《金融经济学原理》课程教学资源（文献资料）Option-implied risk aversion estimation.pdf

Option-implied risk aversion estimation Rihab Bedoui a,b,n , Haykel Hamdi c a Research Laboratory for Economy, Management and Quantitative Finance LaREMFiQ, Tunisia b The Institute of Higher Business Studies of Sousse, Tunisia c The FSEG of Sousse, Tunisia article info Article history: Received 19 April 2015 Received in revised form 20 May 2015 Accepted 18 June 2015 Available online 16 September 2015 JEL classification: C02 C14 C65 G13. Keywords: Risk aversion Subjective density Risk-neutral-density Mixture of log–normal distributions Jump diffusion model Crisis abstract In this paper, following the Jackwerth (2000) work, we estimate the risk aversion function on the French market implied by the joint observation of the cross-section of option prices and time-series of underlying asset returns from a high-frequency CAC 40 index options. We recover risk aversion empirically around the 2007 crisis using two different RiskNeutral densitiy estimation approaches and for different options maturities. We studied simultaneously the impacts of the time-to-maturity, the risk-neutral density estimation model and the time periods of the study on the risk aversion function. Our findings show that the estimated risk-aversion functions that we compute from both mixture of log– normals and jump diffusion models for different maturities and three time periods are unfortunately not positive and monotonically downwards sloping as suggested by the standard assumptions of the economic theory. We note also that the values of the risk aversion are very close to that reported by studies based on consumption data. Moreover, for the three chosen trading days, if we change the time-to-maturity, we note that the risk aversion function shape is not affected for both one and two months maturities which corroborates the findings of Jackwerth (2000) but for the three months maturity, the risk aversion function shape is clearly affected and the U-shaped pattern is more pronounced for the longer maturities. Furthermore, we find that whatever the chosen trading date is, if we have the same time-to-maturity, we keep the same shape of the curve as well as the risk aversion function shape is not substantially affected if we extract the RND from a jump diffusion model or a mixture of log–normals model mainly for the short time-tomaturities. Compared to the time-to-maturity impact on the implied risk aversion, the selected RND model impact is less significant. A noteworthy finding is that the variation interval of the implied risk aversion on the postcrisis, when the market trend is upward, is more remarkable and large than the precrisis and postcrisis for the long-run maturity. Concluding, the risk aversion response is asymmetric depending on the Risk-Neutral density, time to maturity options and the period of study. & 2015 Elsevier B.V. All rights reserved. 1. Introduction The eighties have marked the beginning of the academic research regarding the application of risk aversion levels to areas of risk management and quantitative finance. Among these studies, we find the Blume and Friend (1975) work that attempts to identify the nature of the utility function of households from an analysis of the Heritage composition of 2100 of Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jeca The Journal of Economic Asymmetries http://dx.doi.org/10.1016/j.jeca.2015.06.001 1703-4949/& 2015 Elsevier B.V. All rights reserved. n Corresponding author at: Research Laboratory for Economy, Management and Quantitative Finance LaREMFiQ, Tunisia The Journal of Economic Asymmetries 12 (2015) 142–152

them. Their results led them to conclude that the assumption of the constant relative risk aversion is an acceptable approximation of the reality. Similarly, Szpiro (1986) verifies that the assumption of constant relative risk aversion is true and finds that the coefficient of risk aversion is between about 1.2 and 1.8. In the same context, Nakamura (2007), tests the stability of the risk aversion coefficient to Japanese daily data between 1973 and 1991 and shows that it is invariant. Furthermore, Nishiyama (2007) studied the effect of a change in the coefficient of risk aversion of U.S. banks versus Japanese banks. He noted that the increase in risk aversion of U.S. banks is unambiguously associated with the Asian crisis, while the increase in Japanese banks risk aversion is only weakly associated. Alonso, Ganzala, and Tusell (1990) estimate the relative risk aversion coefficient for the Spanish Stock Market between 1965 and 1984. As Rosenberg and Engle (1997) and Aït-Sahalia and Lo Anderw (2000) use the relationship between the Risk-Neutral and the subjective densities and the risk aversion function to derive the marginal utility function, Coutant (1999) and Enzo, Handel, and Härdle (2006) showed that the risk aversion function varies over time. Jackwerth (1996, 2000) empirically derive risk aversion functions implied by S&P500 index options prices around the 1987 crash. He find that precrash risk aversion functions are positive and decreasing by wealth but postcrash ones are partially negative and partially increasing and irreconcilable with standard assumptions made in economic theory. Other recent research including that of Pérignon. and Villa. (2002). These authors apply the same technique of Aït-Sahlia and Lo (2000) in order to estimate the risk aversion function using the CAC 40 index options. They have defined a new formula for the risk aversion function which is "the geometric risk aversion measure". Finally, we find the study of Bliss and Panigirtzoglou (2004) which covers the estimation of the risk aversion function for different maturities. They use two utilities functions: Exponential and Power and derive the risk aversion coefficient from the S&P 500 and FTSE 100 indexes options. In this paper, following the Jackwerth (2000) work, we estimate the risk aversion function on the French market implied by the joint observation of the cross-section of option prices and time-series of underlying asset returns. Our paper is the first dealing with European options to recover risk aversion empirically around the 2007 crisis, i.e. dividing the period of study into three sub-periods: precrisis, crisis and postcrisis sub-periods, using two different Risk-Neural densitiy estimation approaches and for different options maturities. This paper is the first that studies simultaneously the impacts of the timeto-maturity, the risk-neutral density estimation model and the time periods of the study on the risk aversion function, that is the study of the asymmetric response of risk aversion. The paper is organized as follows. Section 2 presents the fundamental relationship that exists between aggregate riskneutral and subjective probability distributions and risk aversion functions. Section 3 describes our data and procedures used for the empirical work: the implied risk-neutral density estimation methods and subjective density estimation approach. Section 4 presents and analyses the empirical results. Section 5 concludes. 2. 2 Implied risk aversion functions Risk aversion allows not only to understand the agent's risk behaviour but also to specify the shape of the utility function. Two measures of risk aversion have been proposed in the literature: absolute risk aversion (Aa), which is defined in presence of exogenous risks, and relative risk aversion (Ar), defined in presence of endogenous or proportional risks. Pratt (1964) and Arrow (1971) define two measures: A UWU W a = − ′′( ) ′ ( ) / ( ) 1 A WU W U W A W r a = − ′′ ( ) ′( ) = / ( ) 2 where U is the utility function and W is the individual wealth. Arrow (1971) hypothesizes that most investors display decreasing absolute risk-aversion (DARA) and increasing relative risk-aversion (IRRA) with respect to wealth. He points out that the DARA "seems supported by everyday observations"1 but he establishes that IRRA "was not easily confortable with intuitive evidence".2 Theoretically, we can easily see the dependence of Aa and Ar on W. A AW r a = ( ) 3 dA dW A dA dW 4 r r a = + ( ) Therefore, if dAa/dW is negative (DARA), dAr/dW can be positive, negative or null since Aaand W are positive. Besides, usual utility functions impose implicitly decreasing (D), constant (C) or increasing (I) absolute risk aversion (ARA) and relative risk aversion (RRA). For instance, the logarithmic utility function is DARA and CRRA, the power utility function DARA and CRRA and the negative exponential utility function CARA and IRRA. Blume and Friend (1975) specifies that Ar is constant with changing wealth. Lucas model (1978) provides an accurate framework to estimate empirically the relative risk aversion coefficient. The dynamic optimization problems of economic agents typically imply a set of stochastic Euler equations that must be satisfied in equilibrium. These Euler equations imply a 1 See Arrow (1971), p. 96. 2 See Arrow (1971), p. 97. R. Bedoui, H. Hamdi / The Journal of Economic Asymmetries 12 (2015) 142–152 143

