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There are three main issues. As per my comment, one is the lack of specification for the distribution of the jumps (I'll assume that there is a $J_0 = 0$ at time 0 (otherwise, the process doesn't account for no jumps). Unless $P (J \leq -1) = 0$, your price process is problematic, and the Girsanov theorem is not applicable. To see why: $S_t = S_0 e^{\sigma ...


You have to make further assumptions on the distribution of $J_i$s. For example, if $J_i$s are iid normal, your option pricing problem becomes that of Merton (1976) and the solution to it is an infinite sum. If $J_i$s are assumed to be double exponential, you end up with Kou (2004) model and it has an analytical solution. Furthermore, there are three ...


For the general solution in the case where $f$ is not a constant, note that, from the SDE \begin{align*} dx_t = \theta(f(t)-x_t)dt + \sigma dW_t, \end{align*} we obtain that \begin{align*} d\big(e^{\theta t} x_t \big) = \theta e^{\theta t} f(t)dt + \sigma e^{\theta t} dW_t. \end{align*} Then \begin{align*} e^{\theta t} x_t = x_0 + \int_0^t \theta e^{\theta ...


You can just take expectations on both sides of your SDE/corresponding integral equation and obtain an ODE on the expectation function $m_t = \Bbb E[x_t]$: $$ \dot m = \theta(f - m) $$ which you can easily solve using ansatz $m_t = c_t \mathrm e^{-\theta t}$ which brings you to $$ m_t = x_0\mathrm e^{-\theta t} + \theta\cdot\int_0^tf(s)\mathrm ...


Here is another Credit Default Swap database which is rather extensive, daily spreads of roughly 700 entities starting in 2006.


The work of the NYU V-Lab is interesting to me. They try to measure risk in the system as a whole "systemic risk", rather than risk in a single portfolio.


communicate risk in terms of loss to a given portfolio in simple terms so that ANY client would understand.

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