Implied volatility and other implied market parameters

This chapter studies inference of implied parameters of different financial models from observed prices. In particular, it considers inference of a pair (σ_imp(t), ρ_imp(t)) of two unconditionally implied parameters, where σ_imp(t) is the unconditionally implied volatility, and ρ_imp(t) is the unconditionally implied value of ρ(t). The basic pricing rule for models with random volatility is risk-neutral valuation, when the option price is given as the expected value of its future payoff with respect to a risk-neutral measure discounted back to the present time t. The chapter presents an example of a Markovian setting, when maximization over a class of volatilities can be reduced to the solution of some nonlinear parabolic equations. It examines estimation of the future cumulative short-term interest rate and the related problem of bond pricing.