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1

$$\text{Pr}[\tau_1>t,\tau_2\leq t,\tau_3\leq t]=\text{Pr}[\tau_2\leq t,\tau_3\leq t] - \text{Pr}[\tau_1\leq t,\tau_2\leq t,\tau_3\leq t]$$ $$\text{Pr}[\tau_2\leq t,\tau_3\leq t]=C(1,q_2(t),q_3(t))$$


2

We construct a locally risk-free self-financing portfolio $X_t$, at time $t$, with $\Delta_t^1$ share of debt and $\Delta_t^2$ share of equity. That is, \begin{align*} X_t = \Delta_t^1 D_t + \Delta_t^2 E_t. \end{align*} Then, \begin{align*} dX_t &=\Delta_t^1 dD_t + \Delta_t^2 dE_t\\ &=\Delta_t^1\bigg[\Big(\frac{\partial D_t}{\partial t} + \mu ...


0

Use Ito's lemma to get dD Use dE = dV - dD Form a weighted portfolio of D and E, lets call it V, where the weights sum to 1 Use the self-ļ¬nancing to get the dynamics of V Find the weights that makes V risk-free Set the drift of V equal to r


0

We can replicate the security with the bond and the option, and obtain the Black-Scholes PDE. We form the portfolio $A_t=D_t+\alpha E_t$ where $\alpha$ needs to be determined. Applying the self-financing assumption implies that $$dA_t=dD_t+\alpha dE_t$$ so we can write $$\mu A dt+\sigma A dW_t=rD\,dt+\alpha\big(\frac{\partial{E}}{\partial ...


0

You need the derivation of BS eq. https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation#Derivation where in your setup you need S = V and case 1: V = E, case 2: V = D.


0

There are different measures and interpretations of duration. One, as has been pointed out already, has a formula weighting coupons and final contractual cashflow. Other definitions of duration take a broader perspective and relate it to the interest rate sensitivity of the security and not to a particular formula. These go by names such as effective or ...


0

Ditto to Larasing. Any bond's duration is just a matter of coupon, price, discount rate. Credit risk does not factor into this equation.



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