The price of a US stock is given by

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))

Answer 2

(i) The correct selection of Brownian motions W∗1 and W∗2 is:

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

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

e^(-r*(T-t)) * (log(Q^2(t) * S(t)) + (r + 2*A)*(T-t))

Answer 3

To understand how to get the answer to this question, let's break it down step by step:

(i) To make the discounted dollar value of the euro, e^(-rt)Q(t)e^(rf)t, a martingale, we need to find a suitable Brownian motion, W∗1(t), such that the term dQ(t)/Q(t) can be expressed as (βdt+δdW2(t)). In this case, we replace dt with (μ-r)/σ * dt and W1(t) with W∗1(t) to obtain:

dQ(t)/Q(t) = βdt + δdW2(t)
= βdt + δdW∗2(t)

Therefore, W∗1(t) = (μ-r)/σ * t + W1(t).

Next, we want the discounted dollar value of the domestic stock, e^(-rt)S(t), to be a martingale. Using the same reasoning as before, we replace dt with (μ-r)/σ * dt and dW1(t) with dW∗1(t) to obtain:

dS(t)/S(t) = μdt + σdW1(t)
= μdt + σdW∗1(t)

Therefore, the discounted dollar value of the domestic stock is a martingale under the corresponding probability P∗.

(ii) For the exotic payoff C(T) = log(Q^2(T)S(T)) dollars, we already know the values of constants A and B from the previous problem. To determine the domestic price of this claim, we calculate:

C(T) = log(Q^2(T)S(T))
= log((Q(t)e^(A(T-t))+B(W∗2(T)-W∗2(t)))^2 * S(t))
= log((Q^2(t)e^(2A(T-t))+2B(W∗2(T)-W∗2(t))Q(t)e^(A(T-t))+B^2(W∗2(T)-W∗2(t))^2) * S(t))
= log(Q^2(t)) + 2A(T-t)log(e) + 2B(W∗2(T)-W∗2(t))Q(t)e^(A(T-t)) + log(B^2(W∗2(T)-W∗2(t))^2) + log(S(t))

Since log(Q^2(t)), 2A(T-t)log(e), log(B^2(W∗2(T)-W∗2(t))^2), and log(S(t)) are constant terms, we can rewrite the equation as:

C(T) = log(Q^2(t)) + 2A(T-t)log(e) + 2B(W∗2(T)-W∗2(t))Q(t)e^(A(T-t)) + log(S(t))
= log(Q^2(t)S(t)) + 2A(T-t)log(e) + 2B(W∗2(T)-W∗2(t))Q(t)e^(A(T-t))

Finally, we need to express the domestic price of the claim C(T) in terms of e^(-r(T-t)). Applying the exponential discount factor, we have:

e^(-r(T-t)) * (C(T))
= e^(-r(T-t)) * (log(Q^2(t)S(t)) + 2A(T-t)log(e) + 2B(W∗2(T)-W∗2(t))Q(t)e^(A(T-t)))

Using the fact that log(e) = 1, we simplify to:

e^(-r(T-t)) * (log(Q^2(t)S(t)) + 2A(T-t) + 2B(W∗2(T)-W∗2(t))Q(t)e^(A(T-t)))
= e^(-r(T-t)) * (logQ^2(t)S(t) + 2A(T-t)) + 2B(W∗2(T)-W∗2(t))(e^(A(T-t)))Q(t)

Therefore, the domestic price of the claim C(T) is given by:

e^(-r(T-t)) * (logQ^2(t)S(t) + 2A(T-t)) + 2B(W∗2(T)-W∗2(t))(e^(A(T-t)))Q(t)

Please note that this answer assumes that the calculations and substitutions have been done correctly. It is always a good idea to double-check the math and ensure the validity of any assumptions made.