ESTIMATION OF UTILITY FUNCTIONS: MARKET VS. REPRESENTATIVE AGENT THEORY

Transcription

1 ESTIMATION OF UTILITY FUNCTIONS: MARKET VS. REPRESENTATIVE AGENT THEORY Kai Detlefsen Wolfgang K. Härdle Rouslan A. Moro, Deutsches Institut für Wirtschaftsforschung (DIW) Center for Applied Statistics and Economics (CASE), Humboldt-Universität zu Berlin...

2 Motivation - An investor observes the evolution of a stock price in the past and forms his subjective opinion about the future evolution of the price DAX // // // // // Figure : DAX, January - June. Daily observations....

3 Motivation - An opinion on the future value S t of the stock at time t can be described by a density function p which is called subjective density, a.k.a. historical density or physical density. This function can be estimated in many ways (parametric, nonparametric,... ). Examples: Black-Scholes model (Nobel prize 997): log normal distribution GARCH model (Nobel prize, Engle): stochastic volatility non-parametric diffusion model (Ait-Sahalia )...

4 Motivation - We model the logarithmic returns {r t } of the DAX by a GARCH(,) model: r t = σ t Z t σ t = ω + αr t + βσ t From the logarithmic returns r i = log(s i ) log(s i ), i =,..., t and the starting stock price S we can construct the final stock price by S t = S exp( t r i ). i=...

5 Motivation - Figure : Subjective density on April th, for τ = ahead. In order to present the density independent of the starting stock price S we do not plot S t ˆp(S t ) but R t ˆp(R t S ) ( scale)....

6 Motivation - Besides the subjective density there is also a state-price density (SPD) q for the stock price implied by the market prices of options, a.k.a. risk-neutral density. The state-price density differs from the subjective density because it corresponds to replication strategies and hence is a martingale risk neutral measure. A person alone does not use in general a replication strategy but thinks in terms of his subjective density....

7 Motivation -6 We use the Heston continuous stochastic volatility model, which can be regarded as an industry standard for option pricing models. The Heston model is given by ds t S t = rdt + V t dw t where the volatility process is modelled by a square-root process: dv t = ξ(η V t )dt + θ V t dw t, and W and W are Wiener processes with correlation ρ....

8 Motivation -7 Using option prices with time-to-maturity between. and and between and. we get the following estimate for the orisk-neutral state-price density for τ = years ahead. Figure : State-price density q t, r =.6%, April th,....

9 Motivation -8 The pricing kernel m(s t ) is defined as: m(s t ) = exp( rt) q(s t) p(s t ) where r is the interest rate with maturity t. An estimate of the pricing kernel is called. We use the estimate: ˆm(S t ) = exp( rt) ˆq(S t) ˆp(S t ) where ˆq and ˆp are the estimated risk-neutral and subjective densities....

10 Motivation -9 Figure : Empirical pricing kernel for τ =, r =.6%, April th,....

11 Motivation - Problems How to explain the non-monotonicity of the pricing kernel? What type of utility functions can generate observed pricing kernels and prices? What happens if the hypothesis of the existence of the representative investor is abandoned? How can we experimentally estimate individual pricing kernels and utility functions?...

12 Motivation - Outline of the Talk. Motivation. Pricing equation and pricing kernel (stochastic discount factor). Pricing kernel estimation with the Heston and GARCH(,) models. Decomposition of the market utility function. Individual utility functions 6. Market aggregation mechanism 7. Estimation of the distribution of investor types 8. Behavioural experiment design 9. Outlooks...

13 Pricing Equation and Pricing Kernel - Utility Maximisation Problem s.t. C t = e t P t ξ C t+ = e t+ + X t+ ξ max {ξ} U(C t) + E t [βu(c t+ )] () where X t+ a pay-off profile of an asset at t + P t the price of the asset at t ξ portfolio position β discount factor e t, e t+ wages at t and t + E t risk neutral expectation at time t...

14 Pricing Equation and Pricing Kernel - Pricing Equation If the utility function depends only on state variables and the discount factor β = const, the price of any security paying X t+ at time t + is: [ P t = E t β U ] (C t+ ) U (C t ) X t+ = E t [m t X t+ ] () where the pricing kernel (PK) a.k.a. stochastic discount factor or price per chance is: m t (C t, C t+ ) = β U (C t+ ) U (C t ) = const t U (C t+ )...

