I am trying to interpret:
I am having trouble interpreting the replicating strategy:
$\phi$ is a generic payoff function, 0 < S < $\infty$, assumed throughout to be twice differentiable.
$C_0(S_0,K,T) $ represents the model independent price of a European call option with current stock price $S_0$ and strike K and maturity T.
$C_T(S_0,K) $ is $C_0(S_0,K,T) $ At expiry $T \rightarrow 0$
$\phi(S)$ is a the payoff on a portfolio that's the function of the stock price.
This is made up of several components:
1.$\phi(0)$ is some amount of the underlying stock.
- We find some zero coupon bond with interest rate $\phi'(0)$ of which we buy amount S. Hence $\phi'(0)S$
3.$\int_0^\infty n(K)C_T(S,K)dK$ is the payoff if we place a call at every possible strike price with position size n(S)?
Questions: 1. Is my interpretation correct?
For $\phi'(0)S$, why are we multiplying the zero coupon bond rate by the stock price? What does that mean?
Why are we calculating the expectation with respect to the strike price in $\int_0^\infty n(K)C_T(S,K)dK$? What does that give us?