convexity adjustment for pricing mark to market (mtm) cross currency swap

Question

may I know where the convexity adjustment is from and in practice, how is it usually calculated?

is it coming from the correlation between fx and rates ?

am I right that non-mtm cross currency swap in theory does not have this adjustment ?

Answer 1

First, we will write down the payoff of the mark to market basis cross currency swap. Second, we will do some exploring. Third, we hope that our exploration will be fruitful so that we can understand where we need to calculate the convexity adjustment.

The forward curves required are:

• Domestic LIBOR curve $L^\text{d}$, e.g., if the domestic currency is GBP, then this is the GBP LIBOR. Likewise, we also have the foreign LIBOR curve $L^\text{f}$.
• Domestic OIS curve $B^\text{d}$ and foreign OIS curve $B^\text{f}$.
• Domestic basis spread curve $s^\text{d}$, i.e., the real number $s^\text{d}$ such that $L^\text{d}=B^\text{d}+s^\text{d}$, we also have the foreign basis spread curve $s^\text{f}$, i.e., the real number $s^\text{f}$ such that $L^\text{f}=B^\text{f}+s^\text{f}$.

We also need the domestic notional $N^\text{d}$, the FX spot rate $X$ and a coupon $c$ added to the foreign LIBOR rate.

Let $t_\alpha$ be the first reset date and $t_\beta$ be the last payment date, where $\alpha, \beta \in \mathbf{N}$. The discounted cash flow of the mark to market cross currency swap (mtmxccy swap henceforth) will have the discounted cash flow at the first reset date $t_\alpha$ to be $$ \pi_{t_\alpha}^\text{f}=\sum_{i=\alpha+1}^{\beta} \left\lbrace N^\text{d} \left( L^\text{f}(t_{i-1},t_i) +c \right) \tau^\text{f}_i X(t_{i-1}) B^\text{f}(t_\alpha,t_i) \right\rbrace + \sum_{i=\alpha+1}^{\beta} N^\text{d} X(t_{i-1}) \left( B^\text{f}(t_\alpha,t_i) -B^\text{f}(t_\alpha,t_{i-1}) \right) $$ It is worth examining this payoff for a second or two - I recommend setting $\alpha=0,\beta=1$, i.e., the single coupon case, to understand what the cash-flow of the foreign leg is.

A mtmxccy swap will have the PV of the foreign leg at time zero to be $$ \begin{align} \pi_{0}^{\text{f}} & = \mathbb{E}^{ \mathbb{Q}^\text{d} }_{0} \left[ \pi_{t_\alpha}^\text{f} \right] \\ & = N^\text{d} \sum_{i=\alpha+1}^{\beta} \mathbb{E}^{ \mathbb{Q}^\text{d} }_{0} \left[ \left\lbrace \left[ \left( L^\text{f}(t_{i-1},t_i) + c \right) \tau_i^\text{f} +1 \right] B^\text{f}(t_{i-1},t_i) -1 \right\rbrace B^\text{f}(0,t_i) X(t_{i-1}) \right] \\ & = N^\text{d} X(0) \sum_{i=\alpha+1}^{\beta} P^{\text{d}}(0,t_{i-1}) \mathbb{E}^{ \mathbb{Q}^\text{d},t_{i-1} }_{0} \left[ \left\lbrace \left[ \left( L^\text{f}(t_{i-1},t_i) + c \right) \tau^\text{f}_i + 1 \right] B^\text{f}(t_{i-1},t_i)-1 \right\rbrace \right] \\ & = \text{some algebra ...} \\ & = N^\text{d} X(0) \sum_{i=\alpha+1}^{\beta} P^{\text{d}}(0,t_{i-1}) \left[ s^\text{f}(t_{i-1},t_i) + c \right] \mathbb{E}^{ \mathbb{Q}^\text{d},t_{i} }_{0} \left[ B^\text{f} (t_{i-1},t_i) \right] \end{align} $$

I did not show the steps for the "some algebra" part for two reasons - the first, you will need to use the fact that $L^\text{f}=s^\text{f}+B^\text{f}$, which can be written more explicitly as $$ L^\text{f}(t_1,t_2) = s^\text{f}(t_1,t_2)+B^\text{f}(t_1,t_2) = s^\text{f}(t_1,t_2) + \frac{1}{\tau^\text{f}_i} \left[ \frac{1}{B^\text{f}(t_1,t_2)} -1 \right], $$ and secondly, far more importantly, I am extremely lazy. It is good to do some work yourself to verify I have not made a typo.

Now, where does the convexity adjustment come in? The term $\mathbb{E}^{ \mathbb{Q}^\text{d},t_{i} }_{0} \left[ B^\text{f} (t_{i-1},t_i) \right]$ requires a convexity adjustment because the expectation is taken under the domestic measure, but the bond under consideration is naturally expressed in the foreign measure. So you need to switch from the foreign forward measure to the domestic forward measure.

(The domestic leg is even trickier as a time adjustment is also required, but let us leave that for a different day)

It is at this point that you need to specify a model for pricing the bond $B$ - the Vasicek model (or the Hull-White model) usually does the job. You need to introduce the following parameters

1. Domestic bond volatility function
2. Foreign bond volatility function
3. FX spot rate volatility function
4. Correlation between domestic and foreign bond, $\rho^{\text{d,f}}$
5. Correlation FX spot rate and foreign bond, $\rho^{\text{X,f}}$.

1 and 2 are not market quoted - but they can be recovered from the caplet volatility surface, which is market quoted.

Answer 2

This answer may not be complete - there may be a more nuanced and/or other effect that causes another form of convexity adjustment.

The cross-currency swap (xcs) market has a liquid market centered about mtm-xcs. In general practice market pricing is dictated by the prices of the most liquid products, or to put it another way, a convexity adjustment is a pricing adjustment to account for factors that might be generated by using liquid market hedges but not capturing the full extent of market risks.

Non-mtm-xcs are often traded as a result of cross-border corporate issuance and the corporate swaps the cashflows back to native currency. Dealers will hedge with mtm-xcs. In the event the market changes and FX rates moves this will create cashflows which no longer align. While not necessarily creating any xcs basis risk or market delta risk it will create single currency basis risk, which needs to be hedged.

Whilst not directional this adds to hedging requirements and is therefore an unwanted characteristic of the hedge. Therefore a charge (or convexity adj) is added to reflect this and will be proportional to the expected volatility of FX rates.

mtm-xcs hedged by mtm-xcs will not have any extra considerations.