User guide

Models

The models that are currently implemented in fyne are the Black-Scholes, Heston and Wishart models. In order to make notation clear, especially with the naming of the parameters, we state below the discounted underlying price dynamics of the models under the risk-neutral measure.

  • Black-Scholes

\[ dS_t = \sigma S_t dW_t \]

  • Heston
\begin{aligned} dS_t & = \sqrt{V_t} S_t dW_t \\ dV_t & = \kappa(\theta - V_t)dt + \nu \sqrt{V_t}dZ_t \\ d[W, Z]_t & = \rho dt \end{aligned}
  • Wishart
\begin{aligned} dS_t & = S_t \mathrm{Tr} \left( \sqrt{V_t} \left( dW_t R + dZ_t \sqrt{I - RR^T} \right) \right) \\ dV_t & = \left(\beta QQ^T + M V_t + V_t M^T \right) dt + \sqrt{V_t} dW_t Q + Q^T dW_t^T \sqrt{V_t} \\ d[W, Z]_t & = 0 \end{aligned}

Pricing

Each model has its own pricing formula. The available pricing functions are:

  • fyne.blackscholes.formula
  • fyne.heston.formula

These functions return the price of the option in monetary units. If implied volatility is needed, it can be evaluated with fyne.blackscholes.implied_vol{.interpreted-text role="func"}.

Example

In this example, we compute the implied volatility smile according to the Heston model.

import numpy as np
import pandas as pd
import plotly.express as px

from fyne import blackscholes, heston

underlying_price = 100.0
vol = 0.0457
params = (5.07, 0.0457, 0.48, -0.767)
strikes, expiries = np.broadcast_arrays(
    np.linspace(80.0, 120.0, 40)[:, None],
    [0.1, 0.3, 1.0],
)

option_prices = heston.formula(
    underlying_price, strikes, expiries, vol, *params
)
implied_vols = blackscholes.implied_vol(
    underlying_price, strikes, expiries, option_prices
)

data = pd.DataFrame(
    {
        "expirie": expiries.ravel(),
        "strike": strikes.ravel(),
        "implied_vol": implied_vols.ravel(),
    }
)
fig = px.line(
    data,
    x="strike",
    y="implied_vol",
    color="expirie",
    title="Heston volatility smile",
    markers=True,
)

fig.write_html(
    "docs/heston_smile.html",
    full_html=False,
    include_plotlyjs="cdn",
)

Greeks

Greeks are usually associated to the derivatives of the Black-Scholes formula. However, Greeks can be computed according to other models as well. The following are the available Greeks in fyne:

  • fyne.blackscholes.delta
  • fyne.blackscholes.vega
  • fyne.heston.delta
  • fyne.heston.vega

Calibration

In fyne we distinguish two types of calibration:

  • Cross-sectional
  • Calibration from options prices at a single point in time
  • Panel
  • Calibration from options prices at a multiple points in time

Besides, calibration can recover the full parameter set and unobservable state variables or just the unobservable state variables.

The available calibration functions are the following:

  • fyne.heston.calibration_crosssectional
  • fyne.heston.calibration_panel
  • fyne.heston.calibration_vol