William Blog

Interest Rate Modelling - 1

Fundamentals, Notation

Notation Bonds and Forward Rates Zero Conpon Bond and its Forward Price The most basic instrument in the interest rate modelling is zero coupon bond $P(t, T)$ which deliver $1 for certain at ti...

Posted by William on September 26, 2021

Complex Number - 1

Basics

Algebra and Geometry in the Complex Plane The Complex Plane The complex number can be expressed as $z = x + yi$ x is called the real part of z; $x = Re(z)$. y is called the imaginary part...

Posted by William on April 11, 2021

Design Pattern - 2

Visitor

Overview of Visitor Pattern Introduction From the view of design, visitor patter is used to make large class hierarchies more maintainable by seperating the algorithm from the object structure ...

Posted by William on March 10, 2020

Design Pattern - 1

Singleton

Overview of Singleton Introduction The singleton is a pattern that’s used to restrict the number of class instantiations to exactly one. If in the entire program, there is only one object of a ...

Posted by William on February 12, 2020

Local Volatility - 3

Dupire Formula as an Expectation

Derive Dupire Formula by Conditional Expectation Assumption Assume the underlying stock follows the geometric brownian motion under Risk-neutral world: \[dS_t = \mu_tS_tdt+ \sigma_tS_tdw\] Si...

Posted by William on February 7, 2020

Stochastic Calculus Cookbook - 2

Radon-Nikodym Derivative

Radon-Nikodym Derivative Definition RN derivative $Z$ is the link between two equavalent probability measure $P$ and $\tilde{P}$. $Z$ can be defined as below: \[Z = \frac{d\tilde{P}}{dP}\] ...

Posted by William on December 14, 2019

Local Volatility - 1

Fokker Planck Equation

Motivation for Local Volatility In the market, implied volatility can be back out by using obeservable market quotes. Options with different strikes and expirations have different Black-Scholes...

Posted by William on December 3, 2019

Local Volatility - 2

Dupire Formula

Quick Review From the derivation of Last Article - Fokker Planck Equation, below is the final conclusion. \[\frac{\partial p(s,t)}{\partial t} - \frac{1}{2}*\frac{\partial^2 [\sigma^2(s,t) p(s...

Posted by William on December 3, 2019

Equivalent Martingale Measure Result

Numeraire

Market Price of Risk One Factor Suppose the derivatives only dependents on $\theta$ and assume the process of $\theta$ is: \[\frac{d\theta}{\theta} = mdt + sdw\] m and d only depends on $\the...

Posted by William on November 22, 2019

Black Litterman Asset Allocation Framework - 01

Basic Black Litterman

1. Introduction of Black Litterman Framework The Black-Litterman asset allocation model, created by Fischer Black and Robert Litterman, is a sophisticated portfolio construction method that ove...

Posted by William on September 9, 2019