Question 1:

(A) Suppose we have the following Ito process

dS_t = μtdt + dW_t

and let the derivative be the square

v_t = s_t²

Find the Ito process for the derivative in terms of the derivative.

(B) Suppose a share S_t satisfies the Ito process

dS_t = μS_tdt + σS_tdW_t

and let the derivative be the square of the share value

Vt = S_t²

Find the Ito process for the derivative in terms of the derivative.

(C) Suppose a share S_t satisfies the Ito process

dS_t = µS_tdt + σS_tdW_t

and let the derivative be the natural log of the share value

V_t = log(S_t)

Find the Ito process for the derivative in terms of the derivative.

Question 2:

Suppose a share price satisfies the stochastic differential equation

dS_t = S_tµdt + σS_tdWt

An employer wishes to motivate the employees, and decides on a scheme where the employer issues a share derivatives to its employees that can be cashed in after the strike date. The derivative is worth B ($5000) if the share price of the company, S reaches at least X ($15) by the strike date t₀+ τ (τ = 8 months) and 0 (zero) if it does not. The current value of a share price is $12. The annual risk free interest rate r is 8%, which is also the rate of inflation. The annual share volatility is 12%.

Your task is to find the value of this share derivative using three different methods.

(A) Evaluate this derivative at time t₀ by means of repeated simulation of share values using a daily timestep, carefully choosing your daily growth rate and daily share volatility. You will need to run this simulation many times (i.e. 10000) in order to get a single digit of accuracy. In order to get full marks, you need to account for inflation.

(B) Evaluate this derivative by repeated simulation using a single timestep. You will need to run this simulation many times (i.e. 10000) in order to get a single digit of accuracy.

(C) Calculate the solution by hand. Show all of your working, including the calculations to arrive at the mean and standard deviation used to calculate the zscore. Regardless of what you use to evaluate the cumulative distribution of your zscore, (e.g. pnorm() or Standard_normal_table), you need to show both the argument and the result. In order to get full marks, you need to account for inflation.