set of orthogonality conditions that depend on observed variables and on unknown parameters characterizing preferences. Mehra and Prescott (1985) argue that the difference between the average return on equity and the average return on short term default free debt observed on the US market between 1889 and 1978 is not compatible with an Ar comprised between 0 and 10. Their tests show that the value of relative risk aversion parameter for which the model’s averaged risk free rate and equity risk premium match those observed for the US economy over this period is in the order of 55. Ferson and Constantinides (1991) extend the Hansen and Singleton testing procedure by introducing habit persistence in consumption preferences and durability of consumption goods and thus consider a time-non-separable utility function. They test for the Euler equation and estimate the model parameters using the generalized methods of moments (GMM). The estimated values of Ar are between 0 and 12, depending on the instrumental variables and the number of lags considered. Since all these studies require a large macro-economic and financial database, financial economists focus on extracting the implied risk aversion from derivatives markets. The advantage of using an option pricing model in the estimation of risk aversion is that daily or even intraday data can be used. Bartunek and Chowdhury (1997) estimate the implied risk aversion coefficient from the price of SP&500 index option traded on the Chicago Board Options Exchange. In order to estimate Ar a criterion function is minimized which is based on the difference between the observed call values and those generated by the call pricing function, that depends explicitly of the risk aversion coefficient. Unlike the case of equity markets, the risk aversion parameter implied from option prices is between 0 and 1. When the investor is indifferent to risk, the Risk Neutral Density (RND) is his relevant density and if not the corresponding subjective probability would be different. Thus, the higher the risk aversion is, the more different the RND and the subjective probability would be. Hence, the risk aversion can be estimated from the estimation of the both densities. AïtSahalia and Lo (2000) extract a measure of risk-aversion in a standard dynamic exchange economy. There is one single consumption good, no exogenous income, one risky stock, one riskless bond zero-net supply and the riskless rate is assumed constant. In such an economy, financial markets are assumed to be dynamically complete, such that a representative agent can be introduced. The representative agent consumes only at the final date and maximizes the expected utility of the terminal wealth by choosing the amount αs invested in the stock at each intermediary date. The utility function is twice continuously differentiable, increasing and concave. max E UW tST T s [( )] α ≤ ≤ subject to: dW rW r d dZ s SS ss S = + (−) + { } α μ ασ WS ≥ ≤≤ 0, ts T where Ws denotes the wealth at date s, r is the risk-free instantaneous interest rate, m is the instantaneous conditional expected return pert unit time and s is the instantaneous conditional variance per unit time. Using the indirect utility function, the first order condition for the investor is: e JW S s W rs t JW S s W s ss tt , , = ε ∂( ) ∂ −(−) ∂( ) ∂ ξs is the Radon–Nykodim derivative, exp dZ du 1 2 s t s u t s 2 ε θθ = {− − ∫ ∫ and θ is the risk premium, u r θ = σ − The state price density, qt (St), is defined as qt (St)¼ξt pt (St). If we choose s¼T, we get the terminal condition: US e US T 5 rT t ′= ′ ( ) ()t T ε ( ) −(−) T erT t U S U S T t ε = (−) ′( ) ′( ) By taking the derivative of ξT, given by (5), with respect to ST, we get: e U S U S 6 T rT t T t ε ′ = ′ ( ) ′ ( ) ( ) (−) ′ ′ Note that the ratio of (5) and (6), times ST, provides exactly the Arrow–Pratt relative risk-aversion measure, Art (ST) U S U S S 7 T T T T T ε ε ′ = ′′( ) ′′( ) ( ) 144 R. Bedoui, H. Hamdi / The Journal of Economic Asymmetries 12 (2015) 142–152

Plugging T q S p S t T t T ε = ( ) ( )and T q S P S qS P S p S t T t T t T t T t T 2 ε′ = ′ ( ) ( )− ( ) ′ ( ) ( ) into the definition of Art (ST) given by (7), we get a computable expression for the implied relative risk aversion.3 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ AS S p S p S q S q S 8 rt T T t T t T t T t T ( ) ( ) ( ) = ′ − ′( ) ( ) ( ) Jackwerth (2000) proposes a powerful technique by which the absolute implied risk-aversion functions across wealth can be extracted from historical and risk neutral densities implied in option prices. He considers a complete market economy where the representative investor's intertemporal optimization program provides the following expression for the absolute risk aversion function: A S p S p S q S q S . 9 at T t T t T t T t T ( ) ( ) ( ) = ′ − ′( ) ( ) ( ) From (8), Pérignon and Villa (2002) had shown that a geometric measure of the relative risk-aversion can be easily derived. A S pS S p S q S S q S p S tg q S tg . 