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6

First of all it is not clear what exactly you mean by right number, you definitely do not adjust forward swap rate. You probably mean adjusting euro dollar futures contract rates so that you can later use these values to fit the swap/forward libor curve. Reason for adjustment is simple. If you are short ED futures and rates go higher futures price drops ...


4

The change of the price $P(y)$ if the yield changes from $y$ to $y+\Delta y$ is $$ \frac{P(y+\Delta y) - P(y)}{P(y)} = - D \Delta y + \frac12 C \Delta y^2, $$ where $D$ is the duration and $C$ is convexity. For small $\Delta y$ the square is much smaller. Thus the duration component dominates.


4

I dont think you can see convexity in such a plot, since each of these prices are not observed from a single bond deliverable, but from different coupon bond deliveries. If the delivery was always based on same coupon type bond and quite similar maturity (http://www.cmegroup.com/trading/interest-rates/us-treasury/10-year-us-treasury-...


3

I have asked myself the very same question when I first read the book. As far as I can tell, the "scalability" condition is only imposed for technical reasons. It simplifies the subsequent proof of the Fundemental Theorem of Asset Pricing in constrained markets. There are several papers that have shown that the theorem is valid for conic constraints. ...


2

We define a convex risk measure as $$ \rho( \lambda X_1 + (1-\lambda) X_2) \le \lambda \rho( X_1 ) + (1-\lambda) \rho(X_2), $$ for $\lambda \in(0,1) $. A coherent risk measure is subadditive and homogeneous thus for coherent $\rho$ we get: $$ \rho( \lambda X_1 + (1-\lambda) X_2) \le \rho( \lambda X_1) + \rho( (1-\lambda) X_2) $$ by subadditivity and $$ \...


2

Given an index $t \mapsto S(t)$ (this may be a forward swap rate) and some value process $t \mapsto A(t)$ (this may be a swap annuity) we assume that $S/A$ is a traded product (which is true if $S$ is the forward swap rate and A is the corresponding (!) swap annuity. Then the future payoff $S(T) \cdot A(T)$ can be values as $S(t) \cdot A(t)$ (since $S$ is a ...


2

The upper bound for the 80 call is C(90) + 10, or 30. At least assuming no arbitrage. Let's start by assuming the risk-free rate is 0 (this isn't a problem, but the math is clearer without it), so we don't have to discount the price. Then, the call price is given by $C(K) = E_t[(S_T - K)^+]$, which gives: \begin{array} $C(K - 10) &= E_t[max(S_T - (K - ...


1

You're almost right, I think. You shouldn't talk about L(Ti,Ti+1) being a martingale, since this is a RV not observed until Ti. Rather we should talk about LtT, the regular Libor forward rate, and L'tT, the fair rate for an arrears FRA. We have that LtT is a martingale under the ZCB(Ti+1) measure, as you say. But also, L'tT is a martingale under the ZCB(...


1

OK, so I think I have this figured out in my head now in terms of martingale measure theory. Thanks dm63 for pointing me in the right direction! Just for my own peace of mind and perhaps to help others in the future, my understanding is as follows: Vanilla Swap: We observe the LIBOR $L(T_i, T_{i+1})$ at time $T_i$ and payment occurs at $T_{i+1}$. Therefore ...


1

This is indeed just a convention, as you point out. It comes from the fact that zero coupon bonds, by convention, do not have any volatility exposure. Rather, it is assumed the prices of ZCBs are given. Now, you can replicate a regular fra with strike K exactly using ZCBs: Long one ZCB with maturity T(i) and short (1+alpha K) ZCBs with maturity T(i+1)...


1

No you are long convexity. The futures contract has no convexity (since its value is linear as the underlying rate varies, specifically it moves by $25 per bp per contract). Meanwhile, the FRA has positive convexity (it's like a mini bond). The fact that you are long convexity overall is consistent with the fact that the rate on the futures contract is ...


1

well you need to specify dynamics for the rates between $$T_{i+1}$$ and $T_N.$ If you make them log-normal then the standard BGM/LMM drift computation applies and you get a state dependent drift. The expectation does not exist in closed form however. (See eg More Mathematical Finance for detailed discussion.)


1

There is no generic solution. However, the KKT conditions are of the forms \begin{align*} \begin{cases} Qy + \lambda_1 \mu +\lambda_2 Py = 0,\\ \mu^T y = 1,\\ y^TPy \leq k^2 \sigma^2,\\ \lambda_2 \big( y^TPy - k^2 \sigma^2\big) = 0. \end{cases} \end{align*} Here, the condition $$\lambda_2 \big( y^TPy - k^2 \sigma^2\big) = 0 $$ means that two cases need to ...


1

With a dedicated portfolio all interest rate risk has been eliminated, since you hold ZCB's that deliver exactly the cash you need when you need it. So it is a perfect hedge against i.r. risk. With an immunized portfolio you have reduced i.r. risk (by matching duration and convexity) but you still have a probability distribution of outcomes. Yields at ...


1

In our lecture, we were told to omit the proof because it was too difficult. Maybe it will help you though if you can read it here:


1

Note that $$\frac{dQ_{T_p}}{dQ}|_{T_0} = \frac{P(T_0, T_p)}{P(0, T_p)}\frac{A(0, T_0, T_n)}{A(T_0, T_0, T_n)}$$. Then $$E^{Q_{T_p}}\big(S(T_0, T_n)\big) = E^Q\bigg(S(T_0, T_n) \frac{P(T_0, T_p)}{P(0, T_p)}\frac{A(0, T_0, T_n)}{A(T_0, T_0, T_n)}\bigg) \\ = \frac{A(0, T_0, T_n)}{P(0, T_p)} E^Q\bigg(S(T_0, T_n) \frac{P(T_0, T_p)}{A(T_0, T_0, T_n)}\bigg).$$ That ...


1

Most likely the question is about CMS rate convexity adjustment. i.e. today value of a swap rate that fixes at some future time T. Mathematically, the adjustment arises from different measures (annuity versus forward measure). This is a good reference http://www.math.nyu.edu/~alberts/spring07/Lecture4.pdf As a rule of thumb, the size of the adjustment ...


1

Yes, an adjustment has to be made and the reason is that a forward curve now will evolve and not be the same as the future spot curve. For example, a one year forward today is not equal to your spot rate a year hence. So spot curve discount factors have to be adjusted or directly replaced through the forward DFs. Convexity adjustments are already supposed ...


1

Yes, because you're entering into an implicit currency swap. There was an article in "Risk" some time ago on this topic: http://www.risk.net/risk-magazine/technical-paper/1935412/choice-collateral-currency



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