Q$3)(35$ marks$)$

Suppose that a stock price is modelled by a two period recombining binomial model, where each period is six months, with the following parameters: risk free interest rate $r=3\%$ p$.$a$.$; volatility $=10\%$ p$.$a$.$; initial share price $S_0=100.$

The up step is given by $u=e^{(\frac{0\mathrm{.}1}{\sqrt[\phantom{2}]{2}})}.$

Consider a European call option, expiring in one year's time with exercise price $90.$

$($i$)$ Construct a two period recombining tree, showing the share price at each node, and derive the risk neutral probabilities assuming that no dividends are payable. $[5]$

$($ii$)$ Using the binomial tree in $($i$),$ determine the value of the option. $[5]$

Assume now that a dividend of $20$ is payable immediately before the expiry of the option.

$($iii$)($a$)$ Write down the share prices at the final three nodes of the binomial tree in $($ii$)$ assuming the same total return as before.

$5$

$($b$)$ Write down the payoff at each of these three nodes of the option.

$5$

$($iv$)$ Using the risk neutral probabilities from $(1)$ determine the value of the payoff under the option at the end of the first time period, assuming the share price has jumped up in the first time period. $[2]$

$($v$)$ Suppose now that the option is American, not European.

$($a$)$ Determine the payoff at the node described in $($iv$)($i$.$e$.$ at the end of the first time period, assuming the share price has moved up in the first time period$).$

$($b$)$ Using $($a$),$ determine whether the holder of the American option should exercise early at the node described in $($iv$).[5]$

$2$

$($c$)$ State with reasons whether the value at time zero of the American option is either:

$(1)$ strictly less than

$(2)$ greater than or equal to

$(3)$ strictly greater than

the value of the European option. $[3]$

Please do question $3$