Question 1
(a) Put-Call Parity without dividends
We assume that Put-Call Parity does not hold:
CEUt=PEUt+St−PVTt(K)
Now, let’s construct an arbitrage. Buy a European call option, sell a European put option, and borrow the present value of the strike price. The payoff at maturity T is:
(CEUT−PEUT)+(ST−K)−(K−PVTT(K))
The second term simplifies to ST−K, and the last term cancels with the borrowed money. Thus, the payoff is CEUT−PEUT. Since CEUT≥max(ST−K,0) and PEUT≥max(K−ST,0), it follows that CEUT−PEUT≥0. Therefore, there is a guaranteed profit without any possibility of loss, which contradicts the assumption.
(b) Early Exercise of American Call Option
Assume CAMt>St−K does not hold. This implies CAMt≤St−K. If an American call is not exercised early, it is equivalent to holding a European call. Hence, CEUt≤St−K.
Now, buy a European call, sell a stock, and invest the proceeds in a risk-free bond. The payoff at maturity T is CEUT−(ST−K), which is non-negative. This contradicts the fact that CEUt≤St−K, proving that the assumption is false.
Question 2
(a) Pricing European and American Call Options
Use the binomial model to calculate the option prices at each node and each time step. For American options, compare the option value with the intrinsic value at each step to determine if early exercise is optimal.
(b) Pricing European and American Put Options
Similar to part (a), apply the binomial model to calculate put option prices. Compare the option value with intrinsic value for American puts.
(c) Put-Call Parity
Verify Put-Call Parity for European options at =0t=0 and nodes at =1t=1. For American options, consider the possibility of early exercise and adjust accordingly.
(d) Exotic Put Option
Price the exotic put option using the given strike price. No early exercise is allowed.
(e) Derivative Obliging Sale
Price the derivative that obliges selling the stock at =2t=2 for the maximum of stock prices up to that point.