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, o = 0.03 And where Z(t) is a Wiener %3D 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
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, o = 0.03 And where Z(t) is a Wiener %3D 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
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, o = 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 swap
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 =
Where Z(i) is the zero coupon
i3D1
price for maturity i.
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 =
simulated using same steps as Part b above.]
4y and r(t=4y) can be](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8b19c1f-9c17-45b1-8183-3cc393212736%2Fe3e9ff8f-2143-4c0f-9935-8a6a0bf48dd8%2Fa7z42eh_processed.jpeg&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, o = 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 swap
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 =
Where Z(i) is the zero coupon
i3D1
price for maturity i.
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 =
simulated using same steps as Part b above.]
4y and r(t=4y) can be
Transcribed Image Text:2. Your option portfolio has European style put options on stock of ABC
Ltd. with a notional value of $10 million. You are asked to mark-to-
market the option portfolio using the Finite difference schemes under
the Black-Scholes framework.
The trade terms specify the following additional information:
Initial Stock Price (ABC Ltd.) = $40 (as of Today)
Strike = $40
Time to Maturity = 1 Year (12 months expiry)
Risk free rate = 0.10
Volatility = 0.30
For the grid discretization scheme the following parameters can be set
as follows: Smax = 100; Step size for Stock Price (M) = 400; Step size
for time (N) = 400.
a. As shown in class, use implicit Euler method to solve the
above pricing PDE and calculate the price.
b. Comment on the stability of FDS scheme used and its
convergence.
c. The trader added another barrier style (down and out) option to
the portfolio with the same trade terms as above and the
additional barrier level specified at B = $30. You are asked to
price this trade using the finite difference scheme as above.
(Note – you need to apply the appropriate boundary conditions
for barrier style exercise as discussed in the class)
%3D
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