This post explains how to calculate delta sensitivities or delta vector of interest rate swap. 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.
Introduction
For detailed information about the Libor IRS swap pricing and zero curve bootstrapping, refer to the following posts.
- Interest Rate Swap Pricing using R code
- Bootstrapping the Zero Curve from IRS Swap Rates using R code
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 sensitivities or delta vector. 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 (xx is a risk factor). There are, of course, several versions of it but they are all essentially the same.

For ease of notation, let z(t) and s(t) denote the (bootstrapped) zero rates or zero curve and (market observed) swap rates or swap curve 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)),

Bumping up (z(t; ti + 0.5bp) or down (z(t; ti−0.5bp)), delta(ti) is calculated and this process is applied for all ti.

Here, ti, i = 1, 2,…, ni = 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 newly generated zero curve, we can calculate delta sensitivity at time ti as follows.

The following R code calculates delta sensitivities of IRS using these two approaches.
#=========================================================================# # Financial Econometrics & Derivatives, ML/DL using R, Python, Tensorflow # by Sang-Heon Lee # # https://kiandlee.blogspot.com #————————————————————————-# # Calculate Delta Sensitivities of Libor IRS #=========================================================================# graphics.off() # clear all graphs rm(list = ls()) # remove all files from your workspace #========================================================================= # Functions – Definition #========================================================================= #————————————————————– # Calculation of IRS swap price #————————————————————– f_zero_prr_IRS <– function( fixed_rate, # fixed rate vd.fixed_date, vd.float_date, # date for two legs vd.zero_date, v.zero_rate, # zero curve (dates, rates) d.spot_date, no_amt, # spot date, nominal amt save_cf_yn) { # “y” : CF save #———————————————————- # 0) Preprocessing #———————————————————- # convert spot date from date(d) to numeric(n) n.spot_date <– as.numeric(d.spot_date) # Interpolation of zero curve vn.zero_date <– as.numeric(vd.zero_date) f_linear <– approxfun(vn.zero_date, v.zero_rate, method=“linear”) vn.zero_date.inter <– n.spot_date:max(vn.zero_date) v.zero_rate.inter <– f_linear(vn.zero_date) # number of CFs ni <– length(vd.fixed_date) nj <– length(vd.float_date) # output data.frame with CF dates and its interpolated zero df.fixed = data.frame(d.date = vd.fixed_date, n.date = as.numeric(vd.fixed_date)) df.float = data.frame(d.date = vd.float_date, n.date = as.numeric(vd.float_date)) #———————————————————- # 1) Fixed Leg #———————————————————- # zero rate for discounting df.fixed$zero_DC = f_linear(as.numeric(df.fixed$d.date)) # discount factor df.fixed$DF <– exp(–df.fixed$zero_DC* (df.fixed$n.date–n.spot_date)/365) # tau, CF for(i in 1:ni) { ymd <– df.fixed$d.date[i] ymd_prev <– df.fixed$d.date[i–1] if(i==1) ymd_prev <– d.spot_date d <– as.numeric(strftime(ymd, format = “%d”)) m <– as.numeric(strftime(ymd, format = “%m”)) y <– as.numeric(strftime(ymd, format = “%Y”)) d_prev <– as.numeric(strftime(ymd_prev, format = “%d”)) m_prev <– as.numeric(strftime(ymd_prev, format = “%m”)) y_prev <– as.numeric(strftime(ymd_prev, format = “%Y”)) # 30I/360 tau <– (360*(y–y_prev) + 30*(m–m_prev) + (d–d_prev))/360 # cash flow rate df.fixed$rate[i] <– fixed_rate # Cash flow at time ti df.fixed$CF[i] <– fixed_rate*tau*no_amt # day fraction } # Present value of CF df.fixed$PV = df.fixed$CF*df.fixed$DF #———————————————————- # 2) Floating Leg #———————————————————- # zero rate