Python-SciPy: Solving Zero-Coupon Term Structure Using Optimization

This post is presenting how to use Python SciPy Optimization package for solving out zero-coupon rate term structure from a given set of zero-coupon bond prices. Needless to say, we do not need any numerical method to do this, since we have exact analytical formulas for backing out zero-coupon rates from zero-coupon bond prices. However, approach presented in this relatively simple example can be applied into more interesting schemes, such as solving out smooth Libor forward curve from a given set of vanilla swaps. Example of such scheme using Solver and Excel/VBA can be found in here.

Let us assume zero-coupon rate term structure as shown below. Let us also assume, that the only information we can observe is the zero-coupon bond prices. Our task is to solve zero-coupon rates term structure.

In a nutshell, we first set initial guesses for ten zero-coupon rates. After this we minimize sum of squared differences between adjacent zero-coupon rates (in objective function), but subject to constraints. In this case, constraints are differences between observed zero-coupon bond market prices and calculated bond prices, which need to be pushed to zero. When this optimization task has been completed, the resulting zero-coupon term structure will successfully re-price the current zero-coupon bond market, while curve smoothness is maximized. Example program results are shown below.

Thanks for reading this blog.
-Mike

import numpy as np
import scipy.optimize as opt
import matplotlib.pyplot as pl

# sum of squared errors of decision variables
def ObjectiveFunction(x, args):
    return np.sum(np.power(np.diff(x), 2) * args[0])

# zero coupon bond pricing function
def ZeroCouponBond(x, args):
    # return difference between calculated and market price
    return ((1 / (1 + x[args[3]])**args[2]) - args[0]) * args[1]

# zero coupon bond pricing functions as constraints
# args: market price, scaling factor, maturity, index number for rate array
zeroPrices = ({'type': 'eq', 'fun': ZeroCouponBond, 'args': [[0.998801438274071, 1000000.0, 1.0, 0]]},
            {'type': 'eq', 'fun': ZeroCouponBond, 'args': [[0.996210802629012, 1000000.0, 2.0, 1]]},
            {'type': 'eq', 'fun': ZeroCouponBond, 'args': [[0.991943543964159, 1000000.0, 3.0, 2]]},
            {'type': 'eq', 'fun': ZeroCouponBond, 'args': [[0.981028206597786, 1000000.0, 4.0, 3]]},
            {'type': 'eq', 'fun': ZeroCouponBond, 'args': [[0.962851266220459, 1000000.0, 5.0, 4]]},
            {'type': 'eq', 'fun': ZeroCouponBond, 'args': [[0.946534719794057, 1000000.0, 6.0, 5]]},
            {'type': 'eq', 'fun': ZeroCouponBond, 'args': [[0.924997530805076, 1000000.0, 7.0, 6]]},
            {'type': 'eq', 'fun': ZeroCouponBond, 'args': [[0.912584111300984, 1000000.0, 8.0, 7]]},
            {'type': 'eq', 'fun': ZeroCouponBond, 'args': [[0.892632531026722, 1000000.0, 9.0, 8]]},
            {'type': 'eq', 'fun': ZeroCouponBond, 'args': [[0.877098137542374, 1000000.0, 10.0, 9]]})

# initial guesses for ten zero-coupon rates
initialGuess = np.full(10, 0.005)
model = opt.minimize(ObjectiveFunction, initialGuess, args = ([1000000.0]), method = 'SLSQP', constraints = zeroPrices)

# print selected model results
print('Success: ' + str(model.success))
print('Message: ' + str(model.message))
print('Number of iterations: ' + str(model.nit))
print('Objective function (sum of squared errors): ' + str(model.fun))
print('Changing variables (zero-coupon rates): ' + str(model.x * 100))
pl.plot(model.x)
pl.show()