gauss. gauss bayes fhs bayes S&P5 price alpha= N.Sim= 5.5 SPD per unit prob fhs gauss. gauss bayes fhs bayes S&P5 price alpha=.5 N.Sim= 5 T=66 days.5 T=94 days SPD per unit prob fhs gauss. gauss bayes fhs bayes S&P5 price alpha= N.Sim= 5.5 SPD per unit prob fhs gauss. gauss bayes fhs bayes S&P5 price alpha= N.Sim= 5 T=85 days.5 T=76 days

45 Yearly estimates Robustness check Model robustness - α (.) t=9.5 SPD per unit prob fhs gauss. gauss bayes fhs bayes S&P5 price alpha= N.Sim= 5.5 SPD per unit prob fhs gauss. gauss bayes fhs bayes S&P5 price alpha=.5 N.Sim= 5 T=66 days.5 T=94 days SPD per unit prob fhs gauss. gauss bayes fhs bayes S&P5 price alpha= N.Sim= 5.5 SPD per unit prob fhs gauss. gauss bayes fhs bayes S&P5 price alpha= N.Sim= 5 T=85 days.5 T=76 days

46 Table of Contents 4 5 Daily estimates Yearly estimates Robustness check 6

47 Lead by the recent findings on the PK puzzle and due to the fact that most of them share a common error of non-homogeneity due to a non informative and unconditional backward looking objective measure......we proposed a Bayesian non-parametric methodology to update the denominator of the PK We obtained economically valid PKs puzzles due to econometric errors We explained the D/OTM overpricing of put options Compared with puzzling results we obtained faster, less volatile, more robust, wider and economically valid findings Results show robustness also on a daly basis for single time-to-maturities

48 Future works Being a general and flexible method: repeat the same experiment using other methodologies to estimate P and Q Semi-synthetic databases for liquidity issues Exploit the difference in moments estimation to set up trading strategies: skewness and kurtosis trading Use conditional physical measure for entropic value at risk

49 Thank you for your attention.

50 Non-monotonic EPK in the literature / % -7.8% -5.8% -.8% -.8%.%.% 4.% 6.% 8.% S&P 5 return 6 Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 () 9} Pricing kernel Fig.. compatible with the values found using consumption, but no option, data. Is there an equity-premium-like puzzle at the levels of option prices, or do they imply coe$cients of risk aversion that are less extreme than those typically found in the equity-premium literature? This estimate for a is higher than the typical values used in theoretical models (where the range is typically to 5), higher than (μ!r#δ)/σ (the value of a in the Black}Scholes model evaluated at values of μ, δ and σ given by the S&P 5 returns and the riskfree rate r), yet generally lower than the estimates found by studies of the Euler equation in consumption-based asset pricing models. Table 5 reports the range of values of the constant CRRA that have been estimated in the literature. While the implied CRRA coe$cient is informative, it is important to keep in mind that Non-monotonic EPK: Rosengerg and Engle (), Äit Sahalia and Lo (), Jackwert and Brown () EPK GER Return EPK UK Return See Renault and Garcia (996) for a di!erent attempt to confront option data with the Euler equation. However, it should be emphasized that our estimate is based on nominal data, whereas estimates obtained from consumption-based asset-pricing models use real data. For our relatively short time-horizon (less than one year), this distinction may not matter; see footnote 4 for further discussion.

51 Non-monotonic EPK in the literature / Motivation Figure 6: Log Ratios of Risk-Neutral and Physical Distributions from Model Option Prices Figure. Estimated pricing kernels. Figure depicts point estimates of the pricing kernels Notes to Figure: We plot the natural logarithm of the ratio of the risk-neutral conditional estimated Non-monotonic without global restrictions. The point estimates (), are calculated at the mean of the Heston EPK: Dittmar Christoffersen, and densities and the physical conditional histogram. For each year in the option sample, we plot Jacods the ratios for each of the in that year.on On the horizontal axis are log returns in monthly instrumental variables and the support for the graphs is the observed range ofwednesdays the return standard deviations. This figure uses model prices rather than market prices. labor and () the value-weighted index. The coefficients of the pricing kernels are estimated via GMM utilizing the Euler equation condition, GER UK 4 EPK EPK 4 R t!! m t! 6Z t # " N #,.6.8. Return Return.4 where m t! represents a polynomial pricing kernel. The coefficients are estimated using the Hansen and Jagannathan ~997! weighting matrix t!! Z t!~r t!! Z t!' #. The coeffi

52 Correct and non correct PK PK Motivation Neoclassical theory and rational investors: 4 $ in bad states > $ in4.5 good states High payoffs in negative states > High payoffs in positive states. pricing kernel is monotonically decreasing in wealth Density functions Density Density functions BUT ratio Ordinal Density dominance ratio curve 8 Recent literature presents several puzzling 5 PK with different problems: U shaped, one or more flex in the central area, or too high/low values. Correct PK Synthetic PK Return Density functions Density functions Density Density functions ratio Non.8 correct.5 PK. Ordinal Density Density dominance functions ratio curve Density ratio Ordinal dominance curve.5 Ordinal dominance curve.5 Ordinal Density dominance ratio curve Density functions Density ratio XVI Workshop Ordinal Density dominance functions Quantitative curve FinanceDensity ratio Ordi 4.5 Figure.: Density ratios and ordinal dominance curves. 6.5 Ordi