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
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
The PDE for the price of the claim C(T) is given by:
Ct + 1/2 * σ^2 * s^2 * Css + r * (s * Cs - C) + 1/2 * γ^2 * Crr + (b - a * r) * Cr + ρ * γ * σ * s * Crs = 0
where Ct is the partial derivative of C with respect to time t, Cs is the partial derivative of C with respect to stock price s, Cr is the partial derivative of C with respect to short rate r, and Crs is the mixed partial derivative of C with respect to both stock price s and short rate r.
The corresponding boundary condition is:
C(T, s, r) = g*(s, e^(A(T, U) - B(T, U) * r))
where g* is an appropriate function of s and e^(A(T, U) - B(T, U) * r), and A(T, U) and B(T, U) are deterministic functions of maturity T and bond maturity U.
To determine which is the PDE for the price of the claim and the corresponding boundary condition, we can apply the risk-neutral pricing approach.
The first step is to set up the risk-neutral dynamics of the claim, considering the Vasicek model for the short rate and the Black-Scholes-Merton model for the stock.
The risk-neutral dynamics of the claim's price C are given by:
dC(t) = (r(t)Cₛₜ - r(t)C + 1/2 σ²Cₛₛ + (b - a*r(t))Cₛ + ργσsCᵣₛ)dt + γCᵣdW₁(t)
where:
- Cₛₜ is the partial derivative of C with respect to the stock S
- Cₛₛ is the second partial derivative of C with respect to the stock S
- Cᵣₛ is the mixed partial derivative of C with respect to the stock S and the short rate r
- Cᵣ is the partial derivative of C with respect to the short rate r
- γ, σ, a, b are model parameters.
This equation represents the PDE for the price of the claim.
Now, for the corresponding boundary condition, we need to consider the claim's payoff at maturity T. The boundary condition is given by:
C(T, s, r) = g(s, P(T, U))
where g is a function of the stock value S at time T and the price of the risk-free U-bond at time T, P(T, U).
In the boundary condition, we have the deterministic functions A(T, U) and B(T, U) that determine the relationship between the bond price and the short rate. The boundary condition ensures that the claim's price depends on the final values of the stock and the bond.
Therefore, the correct choices for the PDE and the corresponding boundary condition are:
PDE: Ct + 1/2 σ²s²Css + r(sCs - C) + 1/2 γ²Crr + (b - a*r)Cr + ργσsCrs = 0
Boundary condition: C(T, s, r) = g*(s, e^(A(T, U) - B(T, U)r))