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Defaulted bonds valuation

A holder of a defaulted bond expects to receive something of value eventually in the bankruptcy/restructuring process - one or more of maybe some cash, maybe some new debt or equity security, maybe ...
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What is the PDE for this interest rate derivative?

You can work out the specific PDE by applying the multi-dimensional Itô formula to the (sufficiently smooth) value function $\Pi_t = f(t, r, \sigma)$. Follow the steps from the regular Heston PDE ...

what is the rationale behind CTD (Cheapest to Deliver) mechanism in bond futures

Sometimes there is not much liquidity in a particular bond issue and it could become scarce and shoot up in price vis a vis others. Then there would be accusations that someone is manipulating the ...
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Discount factors curve shapes

It might be helpful to think about the forward rates that your curve implies. While instantaneous rates are not very intuitive they are mathematically simple, so if we have a discount curve of $Z(t)$ (...