I need help on my homework for financial engineering, there are 6 questions on black scholes, risk neutral probability, n steps binomial model etc.

I'd like to understand how to calculate the probability of lognormal stock price.

Mathematics of Finance.,UA 250 Homework #6, Spring 2018 7 point for each problem 1 Suppose a stock price is lognormal with drift ,u and volatility a, in other words 803) = s(0)e"'+aw(t). Given K > D , let PK be the probability that 5(T) 2 K. (a) Express PK as the value of a particular integral. (b) Give an expression for PK in terms of the cumulative normal distribution N 0 2 To show that lognormal price dynamics is the continuum limit of a family of binomial trees, we considered the following multiperiod multiplicative tree: 0 the stock price at the initial node is 8(0); I if at a given node the stock price is Snow, then at the next time it can be either MSW", or dSnow, where u 7 .'zt'f-'t'l'rrm and el = EFAWAI; I at each timeJ the stock goes up or down with probability 1/2 for each choice. Fixing NJ consider the stock price after N lilJIlD stepsJ 3(NAt). (a) Suppose that over the course of the N time steps the stock goes upj times and down Nj times. Express lnS(NAt) in terms 3'. (b) What is the mean of lnSUVAt)? (c) What is the variance of lnS(NAt)? 3 Suppose a stock price is lognormal with volatility (I. Consider a derivative with maturity T and payon f(s(T)) : 33(T) . (a) What is its value at time D? (b) What is the Delta of the option considered in this Problem? (Hint: your task is to evaluate erTEE? N331). Recall that under the riskneutral probability distribution, ST is lognormal, and therefore 3'} is also lognormal. Use the fact that if Z is Gaussian with mean m and standard deviation .9 then E[BZ] = em'l'isg. 712) an : 7.44% rm" = 6.17% 1(2):"; = 5.09% r0), = 5.05% T{2)dd = 4.99% Problems 4 and 5 relate to interest rate derivatives (Chapt 9 Section 6) 4 Consider the binomial tree of interest rates shown in the gure (each time interval is one year, and the rates shown are per annum with continuous compounding). Assume the riskneutral probabilities are 1/2 for each branch. (a) Find the values of 3(0, 1), B(0, 2), and B(0,3). (b) Consider the following European call option written on a one year Treasury bill: its maturity is T = 2, and its strike is 0.945, so the payo at time 2 is (3(2, 3) D.945)+. Find the value of this Option at time 0