In this post we do a deep dive on calibration of Heston model using QuantLib Python and Scipy's Optimize package.

`CalibratedModel`

such as `calibrationError`

that we will use in calibrating models using Scipy. QuantLib's strength is all financial models. Scipy's strength is all the solvers and numerical methods. So here, I will show you how you can make the best of both worlds. We will start as usual by importing the modules.

In [1]:

```
import QuantLib as ql
from math import pow, sqrt
import numpy as np
from scipy.optimize import root
```

Let's construct some of the basic dependencies such as the yield and dividend term structures.

In [2]:

```
day_count = ql.Actual365Fixed()
calendar = ql.UnitedStates()
calculation_date = ql.Date(6, 11, 2015)
spot = 659.37
ql.Settings.instance().evaluationDate = calculation_date
risk_free_rate = 0.01
dividend_rate = 0.0
yield_ts = ql.YieldTermStructureHandle(
ql.FlatForward(calculation_date, risk_free_rate, day_count))
dividend_ts = ql.YieldTermStructureHandle(
ql.FlatForward(calculation_date, dividend_rate, day_count))
```

Following is a sample grid of volatilities for different expiration and strikes.

In [3]:

```
expiration_dates = [ql.Date(6,12,2015), ql.Date(6,1,2016), ql.Date(6,2,2016),
ql.Date(6,3,2016), ql.Date(6,4,2016), ql.Date(6,5,2016),
ql.Date(6,6,2016), ql.Date(6,7,2016), ql.Date(6,8,2016),
ql.Date(6,9,2016), ql.Date(6,10,2016), ql.Date(6,11,2016),
ql.Date(6,12,2016), ql.Date(6,1,2017), ql.Date(6,2,2017),
ql.Date(6,3,2017), ql.Date(6,4,2017), ql.Date(6,5,2017),
ql.Date(6,6,2017), ql.Date(6,7,2017), ql.Date(6,8,2017),
ql.Date(6,9,2017), ql.Date(6,10,2017), ql.Date(6,11,2017)]
strikes = [527.50, 560.46, 593.43, 626.40, 659.37, 692.34, 725.31, 758.28]
data = [
[0.37819, 0.34177, 0.30394, 0.27832, 0.26453, 0.25916, 0.25941, 0.26127],
[0.3445, 0.31769, 0.2933, 0.27614, 0.26575, 0.25729, 0.25228, 0.25202],
[0.37419, 0.35372, 0.33729, 0.32492, 0.31601, 0.30883, 0.30036, 0.29568],
[0.37498, 0.35847, 0.34475, 0.33399, 0.32715, 0.31943, 0.31098, 0.30506],
[0.35941, 0.34516, 0.33296, 0.32275, 0.31867, 0.30969, 0.30239, 0.29631],
[0.35521, 0.34242, 0.33154, 0.3219, 0.31948, 0.31096, 0.30424, 0.2984],
[0.35442, 0.34267, 0.33288, 0.32374, 0.32245, 0.31474, 0.30838, 0.30283],
[0.35384, 0.34286, 0.33386, 0.32507, 0.3246, 0.31745, 0.31135, 0.306],
[0.35338, 0.343, 0.33464, 0.32614, 0.3263, 0.31961, 0.31371, 0.30852],
[0.35301, 0.34312, 0.33526, 0.32698, 0.32766, 0.32132, 0.31558, 0.31052],
[0.35272, 0.34322, 0.33574, 0.32765, 0.32873, 0.32267, 0.31705, 0.31209],
[0.35246, 0.3433, 0.33617, 0.32822, 0.32965, 0.32383, 0.31831, 0.31344],
[0.35226, 0.34336, 0.33651, 0.32869, 0.3304, 0.32477, 0.31934, 0.31453],
[0.35207, 0.34342, 0.33681, 0.32911, 0.33106, 0.32561, 0.32025, 0.3155],
[0.35171, 0.34327, 0.33679, 0.32931, 0.3319, 0.32665, 0.32139, 0.31675],
[0.35128, 0.343, 0.33658, 0.32937, 0.33276, 0.32769, 0.32255, 0.31802],
[0.35086, 0.34274, 0.33637, 0.32943, 0.3336, 0.32872, 0.32368, 0.31927],
[0.35049, 0.34252, 0.33618, 0.32948, 0.33432, 0.32959, 0.32465, 0.32034],
[0.35016, 0.34231, 0.33602, 0.32953, 0.33498, 0.3304, 0.32554, 0.32132],
[0.34986, 0.34213, 0.33587, 0.32957, 0.33556, 0.3311, 0.32631, 0.32217],
[0.34959, 0.34196, 0.33573, 0.32961, 0.3361, 0.33176, 0.32704, 0.32296],
[0.34934, 0.34181, 0.33561, 0.32964, 0.33658, 0.33235, 0.32769, 0.32368],
[0.34912, 0.34167, 0.3355, 0.32967, 0.33701, 0.33288, 0.32827, 0.32432],
[0.34891, 0.34154, 0.33539, 0.3297, 0.33742, 0.33337, 0.32881, 0.32492]]
```

