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Question 1. Brownian bridge [12 marks]
Consider the process
where 𝐵𝑡 is a standard Brownian Motion.
(a) Show that 𝑋𝑡 is a standard Brownian bridge [3 marks]. (b)Find 𝐸[exp(2𝑋1/2 − 4𝑋1/3)] [3 marks].
(c) Find 𝐸[𝑋1/4|𝐵5/6] [3 marks].
(d)Using the result from (c), use R or Mathematica to compute 𝑃 (14 < 𝐸[𝑋1/4|𝐵5/6] < 12). 𝑋𝑡 = 𝐵𝑡 − 𝑡𝐵1, 0 ≤ 𝑡 ≤ 1, Question 2. Stationarity [6 marks] (a) Let 𝐺𝑡 and 𝐻𝑡 , 𝑡 ≥ 0, be weakly stationary stochastic processes that are independent from each other. Determine if 𝑍𝑡 =𝐺𝑡𝐻𝑡, 𝑡≥0, (b) Let 𝑋𝑡 , 𝑡 ∈ {... , −1,0,1, ... }, be a stochastic process consisting of a sequence of is weakly stationary [3 marks]. independent RVs with 𝑃(𝑋𝑡 = −1) = 𝑃(𝑋𝑡 = 1) = 12, 𝑡 ∈ {...,−1,0,1,...}. 𝑌𝑡 =(𝑋𝑡 +𝑋𝑡+2)2, 𝑡∈{...,−1,0,1,,...}, Determine if is weakly stationary [3 marks]. Question 3. MC option pricing [12 marks] Under the Black-Scholes model, the prices of two stocks at time 𝑇 > 0 are modelled as 𝛼2
𝑆𝑇 =𝑆0exp((𝑟− 2)𝑇+𝛼𝐵𝑇)
𝑍𝑇 =𝑍0exp((𝑟− 2)𝑇+𝛽𝑊𝑇)
where 𝐵𝑡 and 𝑊𝑡 are standard Brownian motions with cov(𝐵𝑡, 𝑊𝑡) = 𝜌𝑡, |𝜌| ≤ 1.
In this question take 𝑆0 = 15, 𝑍0 = 10, 𝑟 = 1/25, 𝛼 = 1/2, 𝛽 = 1/3 and 𝑇 = 2.
A European spread call option with strike 𝐾𝐶 = 2 has price at time 𝑡 = 0 given by 𝐶 = 𝑒−𝑟𝑇𝐸[max(𝑆𝑇 − 𝑍𝑇 − 𝐾𝐶, 0)].
(a) Taking 𝑛 = 105 paths, use Mathematica to calculate a crude Monte Carlo estimate of 𝐶. Also calculate a 95% two-sided confidence interval for 𝐶. Do this for one of the following two options (making sure to specify your choice):
• 𝜌 = 0 [4 marks]
• 𝜌 = 2/3 [6 marks]. If you choose this option you must use Cholesky decomposition with Mathematica function CholeskyDecomposition to simulate correlated normal RVs (do not use Mathematica functions that generate correlated normal RVs as the additional two marks will not be awarded).

Now consider the European vanilla call with strike 𝐾𝑄 = 12 and price given by the Black-Scholes formula
𝑄=𝑒−𝑟𝑇𝐸[max(𝑆𝑇 −𝐾𝑄,0)] = Φ(𝑑1)𝑆0 − Φ(𝑑2)𝑒−𝑟𝑇𝐾𝑄
and Φ is N(0,1) cumulative distribution function.
𝑑1= (ln𝐾+(𝑟+2)𝑇), 𝑑2=𝑑1−𝛼√𝑇
(b)Again taking 𝑛 = 105 paths, use R to calculate a Monte Carlo estimate of 𝐶 using Q as a control variate. Also calculate a 95% two-sided confidence interval for 𝐶. Do this for one of the following two options (making sure to specify your choice):
• 𝜌 = 0 [4 marks]
• 𝜌 = 2/3 [6 marks]. If you choose this option you must use Cholesky decomposition with R function chol to simulate correlated normal RVs (do not use R functions that generate correlated normal RVs as the additional two marks will not be awarded).
Question 4. Markov processes [12 marks]
You may use R or Mathematica for calculations, but express your answers using proper mathematical notation.
Let 𝑋𝑡 , 𝑡 ≥ 0, be a homogenous Markov chain taking states 𝑋𝑡 ∈ {1,2, … ,5} with generator matrix and initial distribution
−2 1/3 2/3 2/3 1/3 1/4
𝐴= 1 ⁄2 −3 ⁄6 ⁄6 , 𝑝(0)= ⁄2 ,
(1/9 2/9 5/9 1/9 −1 ) (1/4)
respectively.
(a) Find any stationary distributions of 𝑋𝑡 [3 marks]. (b)Calculate 𝐸[𝑋6] [3 marks].