15 Pricing Equation and Pricing Kernel - Pricing Kernel Projection Pricing equation: P t = E t [m t (C t, C t+ )X t+ ] Pricing equation using the projection of the PK onto asset pay-offs X t+ : where the PK projection is: P t = E t [m t (X t+ )X t+ ], () m t (X t+ ) = E t [m t (C t, C t+ ) X t+ ] Since pricing with m t and m t is equivalent, we denote m t (X t+ ) as the pricing kernel, U t (X t+ ) and U t(x t+ ) as a utility and marginal utility function respectively...

16 Pricing Equation and Pricing Kernel - We can write the risk-neutral pricing equation as: P t = β X t+ dq(x t+ ) where Q t (X t+ ) is the observed risk-neutral distribution of returns X t+ at time t +. It is equivalent to P t = β X t+ q t (X t+ ) p t (X t+ ) dp (X t+) where P t (X t+ ) is a subjective distribution, or P t = m t (X t+ )X t+ dp (X t+ ) = E t [m t (X t+ )X t+ ], where the pricing kernel m t (X t+ ) = β q t(x t+ ) p t (X t+ )...

17 Pricing Kernel Estimation - Estimation of the Pricing Kernel The is: ˆm t (X t+ ) = β ˆq(X t+) ˆp(X t+ ), where ˆq and ˆp are the estimated risk-neutral and historical subjective densities; β = e r is a discount factor. PK is estimated with parametric models: the risk neutral density q t from option prices with the Heston model the historical subjective density p t from stock prices with the GARCH(,) model...

18 Pricing Kernel Estimation - Estimation of the Risk Neutral Density q t Risk neutral density q t is estimated from DAX option prices using the stochastic volatility Heston model: ds t S t = rdt + V t dw t where the volatility process is: dv t = ξ (η V t ) dt + θ V t dw t W t, W t Wiener processes with correlation ρ...

19 Pricing Kernel Estimation - The parameters in the Heston model can be interpreted as: ξ mean-reversion speed η long-term variance V short-term variance ρ correlation θ volatility of volatility η and V control the term structure of the implied volatility surface (i.e. time to maturity direction). ρ and θ control the smile/skew (i.e. direction)....

20 Pricing Kernel Estimation - The state-price density is derived from European option prices that may be represented in an implied volatility surface: Volatility Surface (.8,.,.) Vola. Moneyness Maturity..7 (.8,.,.).. (.8,.,.).9. (.,.,.)....

21 Pricing Kernel Estimation S 9. V time Figure : Simulated paths in the Heston model for the parameters V =., η =.8, ξ =, θ =., ρ =.7. S stock process, V variance process....

22 Pricing Kernel Estimation -6 We estimate the parameters of the state-price (objective) density by minimising the MSE of the implied volatilities: n n i= (IV model i IV market i ) where IV model and IV market refer to model and market implied volatilities; n is the number of observations on the surface. Typically, we observe prices of options with the time to maturity τ [.; ] years and K/S [;.]....

23 Pricing Kernel Estimation -7 The plain vanilla prices are calculated by a method of Carr and Madan: C(K, T ) = exp{ α ln(k)} π + exp{ iv ln(k)}ψ T (v)dv for a damping factor α >. The function ψ T is given by ψ T (v) = exp( rt )φ T {v (α + )i} α + α v + i(α + )v where φ T is the characteristic function of log(s T )....

24 Pricing Kernel Estimation -8 Estimation of the Subjective Density p t The logarithmic returns r t of DAX are modelled with the GARCH(,) model: r t = σ t Z t σ t = ω + αr t + βσ t From the logarithmic returns R i = log(s i ) log(s i ), i =,..., t and the starting stock price S we can construct the final stock price as: S t = S exp( t r i ). i= The model is fitted by maximising the likelihood function...

25 Pricing Kernel Estimation -9 We estimate the subjective density p in a forward rolling time window of the length of two years: Fit the GARCH(,) model for DAX returns Simulate N time series of the returns (N=) Compute the final N DAX prices Evaluate ˆp using kernel density estimation with the Gaussian kernel...

26 Pricing Kernel Estimation - Figure 6: Estimated price densities for τ = year, April th,....

27 Pricing Kernel Estimation - Figure 7: Estimated pricing kernel for τ = year, r =.6%, April th,....

28 Pricing Kernel Estimation - Figure 8: Estimated pricing kernel for τ = year, r = %, July th,....

29 Pricing Kernel Estimation - Figure 9: Estimated pricing kernel for τ = year, r =.%, June th,....

30 Pricing Kernel Estimation - Figure : Linear pricing kernel and quadratic utility function (CAPM model). U(X t+ ) = ax t+ + bx t+ + c....

