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Q 1 . Consider a three - period binomial tree model for a stock price process S t , under which the stock price either

Q1.
Consider a three-period binomial tree model for a stock price process St, 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 0.25% per month.
Let S0= R 85.
Consider a European put option on this stock, with maturity in three months (i.e. at time t=3) and strike price R90.
(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
Q2.
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 S0= R 60.
(i) Calculate the price of a European call option with maturity in nine months' time and a strike price of R55.[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%.
Q3
Consider a one-period binomial tree model for the stock price process St.
Let S0= R 100 and assume that in three months' time the stock price is either R125 or R105. 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 1
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