Consider the Vasicek model for the short rate
dr(t)=(b−ar(t))dt+γdW1(t)
and the Black-Scholes-Merton model for a stock S
dS(t)=r(t)S(t)dt+σS(t)dW2(t)
where W1 and W2 are Brownian motions under the risk-neutral probability, and they have correlation ρ. Let P(T,U) be the price at time T of the risk-free U-bond, and let C(T)=g(S(T),P(T,U)) be a claim whose payoff is a function g of the stock value S(T) and of P(T,U), where T is the maturity of the payoff C(T), and U>T.
Which of the following is a PDE for its price?
Ct+1/2*σ^2*s^2*Css+r*(s*Cs−C)+1/2γ^2*Crr+(b−a*r)*Cr+ρ*γ*σ*s*Crs=0-correct
Which of the following is the corresponding boundary condition?
C(T,s,r)=g*(s,e^(A(T,U)−B(T,U)r)) for appropriate deterministic functions A(T,U) and B(T,U)-correct
2 answers
Just wait for another day; the solution will be posted online.
I have the following variation, please help:
Consider the Cox-Ingersoll-Ross (CIR) model for the short rate
dr(t)=(b−ar(t))dt+γr(t)dW1(t)
and the Black-Scholes-Merton model for a stock S
dS(t)=r(t)S(t)dt+σS(t)dW2(t)
where W1 and W2 are Brownian motions under the risk-neutral probability, and they have correlation ρ. Let P(T,U) be the price at time T of the risk-free U-bond, and let C(T)=g(S(T),P(T,U)) be a claim whose payoff is a function g of the stock value S(T) and of P(T,U), where T is the maturity of the payoff C(T), and U>T.
Which of the following statement is true about the CIR model?
a. It's a mean reversion process and the interest rate can be negative
b. It's a mean reversion process and the interest rate cannot be negative
c. It's not a mean reversion process and the interest rate can be negative
d. It's not a mean reversion process and the interest rate cannot be negative
ii) Which of the following is a PDE for its price?
option a. Ct+12σ2s2Css+r(sCs−C)=0
option b. Ct+12σ2s2Css+r(sCs−C)+12γ2rCrr+(b−ar)Cr+ργσsrCrs=0
option c. Ct+12σ2s2Css+r(sCs−C)+12γ2Crr+(b−ar)Cr=0
option d. Ct+12σ2s2Css+r(sCs−C)+12γ2Crr+(b−ar)Cr+ργσsCrs=0
Consider the Cox-Ingersoll-Ross (CIR) model for the short rate
dr(t)=(b−ar(t))dt+γr(t)dW1(t)
and the Black-Scholes-Merton model for a stock S
dS(t)=r(t)S(t)dt+σS(t)dW2(t)
where W1 and W2 are Brownian motions under the risk-neutral probability, and they have correlation ρ. Let P(T,U) be the price at time T of the risk-free U-bond, and let C(T)=g(S(T),P(T,U)) be a claim whose payoff is a function g of the stock value S(T) and of P(T,U), where T is the maturity of the payoff C(T), and U>T.
Which of the following statement is true about the CIR model?
a. It's a mean reversion process and the interest rate can be negative
b. It's a mean reversion process and the interest rate cannot be negative
c. It's not a mean reversion process and the interest rate can be negative
d. It's not a mean reversion process and the interest rate cannot be negative
ii) Which of the following is a PDE for its price?
option a. Ct+12σ2s2Css+r(sCs−C)=0
option b. Ct+12σ2s2Css+r(sCs−C)+12γ2rCrr+(b−ar)Cr+ργσsrCrs=0
option c. Ct+12σ2s2Css+r(sCs−C)+12γ2Crr+(b−ar)Cr=0
option d. Ct+12σ2s2Css+r(sCs−C)+12γ2Crr+(b−ar)Cr+ργσsCrs=0
2 answers