Problem 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. 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) = μx, σ(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 of x(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 M 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.

Problem 2- Computational Problem

Consider the stochastic integral

Y(t) = ₀∫^t e^-k(t-s)dz(s),

where z(s) is a Wiener process.

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

Y_n(t) = _t=0Σ^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. t= 1, n = 100 and M = 1,000.

(ii) Compare the simulated distribution of Y_n(1) with the true distribution of Y(1).

Problem 3 - Computational Problem  

Use the solutions (6.16) and (6.17) to simulate the stock price process x(t) and the return process ln(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 1(b).

Since now it is possible to draw the (z(T) - z(0)) 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 1(b) and the true distribution.

Recalling that x = e^y we may also express in terms of the stock price x to obtain

x (t) = x (0),e(μ-½ σ²)t+σ(z(t)-z(0)).          (6.16)

This last expression (6.16) is the solution to the lognormal stock price stochastic differential equation. This is one of the rare occasions in which we can obtain an analytical solution to a stochastic differential equation. We note that Eq. (6.16) allows us to simulate the process for x(t) up to time t without resorting to discretisation (see Problem 3). Equation (6.16) shows explicitly how the sample paths for the stock price can be viewed as random excursions around a growing trend.

Equation (6.16) may be written

ln (x(t)/x(0)) = (μ - ½σ²)t + σ(z(t)-z(0)).

Problem 4- Consider the expression (8.107) for the price of a European call option, namely

C(S, t) = e-r(T-t)E^~_t[C(S_T, T)],

where E^~t is generated according to the process

dS = rSdt σSdz^~(t).

(i) By simulating M paths for S approximate the expectation with

1/M_i=0Σ^MC(S_T⁽ⁱ⁾, T),

where i indicates the i^th path. Take r =5% p.a., σ = 20%p.a., S = 100, E = 100 and T = 6 months.

(ii) Compare graphically the simulated values for various Al with the true Black-Scholes value.

(iii) Instead of using discretisation to simulate paths for S. use instead the result in equation (6.16). We know from Problem 3 that this involves no discretisation error.

C(S, t) = e^-r(T-t)E_~t[C(S_T, T)]

           = e^-r(T-t) ₀∫∞max[ST - E, 0]p~(ST, T|S, t)dS_T.                               (8.107).