`setup_helpers`

will construct the Heston model helpers and returns an array of these objects. The `cost_function_generator`

is a method to set the cost function and will be used by the Scipy modules. The `calibration_report`

lets us evaluate the quality of the fit. The `setup_model`

method initializes the `HestonModel`

and the `AnalyticHestonEngine`

prior to calibration.

In [4]:

```
def setup_helpers(engine, expiration_dates, strikes,
data, ref_date, spot, yield_ts,
dividend_ts):
heston_helpers = []
grid_data = []
for i, date in enumerate(expiration_dates):
for j, s in enumerate(strikes):
t = (date - ref_date )
p = ql.Period(t, ql.Days)
vols = data[i][j]
helper = ql.HestonModelHelper(
p, calendar, spot, s,
ql.QuoteHandle(ql.SimpleQuote(vols)),
yield_ts, dividend_ts)
helper.setPricingEngine(engine)
heston_helpers.append(helper)
grid_data.append((date, s))
return heston_helpers, grid_data
def cost_function_generator(model, helpers,norm=False):
def cost_function(params):
params_ = ql.Array(list(params))
model.setParams(params_)
error = [h.calibrationError() for h in helpers]
if norm:
return np.sqrt(np.sum(np.abs(error)))
else:
return error
return cost_function
def calibration_report(helpers, grid_data, detailed=False):
avg = 0.0
if detailed:
print "%15s %25s %15s %15s %20s" % (
"Strikes", "Expiry", "Market Value",
"Model Value", "Relative Error (%)")
print "="*100
for i, opt in enumerate(helpers):
err = (opt.modelValue()/opt.marketValue() - 1.0)
date,strike = grid_data[i]
if detailed:
print "%15.2f %25s %14.5f %15.5f %20.7f " % (
strike, str(date), opt.marketValue(),
opt.modelValue(),
100.0*(opt.modelValue()/opt.marketValue() - 1.0))
avg += abs(err)
avg = avg*100.0/len(helpers)
if detailed: print "-"*100
summary = "Average Abs Error (%%) : %5.9f" % (avg)
print summary
return avg
def setup_model(_yield_ts, _dividend_ts, _spot,
init_condition=(0.02,0.2,0.5,0.1,0.01)):
theta, kappa, sigma, rho, v0 = init_condition
process = ql.HestonProcess(_yield_ts, _dividend_ts,
ql.QuoteHandle(ql.SimpleQuote(_spot)),
v0, kappa, theta, sigma, rho)
model = ql.HestonModel(process)
engine = ql.AnalyticHestonEngine(model)
return model, engine
summary= []
```

Solvers such as Levenberg-Marquardt find local minimas and are very sensitive to the initial conditions. Depending on the starting conditions for your solver, you could end up with a good set of parameters with good convergence or not so good set of parameters. We will look at two initial conditions for different solvers and see how the local minima solvers perform. We will compare this with differential evolution that looks for global minima.

