Q$1.$

Consider a three$-$period binomial tree model for a stock price process $S_t,$ under which the stock price either rises by $18\%$ or falls by $15\%$ each month. No dividends are payable.

The continuously compounded risk$-$free rate is $0\mathrm{.}25\%$ per month.

Let $S_0=$ R $85.$

Consider a European put option on this stock, with maturity in three months $($i$.$e$.$ at time $t=3)$ and strike price R$90.$

$($i$)$ Calculate the price of this put option at time $t=0.[4]$

$($ii$)$ Calculate the risk$-$neutral probability that the put option expires out$-$of$-$the money. $[3]$

$($iii$)$ Assess whether the probability calculated in part $($ii$)$ would be higher or lower under the real$-$world probability measure. $[$No further calculation is required.$]3$

Q$2.$

Consider a binomial tree model for the non$-$dividend paying stock with price $St.$ 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 $20\%.$

Q$3$

Consider a one$-$period binomial tree model for the stock price process $S_t.$

Let $S_0=$ R $100$ and assume that in three months' time the stock price is either R$125$ or R$105.$ No dividends are payable on this stock.

Assume also that the continuously compounded risk$-$free rate is $5\%$ per annum.

$($i$)$ Verify that this market is not arbitrage$-$free by considering the relationship between the risk$-$free rate and the stock price movements. $[2]$

$($ii$)($a$)$ Identify a portfolio which would generate an arbitrage profit.

$($b$)$ Calculate this profit.

$4$

Now assume that the continuously compounded risk$-$f rate is $20\%$ per annum. Consider a European put option on this stock, expiring th three months' time and with strike price $K=R120.$

$($iii$)$ Calculate the current price of this put option.

ANS

Please do question $3$