Question 1:

(a). Let Xt= the position (distance) of a particle at time t, with X0=0. Suppose the particle moves one step forward or backward with equal probability at each instant of time and successive steps are independent.

If this movement is repeated many times, find the standard deviations of the distance Xt when t=1,2,3 showing your working.

(b). The binomial tree method for the derivative price uses the formula V= e^-rt[qU+(1-q)D] where q=e^rt-d/u-d with the usual representation of the variables.

The current price of a stock is £20 and over each of the next three periods, the price is expected to go up by 20% or down by 10%. The risk-free interest rate is 3% per period and is compounded continuously. If the strike price of an option is £22:

• show the stock price at each node of the tree;
• calculate the pricing probability;
• draw a tree diagram for the option price clearly giving the price of the American put option at each of the nodes, showing how these values are calculated.