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I'm trying to build Bates model in Python!

$$dS_{t} = \mu S_{t} dt + \sqrt{V_{t}}S_{t}dW_{t}^{1} + J_{t}dQ_{t}$$ $$dV_{t} = \kappa(\theta - V{t})dt + \eta \sqrt{V_{t}}dW_{t}^{2}$$ $$dW_{t}^{1}dW_{t}^{2} = \rho dt$$

Where $dW_{t}^{1}$ and $dW_{t}^{2}$ - normally distributed processes, $dQ_{t}$ - Poisson distribution

import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
sns.set(palette='viridis')
%matplotlib inline
%config InlineBackend.figure_format='retina'

S0 = 62083
r = 0.085
T = 62 / 252
dt = 1 / 252

def MC(S0):

    rho = -0.344558
    kappa = -0.855167
    eta = 0.510156
    theta = 0.643609
    v0 = 0.099067
    a = 0.2
    b = 0.2
    Lamda = 0.25

    Nsim = 100
    Nsteps = 62
    mu = 0.1

    np.random.seed(5)

    dw_1 = np.random.normal(loc = 0, scale = np.sqrt(dt), size = (Nsim, Nsteps))
    dz_1 = np.random.normal(loc = 0, scale = np.sqrt(dt), size = (Nsim, Nsteps))
    dw_2 =  dw_1 * rho + np.sqrt(1 - rho ** 2) * dz_1
    Poisson = np.random.poisson(Lamda * dt, [Nsim, Nsteps])

    St = np.zeros([Nsim, Nsteps+1])
    vol = np.zeros([Nsim, Nsteps+1])

    St[:, 0] = S0
    vol[:, 0] = v0

    for i in range(Nsteps):
        vol[:, i + 1] = np.abs(vol[:, i] + kappa * (theta - vol[:, i]) * \
                               dt + eta * (np.sqrt(vol[:, i]) * dw_2[:, i]))
        St[:, i + 1] = St[:, i] * np.exp((mu - 0.5 * \
                                          vol[:, i]) * dt + np.sqrt(vol[:, i] * dt) \
                                         * dw_1[:, i] + a * Poisson[:, i] \
                                         + np.sqrt(b ** 2) * np.sqrt(Poisson[:,i]))

    return St

There should be a mistake in discretisation of price path (St):

St[:, i + 1] = St[:, i] * np.exp((mu - 0.5 * \
                                          vol[:, i]) * dt + np.sqrt(vol[:, i] * dt) \
                                         * dw_1[:, i] + a * Poisson[:, i] \
                                         + np.sqrt(b ** 2) * np.sqrt(Poisson[:,i]))

I think so, as I get to strange path simulations:

plt.plot(MC(S0).T)
plt.show()

I know the problem is with discretisation of price path (St), not with parameters values as I tested another one. However, I don't know the right way to fix it. Could somebody help me to fix it, please?

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You should review your definition of what a stochastic differential equation is: 

dSt= mu*St*dt + sqrt(vt)*St*dW1t + Jt*dQt

it means simply that 

S_t+1=St + mu*St*dt + sqrt(Vt)*St*(W_t+1-Wt) + Jt*(Q_t+1-Q_t)

which has to be simulated one time step at a time.

It does not follow that the solution is of the form

S_t = S_0 exp( (mu - 0.5 sigma^2) T + other terms )
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  • $\begingroup$ Hey! Thanks for your answer! I want to clarify something. Actually, I know that to simulate Monte Carlo say to Black-Sholes we should you this (it is stuff from Hull book): $$S_{t} = S_{0} e^{\mu - \frac{\sigma^{2}}{2}T + \sigma e \sqrt(T)} $$ Thus, it should be something like this for Bates model, isn't it? $\endgroup$ – David Jan 27 '20 at 21:34
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    $\begingroup$ @David, For the Black-Scholes model you can use a formula similar to the above because the SDE for geometric Brownian motion has a known exact solution. For the Bates model I am not aware of one. For the $V_t$ process, this appears to be equivalent to a CIR process, so this can be exactly simulated as there is a known solution, although this is of limited use if you're after $S_t$. Also the mixing of a Wiener and Poisson process presents it's own difficulties. $\endgroup$ – oliversm Jan 30 '20 at 16:16

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