FINANCIAL MODELS
Homework 8, due Thursday Nov. 1st
Use of Excel spreadsheet or MATHLAB is recommended.

Average (Asian) option:

1- Current price of a non-dividend paying stock is S(0) = 50. Stock’s annual volatility is  σ = 30%  and the continuously compounded interest rate is 5%.  Consider an Average (Asian) call option with strike of 51 that matures in one year. In other words, the payoff at maturity is [Max  (Avearage of Prices During the Past Year – 51 , 0)]. Let Δt = 1/250 (there are approximately 250 trading days in a year) and assume incremental daily price changes have the following process

ΔS/S = ( 0.05 – 0.5 σ2 ) Δt + (σ √Δt ) dW(t) in other words at each increment of time 

[S(t+ Δt) – S(t)]/S(t)  has a normal distribution with mean  = ( 0.05 – 0.5 σ2 ) Δt and Variance =  σ2 Δt 

a)- Use a 1000 realization of the process to estimate the price of the call.

b)-Use a 1000 realization of the process to plot a histogram of the average price of the stock.

c)-Use a 1000 realization of the process to estimate the mean and standard deviation of the average price. 

2- Give and example of random variables Xn that converge to X in distribution but not in  probabilty.

3- If Xn converges to X in probability does it imply that E[Xn] also converges to E[X]? Either prove this statement or give a counter example.