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8

Just a few notes How to make sense of $\text dW_t$ is the entire point of stochastic calculus. It's far beyond the scope of any answer here. You should read some introductory lecture notes/books on stochastic calculus. You could start here. The idea: Riemann-Stieltjes integrals are of the form $\int_0^t f(s)\mathrm{d}g(s)$ and are well-defined if $f$ is ...

3

The present value of a Vanilla Swap (the word Vanilla is used since I am considering the simplest swap, i.e., notional equal to one, contiguous time intervals, constant rate, etc) is given by: \begin{align} V_s(t) &= \mathbb{E}_t^Q \left[ \sum_{i=1}^N D(t, T_{i+1}) \cdot \tau_i \cdot (L(T_i, T_i, T_{i+1}) - k) \right] \end{align} where $T$ describes the ...

7

Ito Lemma (as 'Taylor expansion'): For $X$ an Ito process and $f = f(t, x) ∈ C^{1,2}(\mathbb{R}^2)$ a deterministic function, the stochastic process $$Y_t = f(t,X_t)$$ is an Ito process and we have $$df (t,X_t) = \partial_tf(t,X_t)\,dt + \partial_xf(t,X_t)\,dX_t + \frac{1}{2} \partial_{xx}^2f(t,X_t)(dX_t)^2.$$ Note: Functions $$g(t,x)= \partial_tf(t,x),$$ ...

1

Perhaps it might help if we define the difference between Brownian Motion (BM) and Geometric Brownian Motion (GBM). BM has independent, identically distributed increments while GBM has independent, identically distributed ratios between successive factors. The definition is inherited from that of arithmetic random walks, which are modelled as sums of random ...

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