Jump Diffusion Models

The library allows to create a vast array of jump-diffusion models. The most famous one is the Merton jump-diffusion model.

Merton Model

from quantflow.sp.jump_diffusion import Merton

pr = Merton.create(diffusion_percentage=0.2, jump_intensity=50, jump_skew=-0.5)
pr

Merton(diffusion=WeinerProcess(sigma=0.22360679774997896), jumps=CompoundPoissonProcess[Normal](intensity=50.0, jumps=Normal(mu=-0.01, sigma=0.06324555320336758)))

Marginal Distribution

m = pr.marginal(0.02)
m.std(), m.std_from_characteristic()

(np.float64(0.0714142842854285), np.float64(0.0714142841949521))

from quantflow.utils import plot

plot.plot_marginal_pdf(m, 128, normal=True, analytical=False, log_y=True)

Characteristic Function

plot.plot_characteristic(m)

Option Pricing

We can price options using the OptionPricer tooling.

from quantflow.options.pricer import OptionPricer
pricer = OptionPricer(pr)
pricer

OptionPricer(model=Merton(diffusion=WeinerProcess(sigma=0.22360679774997896), jumps=CompoundPoissonProcess[Normal](intensity=50.0, jumps=Normal(mu=-0.01, sigma=0.06324555320336758))), n=128, max_moneyness_ttm=1.5)

fig = None
for ttm in (0.05, 0.1, 0.2, 0.4, 0.6, 1):
    fig = pricer.maturity(ttm).plot(fig=fig, name=f"t={ttm}")
fig.update_layout(title="Implied black vols", height=500)

This term structure of volatility demostrates one of the principal weakness of the Merton’s model, and indeed of all jump diffusion models based on Lévy processes, namely the rapid flattening of the volatility surface as time-to-maturity increases. For very short time-to-maturities, however, the model has no problem in producing steep volatility smile and skew.

MC paths

pr.sample(20, time_horizon=1, time_steps=1000).plot().update_traces(line_width=0.5)

Exponential Jump Diffusion

This is a variation of the Mertoin model, where the jump distribution is a double exponential, one for the negative jumps and one for the positive jumps.

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