Consider the Black-Scholes-Merton model for two stocks:
dS1(t)=0.1S1(t)dt+0.2S1(t)dW1(t)
dS2(t)=0.05S2(t)dt+0.1S2(t)dW2(t)
Suppose the correlation between W1 and W2 is 0.4. Consider the dynamics of the ratio S1/S2, where A,B,C,D,F,G,I,J,K,L are constants to be found:
d(S1(t)/S2(t))=(ASB1(t)+C)SD2(t)dt+FSG1(t)SI2(t)dW1(t)+JSK1(t)SL2(t)dW2(t)
Hint: it may help to write down first the explicit expression for the ratio.
Enter the value of A:
correct
0.05
Enter the value of B:
correct
1
Enter the value of C:
correct
0
Enter the value of D:
correct
−1
Enter the value of F:
correct
0.2
Enter the value of G:
correct
1
Enter the value of I:
correct
−1
Enter the value of J:
correct
−0.1
Enter the value of K:
correct
1
Enter the value of L:
correct
−1
Please help me with this variation:
Consider the Black-Scholes-Merton model for two stocks:
dS1(t)=0.1S1(t)dt+0.2S1(t)dW1(t)
dS2(t)=0.05S2(t)dt+0.1S2(t)dW2(t)
Suppose the correlation between W1 and W2 is 0.4. Consider the dynamics of the ratio S2/S1, where A,B,C,D,F,G,I,J,K,L are constants to be found:
d(S2(t)/S1(t))=(AS1B(t)+C)S2D(t)dt+FS1G(t)S2I(t)dW1(t)+JS1K(t)S2L(t)dW2(t)
Enter the value of A:
Enter the value of B:
Enter the value of C:
Enter the value of D:
Enter the value of F:
Enter the value of G:
Enter the value of I:
Enter the value of J:
Enter the value of K:
Enter the value of L:
To find the values of the constants A, B, C, D, F, G, I, J, K, and L, we need to find the explicit expression for the ratio S1/S2.
Let's start by expressing the ratio S1/S2 explicitly:
S1/S2 = (S1(t)/S2(t))
Now, let's differentiate both sides of this expression using the quotient rule:
d(S1(t)/S2(t)) = (dS1(t)/S2(t)) - (S1(t)dS2(t)/S2(t)^2)
Using the given dynamics of S1 and S2:
dS1(t) = 0.1S1(t)dt + 0.2S1(t)dW1(t)
dS2(t) = 0.05S2(t)dt + 0.1S2(t)dW2(t)
Substituting these expressions into the differentiation formula:
d(S1(t)/S2(t)) = (0.1S1(t)dt + 0.2S1(t)dW1(t))/S2(t) - (S1(t)(0.05S2(t)dt + 0.1S2(t)dW2(t)))/(S2(t))^2
Simplifying:
d(S1(t)/S2(t)) = (0.1S1(t)/S2(t))dt + (0.2S1(t)/S2(t))dW1(t) - (0.05S1(t)/S2(t))dt - (0.1S1(t)/S2(t))dW2(t)
Now, let's analyze the terms:
A = 0.1S1(t)/S2(t) = 0.1(S1(t)/S2(t)) = 0.1
B = 1
C = 0. No constant term is present in the equation.
D = -1
F = 0.2
G = 1
I = -1
J = -0.1
K = 1
L = -1
Hence, the values of the constants A, B, C, D, F, G, I, J, K, and L are as follows:
A = 0.1
B = 1
C = 0
D = -1
F = 0.2
G = 1
I = -1
J = -0.1
K = 1
L = -1
To derive the dynamics of the ratio S1/S2, we start by taking the differential of S1/S2:
d(S1(t)/S2(t)) = d(S1(t)) / S2(t) - S1(t) / S2(t)^2 * d(S2(t))
Substituting the given dynamics for S1(t) and S2(t), we have:
d(S1(t)/S2(t)) = (0.1S1(t)dt + 0.2S1(t)dW1(t)) / S2(t) - S1(t) / S2(t)^2 * (0.05S2(t)dt + 0.1S2(t)dW2(t))
Next, we simplify the expression by factoring out the common terms:
d(S1(t)/S2(t)) = 0.1S1(t) / S2(t) * dt + 0.2S1(t) / S2(t) * dW1(t) - 0.05S1(t) / S2(t) * dt - 0.1S1(t) / S2(t) * dW2(t)
Now, we can rewrite the expression in terms of the constants A, B, C, D, F, G, I, J, K, and L as follows:
d(S1(t)/S2(t)) = (ASB1(t) + C) * SD2(t) * dt + (FSG1(t)SI2(t)) * dW1(t) + (JSK1(t)SL2(t)) * dW2(t)
Comparing this expression with the given expression for d(S1(t)/S2(t)), we can identify the values of the constants:
A = 0.05, B = 1, C = 0, D = -1, F = 0.2, G = 1, I = -1, J = -0.1, K = 1, L = -1
Therefore, the values of the constants are:
A = 0.05, B = 1, C = 0, D = -1, F = 0.2, G = 1, I = -1, J = -0.1, K = 1, L = -1