We consider a one-period market with the following properties: the current stock priceis S0 = 4. At time T = 1 year, the stock has either moved up to S1 = 8 (with probability0.7) or down towards S1 = 2 (with probability 0.3). We consider a call option on thisstock with maturity T = 1 and strike price K = 5. The interest rate on the money marketis 25% yearly.(a) Find the replicating portfolio (φ, ψ) corresponding to this call option.(b) Find the risk-neutral (no-arbitrage) price of this call option.(c) We now consider a put option with maturity T = 1 and strike price K = 3 onthe same market. Find the risk-neutral price of this put option. Reminder: A putoption gives you the right to sell the stock for the strike price K.1(d) An investor with initial capital X0 = 0 wants to invest on this market. He buysα shares of the stock (or sells them if α is negative) and buys β call options (orsells them is β is negative). He invests the cash balance on the money market (orborrows if the amount is negative).Prove that if the option price is the one found in (a), then the investor will neverhave an arbitrage opportunity. In other words, prove that if one of the possible finalvalues of his investment is positive, then the other has to be negative
We consider a one-period market with the following properties: the current stock price
is S0 = 4. At time T = 1 year, the stock has either moved up to S1 = 8 (with probability
0.7) or down towards S1 = 2 (with probability 0.3). We consider a call option on this
stock with maturity T = 1 and strike price K = 5. The interest rate on the money market
is 25% yearly.
(a) Find the replicating portfolio (φ, ψ) corresponding to this call option.
(b) Find the risk-neutral (no-arbitrage) price of this call option.
(c) We now consider a put option with maturity T = 1 and strike price K = 3 on
the same market. Find the risk-neutral price of this put option. Reminder: A put
option gives you the right to sell the stock for the strike price K.
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(d) An investor with initial capital X0 = 0 wants to invest on this market. He buys
α shares of the stock (or sells them if α is negative) and buys β call options (or
sells them is β is negative). He invests the cash balance on the money market (or
borrows if the amount is negative).
Prove that if the option price is the one found in (a), then the investor will never
have an arbitrage opportunity. In other words, prove that if one of the possible final
values of his investment is positive, then the other has to be negative
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