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Option-implied risk aversion estimation Rihab Bedoui a,b,n, Haykel Hamdi c a

Research Laboratory for Economy, Management and Quantitative Finance LaREMFiQ, Tunisia The Institute of Higher Business Studies of Sousse, Tunisia c The FSEG of Sousse, Tunisia b

a r t i c l e in f o

abstract

Article history: Received 19 April 2015 Received in revised form 20 May 2015 Accepted 18 June 2015 Available online 16 September 2015

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.

JEL classification: C02 C14 C65 G13. Keywords: Risk aversion Subjective density Risk-neutral-density Mixture of log–normal distributions Jump diffusion model Crisis

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

n

Corresponding author at: Research Laboratory for Economy, Management and Quantitative Finance LaREMFiQ, Tunisia

http://dx.doi.org/10.1016/j.jeca.2015.06.001 1703-4949/& 2015 Elsevier B.V. All rights reserved.

R. Bedoui, H. Hamdi / The Journal of Economic Asymmetries 12 (2015) 142–152

143

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 a = − U ′′(W )/ U ′ (W )

(1)

Ar = − WU ′′ (W )/ U ′(W ) = A a W

(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.

Ar = A a W dAr dA a = Ar + dW dW

(3) (4)

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 2

See Arrow (1971), p. 96. See Arrow (1971), p. 97.

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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 [U (WT )]

αs t ≤ S ≤ T

subject to:

dWs = { rWS + αS (μ − r ) } ds + αs σ dZS

WS ≥ 0, t ≤ s ≤ 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: ∂J (Ws , Ss, s ) ∂W

= e−r (s − t )

∂J (Wt , St , s ) εs ∂W

ξs is the Radon–Nykodim derivative, εs = exp { − and

∫t

s

θ dZ u −

1 2

∫t

s

θ 2 du

θ is the risk premium, u−r σ

θ=

The state price density, qt (St), is defined as qt (St) ¼ ξt pt (St). If we choose s ¼T, we get the terminal condition:

U ′ ( ST ) = e−r (T − t ) U ′ ( St ) ε T εT =

(5)

U ′ (S ) er (T − t ) U ′ (ST ) t

By taking the derivative of

ξT, given by (5), with respect to ST, we get:

′

εT′ = er (T − t )

U′ (ST ) U′ ′ (St )

(6)

Note that the ratio of (5) and (6), times ST, provides exactly the Arrow–Pratt relative risk-aversion measure, Art (ST)

εT′ U ′′(ST ) ST = U ′′(ST ) εT

(7)

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145

Plugging

εT = ε′T =

qt (ST ) and pt (ST ) q ′t (ST ) Pt (ST ) − qt (ST ) P ′t (ST ) pt2 (ST )

into the definition of Art (ST) given by (7), we get a computable expression for the implied

relative risk aversion.3

⎡ p′ ( ST ) q ′(ST ) ⎤ ⎥ Art ( ST ) = ST ⎢ t − t ⎢⎣ pt ( ST ) qt (ST ) ⎥⎦

(8)

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 at ( ST ) =

pt′ ( ST ) pt ( ST )

−

qt′(ST ) qt (ST )

.

(9)

From (8),Pérignon and Villa (2002) had shown that a geometric measure of the relative risk-aversion can be easily derived.

Art ( ST ) = pt′ ( ST ) =

( ST ) pt ( ST )

pt′ ( ST ) tg (β )

−

− qt′(ST )

qt′(ST ) tg (α )

.

( ST ) qt ( ST ) (10)

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:

(

Ft, T = ST exp ( rt, T − δt, T ) 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

δt end, T = rt, T −

1 ⎛ Ftend,, T ⎞ ln ⎜ ⎟ τ ⎝ St end ⎠

(12)

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). 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. 4

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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 τ replace the price of each in-the-money call option with P(St, K, τ, rt,T, δt,T)þ(Ft,T K)e rt,T , 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:

C ( St , t ) = e−rτ E⁎ [ max ( ST − K , 0⎤⎦ St , t ] = e − rτ

∫0

∞

max ( ST − K , 0⎤⎦ q ( ST |St , t ) dST

(13)

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 = − e − rτ ∂K

∫K

∞

q ( ST ) dST

(14)

and differentiating twice yields the discounted pd f.