31 Pricing Kernel Estimation - Figure : Power pricing kernel and CRRA utility function. U(X t+ ) = a X γ t+ γ....

32 Pricing Kernel Estimation -6 Figure : Pricing kernel and utility function suggested by Kahneman and Tversky based on behavioural experiments....

33 Decomposition of the Market Utility Function - Estimation of the Market Utility Function Utility function derived from the market data is the market utility function. It requires the assumption about the existence of a representative investor m t (X t+ ) = const t U (X t+ ) () Since a cardinal utility function can be defined up to a linear transformation, the constant can be neglected U t (X t+ ) = Xt+ inf(x t ) m t (s)ds...

34 Decomposition of the Market Utility Function - Figure : Market utility function, DAX, τ = years, June th,....

35 Decomposition of the Market Utility Function - Decomposition of the Utility Function Observation: the portions of the utility function below X t+ = and above X t+ =. are very well approximated with shifted CRRA functions, k =, : U (k) (X t+ c k ) γ k t (X t+ ) = a k γ k + b k, where the shift parameter is c k. The CRRA function becomes infinitely negative for X t+ = c k and is extended as U (k) t (X t+ ) = for X t+ < c k, i.e. investors by all means will avoid the situation when X t+ < c k. For a standard CRRA utility function c k =....

36 Decomposition of the Market Utility Function - Figure : Decomposition of the utility function. DAX, τ = years, June th,....

37 Individual Utility Functions - Individual Utility Functions We abandon the hypothesis of the representative investor: there are many investors in the market. Investor i has a utility function that consists of two CRRA functions: max {U t (X t+, θ, c ); U t (X t+, θ, c,i )}, if X t+ > c U i,t (X t+ ) =, if X t+ c where U t (X t+, θ, c) = a (X t+ c) γ γ + b, θ = (a, b, γ), c,i > c. If a = a =, b = b = and c = c =, we get the standard CRRA utility function....

38 Individual Utility Functions - Parameters θ and θ and c are the same for all investors. Investors differ with the shift parameter c. θ and c are estimated on the lower % of observations, when, assumingly, all investors agree that the market is bad ( bear market). θ is estimated on the upper % of observations, when all investors agree that the state of the world is good ( bull market). The distribution of c that uniquely defines the distribution of switching points is computed with a boosting procedure. a i b i γ i i = (bear market) i = (bull market)

39 Individual Utility Functions - Figure : Individual utility functions, DAX, τ = years, June th,...

40 Individual Utility Functions - Investor Types A change of behaviour from bearish to bullish happens at a switching point Different investors have different perceptional outlooks concerning the future state of economy, i.e. have different boundary between good and bad states Most of investors have switching points in the interval [.9;.], i.e. in the area that corresponds to present unit returns times half-year risk free interest rates...

41 Individual Utility Functions - The individual utility function can be conveniently denoted as: max {U bear (X t+ ); U bull (X t+, c i )}, if X t+ > c U i (X t+ ) =, if X t+ c Switching between U bear and U bull happens at the switching point Z t+, where U bear (Z t+ ) = U bull (Z t+, c i ). The switching point is determined by c i c,i The notations bear and bull have been chosen because U bear is activated when returns are low ( bear market) and U bull when returns are high ( bull market)...

42 Individual Utility Functions -6 Market Conditions and the Switching Point Each investor is characterised with a switching point Z t+ The smoothness of the market utility function is the result of the aggregation of different attitudes U bear characterises more cautious attitudes when returns are low U bull describes the attitudes when the market is booming Both U bear and U bull are concave. However, due to switching the total utility function can be locally convex...

43 Individual Utility Functions -7 Figure 6: Market Relative Risk Aversion Coefficient, DAX, τ = years, June th,...

44 Individual Utility Functions -8 The coefficient of relative risk aversion is: a R (X t+ ) = U (X t+ )X t+ U (X t+ ) We compute it non-parametrically from the estimated pricing kernel, which equals const t U (X t+ )....

45 Market Aggregation Mechanism 6- Naive Utility Aggregation Specify the observable states of the world in the future by returns X t+ Find a weighted average of the utility functions for each state. If the importance of the investors is the same, then the weights are equal Problem: utility functions of different investors cannot be summed up since they are incomparable U t (X t+ ) = N N i= U (i) t (X t+ )...

