The pricing kernel

The conditional probability that expresses the random evolution of a security is called the pricing kernel. It carries all the information required to price any path- independent option. Consider the random evolution of a stock price having stochastic volatility a maturity at time T and a payoff function g. Let p(x, y, T − t; x′, y′) be the (risk-neutral) conditional probability that, given security price x and volatility y at time t, it will have a value of x′ and volatility y′ at time T.  The famous Feynman-Kac formula provides the price for a derivative:

Here the expression p(x,y,τ;x′,y′) is called the pricing kernel. What is the idea of the pricing kernel?
It is the kernel of the transformation that evolves the payoff function g(x′, y′) backwards in time to its current value at time t, and yields the price of the option C(τ; x, y). Next week we will discuss an eigenfunction solution of the pricing kernel.