C

∂ 2C K = S T = e − rτ q ( S T ) . ∂K2

(15)

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 Ncτ strike prices for which we have call options and Npτ strike prices for which we have put options. For date t and horizon τ, observed call and put options prices are denoted Ct , τ , i and Pt , τ , i . 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. τ It should be mentioned that q(.) is the undiscounted RND whereas e r q(.) represents an Arrow–Debreu state price, which is referred to as the RND. 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). 7 8

R. Bedoui, H. Hamdi / The Journal of Economic Asymmetries 12 (2015) 142–152

Ncτ

min θ

∫

∑

ωic

ϑ i=1

2

( Ct,τ,i − Ct ( Ki, τ, θ ) )

N pτ

+

2

∑ ωip ( Pt, τ, i − Pt ( Ki, τ, θ ) )

147

.

i=1

(16)

where and are weights associated with option i and Θ is the domain of θ. These weights could be given by a measure of liquidity of a given option.

ωip

ωic

3.1.3. Mixture of log–normal distributions Bahra (1996), Melick and Thomas (1997)are among the first to describe the RND as a mixture of distributions. For option pricing, the most well-known distribution studied in the literature is the mixture of log–normal densities. The reason is that it appears as a trivial extension of the Black & Scholes model that involves the single log–normal density function.

l ( ST , m, σ ) =

⎛ 2 ⎞⎞ ⎛ 1 log ( ST ) − m ⎟ ⎟ exp ⎜ − ⎜⎜ ⎟ ⎟. ⎜ 2 s 2π s 2 ⎝ ⎠⎠ ⎝

1 ST

(17)

A mixture of such densities yields M

q ( ST ; θ ) =

αi l ( ST , mi , si ).

∑i = 1

(18)

where θ regroups all the unknown parameters αi, mi and si for i¼1, …, M, and M denotes the number of mixtures describing the RND. Obviously, to guarantee that q is a density, we must have αi 40 for all i¼1, …, M, and α1 þ… þ αM ¼1. In other words, q should be a convex combination of various log–normal densities. The option price for such a mixture of log–normal distributions is, for a given strike K and time to maturity τ ¼T–t

CLN (K , θ ) = e−rτ = e − rτ =

e − rτ

∫K

+∞

∫K

+∞

(ST − K ) q ( ST ; θ ) dST M

(ST − K ) ∑ αi l (ST , mi , si ) dST i=1

M

∑i = 1

αi

∫K

+∞

(ST − K ) l ( ST , mi , si ) dST

(19)

where we define the volatility over the horizon of the option as si ¼si √τ, to simplify notations. The last equality is obtained by simply inverting the sum and integral operators. There are various ways to evaluate the integral. For instance, we have9

=

∫K

+∞

(ST − K ) l

( ST ,

(

)

μ , s ) dST = E ( ST |ST > K ) − K Pr ( ST |ST > K ).

⎛ log (k ) − m − s 2 ⎞ ⎤ ⎛ 1 ⎞⎡ log (k ) − μ ⎟ ⎥ − K [1 − φ = exp ⎜ m + s 2⎟ ⎢ 1 − φ ⎜ )] ⎝ ⎠ 2 s S ⎝ ⎠ ⎦⎥ ⎣⎢

(20)

Finally, the option price is given by M

CLN (K; θ ) = e−rτ

⎛

∑ αi {exp ⎜⎝ μi + i=1

=

e − rτ

M

∑i = 1

⎡ ⎛ log (k ) − mi − s 2 ⎞ ⎤ 1 2 ⎞⎡ log (k ) − mi ⎞ ⎤ i ⎥ si ⎟ ⎢ 1 − φ ⎜ ⎟ − K ⎢1 − φ ⎟ ⎥. ⎠⎦ 2 ⎠ ⎢⎣ s si ⎣ ⎝ ⎠ ⎦⎥