for discounting df.float$zero_DC = f_linear(as.numeric(df.float$d.date)) # discount factor df.float$DF <– exp(–df.float$zero_DC* (df.float$n.date–n.spot_date)/365) # tau, forward rate, CF for(i in 1:nj) { date <– df.float$n.date[i] date_prev <– df.float$n.date[i–1] DF <– df.float$DF[i] DF_prev <– df.float$DF[i–1] if(i==1) { date_prev <– n.spot_date DF_prev <– 1 } # ACT/360 tau <– (date – date_prev)/360 # forward rate fwd_rate <– (1/tau)*(DF_prev/DF–1) # cash flow rate df.float$rate[i] <– fwd_rate # Cash flow amount at time ti df.float$CF[i] <– fwd_rate*tau*no_amt # day fraction } # Present value of CF df.float$PV = df.float$CF*df.float$DF # check for cash flows if (save_cf_yn == “y”) { # print(df.float); print(df.fixed) write.csv(df.float, “CF_float.csv”) write.csv(df.fixed, “CF_fixed.csv”) } return(sum(df.float$PV) – sum(df.fixed$PV)) } #————————————————————– # IRS swap zero curve generator #————————————————————– f_zero_maker_IRS <– function( df.mt, # market information data.frame # [d.date, swap_rate, source]] v.unknown_swap_maty_all, # all unknown swap maturity vd.fixed_date, # date for fixed leg vd.float_date, # date for float leg d.spot_date, # spot date no_amt) { # nominal principal amount # convert spot date from date(d) to numeric(n) n.spot_date <– as.numeric(d.spot_date) # for bootstrapped zero curve df.zr <– data.frame( d.date = df.mt$d.date, n.date = as.numeric(df.mt$d.date), tau = as.numeric(df.mt$d.date) – n.spot_date, taui = as.numeric(df.mt$d.date) – n.spot_date, swap_rate = df.mt$swap_rate, zero_rate = rep(0,length(df.mt$d.date)), DF = rep(0,length(df.mt$d.date))) # tau(i) = t(i) – t(i-1) df.zr$taui[2:nrow(df.zr)] <– df.zr$n.date[2:nrow(df.zr)] – df.# semi-annual date[1: (nrow(df.zr)–1)] # divide rows according to its source or instrument type rows_deposit <– which(df.mt$source==“deposit”) rows_futures <– which(df.mt$source==“futures”) rows_swap <– which(df.mt$source==“swap”) #————————————————————– # 3. Bootstrapping – Deposit #————————————————————– for(i in rows_deposit) { # 1) calculate discount factor for deposit df.zr$DF[i] <– 1/(1+df.zr$swap_rate[i]*df.zr$tau[i]/360) # 2) convert DF to spot rate df.zr$zero_rate[i] <– 365/df.zr$tau[i]*log(1/df.zr$DF[i]) } #————————————————————– # 4. Bootstrapping – Futures #————————————————————– # No convexity adjustment is made for(i in rows_futures) { # 1) discount factor from t(i-1) to t(i) df.zr$DF[i] <– 1/(1+df.zr$swap_rate[i]*df.zr$taui[i]/360) # 2) discount factor from spot date to t(i) df.zr$DF[i] <– df.zr$DF[i–1]*df.zr$DF[i] # 3) zero rate from discount factor df.zr$zero_rate[i] <– 365/df.zr$tau[i]*log(1/df.zr$DF[i]) } #————————————————————– # 5. Bootstrapping – Swaps #————————————————————– k <– 1 for(i in rows_swap) { # unknown swap maturity in year swap_maty <– v.unknown_swap_maty_all[k] # 1) find one unknown zero rate for one swap maturity m<–optim(0.01, objf, control = list(abstol=10^(–20), reltol=10^(–20), maxit=50000, trace=2), method = c(“Brent”), lower = 0, upper = 0.1, # for Brent v.unknown_swap_maty = swap_maty, # unknown zero maturity v.swap_rate = df.zr$swap_rate[i], # observed swap rate vd.fixed_date = vd.fixed_date, # date for fixed leg vd.float_date = vd.float_date, # date for float leg vd.zero_date_all = df.zr$d.date[1:i], # all dates for zero curve v.zero_rate_known = df.zr$zero_rate[1: (i–1)], # known zero rates d.spot_date = d.spot_date, no_amt = no_amt) # 2) update this zero curve with the newly found zero rate df.zr$zero_rate[i] <– m$par # 3) convert this new zero rate to discount factor df.zr$DF[i] <– exp(–df.zr$zero_rate[i]*df.zr$tau[i]/365) k <– k + 1 } return(df.zr) } #————————————————————– # objective function to be minimized #————————————————————– objf <– function( v.unknown_swap_zero_rate, # unknown zero curve (rates) v.unknown_swap_maty, # unknown