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This is no means a rigorous proof but I believe it may help. Let's suppose you have Brownian motion under the normal conditions. $dX_t = \mu X_tdt + \sigma X_tdW_t$ The drift $\mu$ and volatility $\sigma$ are constants for each time step. Now let's suppose we 'zoom in' to a smaller time step $s = t/2$. Our drift portion of the equation goes to $\frac{\mu}...


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Overall you are not mistaken, although it is worth revisiting a few steps in your question. We assume $S$ follows the SDE $$ \dfrac{dS}{S} = \mu\:dt+ \sigma\:dW^\mathbb{P}(t) $$ under the physical measure $\mathbb{P}$. If we change to the risk neutral measure $\mathbb{Q}$ (using Girsanov's theorem) then $\mu \to r$ and we have the following SDE $$ \dfrac{dS}...


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There's nothing wrong with your formulation, in my opinion. If you model the rate z_30 with a fixed mean, then indeed the forward ZCB price is long vega. This means that the forward interest rate is short vega (i.e. the 30yr into 10yr forward rate goes down when vol goes up). This is self-consistent. In most textbooks, however, the forward interest ...



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