⎛ − log (k ) − mi ⎞ ⎛ M 1 ⎞ ⎛ − log (k ) − mi − si2 ⎞ ⎟ − e−rτ k ∑i = 1 αi φ ⎜ αi exp ⎜ μi + si2 ⎟ φ ⎜ ⎟. ⎝ ⎠ ⎠ ⎝ 2 Si Si ⎝ ⎠

(21)

Under the risk neutral probability, we have to impose the martingale condition stating that the current price St under the τ RND is equal to the expected discounted price of the underlying asset e r E [ST ], so that10 M St = e−rt E ⎡⎣ ST ⎤⎦ = e−rt ∑

j=1

(

αj exp mj + 1/2S2j

)

(22)

As an alternative, we can directly use the BSM formula and set, in a similar spirit to what precedes, M

CLN (K , θ ) =

∑ ⎡⎣ St Φ ( d1, i ) − Ke−rt Φ ( d2, i ) ⎤⎦ i=1

where d1, i = 9

+ (MI τ +

E (S|S > K ) = exp (μ + 1/2S2)

(log (K ) − m) σ 10

υ=

S [log ( KT )

1/2Si2 )] /Si

and d2, i = d1, i − Si , the martingale condition may be imposed as follows

1 − ϕ (υ − σ ) 1 − ϕ (υ )

In order to estimate the parameters i; mi and si, we use this minimization program m1, m2, S1min, S2, α1, α 2 ∑iN= 1 (Ct , r , i − Ct (Ki, τ , θ ))2 + ∑iN= 1 (Pt , r , i − Pt (Ki, τ , θ ))2 + [e−rt ∑2j = 1 αj exp (mj + 1/2S2j )]2

(23)

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St + CLLN (0, θ )

(24)

This condition just means that, for a strike price of 0, the option will always get exercised and, hence, at maturity we will always get the underlying asset. In practice, imposing K ¼0 in the BSM formula is not a good idea because log (0) is not defined. It is possible to approach this limit case by setting K equal to some very small value. There are several advantages of using this technique. First,the implementation is straightforward, because it suffices to slightly modify the usual BSM formula by introducing the additional parameter m. Second, the implementation via the BSM formula introduces the current price of the underlying asset, St. The parameter m, will therefore be of the magnitude of an annualized asset return. This means that its value can be easily bounded, which is a welcome feature for numerical purposes. Third, the time to maturity τ is explicitly taken into account. This means that mi will be an annualized number which implies that parameters extracted for different maturities are directly comparable. The mixture of distribution approach comes, however, at a cost. A first drawback in fitting a mixture of distributions is the symmetry between the densities. To illustrate this point, assume that a given RND can be correctly described by a mixture involving exactly two normal densities. Using obvious notations, we can write

q ( ST , α, m1, m2 , s1, s2 ) = α l ( ST , m1, s2 ) + (1 − α ) l ( ST , m2 , s2 )

(25)

Obviously, the same density is obtained if we invert the two log–normal distributions, i.e, q (ST, 1 α, m1, m2, s2, s1). Clearly, the order of the parameters plays a critical role here. For an optimization program, this means, however, that several parameter vectors are associated with a same density. This in turn could yield numerically unstable programs where the optimizer cycles in an infinite loop. Another difficulty lies in testing the number of distributions that are involved in the RND. One may be tempted to use a likelihood-ratio type test to check if the ith density should be included in the mixture by testing if αi ¼0. However, if αi ¼0, the parameters mi and si associated with the ith density are unidentified. Stated differently mi and si could take any value, because they would not play a role in the density q. Sometimes, such parameters are called down and specific tests need to be developed. This type of problems appears also in switching regressions. There is no obvious solution to it. Critical values may be obtained by simulations. In partical, one may add distributions up to the point where adding them yields to no further improvement. There exist various solutions to help the optimizer converge in terms of MSE. In the event of two densities, M¼ 2, we start with a grid for α. Since it is known that 0 r α r1, one may subdivide the interval [0, 1] into equally spaced points, where αi ¼iΔ, for i¼0, …, N. In practice, Δ¼0.1 often yields very good results as Jondeau, Ser-Huang Poon and Rockinger (2007) find. Furthermore, to avoid the problem of symmetry mentioned above, it is possible to impose that the densities remain in a given order. One possibility that appears to give satisfactory results is to impose that s1 4s2. The first density will then have a larger standard deviation than the second one. Similar extensions may be given for the case where M¼3. The α parameters may be taken over a simplex.11

