Question: Euler scheme to simulate trajectories of S + Python code to get independent trajectories of S?
For solving a problem I have the following assumptions:
stochastic basis (Ω,FT,(Ft)t≥0,P) where (Ft)t≥0 is a filtration modelling the market information Ft available at time t.
Considering the Black and Scholes model when the volatility may depend on time.
volatility is a deterministic function t → σt which is continuous, strictly positive and defined on the time interval [0,T].
financial market is composed of two assets. The first one is risk-free and its price at time t is St0 = 1, i.e. the risk-free return for the maturity T is zero.
The risky asset price is supposed to admit a risk neutral probability measure Q ∼ P is such a way that, under Q, its (discounted) value S is given by S =Process where B is a standard Brownian motion under Q
I chose a volatility function of the form σt = α(1 + g(t)) where fix α > 0 and a continuous function g satisfying g > −1 on [0,T]
My solution for the Euler scheme: Euler Scheme Can this work out?
My solution for the Python code:
import numpy as np import numpy.random as sim import matplotlib.pyplot as plt T=3; n=50 ;N=5 ;pas=T/n;rootstep=np.sqrt(T/n); W=np.zeros((n+1,N)); for j in range(N): # for loop N values j in (0,1,2...,N-1) for i in range(1+n): W[i,j]=W[i-1,j]+rootstep*sim.randn() dates=np.linspace(0,T,n+1) graph1=plt.plot(dates,W) plt.show(graph1)
Is the problem with the above code solved? What would be improvements? And how can I confirm numerically the value of EQ(ST)? Which mathematical principle to use for this?