Interest rate swaps (IRS) is a contract between two parties to trade cash flows corresponding to different interest rates (Fixed-to-Floating, Floating-to-Fixed, Floating-to- Floating.). Where LIBOR is commonly used as the Floating rate. The following formula gives the value for the swap also known as Net Present Value (NPV) below, where N is the notion amount, RTi-1,Ti is the rate we receive for the period Ti-1 to Ti, there is n periods in total, P(Tj-1,Tj) is the rate we pay for the period Tj-1 to Tj, there is m period in total, period n and m are the time where a payment or payout occurs which is predetermined in the contract. D(t,T) and L(t,T)refer to the discount rate and LIBOR for time t to time T respectively.

The formula can be understood as the difference in the sum of the present value of the receiving rate to the sum of the paying rate in each period, multiplied by the notional amount.

To build a yield curve directly from the market risk free instruments, we first collect numbers of risk free instruments with different maturities. Afterward, the market rate of short term (for example, 1 week, 1 month) instruments can be used directly, for longer maturity we need to bootstrap the corresponding yield. Finally, the yield rate we need can be obtained by linear interpolation.

When trading with an IRS, to avoid the credit risk, where either side fails to make the payment, the margin is usually required, in case any party fails to fulfill its obligation, counterparties can recover part of the loss. If the IRS is traded in an exchange, the margin requirement will follow the exchange’s requirement, for example the London Clearing House apply the LCH’s PAIRS (Portfolio Approach to Interest Rate Scenarios) approach to determine the loss distribution with 10 years of historical data and add-ons for the liquidity and credit risk. For the OTC market, the ISDA Standard Initial Margin Model (SIMM) is commonly used due to its relatively simplified calculation method, SIMM based on 10 days 99% parametric VaR.

SIMM Initial Margin Calculation

The fixed-to-floating IRS initial margin calculation procedure, following ISDA SIMM Methodology, is as follows:

1. To begin with, risk factors of the product have to be identified, in the IRS case the only risk factor is the interest rate. Under the SIMM, the risk factor is defined as r(k,i), with tenor k (2 weeks, 1 month, 3 months, 6 months, 1 year, 2 years, 3 years, 5 years, 10 years, 15 years, 20 years and 30 years) and sub curve i(“OIS”, “Libor1m”, “Libor3m”, “Libor6m”, “Libor12m” ), the risk factor can be approximated with the forward curve.

2. Then, product sensitivity to each risk factor should be determined by the price difference after adding 1 bp to each risk factor separately.

3. Afterward, a weight is assigned to each sensitivity according to the following formula:

WS_{k,i} = RW_{k}s_{k,i}CR_{b}

where RW_{k} is the risk weight according to the product’s maturity set by ISDA where the exact values are listed in appendix 5, s_{k,i} calculated in (2), CR_{b} represents the concentration risk factor which is 1 given normal size of portfolio, see appendix 6 for more detail.

4. Lastly, the sensitivity is aggregated with the following formula:

K^2 = \sum WS_{k,i}^2 + \sum_{i,k}\sum_{j,l}Φ_{i,j}ρ_{k,l}WS_{k,i}WS_{l.j}

where ρk,l is the correlation between tenor predetermined by ISDA and is list in appendix 7, Φi,j represent the correlation between sub curve and is usually 98.6%.

5. K is the initial margin for fixed-to-floating IRS

Margin for the Selected IRS

The margin requirement for our selected IRS can be calculated with the above SIMM, using the yield curve we build earlier as a synthetic instrument to match the SIMM risk factors, we then increase the risk factor with tenor (3,6m and 1,2,3,5yr market rate) and sub yield curve Libor3m by 1bp, one rate at a time and rebootstrap a new yield curve to calculate a new NPV, the difference between the new NPV and the Original NPV is the Delta (Sensitivity) of each risk factor, following the SIMM we can then calculate the weighted sensitivity and aggregated them to get the initial margin for the IRS $ 241,304.93 .

In this session, we use Monte Carlo (MC) simulation and LIBOR Market Model to simulate the random forward rates and thus the VaR and the expected shortfall.

Libor Market Model (LMM)

We follow Hull (2019) for implementing the one factor LMM. In our case, the swap reset quarterly and has a 5-year maturity. Therefore, we set t_k = 0.25k be the reset dates for k = 0, 1, 2, ..., 20. Time difference between the reset dates \delta = t_k and t_{k+1}, i.e. 0.25. See below picture for model specification.

Model Calibration

Since we already have the forward curve, the F_i(t0) are known. The only unknown parameters left is the forward rate volatility. In theory, they can be calibrated through the caplets’ spot volatilities. Define k be the Black volatility of the caplet between period t_k and t_{k+1}, we have the following relationship

\sigma^2_k(t_k) = \sum \Lambda_{k-i}^2\delta.

In practice, i is obtained by minimizing the square error of the flat volatility, which is the implied volatility (IV) of caps quoted by the broker. To calibrate, we first obtain the IV of ATM caps from Bloomberg and compute the caps’ prices through Black formula. Then, we use the caps’ prices recursively from short maturity as well as the zero curve and forward curve to obtain the IV. We report the calibrated result in the following table.

Simulation Result

To simulate 10-day VaR and expected shortfall, we first simulate the forward curves using the LMM. Then we restrip the zero curve and compute the swap new NPV. Thus, we obtain the 10-day return. We repeat it 1000000 times. The VaR and expected shortfall (ES) are computed by the sample quantile of the losses and the sample mean conditional on the losses larger or equal to the estimated VaR respectively. The simulation code and the required data are provided in VaR_sim.R and LMM_data.csv respectively. We further report the simulation result in the following.

10-day 95% VaR $32,835

10-day 99% VaR $47,090

10-day 95% ES $41,570

10-day 99% ES $54,296