Application of Extended Kalman Filter to SPD Estimation

The state price density (SPD) carries important information on the behavior and expectations of the market and it often serves as a base for option pricing and hedging. Many commonly used SPD estimation technique are based on the observation (Breeden and Litzenberger, 1978) that the SPD f(·) may be expressed as (11.1) $$f\left( K \right) = \exp \left\{ {r\left( {T - t} \right)} \right\}\frac{{\partial ^2 C_t \left( {K,T} \right)}}{{\partial K^2 }},$$ where C t (K, T) is a price of European call option with strike price K at time t expiring at time T and r denotes the risk free interest rate. An overview of estimation techniques is given in Jackwerth (1999). Kernel smoothers were in this framework applied by Äit-Sahalia and Lo (1998), Ä it-Sahalia and Lo (2000), or Huynh, Kervella, and Zheng (2002). Some modifications of the nonparametric smoother allowing to apply no-arbitrage constraints were proposed, e.g., by Äit-Sahalia and Duarte (2003), Bondarenko (2003), or Yatchew and Härdle (2006). Apart of the choice of a suitable estimation method, Härdie and Hlávka (2005) show that the covariance structure of the observed option prices carries additional important information that should to be considered in the estimation procedure. Härdie and Hlávka (2005) suggest a simple and easily applicable approximation of the covariance. A more detailed discussion of option price errors may be found in Renault (1997). In this chapter, we will estimate the SPD from observed call option prices using the well-known Kalman filter, invented already in the early sixties and marked by Harvey (1989). Kalman filter may be shortly described as a statistical method used for estimation of the non-observable component of a state-space model and it already became an important econometric tool for financial and economic estimation problems in continuous time finance.