7
The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate.
Let us start with the fair value of a swaption under the annuity measure $\mathcal{A}$ with tenor at time $\tau$:
$$\mathcal{A}(t)\mathbb{E}^\mathcal{A}_t[(\mathcal{S}(\tau)-k)^+]$$
Instead ...
7
The SABR model has an overly fat right tail. If you do the CMS replication using cash-settled swaptions you find that you need ridiculously high strikes.
5
A constant maturity swap (CMS) rate for a given tenor is referenced as a point on the Swap curve. A swap curve itself is a term structure wherein every point on the curve is the effective par swap rate for that tenor. This is analogous to a 3m LIBOR curve represents 3m forward rates for a given tenor.
A swap rate can be considered as a weighted-average of ...
4
A good place to start is Hagan's paper Convexity Conundrum ...available on the web.
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
In a vanilla swap, the IR on the floating leg usually depends on the reset period/swap frequency. If frequency is 6m, 6m LIBOR is used for reset, 3m LIBOR for quarterly resets etc. In a floating CMS leg, the rate used is the CMS rate, regardless of the reset frequency e.g: 10yr CMS leg will use the 10 yr CMS rate, regardless of whether the reset happens semi-...
3
In simple terms: An ordinary swap might be a 10 year swap of Libor vs a fixed rate; this fixed rate is determined in the marketplace every day and is published by Reuters, Bloomberg etc. as the '10 year swap rate'. Once you enter into the swap this rate remains fixed for you, of course, that is why it is called a fixed rate. But every day Reuters publishes a ...
3
The spread are quoted on ICAP or in Bloomberg if you have acess to them or you can refer to the paper of Mercurio where you have some quotes and examples
3
In a cash settled swaption the payoff is settled using the cash annuity contractually computed using the swap rate. Thus is you work out the replication procedure you will find that CMS replication is exact when you replicate on cash settled swaption (at least when $\delta=0$, that is for CMS with fixing in arrears), because Hagan's "street approximation" is ...
3
To lighten notation, we assume a constant accrual factor $\tau$, a swap rate $S_n(T)$ which fixes at $T$ and pays at $T_p$ (e.g. $T_p-T=\text{3 months}$) and a simple CMS payoff of the form:
$$\Phi(S_n(T))=(S_n(T)-K)$$
fixed at time $T=T_m$. We are interested in pricing under a measure for which the underlying risk factor of interest (i.e. the swap rate) is ...
2
From On Valuing Constant Maturity Swap Spread Derivatives
"The CMS tickers are represented as USSWAPyy, where yy is the year
indicator. For Example the tickers for CMS 30 yrs and CMS 2 yrs are
USSWAP30 and USSWAP02 respectively"
2
Let $X_i(t) = 1+\delta_i S_i(t)$. Then $\nu_i$ is the log normal volatility for $X_i(t)$, and because $dX_i(t) = \color{blue}{\delta_i} dS_i(t)$ we get $\nu_i X_i(t) dW_t = \color{blue}{\delta_i} \sigma_i \rho_{n,i} dW_t$ and $\nu_i \approx (\color{blue}{\delta_i} \sigma_i \rho_{n,i} ) / X_i(0)$.
2
Here is a quotation from an interbank broker last week: 1Y 2-10 Str 26-27. This means that a one year at the money straddle on the (10yr cms - 2 yr cms) has a spot price of 0.26% bid, 0.27% offered. So, they are quoted in price terms , not volatility.
To value the above option, most traders would use the volatility of 1y-2y and 1y-10y swaptions , and ...
2
Your intuition is correct and the paper seems to misunderstand the exposure of a swap based on CMS. The term "Constant Maturity Swap" or CMS, refers to the name of an index (the prevailing swap rate at the time of observation). A swap based on the CMS can be versus either a fixed rate or Libor. In the context of this question, consider a USD100mm 5yr swap ...
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
Let $S_t$ the swap-rate and $A_t$ the associated annuity.
You said that the convexity adjustment requires an annuity mapping function. That kind of approach is equivalent to calculate the following term $$E^A\left[G(S_T)\right]$$ where $G$ is the mapping (smooth) function.
One way to calculate that term would be to use a second-order Taylor expansion, and ...
1
A single look CMS spread option is simply an option on the difference between the two forward CMS rates and a chosen strike $K$ on a single expiry date $t$. A CMS spread cap is then a strip of options and pays on each $t_i$ from $t_1$ to $t_n$. Both types are quoted by brokers such as Tullet. Both products allow the investor a view on the shape of the yield ...
1
The cash flows on CMS leg reset based on some index, let's say the 5y swap rate. Duration measures sensitivity to different parts of the yield curve. The amount paid is based on that rate as of the last reset date, so essentially your duration should be close to 5 as your exposure is concentrated at the 5yr point on the yield curve (could be a bit higher/...
1
As far as I can tell, not being an expert in basis swap pricing, this is just algebra -
$$
X_{n,c} = \frac{\sum_{i=1}^n \left( S_{i,c}'(0) + {\rm\bf CA}(S_{i,c}'; \delta)\right) P(0,T_i')}{\sum_{i=1}^n P(0,T_i')} - \frac{1 - P(0,T_n')}{\delta \sum_{i=1}^n P(0,T_i')}
$$
which rearranges to
$$
\sum_{i=1}^n {\rm\bf CA}(S_{i,c}'; \delta) P(0,T_i') = - \sum_{...
1
The question is about the prices of CMS swaps, which are swaps where one side pays CMS and the other side pays Libor + X, where X is the price we are looking for. The payment frequency is usually quarterly or semiannually. These instruments aren't very liquid. As far as I know , historical prices are not available in any database. The best you can do is ...
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