Question 1. Brownian bridge [11 marks]
Consider the process
X_t=B_t-tB_1,0≤t≤1,
where B_t is a standard Brownian Motion.

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

Determine E[exp(〖1/2 X〗_(3⁄10)+〖2/3 X〗_(7⁄10) )] [3 marks].

Show that E[X_(7⁄10)∨X_(3⁄10) ]=3/7 X_(3⁄10) [3 marks].

Using the result from (c), determine E[X_(3⁄70)^2 X_(7⁄10) ] [3 marks].

Question 2. Stationarity [9 marks]
Consider the process
Z_t=〖3X〗_t-1/5 X_(t-1),t∈Z,
where the X_t N(0,1) and independent from each other. Determine if the following processes are strictly stationary and for those that are write down the covariance function.

Y_t=Z_t+Z_(t-1),t∈Z [3 marks].

Y_t=Z_0 Z_(t-1),t∈Z [3 marks].

Y_t=〖cos⁡(t)Z〗_1+〖sin⁡(t)Z〗_3,t∈Z [3 marks].

Question 3. MC option pricing [10 marks]
Consider the geometric Brownian Motion (GBM)
S_t=S_0 exp((r-σ^2⁄2)t+σB_t ),0≤t≤T,
where S_0=10, r=1⁄100, σ=1⁄3, T=2 and B_t is a standard Brownian motion.

An example of a discretely-monitored Asian call option with European payoff has price at t=0 given by
A=e^(-rT) E[max(1/24 ∑_(j=1)^24▒S_(j⁄12) -K,0)]
where K=8.

Taking n=〖10〗^5, use Mathematica to calculate the crude Monte Carlo estimate
A_n=e^(-rT)/n ∑_(k=1)^n▒max(1/24 ∑_(j=1)^24▒S_(j⁄12)^((k) ) -K,0)
e^(-rT)/n ∑_(k=1)^n▒max(1/24 ∑_(k=1)^24▒〖S_0 exp((r-σ^2⁄2)t+σB_(j⁄12)^((k) ) ) 〗-K,0)
where S_t^((k) ), B_t^((k) ) are random samples of S_t and B_t respectively [4 marks]. Also calculate the sample estimate of var(C_n ) [1 mark].

Taking n=〖10〗^5, use R to calculate the control variate Monte Carlo estimate
A ~_n=A_n-a(C_n-C)
where the expected value of the control variate is given by the Black-Sholes European vanilla call price formula
C=Φ(d_1 ) S_0-e^(-rT) Φ(d_2 )K
with
d_1=1/(σ√T) (ln S_0/K+(r+σ^2⁄2)T),
d_2=d_1-σ√T,
Φ(z)=P(Z≤z)forZ N(0,1)
and the crude MC estimate of the control variate
C_n=e^(-rT)/n ∑_(k=1)^n▒max(S_2^((k) )-K,0)
e^(-rT)/n ∑_(k=1)^n▒〖max(S_0 exp((r-σ^2⁄2)t+σB_2^((k) ) )-K,0).〗

Question 4. Markov chains [10 marks]
You may use computational software for calculations, but express your answers using proper mathematical notation.

Let X_t, t∈{1,2,…}, be a homogenous Markov chain taking states X_t∈{1,2,…,5} with one-step transitional probability matrix and initial distribution
P=(■(1/10&3/10&0&2/5&1/5@1/5&0&0&2/5&2/5@1/6&1/3&1/6&1/6&1/6@1/8&1/8&1/4&0&1/2@0&1/3&1/3&0&1/3)),p(0)=(■(1/10@1/4@1/10@1/4@3/10))
respectively.

Calculate var(X_7 ) [2 marks].

Calculate a stationary distribution π [2 marks].

Let Y_t, t≥0, be a homogenous Markov chain taking states Y_t∈{1,2,…,5} with generator matrix
A=(■(0&0&0&0&0@1/4&-1&3/8&1/8&1/4@1&1/2&-3&1/4&5/4@1/6&1/6&1/6&-1&1/2@0&0&0&0&0)).

What is the probability of the state change 3⟶4 when the Markov chain jumps [2 marks]?

Let Y_4=3 and let the waiting time
T_3=min(s>0∨Y_(4+s)≠3).
What is E[T_3^2 ] [2 marks]?

A Markov chain is ergodic if the limit of the state probability vector, 〖lim〗┬(t→∞)⁡p(t), exists and does not depend on p(t). Using this criterion, determine if Y_t ergodic [2 marks]?

Question 5. ARMA processes [10 marks]
Consider the process
X_t-13/12 X_(t-1)-13/24 X_(t-2)+5/8 X_(t-3)=Z_t-67/36 Z_(t-1)-41/72 Z_(t-2)+5/3 Z_(t-3),t∈Z
where Z_t, t∈Z, is a zero-mean white-noise process with variance σ^2.

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

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

Determine if the ARMA(p,q) from (b) is invertible [2 marks].

Plot the ARIMA(p,d,q) and ARMA(p,q) models identified in (a) and (b) respectively with var(Z_t )=σ^2=1 [2 marks]?

Question 6. Diffusion processes [10 marks]
Consider the diffusion process
X_t=(1+t^2 ) B_t,t≥0,
where B_t is a standard Brownian motion.

Using the definition
a(t,x)=〖lim〗┬(h→0)⁡〖E[X_(t+h)-X_t∨X_t=x]/h〗,
find the drift coefficient of X_t [3 marks].

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

Write down the Kolmogorov backward equation for the transition density function p(y,t∨x,s) of the process X_t [2 marks].

Define the process
S_t=e^(X_t ),t≥0,
with X_t as above.

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