Question

The price of a US stock is given by

dS(t)/S(t)=μdt+σdW1(t)

The exchange rate Dollar/Euro is given by

dQ(t)/Q(t)=βdt+δdW2(t)

where W1 has correlation ρ with W2.

>> (i) Select the Brownian motions W∗1 and W∗2 such that the discounted dollar value of the euro e−rtQ(t)erft is a martingale, and the discounted dollar value of the domestic stock e−rtS(t) is also a martingale under the corresponding probability P∗, where r and rf are the US and the Euro risk-free interest rates:

W∗1(t)=(μ−r)/σ*t+W1(t); W∗2(t)=(β+rf−r)/δ*t+W2(t)-correct

Next, consider a (very exotic, unrealistic) payoff C(T) in the amount C(T)=log(Q2(T)S(T)) dollars. Suppose you know from the previous problem the values of constants A,B such that

Q(T)=Q(t)eA×(T−t)+B×(W∗2(T)−W∗2(t))

(ii) The domestic price of the claim C(T) is equal to:

e^−r*(T−t) * (logQ^2(t)*S(t)+(r+2*A)(T−t))-correct

"answered in full"

Answer 1

Please help with this variation:

The price of a US stock is given by

dS(t)/S(t)=μdt+σdW1(t)
The exchange rate Dollar/Euro is given by

dQ(t)/Q(t)=βdt+δdW2(t)
where W1 has correlation ρ with W2.

(i) Select the Brownian motions W1∗ and W2∗ such that the discounted dollar value of the euro e−rtQ(t)erft is a martingale, and the discounted squared dollar value of the domestic stock e−rtS(t) is also a martingale under the corresponding probability P∗, where r and rf are the US and the Euro risk-free interest rates:

Option a. W1∗(t)=μ−rσt+W1(t); W2∗(t)=β−rσt+W2(t)
Option b. W1∗(t)=μ−rσt+W1(t); W2∗(t)=β−rfδt+W2(t)
Option c. W1∗(t)=μ−rf+ρσδσt+W1(t), W2∗(t)=β+rf−rδt+W2(t)
Option d. W1∗(t)=μ−rσt+W1(t); W2∗(t)=β+rf−rδt+W2(t)

Next, consider a (very exotic, unrealistic) payoff C(T) in the amount C(T)=log⁡(Q3(T)) dollars. Suppose you know from the previous problem the values of constants A,B such that

Q(T)=Q(t)eA×(T−t)+B×(W2∗(T)−W2∗(t))

(ii) The domestic price of the claim C(T) is equal to: (you need to solve A, B)

Option a. e−r(T−t)Q3(t)N(d1)
Option b. e−r(T−t)(log⁡Q3(t)+3(μ−12)σ2(T−t)))
Option c. e−r(T−t)(log⁡Q3(t)+3(r−rf−12δ2)(T−t))
Option d. e−r(T−t)(log⁡Q3(t)+3(r−12δ2)(T−t))