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We could have the formula for Heston model for currency as (under the Risk-neutral measure for $r_d$) -

$dS_t = \left( r_d - r_f \right) S_tdt+S_t \sqrt{V_t}dW^S$

$dV_t = a(\bar{V}- V_t)dt + \eta \sqrt{V_t}dW_t^V$

Typically we estimate the model parameters observing the Call and Put options prices with different maturities.

However for the Currency case, where can I see such market tradable option contracts? Like in CME (https://www.cmegroup.com/trading/fx/g10/euro-fx_quotes_globex_options.html?optionProductId=59#optionProductId=8117&strikeRange=ATM), most of the options are traded on the Futures.

So if I want to estimate the model parameters for EUR-USD spot process like the one in Bloomberg terminal https://www.bloomberg.com/quote/EURUSD:CUR, how should I proceed to estimate the model parameters?

Any pointer will be highly appreciated.

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I've been working on this problem a little bit lately. Unfortunately in the FX context, it's not quite as straight-forward as in the equities case, for two reasons:

  1. FX options trade OTC instead of on exchange, so you need access to broker screens to trade them (eg. on BBG)
  2. FX Options are quoted by (delta, tenor, vol) instead of (strike, tenor, price) so we have to do a bit of pre-work to get the options corresponding strikes for our Heston calibration

A EURUSD options screen from BBG looks something like this:

Example BBG vol levels

Trades are done OTC between clients, but many still need to be reported to the DTCC, and BBG has a screen showing some examples of recent OTC options that were traded:

DTCC traded options

The exact procedure required to turn these into (strike, price) pairs depends on the currency pair under consideration, a great reference on the conventions is found in this paper, but it turns out to be relatively simple for EURUSD. As described in the paper, you need a function that looks like this:

import numpy as np
from scipy.stats import norm

def strike_from_fwd_delta(tte, fwd, vol, delta, put_call):
    sigma_root_t = vol * np.sqrt(tte)
    inv_norm = norm.ppf(delta * put_call)

    return fwd * np.exp(-sigma_root_t * put_call * inv_norm + 0.5 * sigma_root_t * sigma_root_t)

strike = strike_from_fwd_delta(tte, fwd, vol, put_call*delta, put_call)

After doing that, I've got two tables (NB this is a different dataset to that shown in the screen image above, because I transcribed and calculated it earlier) - the original table showing the vol for each (delta, tenor) pair, and the new one showing the strike for each pair. The new table looks something like this:

FX option strikes corresponding to (delta, tenor) pairs

Now we have enough to calibrate a Heston vol surface using the (tenor, strike, vol) triples from each observed option (nb. you'll also have to fit domestic and foreign rates curves, but that's another story) - for my options above, the surface looks like this:

Heston vol surface for FX options

Here is a sample of code (the data above is hard-coded at the top) that will generate the vol surface above for you:

import numpy as np
from matplotlib import pyplot as plt
import matplotlib.cm as cm
from mpl_toolkits.mplot3d import Axes3D
import QuantLib as ql

strikes = [1.1787, 1.1788, 1.1794, 1.1804, 1.1815, 1.1846, 1.1873, 1.1909, 1.1978, 1.2046, 1.1833, 1.1854, 1.1891, 1.1942, 1.1995, 1.2092, 1.2178, 1.2263, 1.2426, 1.2574, 1.1741, 1.1725, 1.1702, 1.1673, 1.1646, 1.1619, 1.1598, 1.158, 1.1561, 1.1556, 1.1871, 1.1906, 1.197, 1.2056, 1.2143, 1.2301, 1.2441, 1.2571, 1.2814, 1.3034, 1.1708, 1.1678, 1.1632, 1.1574, 1.1517, 1.1442, 1.1379, 1.1327, 1.1241, 1.1179, 1.192, 1.1977, 1.2078, 1.2214, 1.2351, 1.2605, 1.2834, 1.304, 1.3402, 1.374, 1.1664, 1.1618, 1.1542, 1.1445, 1.1349, 1.1206, 1.1081, 1.0979, 1.0805, 1.0667, 1.1956, 1.2028, 1.2157, 1.233, 1.2506, 1.2839, 1.3147, 1.3419, 1.3876, 1.4314, 1.1635, 1.1577, 1.1479, 1.1354, 1.1231, 1.1035, 1.0859, 1.0718, 1.0483, 1.0288, 1.2012, 1.211, 1.2284, 1.2519, 1.2758, 1.3228, 1.3668, 1.4053, 1.4677, 1.5291, 1.1589, 1.1513, 1.1381, 1.1212, 1.1046, 1.0763, 1.0505, 1.0301, 0.997, 0.9687]
vols = [0.0726, 0.0714, 0.072, 0.0717, 0.076, 0.0728, 0.0727, 0.0728, 0.0749, 0.0759, 0.0743, 0.0733, 0.074, 0.0739, 0.0783, 0.0754, 0.0754, 0.0754, 0.0772, 0.0781, 0.0719, 0.0707, 0.0713, 0.0711, 0.0755, 0.0726, 0.0726, 0.0728, 0.0752, 0.0764, 0.0761, 0.0754, 0.0764, 0.0764, 0.0811, 0.0788, 0.0791, 0.0793, 0.0809, 0.0817, 0.0721, 0.0708, 0.0717, 0.0716, 0.0761, 0.0738, 0.0742, 0.0746, 0.0773, 0.0787, 0.0786, 0.0784, 0.0798, 0.0803, 0.0854, 0.0843, 0.0858, 0.0864, 0.0874, 0.0884, 0.0726, 0.0715, 0.0729, 0.073, 0.078, 0.0767, 0.0782, 0.0789, 0.082, 0.0838, 0.0803, 0.0803, 0.0823, 0.083, 0.0885, 0.0885, 0.0908, 0.0919, 0.0924, 0.0935, 0.0732, 0.0722, 0.0739, 0.0744, 0.0795, 0.0793, 0.0816, 0.0828, 0.0859, 0.0882, 0.083, 0.0834, 0.086, 0.0872, 0.0931, 0.0944, 0.0977, 0.0992, 0.0994, 0.1006, 0.0743, 0.0734, 0.0758, 0.0766, 0.0822, 0.0834, 0.0871, 0.089, 0.0923, 0.0951]
expiries = ['1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y', '1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y', '1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y', '1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y', '1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y', '1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y', '1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y', '1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y', '1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y', '1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y', '1W', '2W', '1M', '2M', '3M', '6M', '9M', '1Y', '18M', '2Y']

