lechgrzelak / financialengineering_ir_xva

Financial Engineering Course

Here you will find materials for the course of "Financial Engineering: Interest rates & xVA"

YouTube lectures you can find at: https://www.youtube.com/ComputationsInFinance

The course is based on the book: "Mathematical Modeling and Computation in Finance: With Exercises and Python and MATLAB Computer Codes", by C.W. Oosterlee and L.A. Grzelak, World Scientific Publishing, 2019.

The website of the book is: https://QuantFinanceBook.com

Personal website of the author (with more courses) is: https://LechGrzelak.com

Content of the course:

Lecture 1- Introduction & Details Regarding the Course
Lecture 2- Understanding of Filtrations and Measures
Lecture 3- The HJM Framework
Lecture 4- Yield Curve Dynamics under Short Rate
Lecture 5- Interest Rate Products
Lecture 6- Construction of Yield Curve and Multi-Curves
Lecture 7- Pricing of Swaptions and Negative Interest Rates
Lecture 8- Mortgages and Prepayments
Lecture 9- Hybrid Models and Stochastic Interest Rates
Lecture 10- Foreign Exchange (FX) and Inflation
Lecture 11- Market Models and Convexity Adjustments
Lecture 12- Valuation Adjustments- xVA (CVA, BCVA and FVA)
Lecture 13- Value-at-Risk and Expected Shortfall
Lecture 14- The Summary of the Course\

financialengineering_ir_xva's Issues

Solution for exercise - Lecture 13

Hi @LechGrzelak great course! really like it
for lecture 13 where is the solution located.

Kind regards and thanks,
Andrew

Vectorized Ho-Lee Solution

Hello Lech!
Here's a vectorized approach for the short-rate models.
Thank you for your lectures & materials!

import typing

def f0T(t,P0T):
    dt = 0.01    
    return -(np.log(P0T(t+dt))-np.log(P0T(t-dt)))/(2*dt)

M: int = 25000 # n_simulations
N: int = 252 # n_steps
T: float = 40 
dt: float = T/N
sigma: float = 0.007
Z: np.ndarray = np.random.normal(0.0, 1.0, (M, N))
P0T: typing.Callable = lambda T: np.exp(-0.017*T)
x0: float = f0T(0.01, P0T) # r0
Ws: np.ndarray = np.cumsum(np.power(dt, 0.5)*Z, axis=1)
Ts: np.ndarray = np.arange(0, T, dt) # np.arange timesteps_vector
theta: typing.Callable = lambda t: (f0T(t+dt,P0T)-f0T(t-dt,P0T))/(2.0*dt) + sigma**2.0*t 

# np.ndarray of (M) short-rates dynamics
Xs = np.cumsum(np.concatenate((np.full(shape=(M, 1), fill_value=x0), 
    theta(Ts)[1:] * dt + sigma * np.diff(Ws)), axis=1), axis=1)

# np.ndarray of (M) money-savings accounts dynamics
Ms = np.cumprod(np.concatenate((np.full(shape=(M, 1), fill_value=1.0), 
    np.exp((Xs[:, 1:] + Xs[:, :-1]) * 0.5 * dt)), axis=1), axis=1)  

# Here we compare the price of an option on a ZCB from Monte Carlo the Market  
P_tMC = np.mean(1/Ms, axis=0)