This post explains how to calculate delta sensitivities or delta vector of interest rate swap, especially delta. delta can be calculated by either 1) zero delta or 2) market delta. To the best of our knowledge, FRTB can use these two methods but SIMM use the market Greeks. We implement R code for two approaches

Delta Sensitivity of LIBOR Interest Rate Swap

For detailed information about the Libor IRS swap pricing and zero curve bootstrapping, refer to the following posts.



In previous posts, we have priced a 5Y Libor IRS swap and generated a zero curve from market swap rates by using bootstrapping. Based on these works, we calculate Greeks of IRS. Since IRS does not have any option characteristics, our focus is to calculate the delta sensitivity. And for convenience, swap value is defined as (floating leg – fixed leg).

Delta Sensitivity

ISDA SIMM uses the following definitions of interest rate risk delta (\(x\) is a risk factor). There are, of course, several versions of it but they are all essentially the same.

\[\begin{align} \text{delta} &= V(x+0.5bp) – V(x-0.5bp) \\ \\ \text{delta} &= \frac{V(x+1bp) – V(x-1bp)}{2} \end{align}\]

For ease of notation, let \(z(t)\) and \(s(t)\) denote the (bootstrapped) zero rate and (market observed) swap rate at time t respectively.

There are two approaches for the calculation of delta: 1) zero delta, 2) market delta.

Zero Delta

Zero delta approach calculates delta sensitivities by bumping up or down zero rates one by one in order.

Once the zero curve (\(z(t)\)) is generated from market swap rates (\(s(t)\)), \[\begin{align} s(t) &= \{s(t_1), …, s(t_i), …, s(t_{ni})\} \\ z(t) &= Bootstrap(s(t)) \\ &= \{z(t_1), …, z(t_i), …, z(t_{ni})\} \end{align}\] Bumping up (\(z(t;t_i+0.5bp)\)) or down (\(z(t;t_i-0.5bp)\)), \(\text{delta}(t_i)\) is calculated and this process is applied for all \(t_i\). \[\begin{align} z(t;t_i+0.5bp) &= \{z(t_1), …, z(t_i)+0.5bp, …, z(t_{ni})\} \\ z(t;t_i-0.5bp) &= \{z(t_1), …, z(t_i)-0.5bp, …, z(t_{ni})\} \\ \text{delta}(t_i) &= V(z(t;t_i+0.5bp)) – V(z(t;t_i-0.5bp)) \end{align}\] Here, \(t_i\), \(i=1,2,…,n_i\) are maturities or dates of market swap rates at which the corresponding zero rates are bootstrapped.

Market Delta

Market delta approach calculates delta sensitivities by bumping up or down market swap rates one by one in order. Unlike the zero delta, every time we bump one market swap rate of a selected maturity, we should run a bootstrapping for finding new zero curve. Using this zero curve, we can calculate delta sensitivity at time \(t_i\) as follows. \[\begin{align} s(t;t_i+0.5bp) &= \{s(t_1), …, s(t_i)+0.5bp, …, s(t_{ni})\} \\ s(t;t_i-0.5bp) &= \{s(t_1), …, s(t_i)-0.5bp, …, s(t_{ni})\} \\ \\ z(t)^{up} &= Bootstrap(s(t;t_i+0.5bp)) \\ &= \{z(t_1)^{up}, …, z(t_i)^{up}, …, z(t_{ni})^{up}\} \\ z(t)^{down} &= Bootstrap(s(t;t_i-0.5bp)) \\ &= \{z(t_1)^{down}, …, z(t_i)^{down}, …, z(t_{ni})^{down}\} \\ \\ \text{delta}(t_i) &= V(z(t)^{up}) – V(z(t)^{down}) \end{align}\]

R code

The following R code calculates delta sensitivities of IRS using these two approaches.

Results

Zero Delta

The following figure and table show zero delta vector along the maturities. A meaningful value of delta is only observed at maturity since delta at maturities less than IRS maturity (3-year) is so small(10~30). But this pattern is not absolute and is subject to the change of market environment because these days shows ultra lower interest rates.

Market Delta

The following figure and table show market delta vector along the maturities. Like zero delta, a meaningful value of delta is only observed at maturity since delta at maturities less than IRS maturity (3-year) is considered a zero. Like zero delta, this pattern is also not absolute and is subject to the change of market environment from the same reason.

Intuition behind IRS delta

In both case of zero and market delta of IRS, we can observe the peak of delta at the IRS maturity. Increase in the interest rate has two effects. Firstly a higher interest rate decreases a discount factor and increases variable cash flows. These two effects have a trade-off.

Secondly, forward rates, which determine future cash flows show the following up and down pattern (we use 25bp up for the clear visual inspection and illustration) because the next swap rate is determined at market value, of which maturity is beyond the bumping maturity. Therefore there are positive and negative effects on future cash flows at the bumping time and the next time.


But at maturity, there is only positive effect on future cashflows because the successive negative effect takes place beyond the maturity of this IRS.

To be more specific, let’s compare the Aqua and Yellow colored line, which represent the forward rate curve with 2-year and 3-year swap rate bumping up respectively. We can observe that as the 2-year swap rate is bumped up, a downward movement of 2-year forward rate at time 2.25-year is following the upward movement of it at time 2-year. But in case of 3-year bumping, there is only upward movement of forward rate at time 3-year.

Conclusion

From this post, we have calculated delta sensitivities of IRS. In this example, two methods do not show some significant differences. But this result is not general because market Greeks permit interactions between market variables but zero Greeks does not (or little).

For example, in case of a Libor 3×6 basis swap, when Libor3M swap rates are changed, Libor6M zero curve is also changed. But there is little or no interaction effect in the case of zero Greeks. Therefore, it is advised for you to investigate the full effect of Greeks calculation in many cases. \(\blacksquare\)