Fisher, parameterisation, efficiency and CLT — It feels like knowledge has advanced, and is continuing to advance, so quickly that in order to keep up there is no time at school to explain how and why things are the way they are. For me, standard deviation is one of those things: something that is simply stated…

Mashing together intuitive derivations littering the web — This is something I always struggled with — half because I didn’t really feel comfortable with the chi-square distribution, half because the idea of ‘degrees of freedom’ seemed incredibly vague. Most derivations are either: symbol heavy with lots of largely unnecessary maths chucked straight in your face end up technically…

Kurtosis, defining fat tails and counter-intuitive results — Option trading is usually associated with the words ‘convexity’ or ‘leverage’ with the assumption being that options are the best instrument to profit from large stock price moves — either positive or negative. This opinion has been spurred on by depictions of optionality in movies such as the Big Short…

Implementing custom distribution sampling in SciPy — In the previous two parts (Part I, Part II) we walked through a quick intro to what sampling entails as well as then digging through the source code of NumPy and SciPy to understand exactly how this is implemented in modern python libraries. All of this can seem like overkill…

Fancy algorithms, source code walkthrough and potential improvements — In Part I we went through the basics of Inverse Transform Sampling (ITS) and created our own ITS pure python implementation to sample numbers from a standard normal distribution. We then compared the speed of our somewhat optimised function to that of the built in SciPy function and found ourselves…

Intro to sampling, writing our own, speed testing — Being able to draw a random sample from a distribution of your choice is very useful. It underlies any kind of stochastic process simulation whether that’s particle diffusion, stock price movements, or modelling any phenomena that displays some kind of randomness through time. For that reason, having access to accurate…

Cutting through excessive probability jargon — When it comes to making things sound and look much more complicated than they are, financial maths really does reign supreme. Fortunately it’s not just me who thinks it’s all overly complicated but even Paul Wilmott thinks the profession takes it just a bit too far and he’s a bit…

Which volatility do I use: implied or realised? — So far we have built up a world in order to analyse the effectiveness of delta hedging for the replication of a vanilla option. We have stated a simple model for how a stock moves around: it drifts a bit has a nice simple normally distributed change every day Based…

Hedging frequency & more pretty scatterplots — We left the previous article having established that miraculously the process of just holding: cash stocks and adjusting this quantity of stocks frequently according to what the Black-Scholes delta tells us. Sure, there are lots of assumptions embedded in the way we have set up our simulations but we can…