3.1.4. Jump diffusion model Modeling the underlying asset with a Black & Scholes model is not consistent. Large values of returns occur too frequently to be consistent with the normality assumption. The rare events may cause brutal assets pricing variations. In order to model such a phenomenon, we should use a Poisson jump process. We assume that the underlying asset price follows a log normal jump diffusion process i.e, the addition of a geometric Brownian motion and a Poisson jump process. This process takes into account the skewness and kurtosis effects. The underlying asset price under the risk neutral world follows the process:

dSt = ( rt − λE (k ) ) St dt + kSt dWt + kSt dqt

(26)

where qt is the Poisson associated probability, λ is the average rate of jump occurrence and k is the size of a jump. Bates (1976) has shown that in case of a diffusion with n jumps, the call price is: ∞

+ C JMP ( St , T , K ) = e−rt τ ∑ P (n jumps) E ⎡⎣ ( ST − K ) /n jumps⎤⎦ i=0

(27)

Ball and Torous (1985) and Malz (1997) proposed a simplified version of the jump-diffusion model, where there will be at most one jump of a constant size. In this Bernoulli version of a jump diffusion model, the equation of the call price is the following: 11

For further details, see Jondeau, Ser-Huang Poon and Rockinger (2007).

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149

1 ⎡ ⎛ (S − K )+ ⎞ ⎤ ⎡ ⎤ St C JMP ( St , t , T , K ) = e−rt τ ∑ P (n jumps) E ⎢ ⎜ t N ( d1 + σ τ ) − Ke−rt τ N (d1) ⎥ ⎟ ⎥ (1 − λτ ) ⎢ ⎣ ⎦ jumps 1 n k ⎝ ⎠ + λ τ ⎦ ⎣ i=0

⎡ ⎤ St N ( d2 + σ τ ) − Ke−rt N ( d2 ) ⎥ + λτ ⎢ ⎣ 1 + λkτ ⎦

(28)

with

log d1 =

( ) − log(1 + λkτ) + ⎛⎝ r − St K

⎜

t

σ2 ⎞ τ⎟ 2 ⎠

(29)

σ τ log

d2 =

( ) − log (1 + λkτ) + log (1 + k) + ⎛⎝ r − St K

⎜

t

σ2 ⎞ τ⎟ 2 ⎠

σ τ

(30)

where (1 λτ) is the probability that no events occur in the option life. We can easily show that the call price is simply a combination of two call prices calculated by Black. (1976) model

⎛ ⎞ St (1 + K ), τ , K |θ ⎟ S JMP ( St , t , T , K |θ ) = (1 − λτ ) C B ⎜ ⎝ 1 + λKτ ⎠

(31)

This technique is a particular case of a mixture of log–normal distributions. The RND is:

q JMP ( ST , St ,t , T , Kθ )=(1 − λτ ) LN ( ST ,α + log (1 + k ),β )

(32)

where:

LN (x)=

⎡ (LN (x)−α 2 ⎤ 1 exp ⎢ − ⎥ ⎣ ⎦ 2β 2π β x

⎛ 1 ⎞2 α = log ( St )+⎜ μ− ⎟ (T − t ) ⎝ 2σ ⎠ the

(33)

(34)

θ ¼(s, λ, κ) vector parameters are estimated using the following program: 1

min n

∑ (C^i − CiJMP

2

θi=1

(35)

3.2. Subjective density The subjective density is estimated using a simulated GARCH whose parameters are estimated using the historical time series of the index. This method was shown by Jackwerth (2000). Aït Sahlia and Lo (2000) propose to estimate the subjective density of ST without making any parametric assumptions. We select seven year time frames. We extract the data from the three months preceding the date of the daily assessment. We fit a GARCH (1,1) model to the three months history of daily asset prices which is described by