46 Market Aggregation Mechanism 6- Investor s Attitude Aggregation Specify perceived states of the world given by utility levels ũ Aggregate the outlooks concerning the returns in the future X t+ for each perceived state...

47 Market Aggregation Mechanism 6- For a subjective state described with the utility level ũ, such that ũ = U () (X () t+ ) = U () (X () t+ ) =... = U (N) (X (N) t+ ) the aggregate estimate of the resulting return is X A t+(ũ) = N N i= X (i) t+ (ũ) if all investors have the same market power. N is the number of investors Important property: the return aggregation procedue is invariant of any monotonic transformation...

48 Market Aggregation Mechanism 6- Figure 7: Inverse market and individual utility functions, DAX, τ = years, June th,...

49 Estimation of the Distribution of Investor Types 7- Estimating the Distribution of Switching Points with a Boosting Algorithm. Generate N realisations of individual utility functions with switching points Z (i), i =,..., N with any prior distribution with a compact support. Add one individual utility function U (i) and delete another U (j) with random switching points Z (i) and Z (j) respectively. Aggregate individual utility functions using subjective state aggregation. If the proximity to estimated market utility function has increased, retain the new swithching point, otherwise do nothing. Repeat steps and until the estimated market and fitted utility functions become close...

50 Estimation of the Distribution of Investor Types 7- The aggregate return in the perceptional state ũ is given by: X A f (ũ) = N N i= U Z i (ũ) where U Z i (ũ) is the inverse individual utility function or: Xf A (ũ) = U Z (ũ)f(z)dz where f(z) is the distribution of switching points, which is derived as the solution of the minimisation problem: {U M (ũ) XA f (ũ) } dp (ũ), min f(z) where U M (ũ) is the inverse of the estimated market utility function....

51 Estimation of the Distribution of Investor Types 7- Distribution of Switching Points...

52 Behavioural Experiment Design 8- Behavioural Experiment Design There are several states of the world ranging from bad (low returns) to good (high returns) There are three groups of participants that are told that the world is more likely to be in the bad, good or approximately the same state in the future, respectively. In this way we expect participants in the three groups to operate with U bear, U bull or in the switching regime Each participant is asked to place markers denoting desired outcomes into the future states, thus building a subjective distribution...

53 Behavioural Experiment Design 8- The prices of putting a marker into a state are given by a risk-neutral distribution estimated from real option market data with the Heston model. Several other distributions of state-prices, such as the log-normal distribution, can also be tested Each participant has an endowment of EUR that he must completely spend building the distribution with markers. In this way the budget constraint is implemented The pricing kernel is computed as the ratio of the risk-neutral density and experimentally derived subjective density times the discount factor, i.e. ˆm t (X t+ ) = β ˆq risk neutral(x t+ ) ˆp experimental (X t+ )...

54 Behavioural Experiment Design 8- Distribution Builder (Sharpe, 6)...

55 Outlooks 9- Claims Representation of individual utility functions as consisting of two parts, activated during perceptionally good and bad states of the world. The perceptional change happens at the swithing point. Investors behave as risk averse individuals in good and bad states but become risk seeking when switching occurs Utility function aggregation procedure based on subjective states of the world Use of DAX data and the Heston model to estimate the market pricing kernel Introduction of a boosting procedure for the estimation of the distribution of switching points...

56 Outlooks 9- Outlooks Extension of the experiment with a trading simulator, so that prices are determined by the participants Testing alternative utility function designs Refining the technique for estimating the distribution of switching points as an inverse problem Study of the dynamics of pricing kernels and individual utility functions...

57 References - References Sharpe, W. F. (6) Investors and Markets: Portfolio Choices, Asset Prices and Investment Advice, Princeton University Press, Princeton, NJ (in print) Rosenberg, J. V. and Engle, R. F. () Empirical Pricing Kernels, Journal of Financial Economics, 6, June, -7. Jackwerth, J. C. () Recovering Risk Aversion from Option Prices and Realized Returns, Review of Financial Studies,, -....

58 References - Cochrane, J. H. () Asset Pricing, Princeton University Press, Princeton, NJ. Aït-Sahalia, Y. and Lo, A. W. () Nonparametric Risk Management and Implied Risk Aversion, Journal of Econometrics, 9, 9-. Kahneman, D and Tversky, A. (979) Prospect Theory: An Analysis of Decision under Risk, Econometrica, 7, March,