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Some terms are not explained in the restricted screenshot provided like $\beta$ and $\gamma$ however, from What I see documented, my suspect is that they are using a Taylor expansion (2nd order) to proxy the generic variation of B after a change in r and t (hint: they indeed assume that B is differentiable at least one time with respect to t). It is also ...

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If interest rates are deterministic (i.e. time-dependent but non-ranom), then \begin{align} B(t,T) &= \exp\left( - \int_t^T r(s)\mathrm{d}s\right) \\ \Leftrightarrow \int_t^T r(u)\mathrm{d}u &= -\ln B(t,T). \end{align} Differentiating both sides with respect to $T$ according to the Leibniz rule yields \begin{align*} r(T) &= -\frac{\partial \ln B(...

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No, commodity futures prices do not always move inline with spot prices. Many commodities (Natural Gas is a good example) have seasonality which reflect supply and demand changes over time. Oil can as well, though not as dramatic, since gasoline (the main consumer byproduct of oil) also has seasonal supply/demand differences. Also, a low spot price can ...

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I am afraid there is no short answer to that question. However there is some literature you can check. In this paper the author gives an overview over different methods and lists a lot of references. One approach is to decompose the volume timeseriies of your checking accounts into two parts: One volatile part: this is money which customers use to cover ...

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Keeping it simple, your payoff at time t is: $S_t-100$ The present value of the stock is $S_0$, it’s current price; and the present value of 100 is its discounted value as you correctly explained in your question.

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