Question 4 This is a example of a two factor model. A financial forward contract is called a quanto product if it is denominated in a currency other than that in which it is traded. AMP, an Australian company has an AUD denominated stock price that we denote by {S}r20. For an USD investor, a quanto forward contract on AMP stock with maturity T has payoff (S - K) converted into USD according to some prearranged exchange rate. This is, the payoff will be E(ST -K) for some preagreed E, where Sr is the AUD asset price at time T. Assume that there is a riskless cash bond in each of the USD and AUD markets, but you have two random processes to model, the stock price St and the exchange rate, that is the value of one AUD in USD which you will denote by {E} Then the Black-Scholes quanto model is USD bondB ert AUD bondD So exp(t+oW) Eo exp(At +pozW AUD asset price S vi- pozW?) Exchange rate E where W and W are independent P-Brownian motions and r, u, v, X, o1, O2 and pare constants. In this model, the volatilities of S, and E are o and o respectively and (W. pW -W?} is a pair of correlated Brownian motions with correlation coefficient p. There is no extra generality in replacing the expressions for St and E by St=So exp(ut+onW} +012W?) E=Eo exp(At+ o1W + 22W?) and W? for independent Brownian motions W (a) What is the value of K that makes the value at time zero of the quanto forward contract zero? (b) A quanto call option write on AMP stock is worth E(Sr - K)+ USD at time T, where Sr is the AUD stock price. Assuming this B-S quanto model, find the time zero price of the option and the replicating portfolio. Question 4 This is a example of a two factor model. A financial forward contract is called a quanto product if it is denominated in a currency other than that in which it is traded. AMP, an Australian company has an AUD denominated stock price that we denote by {S}r20. For an USD investor, a quanto forward contract on AMP stock with maturity T has payoff (S - K) converted into USD according to some prearranged exchange rate. This is, the payoff will be E(ST -K) for some preagreed E, where Sr is the AUD asset price at time T. Assume that there is a riskless cash bond in each of the USD and AUD markets, but you have two random processes to model, the stock price St and the exchange rate, that is the value of one AUD in USD which you will denote by {E} Then the Black-Scholes quanto model is USD bondB ert AUD bondD So exp(t+oW) Eo exp(At +pozW AUD asset price S vi- pozW?) Exchange rate E where W and W are independent P-Brownian motions and r, u, v, X, o1, O2 and pare constants. In this model, the volatilities of S, and E are o and o respectively and (W. pW -W?} is a pair of correlated Brownian motions with correlation coefficient p. There is no extra generality in replacing the expressions for St and E by St=So exp(ut+onW} +012W?) E=Eo exp(At+ o1W + 22W?) and W? for independent Brownian motions W (a) What is the value of K that makes the value at time zero of the quanto forward contract zero? (b) A quanto call option write on AMP stock is worth E(Sr - K)+ USD at time T, where Sr is the AUD stock price. Assuming this B-S quanto model, find the time zero price of the option and the replicating portfolio
