In my previous post, I discussed option Implied Volatility and Binomial Model. In this post, we switch gears and discuss Swap Curve construction. Building a swap curve is so fundamental to interest rate derivatives pricing that it is probably one of most closely guarded proprietary information on the trading desk.

Pricing an interest rate swap involves bootstrapping a blended curve of different instruments based on their maturities and market liquidity. Usually cash deposits, Eurodollar futures are used at the short end and market swap rates are used at the long end. At the long end, however, we have only a subset of market swap rates available and bootstrapping requires all the missing rates to be interpolated from known rates. This makes interpolation methodology a critical part of curve building. Also, since forward rates are a gradient of the discount rates, any misalignment in the latter is magnified in the former.

There is an alternative approach to bootstrapping called global optimization approach, where the short end of the curve is bootstrapped as usual but at the longer end we “guess” the forward rates, compute the par rates of the market swaps and minimize the error between the actual par rates and the computed par rates. We also add a smoothness constraint to the minimization procedure so that overall gradient of the curve is minimized. This approach is illustrated in the excellent book Swaps and Other derivatives 2nd Ed by Richard Flavell

I will use QuantLib to generate swap schedules and to deal with business day conventions. QuantLib can of course generate a fully built swap curve but I will use Scipy’s optimize package for curve building. My objective was to match Spreadsheet 3.9 “Building a blended curve” from the above book. Unfortunately, the spreadsheet does not actually show the equations for Excel’s Solver used for optimization but shows the final result, so that leaves considerable ambiguity in understanding which I hope I will be able to clear.

##### Note on QuantLib and Python

*There are numerous resources online on how to build QuantLib from sources and then build the Python extensions, I would like to point you to the precompiled package for QuantLib-Python maintained by Christoph Gohlke. If you are on windows, you can just install the whl package and get started.*

First some common formulae we will be using:

$$Discount Factor : DF_t = \frac{1}{(1 + r_t * d_t)}$$ where $d_t$ is year fraction and $r_t$ is annual rate

$$Forward Rate : F_{t/T} = \frac{[(DF_t/DF_T)- 1]}{(T-t)}$$ where $DF_t$ is disocunt factor to t and $DF_T$ is disocunt factor to T (both from today)

$$Par Rate of Swap: \frac{(1-D_T)}{\sum_{n=1}^n(\Delta_n * D_n)}$$ where $D_T$ is maturity date discount factor, $\Delta_n$ are the time fractions between 2 reset dates and $D_n$ are the various reset date discount factors.

import QuantLib as ql import numpy as np from scipy.optimize import least_squares from scipy.interpolate import interp1d import math import timeit import matplotlib.pyplot as plt # we are trying to match Flavell's spreadsheet 3.9 today = ql.Date(4, ql.February,2008) calendar = ql.JointCalendar(ql.UnitedStates(), ql.UnitedKingdom()) ql.Settings.instance().evaluationDate = today settle_date = calendar.advance(today, 2, ql.Days)

#define a Swap class to keep state of various dates and year fractions class MySwap(object): def __init__(self, floating_dates, fixed_dates, fixed_rate): super(MySwap, self).__init__() self.fixed_rate = fixed_rate self.floating_dates = floating_dates self.fixed_dates = fixed_dates self.floating_leg_from_dates = floating_dates[:-1] self.floating_leg_to_dates = floating_dates[1:] self.fixed_leg_from_dates = fixed_dates[:-1] self.fixed_leg_to_dates = fixed_dates[1:] self.yf_floating_leg = [ql.Thirty360(ql.Thirty360.BondBasis).yearFraction(d1,d2) for d1,d2 in zip(self.floating_leg_from_dates,self.floating_leg_to_dates)] self.yf_fixed_leg = [ql.Thirty360(ql.Thirty360.BondBasis).yearFraction(d1, d2) for d1, d2 in zip(self.fixed_leg_from_dates, self.fixed_leg_to_dates)] # for sorting 2 swaps by maturity date def __gt__(self, swap2): return self.floating_leg_to_dates[-1] > swap2.floating_leg_to_dates[-1]

#class for Eurodollar futures class MyFuture(object): def __init__(self, begin_date, mkt_yield): super(MyFuture, self).__init__() self.begin_date = begin_date self.mkt_yield= mkt_yield

