In this tutorial we further our investigation into weather derivatives by diving into some real world temperature data. The weather station data goes all the way back to Jan-1859, and we show how to group on any selection/periods using pandas dataframes to extract statistics like extreme temperatures and distributions for specific months.
It is estimated that nearly 30 percent of the US economy is directly affected by the weather [1,3].
Weather derivatives are financial instruments that can be used to reduce risk associated with adverse weather conditions like temperature. Temperature Options are critical for risk management for businesses that can be impacted by extreme temperatures.
In 1978, Breeden and Litzenberger showed how under risk-neutral pricing, that the discounted Risk-Neutral Density (RND) function could be estimated directly from option prices.
In this tutorial we demonstrate how to use the Breeden-Litzenberger formula to derive the risk-neutral density function from European call options with stochastic volatilty under the Heston model. We price the call options using the semi-analytical solution for the Heston model using rectangular integration and compare this to the QuantLib implementation in Python. Once we have the option prices (or possibly market prices) we can implement the Breeden-Litzenberger formula easily using pandas dataframe shift functions. To utilise the estimated pdf within other calculations, we need to make use of scipy’s interpolation function.
In this tutorial we will implement the Monte Carlo Method to simulate a stock portfolio over time.
The method of Importance Sampling involes a change of distribution using the Radon-Nikodym derivative. So we conductr a simulation under a new probability measure and then multiply by the ratio of the two probability density functions (the old over the new), or the Radon Nikodym derivative of one process with respect to the other.
In this tutorial we discuss Monte Carlo convergence and the difference between Pseudo-random numbers and Quasi-random numbers. In previous tutorials will discusses the benefits of combining Monte Carlo Variance Reduction techniques such as antithetic and control variate methods to reduce the standard error of our simulation.
We demonstrate the effectiveness of using quasi-random numbers by compaing the convergence on a pricing a European Call Option by monte carlo simulation using difference methods for creating pseudo and quasi-random variables.
In this tutorial we are pricing a discretely monitored lookback call option with stochastic volatility. The option payoffs are dependent on the extreme values, maximum or minimum, of the underlying asset prices over a certain time period (lookback period). There are two standard lookback options: fixed strike and floating strike. Fixed strike discrete lookback option options pay the difference (if positive) between the max or min of a set of observations, over a lookback period of the asset price and the strike price at the maturity date.
There is no analytical solution for the price of European fixed strike lookback call options with discrete fixings and stochastic volatility under Heston model. However there is a simple analytical formula for the price of a continuously monitored (fixing) fixed strike lookback call with constant volatility.
We will use a combination of Monte Carlo Variance Reduction techniques such as antithetic and control variate methods to reduce the standard error of our simulation. We use the analytical solution for a continuously monitored fixed strike lookback call option to calculate control variates based on delta, gamma and vega sensitivities.
In this tutorial we will be pricing an Average Price Option (APO) otherwise commonly referred to as an Asian Option. We will be pricing a bespoke OTC electricity average rate option in the Australian Electricity Market.
Electricity Average Rate Options in Australia / NZ Energy Markets are settled against the final futures prices. As futures are settled against the arithmetic average of the wholesale energy market price. Therefore final futures prices converge on the arithmetic average of wholesale electricity price. This essentially means listed ASX energy average rate options can be priced using blackscholes until the end of the quarter.
However, to demonstrate pricing of and Average Rate Option, and make things more interesting we will be looking at an OTC Asian Option over Q123 with the underlying as the Q123 NSW futures contract. Let’s say the average rate option prices are determined by the average of teh (closing) daily futures prices across the Q123 period.
In pricing complex or exotic path dependent options, a popular product is the barrier option. These are standard European option expiration style options, however the options cease to exist or only comes into existence if the underlying price crosses a pretermined barrier level.
There is an extremely high false discovery rate in both the academic and financial industry for trading strategies that “produce” alpha. In fact, most of these strategies are false discoveries due to research bias, multiple testing and the true probability of finding a new investment strategy being very low (<< 1%) due to competition. Today we investigate issues of multiple testing and false discovery of a profitable trading strategy. We develop a momentum-based trading strategy on Apple stock and show the issues that can arise from unknowingly completing multiple testing on the same dataset.
