-1
$\begingroup$

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?

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.