Derivative markets offer a rich source of information to extract the market's expectations of the future price of an asset. Using option prices, one may derive the whole risk-neutral probability distribution of the underlying asset price at the maturity date of the options. Once this distribution also called State-Price Density (SPD) is estimated, it may serve for pricing new, complex or illiquid derivative securities.
There exist numerous methods to recover the SPD empirically. They can be separated in two classes:
The first class includes methods which consist in estimating the parameters of a mixture of log-normal densities to match the observed option prices, Melick and Thomas (1997). Another popular approach in this class is the implied binomial trees method, see Rubinstein (1994), Derman and Kani (1994) and Chapter 7. Another technique is based on learning networks suggested by Hutchinson et al. (1994), a nonparametric approach using artificial neural networks, radial basis functions, and projection pursuits.
The second class of methods is based on the result of
Breeden and Litzenberger (1978). This methodology is based on European
options with identical time to maturity, it may therefore be
applied to fewer cases than some of the techniques in the first
class. Moreover, it also assumes a continuum of strike prices on
which can not be found on any stock exchange.
Indeed, the strike prices are always discretely spaced on a finite
range around the actual underlying price. Hence, to handle this
problem an interpolation of the call pricing function inside the
range and extrapolation outside may be performed. In the
following, a semiparametric technique using nonparametric
regression of the implied volatility surface will be introduced to
provide this interpolation task. A new approach using constrained
least squares has been suggested by Yatchew and Härdle (2002) but will
not be explored here.
The concept of Arrow-Debreu securities is the building block for
the analysis of economic equilibrium under uncertainty.
Rubinstein (1976) and Lucas (1978) used
this concept as a basis to construct dynamic general equilibrium
models in order to determine the price of assets in an economy.
The central idea of this methodology is that the price of a
financial security is equal to the expected net present value of
its future payoffs under the risk-neutral probability density
function (PDF). The net present value is calculated using the
risk-free interest rate, while the expectation is taken with
respect to the weighted-marginal-rate-of-substitution PDF of the
payoffs. The latter term is known as the state-price density
(SPD), risk-neutral PDF, or equivalent martingale measure. The
price of a security at time (
) with a single
liquidation date
and payoff
is then:
Rubinstein (1985) shows that if one has two of the three following pieces of information: