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Suppose the short rate $r$ follows the diffusive process $$dr=\mu dt+\sigma dB$$ where $B$ is the standard Brownian motion. The price of a bond portfolio $P(r,t,T)$ at time $t$ maturing at time $T$ follows dP=\frac{\partial P}{\partial t} dt+\frac{\partial P}{\partial r}dr+\frac12\frac{\partial^2 P}{\partial r^2}dr^2=\Big(\frac{\partial P}{\partial t}+ ... 1 He meant to compute the swaption price given by (2.4a), that is, \begin{align*} V_{opt} = L_0 E\left((R_s(\tau)-R_{fix})^+ \mid \mathcal{F}_0 \right). \end{align*} Under the swap measure (i.e., with L_t as the numeraire), the swap rate process \{R_s(t), t \ge 0\} is a martingale, and is assumed to be of the form \begin{align*} R_s(\tau) = R_s^0 e^{\sigma ... 0 I think you have to remember that the value is where it's trading. I know that might not be as deep as what you are looking for. But when you start to get into CDS you are getting into what the right spread is. You can trade forwards on every point on the curve with JPYUSD, so you can compare it pretty easily to a similar USD denominated bond. So I ... 1 Let's recall the definition of a Martingale first: it is a stochastic process X(t) that has the following property: let 0 \leq t < T two real numbers. Let \mathcal{F}_t be a filtration for the process X at time t. We have then: \mathbb{E}[X(T)|\mathcal{F}_t] = X(t) $$Now, if you use Black's model, you describe your asset price using a ... 1 This is pretty much impossible to do, but if you must, you'll have to make some assumptions. You can assume that the yields given are par yields. In other words, they represent both the yield AND the coupon rates of bonds trading at par. And assuming you also have short-term interest rates, you can compute forward price on this hypothetical par bond and use ... 3 Assume : R a recovery rate, a continuous payment a flat intensity \lambda i.e$$\mathbb{P}(\tau>t)=e^{-\lambda t}$$a flat discount rate r With bonds prices Assuming JPM bond pays a coupon rate of \kappa the risk free bond (being US bonds) pays a coupon rate of \kappa^{risk~free} you have :$$\text{PV}(\text{Bond}_{JPM}) = \int_{0}^T ...

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One more thing that must be considered is the expected recovery rate. A model that ignores this rate is not tied to the real world. To estimate the probability of default, you would need to find the rate that needs to be applied to each time step/payment such that risk free discounting of payments yields the price of the bond. Specifically, Price = ...

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I think it depends on your goals and how sophisticated you wish to be. At the lowest level, one can just take the spread of JPM over some relatively risk free rate (Treasurys or swaps) and declare that is the probability of default. Others (e.g. Elton, Gruber, et al in Explaining the Rate Spread on Corporate Bonds) try to measure the components. While ...

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I agree with the assertion in the OP. If two bonds are identical then the interest rate sensitivity of the one with higher credit risk is lower. That's because the expected cash flows are smaller due to credit risk.

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If you have a 0 coupon junk bond with the same time to maturity as a investment grade bond that pays coupons, the junk bond will have higher duration and visa versa. Calling it a junk or an ig bond doesn't change the duration, the formula is still the same so you can't say a ig bond always has a larger or smaller duration.

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Duration is technically independent of credit risk. ANY bond's duration is just a matter of coupon, price, discount rate. However, many issued high yield bond ARE typically shorter, because of a. high coupon (all else equal makes duration shorter) b. they can't issue too long: they themselves don't want to finance expensively, and investors don't want to ...

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The formula you quote (forward minus spot) is the yield carry for a financed position. The problem is that different people use the word carry to mean different things. The most commonly used convention, at least when we prepare analytical reports and quote sheets, is to use the word "Carry" to refer to the breakeven measure – it tells us how much yield ...

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@AlexC has already provided the correct answer, but I thought I'd provide a bit more details. The breakeven inflation (still the mostly widely used practitioner terminology) is defined as follows: $$\text{breakeven inflation} = \text{nominal yield} - \text{TIPS yield}.$$ It is called the breakeven inflation ("BEI") because if ex-post realized inflation is ...

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The difference represents "inflation compensation", or the amount that fixed bond investors must receive over and above the TIPS rate to make them accept the risk of inflation. The inflation compensation is thought to consist of the expected inflation plus a risk premium which varies over time. ...

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