rate = 0.0
today = ql.Date(1, 9, 2020)
spot = 1.1786
usd_calendar = ql.NullCalendar()

# Set up the flat risk-free curves
usd_curve = ql.FlatForward(today, 0.0, ql.Actual365Fixed())
eur_curve = ql.FlatForward(today, 0.0, ql.Actual365Fixed())

usd_rates_ts = ql.YieldTermStructureHandle(usd_curve)
eur_rates_ts = ql.YieldTermStructureHandle(eur_curve)

v0 = 0.005; kappa = 0.01; theta = 0.0064; rho = 0.0; sigma = 0.01

heston_process = ql.HestonProcess(usd_rates_ts, eur_rates_ts, ql.QuoteHandle(ql.SimpleQuote(spot)), v0, kappa, theta, sigma, rho)
heston_model = ql.HestonModel(heston_process)
heston_engine = ql.AnalyticHestonEngine(heston_model)

# Set up Heston 'helpers' to calibrate to
heston_helpers = []

for strike, vol, expiry in zip(strikes, vols, expiries):
    tenor = ql.Period(expiry)

    helper = ql.HestonModelHelper(tenor, usd_calendar, spot, strike, ql.QuoteHandle(ql.SimpleQuote(vol)), usd_rates_ts, eur_rates_ts)
    helper.setPricingEngine(heston_engine)
    heston_helpers.append(helper)
    
lm = ql.LevenbergMarquardt(1e-8, 1e-8, 1e-8)
heston_model.calibrate(heston_helpers, lm,  ql.EndCriteria(5000, 100, 1.0e-8, 1.0e-8, 1.0e-8))
theta, kappa, sigma, rho, v0 = heston_model.params()
feller = 2 * kappa * theta - sigma ** 2

print(f"theta = {theta:.4f}, kappa = {kappa:.4f}, sigma = {sigma:.4f}, rho = {rho:.4f}, v0 = {v0:.4f}, spot = {spot:.4f}, feller = {feller:.4f}")

# Plot the vol surface ...
heston_handle = ql.HestonModelHandle(heston_model)
heston_vol_surface = ql.HestonBlackVolSurface(heston_handle)

def plot_vol_surface(vol_surface, plot_years=np.arange(0.1, 3, 0.1), plot_strikes=np.arange(70, 130, 1), funct='blackVol'):
    if type(vol_surface) != list:
        surfaces = [vol_surface]
    else:
        surfaces = vol_surface

    fig = plt.figure(figsize=(10, 6))
    ax = fig.gca(projection='3d')
    X, Y = np.meshgrid(plot_strikes, plot_years)
    Z_array, Z_min, Z_max = [], 100, 0

    for surface in surfaces:
        method_to_call = getattr(surface, funct)

        Z = np.array([method_to_call(float(y), float(x)) 
                      for xr, yr in zip(X, Y) 
                          for x, y in zip(xr, yr)]
                     ).reshape(len(X), len(X[0]))

        Z_array.append(Z)
        Z_min, Z_max = min(Z_min, Z.min()), max(Z_max, Z.max())

    # In case of multiple surfaces, need to find universal max and min first for colourmap
    for Z in Z_array:
        N = (Z - Z_min) / (Z_max - Z_min)  # normalize 0 -> 1 for the colormap
        surf = ax.plot_surface(X, Y, Z, rstride=1, cstride=1, linewidth=0.1, facecolors=cm.coolwarm(N))

    m = cm.ScalarMappable(cmap=cm.coolwarm)
    m.set_array(Z)
    plt.colorbar(m, shrink=0.8, aspect=20)
    ax.view_init(30, 300)

plot_vol_surface(heston_vol_surface, plot_years=np.arange(0.1, 2.0, 0.1), plot_strikes=np.linspace(1.0, 1.5, 30))
| improve this answer | |
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  • $\begingroup$ Thanks. What instrument code you apply to get the vol data from BBG? And what exactly is the underlying for these options? is it CBOE quote? $\endgroup$ – Bogaso Sep 7 at 14:38
  • $\begingroup$ These are indicative levels based on OTC quotes from brokers (mostly the big IBs), but you would need to RFQ them to get a firm tradable price if you wanted to trade (eg via BBG chat) - I'm away from the terminal today but I'll take some screenshots tomorrow. FX options do not trade on exchange. $\endgroup$ – StackG Sep 7 at 14:54
  • $\begingroup$ Thanks. Those screenshots will be really helpful. I am also curious to understand the underlying EURUSD rate's source for BBG. Btw, what is RFQ? $\endgroup$ – Bogaso Sep 7 at 15:00
  • $\begingroup$ Request-For-Quote. Basically, asking a dealer to quote you! $\endgroup$ – StackG Sep 7 at 22:08
  • $\begingroup$ Screenshots added! $\endgroup$ – StackG Sep 8 at 23:10

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