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Consider the jump diffusion
𝑌𝑡=𝜇𝑡+𝜎𝐵𝑡+∑𝐻𝑘, 𝑡≥0, 𝑘=1
where 𝜇 ∈ R, 𝜎 > 0, 𝐵𝑡 is a standard Brownian motion, (𝑁𝑡)𝑡≥0 is a Poisson process with intensity 𝜆 > 0 and 𝐻𝑘 ~Exp(𝛼) for 𝑘 ∈ {1,2, … , 𝑁𝑡 }. Assume the SPs and RVs appearing in 𝑌𝑡 are independent of each other.
(c) Find 𝐸[𝑒𝑖𝑢𝑌𝑡 ] where 𝑖2 = −1 and 𝑢 ∈ R [3 marks]. (d)Noting that 𝐸[𝐻𝑘] = 1/𝛼, find 𝐸[𝑌8|𝑌6 = 2] [3 marks].
Question 5. ARMA processes [12 marks]
Consider the process
𝑋𝑡 −2𝑋𝑡−1 −3𝑋𝑡−2 + 5 𝑋𝑡−3 =𝑍𝑡 +17𝑍𝑡−1 −23𝑍𝑡−2 − 65 𝑍𝑡−3, 𝑡∈Z, 3 4 12 42 28 168
where 𝑍𝑡, 𝑡 ∈ Z, is a zero-mean white-noise process with variance 𝜎2.
(a)The process above is not stationary. Explain why and identify an appropriate
ARIMA(𝑝, 𝑑, 𝑞) model [3 marks].
(b) Describe how the ARIMA(𝑝, 𝑑, 𝑞) model from (a) could be converted to an ARMA(𝑝, 𝑞)
model [3 marks].
(c) Determine if the ARMA(𝑝, 𝑞) from (b) is invertible [3 marks].
(d) Plot the ARIMA(𝑝, 𝑑, 𝑞) and ARMA(𝑝, 𝑞) models identified in (a) and (b) respectively withvar(𝑍𝑡)=𝜎2 =2for𝑡∈{0,1,2,…,500}.UseMathematicaorRforthis[3marks].
Question 6. Diffusion processes [12 marks]
Consider the diffusion process
𝑋𝑡 =exp(𝑡+𝐵𝑡2), 𝑡≥0, where 𝐵𝑡 is a standard Brownian motion.
(a) Using the Ito formula, write down the stochastic integral equation of the process 𝑋𝑡 [3 marks].

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(b)Write down the Kolmogorov backward equation for 𝑋𝑡 [3 marks].
Now consider the standard geometric OU process
𝑍𝑡 =exp(𝑒−𝑡+𝑒−𝑡∫𝑒𝑢𝑑𝐵𝑢), 𝑡≥0, 0
where 𝐵𝑡 is a standard Brownian motion.
(c) Show that the transition density function of 𝑍𝑡 is given by
1 (log(𝑦) − 𝑒−(𝑡−𝑠)log (𝑥))2 𝑓(𝑦, 𝑡|𝑥, 𝑠) = 𝑦√𝜋(1 − 𝑒−2(𝑡−𝑠)) exp (− 1 − 𝑒−2(𝑡−𝑠) ).
(d)Show that the stochastic differential equation of the process 𝑍𝑡 is given by 𝑑𝑍𝑡 =(12−log(𝑍𝑡))𝑍𝑡𝑑𝑡+𝑍𝑡𝑑𝐵𝑡.

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