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1

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 ...


0

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 ...


2

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 ...


2

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 ...


2

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


1

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.


-1

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



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