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I am looking at the valuation of an Interest Rate Swap (IRS thereafter) which is pretty much vanilla with one small tweak. Floating leg pays 3 months LIBOR in monthly intervals. To be precise: floating leg resets every month, and the 3M LIBOR prevailing at the reset date is paid out at the end of the monthly interval. Payment is of course scaled to 1 month period (multiplied by year fraction equivalent to this monthly period). I feel that I should use convexity adjustment similarly as in the case of the in arrears IRS (but the adjustment will be different this time around). Can anyone guide me to the appropriate convexity adjustment for this case?

Maturity: 5 years. Floating leg: monthly payments based on the 3M LIBOR prevailing on the reset date (reset dates occur monthly 2 business days before the start of each monthly coupon period). Fixed leg: annual fixed payments.

The closest case that I've found was in great Brigo and Mercurio book "Interest Rate Models - Theory and Practice" 13.8.5 page 566 "Forward Rates Resettung Unnaturally and Average-Rate Swap". However Brigo and Mercurio discuss contract that pays after natural payment date (and in my case it is before natural paymen date).

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  • $\begingroup$ You should add the link to the reference paper you cite. $\endgroup$ – SRKX Dec 4 '16 at 14:30
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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)}-1 \bigg), \end{align*} where $P(t, \mu)$ is the price at time $t$ of a zero-coupon bond with maturity $\mu$ and unit face value.

Let $Q^{T_e}$ denote the $T_e$-forward measure and $E^{T_e}$ denote the corresponding expectation operator. Moreover, let $Q^{T_p}$ denote the $T_p$-forward measure and $E^{T_p}$ denote the corresponding expectation operator. We seek to compute the value defined by \begin{align*} P(t_0, T_p)E^{T_p}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big). \end{align*} Note that, for $0 \le t \le T_p$, \begin{align*} \eta_t \equiv \frac{dQ^{T_p}}{dQ^{T_e}}\big|_{t} = \frac{P(t, T_p)P(0, T_e)}{P(0, T_p)P(t, T_e)}. \end{align*} Therefore, \begin{align*} &\ P(t_0, T_p)E^{T_p}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big) \\ =&\ P(t_0, T_p)E^{T_e}\Big(\frac{\eta_{T_p}}{\eta_{t_0}}L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_p)E^{T_e}\Big(\frac{P(t_0, T_e)}{P(t_0, T_p)P(T_p, T_e)} L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_e)E^{T_e}\Big(\frac{1}{P(T_p, T_e)} L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_e)E^{T_e}\Big(\big(1+ \Delta_p^e L(T_p, T_p, T_e) \big) L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_e)E^{T_e}\Big(\big(1+ \Delta_p^e L(T_s, T_p, T_e) \big) L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big),\tag{1} \end{align*} where $\Delta_p^e = T_e-T_p$.

We assume that under the $T_e$-forward measure $Q^{T_e}$, \begin{align*} dL(t, T_s, T_e) &= \sigma_s L(t, T_s, T_e) d W_t^s,\\ dL(t, T_p, T_e) &= \sigma_p L(t, T_p, T_e)d\Big(\rho W_t^s + \sqrt{1-\rho^2}W_t^p\Big), \end{align*} where $\sigma_s$, $\sigma_p$, and $\rho$ are some constants, and $\{W_t^s, 0 \le t \le T_s\}$ and $\{W_t^p, 0 \le t \le T_s\}$ are two independent standard Brownian motions. Then, \begin{align*} &\ L(T_s, T_p, T_e) L(T_s, T_s, T_e) \\ =&\ L(t_0, T_p, T_e) L(t_0, T_s, T_e) e^{-\frac{\sigma_s^2}{2}(T_s-t_0) -\frac{\sigma_p^2}{2}(T_s-t_0) + \sigma_s\big(W_{T_s}^s -W_{t_0}^s\big) + \sigma_p\Big(\rho \big(W_{T_s}^s - W_{t_0}^s\big) + \sqrt{1-\rho^2}\big(W_{T_s}^p - W_{t_0}^p\big)\Big)}. \end{align*} Moreover, \begin{align*} E^{T_e}\big(L(T_s, T_p, T_e) L(T_s, T_s, T_e) \big) = L(t_0, T_p, T_e) L(t_0, T_s, T_e)e^{\rho \sigma_s\sigma_p (T_s-t_0)}. \end{align*} From $(1)$ above, \begin{align*} &\ P(t_0, T_p)E^{T_p}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big) \\ =&\ P(t_0, T_e)E^{T_e}\Big(\big(1+ \Delta_p^e L(T_s, T_p, T_e) \big) L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_e)\left(L(t_0, T_s, T_e) + \Delta_p^e L(t_0, T_p, T_e) L(t_0, T_s, T_e)e^{\rho \sigma_s\sigma_p (T_s-t_0)}\right)\\ =&\ P(t_0, T_e)L(t_0, T_s, T_e)\left(1 + \Delta_p^e L(t_0, T_p, T_e) e^{\rho \sigma_s\sigma_p (T_s-t_0)}\right)\\ =&\ P(t_0, T_p)L(t_0, T_s, T_e)\frac{1 + \Delta_p^e L(t_0, T_p, T_e) e^{\rho \sigma_s\sigma_p (T_s-t_0)}}{1+\Delta_p^e L(t_0, T_p, T_e)}\\ =&\ P(t_0, T_p)\left[L(t_0, T_s, T_e) + \frac{\Delta_p^eL(t_0, T_s, T_e)L(t_0, T_p, T_e)\left(e^{\rho \sigma_s\sigma_p (T_s-t_0)}-1 \right)}{1+\Delta_p^e L(t_0, T_p, T_e)}\right]. \end{align*} Here, the term \begin{align*} \frac{\Delta_p^eL(t_0, T_s, T_e)L(t_0, T_p, T_e)\left(e^{\rho \sigma_s\sigma_p (T_s-t_0)}-1 \right)}{1+\Delta_p^e L(t_0, T_p, T_e)} \end{align*} is the convexity adjustment. In practice, you may assume that $\rho\approx 1$ and $\sigma_p\approx\sigma_s$.

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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.

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  • $\begingroup$ I've expanded contract description in my post. It seems that what you discussed in your referenced paper differs slightly from my case. $\endgroup$ – jakub Sep 5 '16 at 8:23
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What you need is the convexity adjustment for 3 month libor when the payment is made 1 month after the reset date (ie 2 months before the natural date). As an approximation, this will be approximately 2/3 of the convexity adjustment for an arrears swap (paid 3 months before the natural date) and it will be approximately 4/3 of the convexity adjustment for an averaging swap (paid an average of 1.5 months before the natural date).

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