#class for Cash deposits class MyDeposit(object): def __init__(self, begin_date, end_date, mkt_yield): super(MyDeposit, self).__init__() self.begin_date = begin_date self.end_date = end_date self.mkt_yield= mkt_yield self.yf = ql.Thirty360(ql.Thirty360.BondBasis).yearFraction(begin_date, end_date) self.disc_factor = 1/(1+(self.yf*mkt_yield))

# A utility function to recursively calculate a series of discount factors from year fractions # and forward rates def get_disc_factors(yf, rates): df = [0] * len(yf) df[0] = 1.0 / (1.0 + (yf[0] * rates[0])) for i in range(1, len(yf)): df[i] = 1.0 / (1.0 + (yf[i] * rates[i])) * df[i - 1] return df

# this is the objective function which will be minimized using least squares method # we calculate the par swap rate from the "guessed" forward rates # and minimize the difference between this par swap rate and the market quoted par swap rate # the first 4 forward rates have been derived from deposit and Eurodollar futures and # are known #@profile def scipy_func_to_minimize(fwdGuesses): x = np.concatenate([known_fwds,fwdGuesses]) df_float_leg_longest_swap = get_disc_factors(swaps[-1].yf_floating_leg, x) def par_rate(yf,df): up = 1 - df[-1] down = np.sum([y*d for y,d in zip(yf,df)]) return up/down def swap_rate(swap): df = np.interp([d.serialNumber() for d in swap.fixed_leg_to_dates], [dl.serialNumber() for dl in swaps[-1].floating_leg_to_dates], df_float_leg_longest_swap) return par_rate(swap.yf_fixed_leg, df) fi = [swap_rate(s) - s.fixed_rate for s in swaps] sq = smoothness_func(x) fi.append(sq) return fi

# This function ensures a smooth forward curve by minimizing the gradient of the # forward rates # This function is added as an additional constraint to the above # objective function def smoothness_func(fwdGuesses): sq = 0.0 for i, f in enumerate(fwdGuesses[:-1]): sq += math.pow((fwdGuesses[i] - fwdGuesses[i + 1]), 2) return sq

# A utility function to define a vanilla IRS, specifically USD-3M-Libor # fixed-float swap def makeVanilla1Y3MSwap(maturity, fixedRate, index): end = calendar.advance(settle_date, maturity) fixedLegTenor = ql.Period("1y") fixedLegBDC = ql.ModifiedFollowing fixedLegDC = ql.Thirty360(ql.Thirty360.BondBasis) spread = 0.0 fixedSchedule = ql.Schedule(settle_date, end, fixedLegTenor, ql.TARGET(), fixedLegBDC, fixedLegBDC, ql.DateGeneration.Forward, False) floatSchedule = ql.Schedule(settle_date, end, index.tenor(), index.fixingCalendar(), index.businessDayConvention(), index.businessDayConvention(), ql.DateGeneration.Forward, False) swap = ql.VanillaSwap(ql.VanillaSwap.Payer, 1.0, fixedSchedule, fixedRate, fixedLegDC, floatSchedule, index, spread, index.dayCounter()) return swap

# Market data as of 2/4/2008 libor3m = ql.USDLibor(ql.Period(3, ql.Months)) deposits = [(calendar.advance(settle_date, ql.Period(1, ql.Weeks)),0.032175), (calendar.advance(settle_date, ql.Period(1, ql.Months)),0.031813), (calendar.advance(settle_date, ql.Period(3, ql.Months)),0.03145)] depos = [MyDeposit(settle_date, t[0], t[1]) for t in deposits] # ED Futures start dates : Mar-08,Jun-08,Sep-08,Dec-08,Mar-09 fQuotes = [97.000,97.410,97.520,97.495,97.395] futures = [] date_holder = settle_date for fz in fQuotes: fut_begin_date = ql.IMM_nextDate(date_holder) futures.append(MyFuture(fut_begin_date, (100.-fz)/100.)) date_holder = fut_begin_date ql_swaps = [makeVanilla1Y3MSwap(ql.Period(2, ql.Years),2.795e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(3, ql.Years),3.035e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(4, ql.Years),3.275e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(5, ql.Years),3.5050e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(6, ql.Years),3.715e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(7, ql.Years),3.885e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(8, ql.Years),4.025e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(9, ql.Years),4.155e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(10, ql.Years),4.265e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(12, ql.Years),4.435e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(15, ql.Years),4.615e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(20, ql.Years),4.755e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(25, ql.Years),4.805e-2,libor3m), makeVanilla1Y3MSwap(ql.Period(30, ql.Years),4.815e-2,libor3m)] swaps = [MySwap([d for d in x.floatingSchedule()], [d for d in x.fixedSchedule()], x.fixedRate()) for x in ql_swaps] swaps.sort()