swap maturity v.swap_rate, # fixed rate vd.fixed_date, # date for fixed leg vd.float_date, # date for float leg vd.zero_date_all, # all dates for zero curve v.zero_rate_known, # known zero curve (rates) d.spot_date, # spot date no_amt) { # nominal principal amount # zero curve augmented with zero rates for swaps v.zero_rate_all <– c(v.zero_rate_known, v.unknown_swap_zero_rate) v.swap_pr <– NULL # vector of swap prices k <– 1 for(i in v.unknown_swap_maty) { # calculate IRS swap price swap_pr <– f_zero_prr_IRS( v.swap_rate[k], # fixed rate, vd.fixed_date[1: (2*i)], # semi-annual date vd.float_date[1: (4*i)], # quarterly date vd.zero_date_all, # zero curve (dates) v.zero_rate_all, # zero curve (rates) d.spot_date, no_amt, “n”) # concatenate swap prices v.swap_pr <– c(v.swap_pr, swap_pr) k <– k + 1 } return(sum(v.swap_pr^2)) } #========================================================================= # Main #========================================================================= #————————————————————– # 1. Market Information #————————————————————– # Zero curve from Bloomberg as of 2021-06-30 until 5-year maturity df.mt <– data.frame( d.date = as.Date(c(“2021-10-04”,“2021-12-15”, “2022-03-16”,“2022-06-15”, “2022-09-21”,“2022-12-21”, “2023-03-15”,“2023-07-03”, “2024-07-02”,“2025-07-02”, “2026-07-02”)), # we use swap rate not zero rate. swap_rate= c(0.00145750000000000, 0.00139609870272047, 0.00203838571440434, 0.00197747863867587, 0.00266249271921742, 0.00359490949297661, 0.00512603194652204, 0.00328354999423027, 0.00571049988269806, 0.00793000012636185, 0.00964949995279312 ), source = c(“deposit”, rep(“futures”,6), rep(“swap”, 4)) ) #————————————————————– # 2. Libor Swap Specification #————————————————————– d.spot_date <– as.Date(“2021-07-02”) # spot date (date type) n.spot_date <– as.numeric(d.spot_date) # spot date (numeric type) no_amt <– 10000000 # notional principal amount # swap cash flow schedule from Bloomberg lt.cf_date <– list( fixed = as.Date(c(“2022-01-04”,“2022-07-05”, “2023-01-03”,“2023-07-03”, “2024-01-02”,“2024-07-02”, “2025-01-02”,“2025-07-02”, “2026-01-02”,“2026-07-02”)), float = as.Date(c(“2021-10-04”,“2022-01-04”, “2022-04-04”,“2022-07-05”, “2022-10-03”,“2023-01-03”, “2023-04-03”,“2023-07-03”, “2023-10-02”,“2024-01-02”, “2024-04-02”,“2024-07-02”, “2024-10-02”,“2025-01-02”, “2025-04-02”,“2025-07-02”, “2025-10-02”,“2026-01-02”, “2026-04-02”,“2026-07-02”)) ) #————————————————————– # 3. 5-year swap price : base #————————————————————– i = 5 # 5-year swap # zero pricing df.zr <– f_zero_maker_IRS( df.mt, c(2,3,4,5), lt.cf_date$fixed, lt.cf_date$float, d.spot_date, no_amt) pr <– f_zero_prr_IRS( df.mt$swap_rate[i+6], lt.cf_date$fixed[1: (2*i)], lt.cf_date$float[1: (4*i)], df.zr$d.date, df.zr$zero_rate, d.spot_date,no_amt, save_cf_yn = “y”) print(paste0(i,“-year Swap price at spot date = “, pr)) df.zr_delta <– df.mt_delta <– df.zr[,–c(2,3,4)] df.zr_delta$pr <– df.mt_delta$pr <– pr #————————————————————– # 3. Bump and Reprice for Market Greeks #————————————————————– df.mt_delta$delta <– df.mt_delta$pr_up <– df.mt_delta$pr_dn <– NA # iteration for all market maturities for(r in 1:11) { #——————— # bump up (1bp up) #——————— df.mt_bump <– df.mt # initialization df.mt_bump$swap_rate[r] <– df.mt_bump$swap_rate[r] + 0.0001 # zero pricing df.zr <– f_zero_maker_IRS(df.mt_bump, c(2,3,4,5), lt.cf_date$fixed, lt.cf_date$float, d.spot_date, no_amt) pr <– f_zero_prr_IRS(df.mt$swap_rate[i+6], lt.cf_date$fixed[1: (2*i)], lt.cf_date$float[1: (4*i)], df.zr$d.date, df.zr$zero_rate, d.spot_date, no_amt, “n”) # save price with bumping up df.mt_delta$pr_up[r] <– pr # check whether swap prices at spot date is at par pr <– f_zero_prr_IRS(df.mt_bump$swap_rate[i+6], lt.cf_date$fixed[1: (2*i)], lt.cf_date$float[1: (4*i)], df.zr$d.date, df.zr$zero_rate, d.spot_date,no_amt, “n”) print(paste0(i,“-year Swap price at spot date = “, pr)) #——————— # bump down (1bp down) #——————— df.mt_bump <– df.mt # initialization df.mt_bump$swap_rate[r] <– df.mt_bump$swap_rate[r] – 0.0001 # zero pricing df.zr <– f_zero_maker_IRS(df.mt_bump, c(2,3,4,5), lt.cf_date$fixed, lt.cf_date$float, d.spot_date, no_amt) pr <– f_zero_prr_IRS(df.mt$swap_rate[i+6], lt.cf_date$fixed[1: (2*i)], lt.cf_date$float[1: (4*i)], df.zr$d.date, df.zr$zero_rate, d.spot_date,no_amt, “n”) # save price with bumping down df.mt_delta$pr_dn[r] <– pr # check whether swap prict at spot date is at par pr <– f_zero_prr_IRS(df.mt_bump$swap_rate[i+6], lt.cf_date$fixed[1: (2*i)], lt.cf_date$float[1: (4*i)], df.zr$d.date, df.zr$zero_rate, d.spot_date,no_amt, “n”) print(paste0(i,“-year Swap price at spot date = “, pr)) } # Market Greeks : Delta calculation df.mt_delta$delta <– (df.mt_delta$pr_up – df.mt_delta$pr_dn)/2 df.mt_delta x11(width = 5, height = 3.5) barplot(delta ~ substr(d.date,1,7), data = df.mt_delta, width = 0.5, col = “blue”) x11(width = 5, height = 3.5) barplot(delta ~ substr(d.date,1,7), data = df.mt_delta[1:10,], width = 0.5, col = “green”) #————————————————————– # 4. Bump and Reprice for Zero Greeks #————————————————————– df.zr_delta$delta <– df.zr_delta$pr_up <– df.zr_delta$pr_dn <– NA # zero pricing df.zr <– f_zero_maker_IRS(df.mt, c(2,3,4,5), lt.cf_date$fixed, lt.cf_date$float, d.spot_date, no_amt) for(r in 1:11) { #——————— # bump up (1bp up) #——————— df.zr_bump <– df.zr # initialization df.zr_bump$zero_rate[r] <– df.zr_bump$zero_rate[r] + 0.0001 # zero pricing pr <– f_zero_prr_IRS(df.mt$swap_rate[i+6], lt.cf_date$fixed[1: (2*i)], lt.cf_date$float[1: (4*i)], df.zr_bump$d.date, df.zr_bump$zero_rate, d.spot_date, no_amt, “n”) # save price with bumping up df.zr_delta$pr_up[r] <– pr #——————— # bump down (1bp down) #——————— df.zr_bump <– df.zr # initialization df.zr_bump$zero_rate[r] <– df.zr_bump$zero_rate[r] – 0.0001 # zero pricing pr <– f_zero_prr_IRS(df.mt$swap_rate[i+6], lt.cf_date$fixed[1: (2*i)], lt.cf_date$float[1: (4*i)], df.zr_bump$d.date, df.zr_bump$zero_rate, d.spot_date,no_amt, “n”) # save price with bumping down df.zr_delta$pr_dn[r] <– pr } # Market Greeks : Delta calculation df.zr_delta$delta <– (df.zr_delta$pr_up – df.zr_delta$pr_dn)/2 df.zr_delta x11(width = 5, height = 3.5) barplot(delta ~ substr(d.date,1,7), data = df.zr_delta, width = 0.5, col = “blue”) x11(width = 5, height = 3.5) barplot(delta ~ substr(d.date,1,7), data = df.zr_delta[1:10,], width = 0.5, col = “green”) Colored by Color Scripter
Stay tuned for the next installment to learn about the output demonstrating zero delta vector along the maturities.
For additional insight on this topic and to download the R scripts, visit https://kiandlee.blogspot.com/2021/07/delta-sensitivity-of-interest-rate-swap.html.
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This material is from SHLee AI Financial Model and is being posted with its permission. The views expressed in this material are solely those of the author and/or SHLee AI Financial Model and Interactive Brokers is not endorsing or recommending any investment or trading discussed in the material. This material is not and should not be construed as an offer to buy or sell any security. It should not be construed as research or investment advice or a recommendation to buy, sell or hold any security or commodity. This material does not and is not intended to take into account the particular financial conditions, investment objectives or requirements of individual customers. Before acting on this material, you should consider whether it is suitable for your particular circumstances and, as necessary, seek professional advice.
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