We will setup the Heston model with two different initial conditions:

```
- theta, kappa, sigma, rho, v0 = (0.02, 0.2, 0.5, 0.1, 0.01)
- theta, kappa, sigma, rho, v0 = (0.07, 0.5, 0.1, 0.1, 0.1)
```

`theta, kappa, sigma, rho, v0 = (0.02,0.2,0.5,0.1,0.01)`

In [5]:

```
model1, engine1 = setup_model(
yield_ts, dividend_ts, spot,
init_condition=(0.02,0.2,0.5,0.1,0.01))
heston_helpers1, grid_data1 = setup_helpers(
engine1, expiration_dates, strikes, data,
calculation_date, spot, yield_ts, dividend_ts
)
initial_condition = list(model1.params())
```

In [6]:

```
%%time
lm = ql.LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model1.calibrate(heston_helpers1, lm,
ql.EndCriteria(500, 300, 1.0e-8,1.0e-8, 1.0e-8))
theta, kappa, sigma, rho, v0 = model1.params()
print "theta = %f, kappa = %f, sigma = %f, rho = %f, v0 = %f" % \
(theta, kappa, sigma, rho, v0)
error = calibration_report(heston_helpers1, grid_data1)
summary.append(["QL LM1", error] + list(model1.params()))
```

`theta, kappa, sigma, rho, v0 = (0.07,0.5,0.1,0.1,0.1)`

In [7]:

```
model1, engine1 = setup_model(
yield_ts, dividend_ts, spot,
init_condition=(0.07,0.5,0.1,0.1,0.1))
heston_helpers1, grid_data1 = setup_helpers(
engine1, expiration_dates, strikes, data,
calculation_date, spot, yield_ts, dividend_ts
)
initial_condition = list(model1.params())
```

In [8]:

```
%%time
lm = ql.LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model1.calibrate(heston_helpers1, lm,
ql.EndCriteria(500, 300, 1.0e-8,1.0e-8, 1.0e-8))
theta, kappa, sigma, rho, v0 = model1.params()
print "theta = %f, kappa = %f, sigma = %f, rho = %f, v0 = %f" % \
(theta, kappa, sigma, rho, v0)
error = calibration_report(heston_helpers1, grid_data1)
summary.append(["QL LM2", error] + list(model1.params()))
```

We see that the solver produces a 11% average of absolute error. This is not particularly great.

In [9]:

```
model2, engine2 = setup_model(
yield_ts, dividend_ts, spot,
init_condition=(0.02,0.2,0.5,0.1,0.01))
heston_helpers2, grid_data2 = setup_helpers(
engine2, expiration_dates, strikes, data,
calculation_date, spot, yield_ts, dividend_ts
)
initial_condition = list(model2.params())
```

In [10]:

```
%%time
cost_function = cost_function_generator(model2, heston_helpers2)
sol = root(cost_function, initial_condition, method='lm')
theta, kappa, sigma, rho, v0 = model2.params()
print "theta = %f, kappa = %f, sigma = %f, rho = %f, v0 = %f" % \
(theta, kappa, sigma, rho, v0)
error = calibration_report(heston_helpers2, grid_data2)
summary.append(["Scipy LM1", error] + list(model2.params()))
```

`theta, kappa, sigma, rho, v0 = (0.07,0.5,0.1,0.1,0.1)`

In [11]:

```
model2, engine2 = setup_model(
yield_ts, dividend_ts, spot,
init_condition=(0.07,0.5,0.1,0.1,0.1))
heston_helpers2, grid_data2 = setup_helpers(
engine2, expiration_dates, strikes, data,
calculation_date, spot, yield_ts, dividend_ts
)
initial_condition = list(model2.params())
```

In [12]:

```
%%time
cost_function = cost_function_generator(model2, heston_helpers2)
sol = root(cost_function, initial_condition, method='lm')
theta, kappa, sigma, rho, v0 = model2.params()
print "theta = %f, kappa = %f, sigma = %f, rho = %f, v0 = %f" % \
(theta, kappa, sigma, rho, v0)
error = calibration_report(heston_helpers2, grid_data2)
summary.append(["Scipy LM2", error] + list(model2.params()))
```

If you want to use a simpler approach like least squares, you can do that with Scipy. Here is how you would use it.