ϵt =σt Zt σt2=ψ +α ϵ2t −1+βσt2−1

(36)

where Zt is an independent identically distributed innovation with a standard normal distribution. The logarithmic returns of the daily asset prices are calculated according to Eτ = ln (ST /St ) with τ ¼ T t. Then, this time series and its daily standard deviation sigmat are the input of the GARCH estimation. The parameters ψ, α and β are estimates using the quasi maximum likelihood method. After simulating the GARCH process parameters, a simulation of a new GARCH process is conducted, starting of the daily assessment using the Eq. (36) but this time the unknown variables are the time series st and Et, whereas the parameters ψ, α and β are the ones estimated from the historical data. The simulation creates a T days long time series, and is run N times. The simulated index price is calculated as

St = St − 1e ϵt ∀ t ϵ{1, …, T}

(37)

where S0 is the current level of the index on the day of the daily assessment. Therefore after the simulation has been completed, we calculate the τ-period continuously compounded returns, uτ≡

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ln(ST/St), and construct a kernel estimator of the density function g(.) of these returns from our four year sample. We derive the kernel density with a Gaussian kernel and bandwidth h12:

h=

1. 8σ n

g ( υτ )=

(38)

1 Thυ i

T

∑ Kυ (

υτ − υti, τ

i=1

hυ

)

(39)

As Aït-Sahalia and Lo Anderw (2000), from the density of the continuously compounded returns we can calculate

⎛ ⎛ S ⎞⎞ Pr ( ST ≤S )=Pr ( St e υτ ≤S )=Pr ⎜ υτ ≤ln ⎜ ⎟ ⎟= ⎝ St ⎠ ⎠ ⎝

ln ( S ) St

∫−∞

g (υτ ) dυτ

(40)

and recover the price density pt (: ) corresponding to return density as S

g (log ( S )) ∂ t pt (S )= Pr ( ST ≤S )= ∂S S

(41)

Our estimator of the subjective density of the index price is

pt ( ST )=

g (log (S/St )) ST

(42)

which can be computed directly from the estimator (39) of the density function g(.).

4. Empirical results In this section, we estimate the risk aversion function from mixture of log–normals RND and jump diffusion model RND of CAC 40 index options between January 1st 2007 and December 31st 2007. The estimation of the subjective density is based on the time-series of CAC 40 index returns over a seven-year period.13 We provide in Figs. 1, 3 and 4 the risk neutral densities estimated through the jump diffusion model and the mixture of log–normal distributions and the subjective density observed respectively on the trading days January 10th 2007, July 10th 2007 and October 17th 2007 for three various options maturities: one month, two months and three months. Figs. 1, 3 and 4 show that both the risk neutral distributions and the subjective distribution look about lognormal for short maturity (1 month maturity) during the precrisis, crisis and postcrisis periods. During the crisis and postcrisis periods, we remark that the risk neutral distributions of the two chosen models and for the various maturities are left-skewed and leptokurtic however the subjective distribution does not change in shape as Jackwerth and Rubinstein (1996) and Jackwerth (2000) document we could conclude in this case that the risk aversion function changed too around the crisis which we empirically investigate. Moreover, since the subjective distribution does not change in shape around the crisis and postcrisis periods contrary to the risk neutral ones, this implies big changes in the risk premium. We report in Figs. 2, 5 and 6 the estimated risk-aversion function that we compute from both mixture of log–normals and jump diffusion models RNDs, observed respectively on the trading days January 10th 2007, July 10th 2007 and October 17th for three various options maturities: one month, two months and three months. We obtain for each value of the cash price at expiration ST an implied value for the relative risk aversion. Note that the estimated risk aversion is positive for ST oS0, implying a concave utility function although it is negative for ST 4S0. The empirically observed risk aversion functions are unfortunately not positive and monotonically downwards sloping as suggested by the standard assumptions of the economic theory. Thereby, the implied preferences are of decreasing relative risk aversion class and the risk aversion function exhibits a U-shaped pattern with respect to ST. In other words, the market prices of CAC 40 options and the market returns on the CAC 40 index are such that the representative agent becomes more averse as the index goes down in value, as well as for very high values of the index so the Constant Relative Risk Aversion preferences cannot capture this phenomenon accentuating thereby the differences between market prices and Black & Scholes model. These findings are consistent with Aït-Sahlia and Lo (2000), Jackwerth (2000) and Pérignon and Villa (2002) findings. We note also that the values of the risk aversion are very close to that reported by studies based on consumption data, such as Mehra and Prescott (1985). 12