10 rt T t T T t T t T T t T t T t T ( ) () ( ) ( ) ( ) ( ) ( ) β α = ′ − ′( ) = ′ ( ) − ′( ) ( ) ( ) where α is the angle bounded by the x-axis and the line linking the axes origin to the point with coordinates (ST; q (ST)) and β the angle bounded by the x-axis and the line linking the axes origin to the point with coordinates (ST; p (ST)).3 3. Data and estimation Our data is provided by the SBF-Paris Bourse4 and includes intraday values of the CAC 40 stock index and intraday transaction prices of CAC 40 options over the period January 1st 2007–December 31st 2007. CAC 40 options are traded on the MONEP, the French derivatives market. Trading covers eight open maturities: three spot months, four quarterly maturities (March, June, September, December) and two half-yearly maturities (March, September). The maturity date is the last trading day of each month. Trading takes place on a continuous basis between 9:00 am and 5:30 pm. The future underlying asset Ft,T cannot be observed exactly at time t, where t denotes a given day and time and T is the maturity. Aït-Sahalia and Lo (1998) suggest extracting an implied future index price from the put-call parity. Given our intraday data, we cannot get contemporaneous trades concerning a call and a put with similar strike price and maturity. Thus, for each maturity, we compute: FS r tT T tT tT , , = −) ( exp( ) δ τ ( ) 11 To circumvent the unobservability of the dividend rate δt,T, we extract an implied value of the dividend rate between the end of day t and T from the daily closing price of the index, Stend and the settlement future index price, Ftend, observed at the end of each day. The obtained dividend rate is the dividend rate expected by the market between tend and T. 5 ⎛ ⎝ ⎜ ⎞ ⎠ T r ⎟ F T S , 1 ln , 12 t end t T t t end , , end δ τ = − ( ) Ft,T is then computed using the dividend rate and the riskfree interest rate proxied by Euribor. Since some CAC 40 options are not traded actively, we need to filter the data carefully. Five filters are applied to the initial data. We omit the quotes of the first and the last 15 min each day and the option quotes characterized by a price lower or equal to one tick. We only 3 See Aït-Sahalia and Lo (2000). 4 SBF-Paris Bourse provides a monthly CD-ROM including intraday values of the CAC 40 stock index and intraday transaction prices of CAC 40 options traded on the MONEP. 5 The constant dividend hypothesis passes over the clustering of dividends paid by most firms during specific months. However, as far as we are concerned, the bias is not relevant since we use market prices of futures to estimate an implied dividend rate. Nevertheless, this formula requires the prediction of all dividends amounts and payment dates paid by the 40 CAC 40 index shares and the prediction of the forward interest rates. R. Bedoui, H. Hamdi / The Journal of Economic Asymmetries 12 (2015) 142–152 145

146 R Bedoui,H.Hamdi The Journal of Economic Asymmetries 12(2015)142-152 with a and 115.This far-away-from the money violating general no-arbitrage conditions,that is put-call parity.are eliminated as well as we replace the price of all liquid options. money options.with the price implied by put-ca party at the evant s ike prices.Specially.w with price SK.)is out-of-the money and therefore liquid.After this procedure.all the information incudedin timated. any loss n can b 3.1.Risk neutral density function lationship between option prices and the RND.In order to extract the RND functions we find in literature methodological tural approaches s well as s structural methods of RND function We find parametri semi-para tion-based ap oaches:the numerical approximation of the kND pased on the second der ative of opi on prices wit nberger(1978 (the normal distribution of Jarrow and Rudd (192)(iv)the Hermite polynomials,sug sted by Madan and Mline (1994) (v)Hes derity model (1993)and(vi)the umpctral and parametric method.that is the mixture ton's stochast log normal distributions and a structural method namely the jumpdiffusion model the first to derive the RND using the following price of a call option formula C(St.t)=e-mE-[max(ST-K.0]S:.t] -e-"max(Sr-K.o]q(SrIS.t)ds 13) d()is the undiscounted ntiating this equation with respectt exercise price K yields the discounted cdf -e"f"a( (14 and differentiating twice yields the discounted pdf (15) These equations show that the second derivative of the call price yields the discounted RND.This suggests that a first eless,thi practice.Also,it has been shown that RNDs estimated in thi s manner are very unstable.In fact.twice ex h de of these models are estimated. 312ecawteioesmaieagenmodewihrespetoastopanmctesaAgnethatoteraebe Nstrike pricesfor which we have call options and N strike prices for which we have put options.For datetand horion [.t. nd hne parameter vector is typically estimated by non-linear least squares,by minimizing for each day and each maturity

