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When the distance between the $T_i$ is small then \begin{align}\tag{1} \frac{R(T)}{R(S)}=\prod_{i=m(S)}^{m(T)}1+L(T_i,T_{i+1};T_i)\Delta_i\approx\exp\left(\int_S^Tr(u)\,du\right)\,. \end{align} On the other hand, $$\tag{2} 1+L(S,T,S)\Delta=\frac{1}{P(S,T)} $$ where $P(S,T)$ is the conditional zero bond price $$ P(S,T)=\mathbb E\Big[\exp\Big(-\int_S^Tr(u)\,du\...


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Ok I was having the same question while studying Financial Mathematics. After some careful thinking, I came to the obvious conclusion that this is nothing weird. To explain, the risk-neutral price of 0.5$ is actually the cost of the replicating portfolio. That is, you can completely hedge your option by buying 0.5 of the risky asset for 0.5\$ and have the ...


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