#Starting to build fwd curve, add deposits first fwd_curve = [(dep[0], dep[1]) for dep in deposits] # get next 3 forward rates from futures rates by interpolating on the # provided rates fut_curve_dates = [calendar.advance(settle_date, ql.Period(3*i, ql.Months)) for i in range(1, 5)] fut_interpolation = interp1d([f.begin_date.serialNumber() for f in futures], [f.mkt_yield for f in futures]) fwd_curve.extend([(fut_curve_dates[idx], fut_interpolation(fut_curve_dates[idx-1].serialNumber())[()]) for idx, f in enumerate(fut_curve_dates) if idx > 0])

# At this point we know first 4 forward rates # and we guess the next 116 forwards to be same as the 4th forward rate known_fwds = [] for d in fut_curve_dates: known_fwds.extend([f[1] for f in fwd_curve if f[0] == d]) fwd_guesses = [known_fwds[-1]] * (len(swaps[-1].floating_leg_to_dates)-len(known_fwds))

# Now we proceed with optimization and get the optimized forwards # that provide the arbitrage-free and smooth forward curve start_time = timeit.default_timer() x = least_squares(scipy_func_to_minimize, fwd_guesses, verbose=1).x print('### Scipy optimization took:%s seconds' % str(timeit.default_timer() - start_time)) optimized_fwds = np.concatenate([known_fwds, x])

`gtol` termination condition is satisfied.

Function evaluations: 20, initial cost: 1.7812e-03, final cost 1.5585e-09, first-order optimality 1.04e-08.

### Scipy optimization took:2.5262434 seconds

#Start building a discount curve as a list of tuples (date,disc_factor), # first discount factor = 1.0 dates_df = [(settle_date, 1.0)] for idx, fwd in enumerate(optimized_fwds): temp_tuple = (swaps[-1].floating_leg_to_dates[idx], dates_df[idx][1]/(1 + (optimized_fwds[idx] * swaps[-1].yf_floating_leg[idx]))) dates_df.append(temp_tuple)

#Start building a zero curve zc_rates = [] cumulative_floating_yf = np.cumsum(swaps[-1].yf_floating_leg) cumulative_floating_yf = np.insert(cumulative_floating_yf, 0, 0.) entire_date_range = swaps[-1].floating_leg_from_dates + [swaps[-1].floating_leg_to_dates[-1]] zc_rates = [ (dt, -np.log(dc[1])/yf) for idx, (dc, yf, dt) in enumerate(zip(dates_df,cumulative_floating_yf, entire_date_range)) if idx > 0] zc_temp_tup = (swaps[-1].floating_leg_from_dates[0], zc_rates[0][1]) zc_rates.insert(0, zc_temp_tup)

#We already have forward rates, just put them in a list of tuples fwd_rates = [(d,fwd) for d, fwd in zip(swaps[-1].floating_leg_from_dates, optimized_fwds)]

# Plot all 3 curves _ = plt.figure(figsize=(10, 8.3)) _ = plt.subplot(221) x_axis = cumulative_floating_yf[1:] y_axis = list(f[1] for f in fwd_rates) _ = plt.plot(x_axis, y_axis) plt.grid() _ = plt.xlabel('Time') _ = plt.ylabel('Forward Rates') _ = plt.title('Forward Curve') _ = plt.subplot(222) x_axis = cumulative_floating_yf y_axis = list(d[1] for d in dates_df) _ = plt.plot(x_axis, y_axis) plt.grid() _ = plt.xlabel('Time') _ = plt.ylabel('Discount Factors') _ = plt.title('Discount Curve') _ = plt.subplot(223) x_axis = cumulative_floating_yf y_axis = list(z[1] for z in zc_rates) _ = plt.plot(x_axis, y_axis) plt.grid() _ = plt.xlabel('Time') _ = plt.ylabel('Zero Rates') _ = plt.title('Zero Curve') plt.show()