In [13]:

```
from scipy.optimize import least_squares
```

In [14]:

```
model3, engine3 = setup_model(
yield_ts, dividend_ts, spot,
init_condition=(0.02,0.2,0.5,0.1,0.01))
heston_helpers3, grid_data3 = setup_helpers(
engine3, expiration_dates, strikes, data,
calculation_date, spot, yield_ts, dividend_ts
)
initial_condition = list(model3.params())
```

In [15]:

```
%%time
cost_function = cost_function_generator(model3, heston_helpers3)
sol = least_squares(cost_function, initial_condition)
theta, kappa, sigma, rho, v0 = model3.params()
print "theta = %f, kappa = %f, sigma = %f, rho = %f, v0 = %f" % \
(theta, kappa, sigma, rho, v0)
error = calibration_report(heston_helpers3, grid_data3)
summary.append(["Scipy LS1", error] + list(model3.params()))
```

With the second initial condition:

In [16]:

```
model3, engine3 = setup_model(
yield_ts, dividend_ts, spot,
init_condition=(0.07,0.5,0.1,0.1,0.1))
heston_helpers3, grid_data3 = setup_helpers(
engine3, expiration_dates, strikes, data,
calculation_date, spot, yield_ts, dividend_ts
)
initial_condition = list(model3.params())
```

In [17]:

```
%%time
cost_function = cost_function_generator(model3, heston_helpers3)
sol = least_squares(cost_function, initial_condition)
theta, kappa, sigma, rho, v0 = model3.params()
print "theta = %f, kappa = %f, sigma = %f, rho = %f, v0 = %f" % \
(theta, kappa, sigma, rho, v0)
error = calibration_report(heston_helpers3, grid_data3)
summary.append(["Scipy LS2", error] + list(model3.params()))
```

`differential_evolution`

methodology.

In [18]:

```
from scipy.optimize import differential_evolution
```

In [19]:

```
model4, engine4 = setup_model(yield_ts, dividend_ts, spot)
heston_helpers4, grid_data4 = setup_helpers(
engine4, expiration_dates, strikes, data,
calculation_date, spot, yield_ts, dividend_ts
)
initial_condition = list(model4.params())
bounds = [(0,1),(0.01,15), (0.01,1.), (-1,1), (0,1.0) ]
```

In [20]:

```
%%time
cost_function = cost_function_generator(
model4, heston_helpers4, norm=True)
sol = differential_evolution(cost_function, bounds, maxiter=100)
theta, kappa, sigma, rho, v0 = model4.params()
print "theta = %f, kappa = %f, sigma = %f, rho = %f, v0 = %f" % \
(theta, kappa, sigma, rho, v0)
error = calibration_report(heston_helpers4, grid_data4)
summary.append(["Scipy DE1", error] + list(model4.params()))
```

`maxiter`

in order to limit the time taken. In production scenarios, you may want to try a larger number or not provide any value and default to 1000. This can help search a larger area of the parameter space.

In [21]:

```
model4, engine4 = setup_model(yield_ts, dividend_ts, spot)
heston_helpers4, grid_data4 = setup_helpers(
engine4, expiration_dates, strikes, data,
calculation_date, spot, yield_ts, dividend_ts
)
initial_condition = list(model4.params())
bounds = [(0,1),(0.01,15), (0.01,1.), (-1,1), (0,1.0) ]
```

In [22]:

```
%%time
cost_function = cost_function_generator(
model4, heston_helpers4, norm=True)
sol = differential_evolution(cost_function, bounds, maxiter=100)
theta, kappa, sigma, rho, v0 = model4.params()
print "theta = %f, kappa = %f, sigma = %f, rho = %f, v0 = %f" % \
(theta, kappa, sigma, rho, v0)
error = calibration_report(heston_helpers4, grid_data4)
summary.append(["Scipy DE2", error] + list(model4.params()))
```

Here we will use the Basin Hopping (annealing like) method to solve for the parameters. A couple things to make note here. The Basin Hopping method works best when used wiht a minimizer. Here I played with various minimizers and finally decided to use something that supports bounds checking. Without bounds checking, I often ended with `nan`

and did not have a meaningful solution in the end.

