A. Consider a two-period binomial model for a non-dividend-paying share whose current price is 0 = 100. Over each six-month period, the share price can either move up by a factor = 1.2 or down by a factor of = 0.8. The risk-free rate is = 5% per six- month period.

1A) Verify there is no arbitrage in the market

2A) Consider a modified call option where, the underlying asset price at maturity is the arithmetic average of share prices, denoted , at times 0, 0.5 and 1 measured in years, with a strike price of 100. That is, the payoff at maturity is given by. max{ 100, 0}

Calculate the initial price of this call option, assuming it can only be exercised at maturity.

B. Let follow the Ornstein-Uhlenbeck process defined by

_t =0.1(0.2 _t )+0.4_t,

where is a standard Brownian motion. Let _t = 1/X_t . Given that _t follows the stochastic differential equation

_t = (_t)+ (_t)_t

for some functions () and (). Calculate the value of (1) and (1).

C. State the SDEs under the risk-neutral measure for (), the default-free instantaneous rate of interest at time , under the two models, defining all notation used: (1)Hull and White and (2)Two-factor Vasicek. Also, state the advantages of the Hull-White model over the single factor Vasicek model.