Econometric specification of the risk neutral valuation model
Introduction
In complete markets the assumption of no arbitrage opportunity implies the existence of a unique risk neutral measure, which may be used for pricing derivative assets (Harrison and Kreps, 1979). As a by-product the derivative prices satisfy deterministic relationships, which are incompatible with a standard statistical analysis based on observed derivative prices. An illustration is given by the Black–Scholes model (Black and Scholes, 1973). The price at date t of a European call with strike K and maturity date t+H is a function of the current price St of the underlying asset and of the short-term interest rate rt:where σ is the volatility parameter. If at date t we observe the prices and one derivative price the implied Black–Scholes volatility, defined as the solution ofis an estimator of σ which is infinitely accurate. If now we observe an additional derivative price , we get another infinitely accurate estimator In practice the two estimates and are different and the underlying model is rejected with probability one.1
The empirical literature on derivative assets proposed pragmatic approaches to circumvent this basic difficulty. They are of different kinds.
(1) The complete market modelling may be considered only as a benchmark used for regression models. This approach is followed when the volatility in the Black–Scholes model is estimated by ordinary least squares from derivative prices:
However the underlying modelcontains additional error terms uj,t, i.e. some extra randomness which creates incompleteness and is incompatible with Black–Scholes derivations and the assumption of a unique valuation [see Malz 1995, Malz 1996 for an extended Black–Scholes model corrected for the smile, Bahra (1996) for a model based on a mixture of log-normal distributions].
(2) Another approach consists in increasing the number of parameters in such a way that this number is always larger than the number of observed derivative prices. This literature includes all the studies on nonparametric estimation of the risk neutral density based on Hermite Polynomials, Edgeworth expansion, or kernel methods [see, e.g. Hutchinson et al. (1994), Madan and Milne (1994), Ait-Sahalia and Lo (1998), Abken et al. (1996), Corrado and Su (1996), Stutzer (1996), Jondeau and Rockinger (1997), Söderlind and Svensson (1997)]. However, although it is possible to prove the consistency of the nonparametric estimator when the number of derivative prices increases, the distributional properties cannot be derived without introducing extra randomness, i.e. market incompleteness.
(3) More flexibility may also be introduced by suppressing the complete market assumption. Then there exists an infinite number of admissible valuation measures. A part of the literature derives the deterministic bounds on derivative prices implied by the assumption of no arbitrage opportunity [see Merton (1973) and Hodges (1996) who gives the constraints on the implied Black–Scholes volatilities, and Gourieroux et al. (1997, Chapter 8) for a general discussion]. Nevertheless this approach often provides large derivative price intervals not enough informative to be used in practice. This approach may be completed by introducing some prior distribution on the location of the valuation measure and deriving the posterior distribution of the admissible derivative price inside the previous interval (Ncube, 1993). Some authors select one of the admissible neutral probability measure (Magnien et al., 1996).
(4) Finally many econometric papers have directly specified descriptive dynamic models for the derivative prices or for these prices corrected by strike and maturity effects, i.e. for the implied Black–Scholes volatilities (Dupire 1994 and Engle and Mustafa, 1992). The corresponding estimation methods and the predicted derivative prices are generally not compatible with the no arbitrage implications. For instance it is known that long memory modelling for prices allows for perfect arbitrage (Rogers, 1997).
In this paper we propose to reconcile the complete market hypothesis and the statistical inference.
In Section 2 we explain that the notion of market incompleteness refers to the information held and used by the market participants whereas statistical inference is concerned with the econometrician's information. We deduce that, under asymmetric information, the visible implication of the no arbitrage condition is the ability to obtain the derivative prices as expected discounted cash-flows with respect to a stochastic valuation measure. Therefore the extra randomness necessary to perform statistical inference is introduced via the difference between the informations.
In Section 3, we study different consequences of such a modelling. We give the first- and second-order stochastic properties of the derivative prices. Then we predict the derivative prices and check that the predicted values satisfy the no arbitrage condition.
In Section 4, we use gamma measures as a basis for specifying parametric models, and particularize this specification in Section 5 to extend Black–Scholes models. It is seen that the parameters can be estimated by a generalized least-squares method taking into account the correlation and heteroscedasticity corresponding to the cash-flow patterns.
Finally in Section 6 we consider the case of derivative assets with different horizons or with path dependent cash-flows.
Section snippets
Stochastic risk neutral measure
In this section and the two following ones, we consider an underlying index St, t varying, and derivative assets based on S. S may correspond to the price of a tradable financial asset, to a market index or to some other statistical index such as the aggregate result of the US insurance companies. At date t, we only consider derivatives with a given residual maturity H, delivering an indexed cash-flow g(St+H) (say) at the maturity date.
Stochastic properties of derivative prices
Let us fix the date t, and consider only the valuation problem at this date and horizon H. Then the random measure Qt is indexed by s only. We assume the existence of the infinitesimal first- and second-order moments of Qt, conditional to the exogenous information available at t, for the statistician (i.e. not including the current observed derivative prices); the expectation, covariance and variance operators are:
The gamma specification
It remains to propose a tractable specification for the distribution of the stochastic valuation measure. The gamma specification seems a good candidate for at least three reasons:
(i) it leads to a clear factorization of the distribution into the parts corresponding to the zero-coupon price and to the risk neutral probability; (ii) it is easily located around deterministic valuation formula, such as the Black–Scholes’ one; (iii) the estimation and computation steps can be easily performed by
A parametric specification
The gamma modelling may be particularized by selecting the average measure νt in a parametric family νt=νt(θt). Then the distribution of the derivative prices will only depend on a finite number of parameters, i.e. . In particular we can select this specification in order to calibrate the expected derivative prices on some deterministic valuation formula, such as the Black–Scholes or Hull–White ones. We describe the consequence of these choices and we discuss statistical inference.
Extension to the multihorizon case
The number of liquid derivatives with a given maturity is in practice rather small and it is preferable to base the estimation of the structural parameters on all the observed liquid derivatives. Therefore, it is important to extend the previous approach to the case of derivative assets with different maturities or with path dependent cash-flows. We first recall the constraints implied by the no arbitrage opportunity condition; then we discuss different specifications for the stochastic risk
Concluding remarks
In this paper we have tried to reconcile the standard pricing theories based on the complete market assumption and the econometric implementation, which requires enough randomness to derive nondegenerate distributions for the estimators. The proposed solution is to suppose asymmetric information where the market participants are more informed than the econometrician. It leads to a pricing formula with a stochastic pricing measure, which may be easily located around some standard deterministic
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