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9

Let us denote $\delta$, the Libor's tenor (e.g. 3M), $P(t, T)$ the price of a zero coupon bond price paying 1 unit of currency at $T$, and $L_t(T, T + \delta)$ the forward 3M Libor starting at $T$ and ending at $T+\delta$, seen from $t$: $$L_0(T, T + \delta) = \frac{1}{\delta} \left(\frac{P(0, T)}{P(0, T + \delta)} - 1 \right)$$ The vanilla case: payment ...

7

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 ...

6

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-...

5

Consider a date sequence \begin{align*} 0 \leq t_0 \leq T_s < T_p < T_e, \end{align*} where $t_0$ is the valuation date, $T_s$ is the Libor start date, $T_p$ is the payment date, and $T_e$ is the Libor end date. Let $\Delta_s^e = T_e-T_s$. For $0\le t \le T_s$, define \begin{align*} L(t, T_s, T_e) = \frac{1}{\Delta_s^e}\bigg(\frac{P(t, T_s)}{P(t, T_e)}-...

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

The chart you posted does not give a correct visual representaion of convexity . Convexity is not $\frac{\partial^2 P}{\partial y^2}$ but $\frac{1}{P}\frac{\partial^2 P}{\partial y^2}$. So you have to normalize for P. The 4 curves you plot have very different P. When the curves are redrawn normalized so they go through the same point $(y_0,P_0)$ you will ...

4

To directly answer the question: bond A= one day to maturity , price 100, yield 2%. Bond B: 10 years to maturity, price 100 yield 2%. This is perfectly possible. Bond B has greAter convexity but it also has substantially more risk.

4

Consider a date sequence \begin{align*} 0 \leq t_0 \leq T_s < T_e < T_p, \end{align*} where $t_0$ is the valuation date, $T_s$ is the Libor start date, $T_e$ is the Libor end date, and $T_p$ is the payment date. Let $\Delta_s^e = T_e-T_s$. For $0\le t \le T_s$, define \begin{align*} L^e(t, T_s, T_e) = \frac{1}{\Delta_s^e}\bigg(\frac{P(t, T_s)}{P(t, T_e)...

4

This has been posted a few times now, so I will invest the time on a full response. FRA / Futures convexity has nothing to do with profits/losses being immediately recognised on the future through margin settlement, whilst deferred on the FRA. Although this seems to be a very common belief amongst many practitioners it is not correct. Let me ...

4

No, and this is wrong. The implied vols (from market prices) are actually not necessarily convex but yet may be still arbitrage-free, there are many examples of this for various equities. Furthermore, preserving convexity is not necessarily enough either. In terms of implied variance $w(y)=\sigma^2 T$ as a function of log-moneyness $y=\ln\frac{K}{F}$, the no ...

4

I think it is far easier to understand by just drawing the payoffs. You have two put options: A European put option on a non-dividend paying stock with strike price 80 is priced at 8 dollars, and a put option on the same stock with strike price 90 dollars is priced at 9 dollar The difference between the two payoffs is equal to 10 dollars (90 strike puts ...

3

This seems to be a (short term, only 3 months) CMS swap. I wrote a paper about the different approaches to price them, available here. You can pick the one best fitted for your needs.

3

Well, you need to know what is the stochashtic model you are using for $y_T$, if you assume it's a geometric brownian motion you have this process : $y_T = y_0 e^{\sigma W_T - \frac{1}{2} \sigma^2T}$ If you compute the expectation and variance you get $\mathbb{E}(y_T) = y_0$ and $Var(y_T) = {y_0}^2( e^{\sigma^2 T }-1)$ As $y_0$ is constant you ...

3

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)...

3

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 ...

3

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 ...

3

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 ...

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. ...

3

Let $B_t= e^{\int_0^t r_sds}$ be the money-market account value at time $t$, and $P(t, T)$ be the value of the zero-coupon bond with maturity $T$ and unit face amount. Moreover, let $Q$ be the risk-neutral measure and $Q_T$ be the $T$-forward measure. If the interest rate $r_t$ is deterministic, then \begin{align*} P(t, T) &= E\left(e^{-\int_t^T r_s ds} \...

3

You can think of both ( difference and ratio ) indicators as some aggregated measure of difference between flat vol (ATM vol) and "total vol" than includes skew and kurtosis effects.

3

The other two answers do a good job of explaining, within the context of mathematical financial models, why a convexity adjustment is necessary, but I think a more tangible perspective can also be useful. Consider two forward rate agreements (FRA) to receive fixed and pay floating, with the same fixing date $T_s$ and end date $T_e$. The first pays on the ...

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 - ... 2 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 ... 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 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 2 You get a convexity adjustment from forward correlations only if you model separately the forwards and they are not perfectly correlated on the time interval$[0, T_1]$, as is the case in inflation market models where each forward CPI index is modelled separately from the others, with a global instantaneous correlation structure, not set to identity, similar ... 2 If you have done your simulation under the payment date forward measure then you only need to take the expectation of the indicator of the swap rate being between$K_1$and$K_2$. If you have done your simulation under the risk neutral measure (which is associated with the savings account as numeraire) then you take the expectation of the indicator of the ... 2 Short Futures and Long Puts are the main hedging strategies for hedging an equity p'folio against a drop, so it is natural that a proposed new technique, going Long a VarSwap, is being compared to the 2 traditional techniques. How do the 2 traditional techniques differ ? When you hedge an equity p'folio with short futures, you have a constant delta. As S ... 2 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 ...

2

The problem you are proposing has a non-convex feasible set, so you can't formulate it in a DCP-conforming way. To see this, consider a 2-element benchmark portfolio [1, 1]. You are optimizing over portfolios [x, y] such that |x-1| + |y-1| >= 1.6. Note that [2, 0] and [0, 2] are both feasible portfolios, but the convex combination 0.5[2, 0] + 0.5[0, 2] = [1, ...

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