1. You are using Vasicek model as the short rate process that is calibrated as: dr(t) = a(b-r(t)) dt + odz(t) Where a = 0.3, b = 0.05,0 = 0.03 And where Z(t) is a Wiener process under the risk neutral framework modelling the random market risk factor. a. Use the above model to calculate the analytical price of a 5-year Zero-coupon (with a face value of $1) bond assuming that instantaneous short rate r(0) = 0.08. b. Using Euler discretization (as taught in class) and Monte Carlo simulation technique, calculate the above 5-year Zero-coupon bond price assuming that instantaneous short rate r(0) = 0.08. Use appropriate time step and number of simulations to minimize the MC error. c. Calculate the swap rate (fixed rate leg) of an interest rate siwap that has a maturity of 5 years and the floating leg has an annual coupon payment frequency linked to LIBOR. [Hint: Note the price of a swap is given as: 1-Z(N) Swap Rate=N EZ(i) price for maturity i. Where Z(i) is the zero coupon d. Calculate the price of a European Call option on 5-year zero- coupon bond (with a face value of $1000) and a maturity of 4 years and Strike price of $900 using the Vasicek model that has been calibrated as above. Note this solution allows for simulation of negative interest rates that is a known limitation of Vasicek model. [Hint: The payoff of call option is: Max (Z [t,T] - K,0), and you may ignore the discounting of payoff or use the 4 year zero spot rate to discount the payoff. And also note that Z [t (= 4y), T (= 5y)] can be computed using exponential affine formula where the t = 4y and r(t-4y) can be simulated using same steps as Part b above.]
1. You are using Vasicek model as the short rate process that is calibrated as: dr(t) = a(b-r(t)) dt + odz(t) Where a = 0.3, b = 0.05,0 = 0.03 And where Z(t) is a Wiener process under the risk neutral framework modelling the random market risk factor. a. Use the above model to calculate the analytical price of a 5-year Zero-coupon (with a face value of $1) bond assuming that instantaneous short rate r(0) = 0.08. b. Using Euler discretization (as taught in class) and Monte Carlo simulation technique, calculate the above 5-year Zero-coupon bond price assuming that instantaneous short rate r(0) = 0.08. Use appropriate time step and number of simulations to minimize the MC error. c. Calculate the swap rate (fixed rate leg) of an interest rate siwap that has a maturity of 5 years and the floating leg has an annual coupon payment frequency linked to LIBOR. [Hint: Note the price of a swap is given as: 1-Z(N) Swap Rate=N EZ(i) price for maturity i. Where Z(i) is the zero coupon d. Calculate the price of a European Call option on 5-year zero- coupon bond (with a face value of $1000) and a maturity of 4 years and Strike price of $900 using the Vasicek model that has been calibrated as above. Note this solution allows for simulation of negative interest rates that is a known limitation of Vasicek model. [Hint: The payoff of call option is: Max (Z [t,T] - K,0), and you may ignore the discounting of payoff or use the 4 year zero spot rate to discount the payoff. And also note that Z [t (= 4y), T (= 5y)] can be computed using exponential affine formula where the t = 4y and r(t-4y) can be simulated using same steps as Part b above.]
Computer Networking: A Top-Down Approach (7th Edition)
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Chapter1: Computer Networks And The Internet
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![1. You are using Vasicek model as the short rate process that is
calibrated as:
dr(t) = a(b-r(t)) dt + odz(t)
Where a = 0.3, b = 0.05,0 = 0.03
And where Z(t) is a Wiener
process under the risk neutral framework modelling the random market
risk factor.
a. Use the above model to calculate the analytical price of a 5-year
Zero-coupon (with a face value of $1) bond assuming that
instantaneous short rate r(0) = 0.08.
b. Using Euler discretization (as taught in class) and Monte Carlo
simulation technique, calculate the above 5-year Zero-coupon
bond price assuming that instantaneous short rate r(0) = 0.08.
Use appropriate time step and number of simulations to
minimize the MC error.
c. Calculate the swap rate (fixed rate leg) of an interest rate siwap
that has a maturity of 5 years and the floating leg has an annual
coupon payment frequency linked to LIBOR. [Hint: Note the
price of a swap is given as:
1-Z(N)
Swap Rate=N
EZ(i)
price for maturity i.
Where Z(i) is the zero coupon
d. Calculate the price of a European Call option on 5-year zero-
coupon bond (with a face value of $1000) and a maturity of 4
years and Strike price of $900 using the Vasicek model that has
been calibrated as above. Note this solution allows for
simulation of negative interest rates that is a known limitation of
Vasicek model.
[Hint: The payoff of call option is: Max (Z [t,T] - K,0), and you
may ignore the discounting of payoff or use the 4 year zero spot rate
to discount the payoff.
And also note that Z [t (= 4y), T (= 5y)] can be computed using
exponential affine formula where the t = 4y and r(t-4y) can be
simulated using same steps as Part b above.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b0d6095-338e-4784-91dd-2e1b0b24a419%2F1760c536-94af-4e53-8cd7-6b9cb586ea69%2F8zmsia_processed.png&w=3840&q=75)
Transcribed Image Text:1. You are using Vasicek model as the short rate process that is
calibrated as:
dr(t) = a(b-r(t)) dt + odz(t)
Where a = 0.3, b = 0.05,0 = 0.03
And where Z(t) is a Wiener
process under the risk neutral framework modelling the random market
risk factor.
a. Use the above model to calculate the analytical price of a 5-year
Zero-coupon (with a face value of $1) bond assuming that
instantaneous short rate r(0) = 0.08.
b. Using Euler discretization (as taught in class) and Monte Carlo
simulation technique, calculate the above 5-year Zero-coupon
bond price assuming that instantaneous short rate r(0) = 0.08.
Use appropriate time step and number of simulations to
minimize the MC error.
c. Calculate the swap rate (fixed rate leg) of an interest rate siwap
that has a maturity of 5 years and the floating leg has an annual
coupon payment frequency linked to LIBOR. [Hint: Note the
price of a swap is given as:
1-Z(N)
Swap Rate=N
EZ(i)
price for maturity i.
Where Z(i) is the zero coupon
d. Calculate the price of a European Call option on 5-year zero-
coupon bond (with a face value of $1000) and a maturity of 4
years and Strike price of $900 using the Vasicek model that has
been calibrated as above. Note this solution allows for
simulation of negative interest rates that is a known limitation of
Vasicek model.
[Hint: The payoff of call option is: Max (Z [t,T] - K,0), and you
may ignore the discounting of payoff or use the 4 year zero spot rate
to discount the payoff.
And also note that Z [t (= 4y), T (= 5y)] can be computed using
exponential affine formula where the t = 4y and r(t-4y) can be
simulated using same steps as Part b above.]
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