# Stochastic Processes

Brownian bridge [11 marks]

Consider the process

where  is a standard Brownian Motion.

• Show that is a standard Brownian bridge [2 marks].

• Determine [3 marks].

• Show that [3 marks].

• Using the result from (c), determine [3 marks].

Question 2. Stationarity [9 marks]

Consider the process

where the  and independent from each other. Determine if the following processes are strictly stationary and for those that are write down the covariance function.

• [3 marks].

• [3 marks].

• [3 marks].

## Question 3. MC option pricing [10 marks]

Consider the geometric Brownian Motion (GBM)

where , , ,  and  is a standard Brownian motion.

An example of a discretely-monitored Asian call option with European payoff has price at  given by

where .

• Taking , use Mathematica to calculate the crude Monte Carlo estimate

where ,  are random samples of  and  respectively [4 marks]. Also calculate the sample estimate of  [1 mark].

• Taking , use R to calculate the control variate Monte Carlo estimate

where the expected value of the control variate is given by the Black-Sholes European vanilla call price formula

with

and the crude MC estimate of the control variate

Question 4. Markov chains [10 marks]

You may use computational software for calculations, but express your answers using proper mathematical notation.

Let , , be a homogenous Markov chain taking states  with one-step transitional probability matrix and initial distribution

respectively.

• Calculate [2 marks].

• Calculate a stationary distribution [2 marks].

Let , , be a homogenous Markov chain taking states  with generator matrix

• What is the probability of the state change when the Markov chain jumps [2 marks]?

• Let and let the waiting time

What is  [2 marks]?

• A Markov chain is ergodic if the limit of the state probability vector, , exists and does not depend on . Using this criterion, determine if ergodic [2 marks]?

Question 5. ARMA processes [10 marks]

Consider the process

where , , is a zero-mean white-noise process with variance .

• The process above is not stationary. Explain why [2 marks] and identify an appropriate ARIMA() model [2 marks].

• Describe how the ARIMA() model from (a) could be converted to an ARMA() model [2 marks].

• Determine if the ARMA() from (b) is invertible [2 marks].

• Plot the ARIMA() and ARMA() models identified in (a) and (b) respectively with [2 marks]?

Question 6. Diffusion processes [10 marks]

Consider the diffusion process

where  is a standard Brownian motion.

• Using the definition

find the drift coefficient of  [3 marks].

• Find the diffusion coefficient of using the Ito formula [2 marks].

• Write down the Kolmogorov backward equation for the transition density function of the process [2 marks].

Define the process

with  as above.

• Using the Ito formula, write down the stochastic differential equation of the process [3 marks]. 