I have chosen bounds based on a very basic reasoning. One needs careful reasoning to use appropriate bounds for the problem at hand.

In [23]:

```
from scipy.optimize import basinhopping
```

In [24]:

```
class MyBounds(object):
def __init__(self, xmin=[0.,0.01,0.01,-1,0], xmax=[1,15,1,1,1.0] ):
self.xmax = np.array(xmax)
self.xmin = np.array(xmin)
def __call__(self, **kwargs):
x = kwargs["x_new"]
tmax = bool(np.all(x <= self.xmax))
tmin = bool(np.all(x >= self.xmin))
return tmax and tmin
bounds = [(0,1),(0.01,15), (0.01,1.), (-1,1), (0,1.0) ]
```

In [25]:

```
model5, engine5 = setup_model(
yield_ts, dividend_ts, spot,
init_condition=(0.02,0.2,0.5,0.1,0.01))
heston_helpers5, grid_data5 = setup_helpers(
engine5, expiration_dates, strikes, data,
calculation_date, spot, yield_ts, dividend_ts
)
initial_condition = list(model5.params())
```

In [26]:

```
%%time
mybound = MyBounds()
minimizer_kwargs = {"method": "L-BFGS-B", "bounds": bounds }
cost_function = cost_function_generator(
model5, heston_helpers5, norm=True)
sol = basinhopping(cost_function, initial_condition, niter=5,
minimizer_kwargs=minimizer_kwargs,
stepsize=0.005,
accept_test=mybound,
interval=10)
theta, kappa, sigma, rho, v0 = model5.params()
print "theta = %f, kappa = %f, sigma = %f, rho = %f, v0 = %f" % \
(theta, kappa, sigma, rho, v0)
error = calibration_report(heston_helpers5, grid_data5)
summary.append(["Scipy BH1", error] + list(model5.params()))
```

In [27]:

```
model5, engine5 = setup_model(
yield_ts, dividend_ts, spot,
init_condition=(0.07,0.5,0.1,0.1,0.1))
heston_helpers5, grid_data5 = setup_helpers(
engine5, expiration_dates, strikes, data,
calculation_date, spot, yield_ts, dividend_ts
)
initial_condition = list(model5.params())
```

In [28]:

```
%%time
mybound = MyBounds()
minimizer_kwargs = {"method": "L-BFGS-B", "bounds": bounds}
cost_function = cost_function_generator(
model5, heston_helpers5, norm=True)
sol = basinhopping(cost_function, initial_condition, niter=5,
minimizer_kwargs=minimizer_kwargs,
stepsize=0.005,
accept_test=mybound,
interval=10)
theta, kappa, sigma, rho, v0 = model5.params()
print "theta = %f, kappa = %f, sigma = %f, rho = %f, v0 = %f" % \
(theta, kappa, sigma, rho, v0)
error = calibration_report(heston_helpers5, grid_data5)
summary.append(["Scipy BH2", error] + list(model5.params()))
```

Here is a summary of all the results with the calibration error overall, and the respective parameters. All the local minima methods give parameters that are very different based on the initial condition that we start with. This is different in contrary with the global minimization methods that all end up in more or less the same proximity of each other.

The global solvers such as Differential Evolution and Basin Hopping are capable of finding the global minima and it is sometimes a question of computation resources. Here, I have lower "iterations" set for these routines for faster solving. Even with such a short threshold, we get fairly good solution set. I think it is premature to compare the effectiveness of different global solvers just based on the results here. The scipy optimize package has detailed documentation with various tuning parameters. I haven't exploited the nuances much, and is left as an exercise for the reader.

Hope you find this useful!

In [29]:

```
from pandas import DataFrame
DataFrame(
summary,
columns=["Name", "Avg Abs Error","Theta", "Kappa", "Sigma", "Rho", "V0"],
index=['']*len(summary))
```

Out[29]:

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