consider options with a moneyness6 comprised between 0.85 and 1.15. This procedure eliminates far-away-from the money observations, which are unreliable due to their low volume and low sensitivity towards volatility. Besides, options quotes violating general no-arbitrage conditions, that is put-call parity, are eliminated as well as we replace the price of all illiquid options, that is in-the-money options, with the price implied by put-call parity at the relevant strike prices. Specially, we replace the price of each in-the-money call option with P(St, K, τ, rt,T, δt,T)þ(Ft,TK)ert,Tτ , where, by construction, the put with price P(St, K, τ, rt,T, δt,T) is out-of-the money and therefore liquid. After this procedure, all the information included in liquid put prices is extracted and resides in corresponding call prices through the put-call parity. Put prices may now be splayed without any loss of reliable information. Since our data include only call options, a single volatility function can be estimated. 3.1. Risk neutral density functions A huge literature arose from the early 1990s on the most appropriate way to estimate the RND. At the origin of all the methods, we find the famous work of Breeden and Litzenberger (1978), who are the pioneers who determined a relationship between option prices and the RND. In order to extract the RND functions, we find in literature methodological non-structural approaches as well as structural methods of RND functions estimation. We find parametrically, semi-parametrically and nonparametrically estimation methods of the RND, that is the kernel and the tree-based methods as well as six parametric and semi-parametric option-based approaches: (i) the numerical approximation of the RND based on the second derivative of option prices with respect to the strike price, as suggested by Breeden and Litzenberger (1978), (ii) the mixture of log–normal distributions following Melick and Thomas (1997), (iii) the Edgeworth expansion around the lognormal distribution of Jarrow and Rudd (1982), (iv) the Hermite polynomials, suggested by Madan and Mline (1994), (v) Heston's stochastic volatility model (1993) and (vi) the jump diffusion model following Bates (1976). In this paper, in order to extract the RND density, we use a non-structural and parametric method, that is the mixture of log–normal distributions and a structural method namely the jump diffusion model. 3.1.1. The Breeden and Litzenberger relation Breeden and Litzenberger (1978) were the first to derive the RND using the following price of a call option formula: ⎤⎦ ⎤⎦ CS t e E S K S t e S K q S S t dS , max , 0 , max , 0 , 13 t r T t r T Tt T 0 ∫ () ( ( () =[ − ] = −| ( ) τ τ − ⁎ − ∞ where for date t and maturity date T, we denote C the call price, r is the risk-free interest rate, S is the underlying asset price, K is the strike price and q(.) is the undiscounted RND. Differentiating this equation with respect to the exercise price K yields the discounted cd f C K e q S dS 14 r K ∫ ( ) T T ∂ ∂ = − ( ) − τ ∞ and differentiating twice yields the discounted pd f. C C K K S e qS . 15 T r T 2 2 ( ) ∂ ∂ = = ( ) − τ These equations show that the second derivative of the call price yields the discounted RND. 7 This suggests that a first method to extract RND is to approximate it numerically applying the finite difference approach to (15). Nevertheless, this method relies on the hypothesis that there exist traded option prices for many strikes. This is not likely to be the case in practice. Also, it has been shown that RNDs estimated in this manner are very unstable. In fact, differentiating twice exacerbates even tiny errors in the prices and may be difficult.8 That is why it is necessary to extract RND using alternative methods that put more structure on the option prices. Before describing such methods, we briefly show how the parameters of these models are estimated. 3.1.2. RND parameters estimation Suppose that we have to estimate a given model with respect to a set of parameters θ. Assume that for horizon τ, we have Nc τ strike prices for which we have call options and Np τ strike prices for which we have put options. For date t and horizon τ, observed call and put options prices are denoted Ct i , , τ and Pt i , , τ . Theoretical call and put implied by the assumed model, are denoted Ct(K, τ, θ) and Pt(K, τ, θ), respectively, for strike price K and horizon τ. Then, the parameter vector θ∈Θ is typically estimated by non-linear least squares, by minimizing for each day and each maturity. 6 Strike price divided by the future index level. 7 It should be mentioned that q(.) is the undiscounted RND whereas erτ q(.) represents an Arrow–Debreu state price, which is referred to as the RND. 8 For instance, if option prices suffer from non-synchronicity bias (that is, the underlying asset price is not observed at the same time as the option price), if the option price is fudged because of some microstructure reason (for instance, due to the bid-ask spread). 146 R. Bedoui, H. Hamdi / The Journal of Economic Asymmetries 12 (2015) 142–152