See Jackwerth (2000) for the choice of the bandwidth h. Jackwerth (2000) showed that the general shape of the risk aversion function is stable with respect to different length for the historical sample of 2, 4 or 10 years. 13

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4.1. Time to maturity impact For the three trading days that we have chosen, we change the time-to-maturity from 1 month to 2 and 3 months, we note that the risk aversion function shape is not affected for both the 1 and 2 months maturities which cooroborates the findings of Jackwerth (2000) but for the 3 months maturity, the risk aversion function shape is clearly affected. Moreover, the U-shaped pattern is more pronounced for the longer maturities. Note that whatever the chosen trading date is (representing the precrisis, crisis or postcrisis period), if we have the same time-to-maturity, we keep the same shape of the curve. 4.2. The selected RND model impact We note that 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-to-maturities. Compared to the time-to-maturity impact on the implied risk aversion, the selected RND model impact is less significant. Therefore, the risk aversion function asymmetry is more pronounced for the maturity of the options than the risk neutral density model estimation. 4.3. The selected modeling date impact (during a precisis, a crisis or postcrisis period) Unlike other empirical studies, Jackwerth (2000) estimates absolute risk aversion functions on the US market for different time periods (Precrash and Postrcarsh around the 1987 crash). In our study, we choose the three trading days January 10th 2007, July 10th 2007 and October 17th. All things being equal, 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. Indeed, we note that the variation interval widen from [ 40, 40] to [ 50, 100] for the 3 month time-to-maturity. Moreover, we note that the selected modeling date impact is more remarkable than option maturity and RND selected model impacts. The risk aversion asymmetry is more observable when considering the selected modeling date impact. A noteworthy finding is that the variation interval of the implied risk aversion, using a jump diffusion model for the RND estimation, is equal to [ 50, 50] for the precrisis time period and is the same [ 100, 100] for the crisis and postcrisis time periods nevertheless when using a mixture of log–normals for the RND estimation, the variation interval of the implied risk aversion is the same [ 100, 50] whatever the market trend is and this is observed for a two month time-to-maturity.

5. Conclusion 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 highfrequency CAC 40 index options. We 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 density estimation approaches and for different options maturities. We studied simultanously the impacts of the time-to-maturity, the riskneutral density estimation model and the time periods of the study on the risk aversion function. We report that both the risk neutral distributions and the subjective distribution look about lognormal for short maturity (1 month maturity) during the precrisis, crisis and postcrisis periods. During the crisis and postcrisis periods, we remark that the risk neutral distributions of both the mixture of log–normals and jump diffusion models and for the various maturities are left-skewed and leptokurtic however the subjective distribution does not change in shape as Jackwerth and Rubinstein (1996) and Jackwerth (2000) document. Moreover, the estimated risk-aversion function that we compute from both mixture of log–normals and jump diffusion models for different maturities and three time periods is positive for ST oS0, implying a concave utility function although it is negative for ST 4S0. The empirically observed risk aversion functions 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. For the three trading days that we have chosen, we change the time-to-maturity from 1 month to 2 and 3 months, we note that the risk aversion function shape is not affected for both the 1 and 2 months maturities which cooroborates the findings of Jackwerth (2000) but for the 3 months maturity, the risk aversion function shape is clearly affected. Furthermore, the U-shaped pattern is more pronounced for the longer maturities. We note also that whatever the chosen trading date is (representing the precrisis, crisis or postcrisis period), if we have the same time-to-maturity, we keep the same shape of the curve. We note that 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-to-maturities. 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.

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