InterestRate

Assignment 1

Reports

• Develop a small library implementing B-splines. This library should be capable of calculating the derivatives and integrals of the basis functions. Make sure that you take a full advantage of the recursive properties of the basis functions.

  • Please refer to python script: Assignment1/splines.py
  • Spline Class: with methods splrep(), splint(), spldev(), be capable of the derivatives and integrals of the basis functions.
  • Use cache in the function to take a full advantage of the recursive properties of the basis functions.

• Build a function that computes the discount factor between any two dates.

  • Please refer to python script: Assignment1/curves.py
  • Inside the Curve Class, disc_factor() is implemented, and both OIS and LIBOR classes will be derived from Curve.

• Build a function that computes the forward LIBOR rate for any settlement and underlying tenor.

  • Please refer to python script: Assignment1/curves.py
  • Inside the Curve Class, forwards() is implemented, and LIBOR Class will inheritant this method.

• Build a function that computes the (spot or forward) swap rate for any settlement and underlying tenor.

  • Please refer to python script: Assignment1/swaps.py
  • Inside the Swap Class, SwapRates() is to compute the swap rate for given settlement and underlying tenor.

• Use the enclosed market data sheet and the method described in class to build the instantaneous OIS and LIBOR curves.

  • Please refer to python script: Assignment1/curves.py
  • Inside the LIBOR and OIS classes, they both inheritant r() method in Curve base class, and this method is building the instantaneous curves.

• Build a function that calculates the PV of any spot or forward starting swap based on your curves.

  • Please refer to the main file: Assignment1/main.py

Usage

• Step 1: Please make sure your computer has installed python2 and required modules, if not and you want to test this assignment by running the program, please go https://github.com/weiyialanchen/MacInstallation and follow the installation guide.

• Step 2: Download the repository by git clone https://github.com/weiyialanchen/InterestRate.git

• Step 3: Go to Assignment 1 directory and run the main file

  cd ~/InterestRate/Assignment1
  python main.py
  

Assignment 2

Reports

• Implement the model using Euler’s scheme (note that for the normal LMM, Euler’s and Milstein’s schemes are identical). For drift term calculations, implement the ability to do both: the exact calculation and the frozen curve approximation.

  • Please refer to python script: Assignment2/libor_market.py
  • The exact calculation: set the parameter b_frozenCurve as False
  • The frozen curve approximation: set the paramter b_frozenCurve as True

• Apply your model to a spot starting 10 year knock-out swap. A knock-out swap is an interest rate swap with a special termination feature. Namely, if, on a fixed leg coupon date (or more precisely, two business days before), the 10 year swap rate sets below a preset barrier B, the swap is terminated. Notice that this is a path dependent derivative and Monte Carlo simulations are an appropriate approach to pricing this product. Use 2000 simulated paths to carry out the calculation. As a variance reducing method, you may also consider using antithetic variables.

  • Knock-out swap: Knock_Out_Swap class in Assignment2/swaps.py derived from Swap class implemented in Assignment1
  • Monte Carlo Simulation: please refer to Knock_Out_Swap.simulate()

• 2000 simulated paths calculation, assuming B = 0.95%, determine the break-even rate on the fixed leg of the swap.

  • Please refer to calc_swapRate() in Assignment2/main.py, see output for 2000 paths below

  Use 2,000 simulated paths to carry out the calculation (the exact calculation) - 
  The break-even rate on the fixed leg of the swap: 	0.0222146461996 
  
  Use 2,000 simulated paths to carry out the calculation (the frozen curve approximation) - 
  The break-even rate on the fixed leg of the swap: 	0.0221701123552
  

• How accurate is your calculation? Compare against a run with 5,000 simulated paths.

  • Please refer to calc_swapRate() in Assignment2/main.py, see output for 5000 paths below

  Compare against a run with 5,000 simulated paths - 
  The break-even rate on the fixed leg of the swap: 	0.0227748459579
  

  It's very accurate, with error around 2%.

• Analyse the performance of each of the drift terms calculation methods, and the accuracy of the frozen curve approximation.

  • The performance between different methods is very close, though the exact calculation is more accurate in theory.
  • But the error of frozen curve approximation method is smaller than 0.01 digits / 2% error, a good enough approximation.

Usage

• Step 1: Please make sure your computer has installed python2 and required modules, if not and you want to test this assignment by running the program, please go https://github.com/weiyialanchen/MacInstallation and follow the installation guide.

• Step 2: Download the repository by git clone https://github.com/weiyialanchen/InterestRate.git

• Step 3: Go to Assignment 2 directory and run the main file

  cd ~/InterestRate/Assignment2
  python main.py