Geometric brownian motion for the stock price

Question

Question 1: Computational Problem:

Consider the process x(t), driven in discrete time by

Δx(t) = x(t + Δt) - x(t) = μ(x,t)Δt + σ(x,t)Δz(t)

where z(t) is a Wiener process. This is equation before passing to the limit. Here 0 < t ≤ T and Δt=T/n where n is user specified.

Write a program to simulate this process on [0, T]. The program structure should allow the user to specify T, n, the functions μ(x,t) and σ(x,t). As well the number of simulations M should be an input variable.

(a) Take x(0) = 1, μ(x,t) = 0, σ(x,t) = 1 and T = 1 so that x(t) is a pure Wiener process on [0, 1]. Initially take n = 100 and simulate M = 1,000 paths.

(i) Use the output to compare graphically the distribution of x(t) at t = 0.5 and t = 1.

(ii) Experiment with n and M to try to obtain better approximations to the known distributions.

(iii) Constructing a table to compare these with the known theoretical distributions for x(t) and comment on the effect of n and M.

(b) Take x(0) = 1, µ(x,t) = σx and T = 2 so that x(t) is the geometric Brownian motion for the stock price.

(i) Use µ = 0.15 and σ = 0.20 and take n = 100 and simulate M= 1,000 paths to obtain better approximations.

(ii) Use the output to graph the distributions ofx(t) and ln(x(t)/x(0)) at t= 2.

(iii) Constructing a table to compare these with the known true distributions for x(T) and ln(x(T)/x(0)). As in the previous question play with the values of n and Al and comment on the effect of n and M.

(c) Repeat the exercise by taking x(0) = 0.06,

µ(x,t)= k(x—x)

with k = 0.5, x‾ = 0.065 and σ(x,t) = σ with  σ = 0.02,n = 100 and M = 1,000. Calculate and graph the distribution of x at T = 6 months and T = 12 months, and compare these with the theoretical distributions. Comment on the effect of n and M.

Question 2: Computational Problem - Consider the stochastic integral

Y(t) = ∫₀^t e-^K(t-s)dz(s),

where :(s) is a Wiener process.

(i) Write a program that will approximate Y(t) by

Y_n(t) = Σ_i=0 ^n-1 e^{-k(nΔt - iΔt)} Δz_i

where Δt = t/n and Δz_i = z((i + 1)Δt) - z(iΔt). The values of k, t, n and the number M of simulated paths should be user defined inputs. Initially take k = 0.5, = 1, n = 100 and M = 1.000.

(ii)  Compare the simulated distribution of 4(1) with the true distribution of Y(1) [see Section 5.2 if you have forgotten how to calculate this]. Experiment with the values of n and Al.

Problem 3: Computational Problem - Use the solutions to simulate the stock price process x(r) and the return process In(x(t)/x(0)) in the interval (0, T) and in particular obtain the simulated distribution for these quantities. Use the same values for x(0), μ, σ and T as used in Problem.

Since now it is possible to draw the directly from a normal distribution, discretisation error is avoided. Gauge the impact of the discretisation error by comparing the distributions obtained here with the ones obtained in Problem and the true distribution.

Change of Measure—Let x follow a (μ, σ) process

dx = μdt + σdz,

where µ and σ are constant.

(a) Write done the process for x(t) and the probability density function p(x, t|x_o, 0);

(b) For a constant λ, define ξ(t) as in equation in the note and work out ξ(t) p(x,t)|x_o, 0);

(c). Using (a) and (b) to show that p‾(x, t|x₀,0) = ξ(t)p(x,t)x_o, 0) is the probability density function of a process with mean of μ – λσ and variance of σ² process.