Q$2.$

Consider a binomial tree model for the non$-$dividend paying stock with price $S_t.$ Assume this price either rises by $30\%$ or falls by $20\%$ each quarter $(3$ months$)$ for the next three quarters. Assume also that the risk$-$free rate is $2\%$ per annum continuously compounded. Let $S_0=$ R$60.$

$($i$)$ Calculate the price of a European call option with maturity in nine months' time and a strike price of R$55.[3]$

$($ii$)$ Calculate the price of a European put option with the same maturity and strike price as the contract in part $($i$).[1]$

Assume the investor has a portfolio formed by a short position in the call option given in part $($i$)$ and a long position in the put option given in part $($ii$).$

$($iii$)$ Determine how the value of the portfolio would differ if the possible change in the stock price was a fall of $30\%$ instead of $20\%.$