Consider a European call option X and the corresponding put option Y on a stock with...

 

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Consider a European call option X and the corresponding put option Y on a stock with price process S,. Let II(t; X) and II(t;Y) be separately the prices of the two options at time t. Both options have maturity T and strike-price K. Let the price at time t of the bond that pays 1 at maturity T be p(t,T). We assume that there exists a sufficiently rich and regular bond market. (a) Prove the put-call parity: II(t;Y)=[(t;X) + p(t,T)K-S. (b) Assume that today (t =0), two persons A and B sign a contract: A borrows 100 from B in one year (at t = 1) and repays 104 to B one more year later (at t = 2). On the bond market, we can find the prices: p(0,1)=0.967, p(0,2)=0.935. What is the value of this contract for lender B at t=0 ? (c) Consider a type of coupon bond: at time T, i = 1, n, the owner (buyer) of the coupon bond receives the coupon payment with fixed return r (which means, coupons are determined in terms of return) and the face value is K. At time T, the owner receives also a face value K, in addition to the coupon payment. Determine r so that the price of this coupon bond at time To is equal to the face value K. Consider a European call option X and the corresponding put option Y on a stock with price process S,. Let II(t; X) and II(t;Y) be separately the prices of the two options at time t. Both options have maturity T and strike-price K. Let the price at time t of the bond that pays 1 at maturity T be p(t,T). We assume that there exists a sufficiently rich and regular bond market. (a) Prove the put-call parity: II(t;Y)=[(t;X) + p(t,T)K-S. (b) Assume that today (t =0), two persons A and B sign a contract: A borrows 100 from B in one year (at t = 1) and repays 104 to B one more year later (at t = 2). On the bond market, we can find the prices: p(0,1)=0.967, p(0,2)=0.935. What is the value of this contract for lender B at t=0 ? (c) Consider a type of coupon bond: at time T, i = 1, n, the owner (buyer) of the coupon bond receives the coupon payment with fixed return r (which means, coupons are determined in terms of return) and the face value is K. At time T, the owner receives also a face value K, in addition to the coupon payment. Determine r so that the price of this coupon bond at time To is equal to the face value K.

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a To prove the putcall parity we start with the portfolio consisting of a long call option t X and a ... View the full answer