# Replicating portfolio with stock, bond and call option

I am trying to interpret:

I am having trouble interpreting the replicating strategy:

Context:

$$\phi$$ 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 $$T \rightarrow 0$$

The strategy: My interpretation:

$$\phi(S)$$ 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.

1. 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?

1. For $$\phi'(0)S$$, why are we multiplying the zero coupon bond rate by the stock price? What does that mean?

2. 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?

• The term containing $\phi(0)$ is the bond holding, the term $\phi'(0)S$ means a holding in the underlying stock. Take a look at this thread: quant.stackexchange.com/questions/26126/… Oct 22 '19 at 11:36