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1) Gatheral expresses everything in forward terms: forward value of the spot and of the call. Consider an asset $A$. You need to hold $A$ at time $T$ but since you don't need it now you don't want to buy it now. Instead you enter a forward contract with someone that says that at time $T$ you will pay the amount $K$ and get the asset in exchange. What ...


Both the quantity $\frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+$ and the quantity $(A-p(T_{i-1},T_i))^+$ are known at time $T_{i-1}$. Then the payment $\frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+$ at time $T_i$ discount back to time $T_{i-1}$ is the equivalent payment. That is \begin{align*} \bigg[\frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+\bigg] \times ...

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