1) (35 pts) Consider the Brownian bridge process (BBP) of the form X = W-tW At t=0.N calculate the probabilities a) Pr(x < 0.2) b) Pr(0.15 < x < 0.25) c) Pr(rt > 0.3) 2) (35pts) The Vasicek model of stochastic interest rate r, is given by stochastic differential equation Tt Let drt B (a-r) dt + odWt = a = 1, To = 0.2, B = 0.N and = 0.5. At t = 1 calcu- late the probabilities a) Pr(r 0.1) b) Pr(0.15 < r < 0.2) c) Pr(rt > 0.25) 3)The Black-Karasinski model of stochastic interest gion by stochastic differential equation (SDE) 3)The Black-Karasinski model of stochastic interest rater is given by stochastic differential equation drt=rt A+ 4+ 1 B - C' lnr.) dt + BrdW where W, is a Wiener process and A, B and C are con- stants. Find the interest rate rt as a function of HINT Use H = lnr. AdW. QUESTION 4 Not yet answered Marked out of 1.00 Flag question You model a stock price S(t) using a stochastic process, with t measured in years. Your model implies that the risk-neutral distribution for the stock price at t = 3 is 45 with probability 0.5, S(3) 55 80 with probability 0.3, with probability 0.2. Assume that interest is compounded annually at nominal rate r = 2%. Calculate the no-arbitrage price of a European call option with strike price 50 and expiry date T = 3. State your answer to the nearest pence. Answer: