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

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

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.

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