8.1 Introduction

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 $ \mathbb{R^{+}}$ 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 $ t$ ($ P_{t}$) with a single liquidation date $ T$ and payoff $ Z(S_{T})$ is then:

$\displaystyle P_{t} = e^{-r_{t,\tau} \tau} \textrm{E}_{t}^{*}[Z(S_{T})] = e^{-r_{t,\tau} \tau} \int_{-\infty}^{\infty} Z(S_{T}) f_{t}^{*} (S_{T}) dS_{T}$ (8.1)

where $ \textrm{E}_{t}^{*}$ is the conditional expectation given the information set in $ t$ under the equivalent martingale probability, $ S_{T}$ is the state variable, $ r_{t,\tau}$ is the risk-free rate at time $ t$ with time to maturity $ \tau$, and $ f_{t}^{*} (S_{T})$ is the SPD at time $ t$ for date $ T$ payoffs.

Rubinstein (1985) shows that if one has two of the three following pieces of information:

then one can recover the third. Since the agent's preferences and the true data-generating process are unknown, a no-arbitrage approach is used to recover the SPD.